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Oikonomou1,2 +1 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece +2 Institut für Theoretische Physik, Goethe Universität Frankfurt, Max-von-Laue-Str.1, 60438 Frankfurt am Main, Germany +31 January 2023 +ABSTRACT +In this work we study the neutron star phenomenology of 𝑅𝑝 attractor theories in the Einstein frame. The Einstein frame 𝑅𝑝 +attractor theories have the attractor property that they originate from a large class of Jordan frame scalar theories with arbitrary +non-minimal coupling. These theories in the Einstein frame provide a viable class of inflationary models, and in this work we +investigate their implications on static neutron stars. We numerically solve the Tolman-Oppenheimer-Volkoff equations in the +Einstein frame, for three distinct equations of state, and we provide the mass-radius diagrams for several cases of interest of the +𝑅𝑝 attractor theories. We confront the results with several timely constraints on the radii of specific mass neutron stars, and as +we show, only a few cases corresponding to specific equations of state pass the stringent tests on neutron stars phenomenology. +Key words: stars: neutron; Physical Data and Processes, cosmology: theory +INTRODUCTION +The direct gravitational wave observation GW170817 LIGO & +Virgo Collaboration, et al. (2017, 2020) initiated what is nowadays +known as gravitational wave astronomy. Neutron stars (NS) Haensel, +Potekhin & Yakovlev (2007); Friedman & Stergioulas (2013); Baym, +et al. (2018); Lattimer & Prakash (2004); Olmo, Rubiera-Garcia & +Wojnar (2020) are at the core of astrophysical gravitational wave +observations, and numerous scientific areas are jointly studying NS +from their perspective, for example nuclear theory Lattimer (2012); +Steiner & Gandolfi (2012); Horowitz, et al. (2005); Watanabe, Iida +& Sato (2000); Shen, et al. (1998); Xu, et al. (2009); Hebeler, et +al. (2013); Mendoza-Temis, et al. (2014); Ho, et al. (2015); Kanakis- +Pegios, Koliogiannis & Moustakidis (2020); Tsaloukidis et al. (2022), +high energy physics Buschmann, et al. (2021); Safdi, Sun & Chen +(2019); Hook, et al. (2018); Edwards, et al. (2020); Nurmi, Schi- +appacasse & Yanagida (2021), modified gravity Astashenok, et al. +(2020, 2021); Capozziello, et al. (2016); Astashenok, Capozziello & +Odintsov (2015, 2014, 2013); Arapoˇglu, Deliduman & Eksi (2011); +Panotopoulos et al. +(2021); Lobato et al. +(2020); Numajiri et +al. (2022) and astrophysics Altiparmak, Ecker & Rezzolla (2022); +Bauswein, et al. (2020b); Vretinaris, Stergioulas & Bauswein (2020); +Bauswein, et al. (2020a, 2017); Most, et al. (2018); Rezzolla, Most & +Weih (2018); Nathanail, Most & Rezzolla (2021); Köppel, Bovard & +Rezzolla (2019); Raaijmakers et al. (2021); Most, et al. (2021); Ecker +& Rezzolla (2022); Jiang, et al. (2022). The perspective of modified +gravity implications on NS has been for a long time in the mainstream +of NS works, see for example Astashenok, Capozziello & Odintsov +(2015, 2014) and also Refs. Pani & Berti (2003); Staykov, et al. +(2014); Horbatsch, et al. (2015); Silva, et al. (2015); Doneva, et al. +(2013); Xu, Gao & Shao (2020); Salgado, Sudarsky & Nucamendi +(1998); Shibata, et al. (2014); Arapoğlu, Ekşi & Yükselci (2019); +Ramazanoğlu & Pretorius (2016); Motahar, et al. (2019); Chew, et +al. (2019); Blázquez-Salcedo, Scen Khoo & Kunz (2020); Motahar, +et al. (2017); Odintsov & Oikonomou (2021, 2022a); Oikonomou +(2021); Pretel et al. (2022); Pretel & Duarte (2022); Cuzinatto et +al. (2016) for scalar-tensor descriptions of NS phenomenology. The +main effect of modified gravity descriptions of NS is the significant +elevation of the maximum NS masses, with modified gravity bring- +ing this maximum mass near or inside the mass-gap region with +𝑀 ≥ 2.5 𝑀⊙. Regarding non-minimally coupled scalar field theo- +ries, there exists a vast class of viable inflationary potentials which +have the remarkable property of being attractors Kallosh, Linde +& Roest (2014a); Kallosh & Linde (2013); Ferrara, et al. (2013); +Kallosh, Linde & Roest (2013); Linde (2015); Cecotti & Kallosh +(2014); Carrasco, Kallosh & Linde (2015); Carrasco, et al. (2015); +Kallosh, Linde & Roest (2015); Roest & Scalisi (2015); Kallosh, +Linde & Roest (2014b); Ellis, Nanopoulos & Olive (2013); Cai, +Gong & Pi (2014); Yi & Gong (2016); Akrami, et al. (2018); Qum- +mer, Jawad & Younas (2020); Fei, Yi & Yang (2020); Kanfon, Mavoa +& Houndjo (2020); Antoniadis, et al. (2020); García-García, et al. +(2019); Cedeño, et al. (2019); Karamitsos (2019); Canko, Gialamas +& Kodaxis (2020); Miranda, et al. (2019); Karam, Pappas & Tam- +vakis (2019); Nozari & Rashidi (2018); García-García, et al. (2018); +Rashidi & Nozari (2018); Gao, Gong & Fei (2018); Dimopoulos, +Wood & Owen (2018); Miranda, Fabris & Piattella (2017); Karam, +Pappas & Tamvakis (2017); Nozari & Rashidi (2017); Gao & Gong +(2018); Geng, Lee & Wu (2017); Odintsov & Oikonomou (2020, +2016, 2017); Järv, et al. (2020). The attractor terminology is justi- +fied due to the fact that distinct non-minimally coupled scalar-tensor +inflationary theories, lead to the same Einstein frame inflationary +phenomenology, which is compatible with the latest Planck data +Planck Collaboration (2020). The question always when studying +these attractor models is whether these models can be distinguished +in some way, phenomenologically. From an inflationary point of +view, and regarding the large wavelength Cosmic Microwave Back- +ground modes, a discrimination between these models is impossible. +However, this discrimination is possible if NS are studied. Indeed, +the phenomenologically indistinguishable attractor models can be +discriminated in NS and vice versa, with the latter feature being phe- +© 0000 The Authors +arXiv:2301.12136v1 [gr-qc] 28 Jan 2023 + +2 +Oikonomou +nomenal. That is, if some models are indistinguishable with respect +to their NS phenomenology, they can be distinguished if their infla- +tionary properties are studied. To address these issues in a concrete +way, in this work we shall study 𝑅𝑝 attractor theories. The inflation- +ary phenomenology of these theories is studied in the recent literature +Odintsov & Oikonomou (2022b) see also Motohashi (2015); Renzi, +Shokri & Melchiorri +(2009) for subcases of the original 𝑅𝑝 at- +tractors theories. For a spherically symmetric metric we derive and +solve numerically the Einstein frame Tolman-Oppenheimer-Volkoff +(TOV) equations, using an LSODA based double shooting python 3 +numerical integration Stergioulas (2019). We derive the Jordan frame +𝑀 −𝑅 graphs for the 𝑅𝑝 attractors, for three different piecewise poly- +tropic Read, et al. (2009a,b) equations of state (EoS), WFF1 Wiringa, +Fiks & Fabrocini (1988), the SLy Douchin & Haensel (2001), and +the APR EoS Akmal, Pandharipande & Ravenhall (1998), using the +Arnowitt-Deser-Misner (ADM) definition of Jordan frame masses +of NS Arnowitt, Deser & Misner (1960). The NSs temperature is +significantly lower than the Fermi energy of the constituent particles +of NSs, thus NS matter can be in principle described by a single- +parameter EoS that may describe perfectly cold matter at densities +higher than the nuclear density. However, a serious problem emerges, +having to do with the uncertainty in the EoS, which is larger, and +the pressure as a function of the baryonic mass density cannot be +accurately defined and is uncertain to one order of magnitude at least +above the nuclear density. Moreover, the exact nature of the phase of +matter at the NSs core is highly uncertain. Hence, a parameterized- +type EoS at high densities is an optimal choice for an EoS, thus +rendering the piecewise polytropic EoS a suitable choice. In order +to construct the piecewise polytropic EoS, astrophysical constraints +are taken into account, both observational and theoretical, like the +causality constraints, see Read, et al. (2009a,b), to also confirm the +causality fulfilment for all the piecewise polytropic EoS we shall use +in this paper. For the construction of the piecewise polytropic EoS +one uses a low-density part with 𝜌 < 𝜌0, which is basically chosen to +be a tabulated and well-known EoS for the crust, and furthermore, the +piecewise polytropic EoS also has a large density part with 𝜌 ≫ 𝜌0. +We finally confront the resulting NS phenomenologies with several +recent constraints on the radii of specific mass NS Altiparmak, Ecker +& Rezzolla (2022); Raaijmakers et al. (2021); Bauswein, et al. (2017) +and as we show, only a few scenarios and EoS are compatible with +the constraints on NS radii. Obviously, the gravitational wave astron- +omy era has changed the way of thinking on theoretical astrophysics, +since several models of scalar-tensor gravity which in the recent past +could be considered as viable, nowadays may no longer be valid. +1 INFLATIONARY PHENOMENOLOGY OF 𝑅𝑃 +ATTRACTORS +The full analysis of the generalized 𝑅𝑝 attractors is given in Ref. +Odintsov & Oikonomou (2022b), so we refer the reader for details. +Here we shall briefly discuss the inflationary phenomenological prop- +erties of 𝑅𝑝 attractors in order to stress their importance among other +cosmological attractors Kallosh, Linde & Roest (2014a); Kallosh & +Linde (2013); Ferrara, et al. (2013); Kallosh, Linde & Roest (2013); +Linde (2015); Cecotti & Kallosh (2014); Carrasco, Kallosh & Linde +(2015); Carrasco, et al. (2015); Kallosh, Linde & Roest (2015); Roest +& Scalisi (2015); Kallosh, Linde & Roest (2014b); Ellis, Nanopou- +los & Olive (2013); Cai, Gong & Pi (2014); Yi & Gong (2016); +Akrami, et al. (2018); Qummer, Jawad & Younas (2020); Fei, Yi +& Yang (2020); Kanfon, Mavoa & Houndjo (2020); Antoniadis, +et al. (2020); García-García, et al. (2019); Cedeño, et al. (2019); +Karamitsos (2019); Canko, Gialamas & Kodaxis (2020); Miranda, +et al. (2019); Karam, Pappas & Tamvakis (2019); Nozari & Rashidi +(2018); García-García, et al. (2018); Rashidi & Nozari (2018); Gao, +Gong & Fei (2018); Dimopoulos, Wood & Owen (2018); Miranda, +Fabris & Piattella (2017); Karam, Pappas & Tamvakis (2017); Nozari +& Rashidi (2017); Gao & Gong (2018); Geng, Lee & Wu (2017); +Odintsov & Oikonomou (2020, 2016, 2017); Järv, et al. (2020). The +𝑅𝑝 attractors constitute a class of their own among other attrac- +tors, and all the 𝑅𝑝 attractors in the Einstein frame correspond to +generalizations of the following Einstein frame potential, +𝑉(𝜑) = 𝑉0 𝑀4 +𝑝𝑒−2 +√︃ +2 +3 𝜅 𝜑 +� +𝑒 +√︃ +2 +3 𝜅 𝜑 − 1 +� +𝑝 +𝑝−1 +, +(1) +where 𝑀𝑝 = +1 +√ +8𝜋𝐺 is the reduced Planck mass and 𝐺 is Newton’s +gravitational constant. The inflationary properties of the above theory +have been addressed in the recent literature, see for example Moto- +hashi (2015); Renzi, Shokri & Melchiorri (2009). The scalar-tensor +theory with the potential (1) corresponds to the Jordan frame 𝐹(𝑅) +gravity, +𝐹(𝑅) = 𝑅 + 𝛽𝑅𝑝 , +(2) +with 𝛽 is a free parameter with its physical dimensions in natural +units being [𝛽] = [𝑚]2−2𝑝. The 𝑅𝑝 attractors have the following +scalar potential in the Einstein frame, +𝑉(𝜑) = 𝑉0 𝑀4 +𝑝𝑒−2 +√︃ +2 +3𝛼 𝜅 𝜑 +� +𝑒 +√︃ +2 +3𝛼 𝜅 𝜑 − 1 +� +𝑝 +𝑝−1 +, +(3) +where 𝑀𝑝 is the reduced Planck mass, and for 𝛼 = 1 we obtain +the scalar theory with scalar potential (3). Now the question is why +these models are classified as attractor models, what justifies the +terminology attractors? It is the class of scalar-tensor Jordan frame +theories which correspond to the Einstein frame potential (3) that +justify the use of the terminology attractors. Basically, the potential +(3) can be the Einstein frame potential for a large class of Jordan +frame scalar-tensor theories, as we now evince. The 𝜙-Jordan frame +action is, +S𝐽 = +∫ +𝑑4𝑥 +� Ω(𝜙) +2𝜅2 𝑅 − 𝜔(𝜙) +2 +𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 − 𝑉𝐽 (𝜙) +� +, +(4) +with the scalar field describing a non-canonical scalar field in +the Jordan frame, and the coupling function has the general form +Ω(𝜙) = 1+𝜉 𝑓 (𝜙) with 𝜉 and 𝑓 (𝜙) being the arbitrary dimensionless +coupling and an arbitrary dimensionless function respectively. The +𝑅𝑝 attractors have the following 𝜙-Jordan frame scalar potential, +𝑉𝐽 (𝜙) = 𝑉0 (Ω(𝜙) − 1) +𝑝 +𝑝−1 , +(5) +and more importantly, the kinetic term function 𝜔(𝜙) has the follow- +ing form, +𝜔(𝜙) = 1 +4𝜉 +� 𝑑Ω(𝜙) +𝑑𝜙 +�2 +Ω(𝜙) +. +(6) +Hence the large class of the 𝑅𝑝-attractors correspond to the Jordan +frame theories which are described by Eqs. (5) and (6). Notice that +the Jordan frame functions 𝑓 (𝜙) are arbitrary and we shall not need +to specify these. By performing the conformal transformation of the +Jordan frame metric 𝑔𝜇𝜈, +˜𝑔𝜇𝜈 = Ω(𝜙)𝑔𝜇𝜈 , +(7) +MNRAS 000, 1–8 (0000) + +𝑅𝑝 Attractors Static Neutron Star Phenomenology +3 +Figure 1. The constraints CSI, CSII and CSIII. This figure is inspired and +based after editing on Credit: ESO/L.Calçada: https://www.eso.org/ +public/images/eso0831a/. +we get the Einstein frame action, +S𝐸 = +√︁ +− ˜𝑔 +� +˜𝑅 +2𝜅2 − ˜𝑔𝜇𝜈𝜕𝜇𝜑𝜕𝜈𝜑 − 𝑉(𝜑) +� +, +(8) +with ˜𝑔𝜇𝜈 denoting the Einstein frame metric tensor, and the “tilde” +indicates Einstein frame quantities. Also the Einstein frame potential +𝑉(𝜙) and the Jordan frame potential 𝑉𝐽 (𝜙) are related as follows, +𝑉(𝜑) = Ω−2(𝜙)𝑉𝐽 (𝜙) . +(9) +Notice that the general relation which connects the Jordan frame +scalar field 𝜙 with the canonical Einstein frame scalar field 𝜑 is, +� 𝑑𝜑 +𝑑𝜙 +�2 += 3 +2 +� 𝑑Ω(𝜙) +𝑑𝜙 +�2 +Ω(𝜙) ++ 𝜔(𝜙) +Ω(𝜙) , +(10) +hence for the 𝑅𝑝 attractors, in which case the kinetic term function +𝜔(𝜙) is chosen to be that of Eq. (6), we finally have the important +relation of the non-minimal scalar coupling function to gravity, +Ω(𝜙) = 𝑒 +√︃ +2 +3𝛼 𝜑 , +(11) +with the parameter 𝛼 being defined to be, +𝛼 = 1 + 1 +6𝜉 . +(12) +Notice that by substituting Eq. (11) in Eq. (9) we obtain the gen- +eralized 𝑅𝑝-attractor potential of Eq. (3). Furthermore, the impor- +tant case with 𝛼 = 1 is realized when 𝜉 → ∞, or similarly when +Ω(𝜙) ≪ 3 +2 +� +𝑑Ω(𝜙) +𝑑𝜙 +�2 +𝜔(𝜙) +. The 𝑅𝑝 attractors yield a viable inflationary +phenomenology, see Ref. Odintsov & Oikonomou (2022b), with the +spectral index of the primordial scalar perturbations as a function of +the canonical scalar field being, +𝑛𝑠 = +� � +3𝛼 + (3𝛼 − 2)𝑝2 + (8 − 6𝛼)𝑝 − 8 +� +𝑒2 +√︃ +2 +3 +√︃ +1 +𝛼 𝜅 𝜑 +(13) +− 2(𝑝 − 1)(−3𝛼 + (3𝛼 − 2)𝑝 + 8)𝑒 +√︃ +2 +3 +√︃ +1 +𝛼 𝜅 𝜑 + (3𝛼 − 8)(𝑝 − 1)2� +× 3𝛼(𝑝 − 1)2 +� +𝑒 +√︃ +2 +3 +√︃ +1 +𝛼 𝜅 𝜑 − 1 +�2 +, +and the tensor-to-scalar ratio is, +𝑟 = +16 +� +(𝑝 − 2)𝑒 +√︃ +2 +3 +√︃ +1 +𝛼 𝜅 𝜑 − 2𝑝 + 2 +�2 +3𝛼(𝑝 − 1)2 +� +𝑒 +√︃ +2 +3 +√︃ +1 +𝛼 𝜅 𝜑 − 1 +�2 +. +(14) +Also the free parameter 𝑉0 of the potential is constrained to have +values +𝑉𝑠 ∼ 9.6 × 10−11 , +(15) +a results which originates from the constraints of the Planck data on +the Einstein frame amplitude Δ2𝑠 of the scalar perturbations, +Δ2 +𝑠 = +1 +24𝜋2 +𝑉(𝜑 𝑓 ) +𝑀4𝑝 +1 +𝜖(𝜑 𝑓 ) . +(16) +For the purposes of this paper, we shall consider several limiting +cases for the values of the parameter 𝛼, mainly the cases 𝛼 ≠ 1, +and the case 𝛼 = 1, which corresponds to the strong 𝜉 coupling +theory. Also in order to have a viable inflationary phenomenology, +the parameter 𝑝 which is the exponent in the 𝑅𝑝 attractors potential, +has to take values in the range 1.91 ≤ 𝑝 ≤ 1.99. It proves that this is +irrelevant for NS studies, so we shall assume that 𝑝 = 1.91 without +loss of generality. In the next section we shall specify the values of +the various functions involved in the TOV equations of NS. +2 NEUTRON STARS WITH 𝑅𝑃 ATTRACTORS +For the purpose of studying NS in Einstein frame, we shall use the +Geometrized physical units system 𝐺 = 𝑐 = 1, and we shall adopt +the notation of Ref. Pani & Berti (2003). +The Jordan frame scalar-tensor theory has the following form, +S = +∫ +𝑑4𝑥 +√−𝑔 +16𝜋 +� +Ω(𝜙)𝑅 − 1 +2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 −𝑈(𝜙) +� ++ 𝑆𝑚(𝜓𝑚, 𝑔𝜇𝜈) , +(17) +and by performing the following conformal transformation, +˜𝑔𝜇𝜈 = 𝐴−2𝑔𝜇𝜈 , 𝐴(𝜙) = Ω−1/2(𝜙) , +(18) +we obtain the Einstein frame action, +S = +∫ +𝑑4𝑥 +√︁ +− ˜𝑔 +� +˜𝑅 +16𝜋 −1 +2 ˜𝑔𝜇𝜈𝜕𝜇𝜑𝜕𝜈𝜑−𝑉(𝜑) +16𝜋 +� ++𝑆𝑚(𝜓𝑚, 𝐴2(𝜑)𝑔𝜇𝜈) , +(19) +with 𝜑 denoting the Einstein frame canonical scalar field as in the +previous section, and +𝑉(𝜑) = 𝑈(𝜙) +Ω2 +. +(20) +For the 𝑅𝑝 attractors with general 𝛼, the important function 𝐴(𝜑) +has the following form, +𝐴(𝜑) = 𝑒− 1 +2 +√︃ +2 +3𝛼 𝜑 , +(21) +therefore, the function 𝛼(𝜙) which is defined as follows, +𝛼(𝜑) = 𝑑 ln 𝐴(𝜑) +𝑑𝜑 +, +(22) +takes the form, +𝑎(𝜑) = −1 +2 +√︂ +2 +3𝛼 . +(23) +MNRAS 000, 1–8 (0000) + +CS I +-0.99 +CS II +R1.4Mo +-0.81 +CS III +-0.04 +-0.034 +Oikonomou +Table 1. CSI vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for +NS Masses 𝑀 ∼ 2𝑀⊙ +𝑅𝑝 Attractor Model +APR +SLy +WFF1 +𝛼 = 1 +𝑀 = 2.00 𝑀⊙ +𝑀 = 2.01 𝑀⊙ +𝑀 = 0.31 𝑀⊙ +𝛼 = 1 +𝑅 = 11.10km +𝑅 = 11.17km +𝑅 = 11.06km +𝛼 = 0.1 +𝑀 = 2.02 𝑀⊙ +𝑀 = 2.00 𝑀⊙ +𝑀 = 2.00 𝑀⊙ +𝛼 = 0.1 +𝑅 = 11.52km +𝑅 = 11.818km +𝑅 = 11.012km +𝛼 = 8 +𝑀 = 2.00 𝑀⊙ +𝑀 = 2.09 𝑀⊙ +𝑀 = 0.32 𝑀⊙ +𝛼 = 8 +𝑅 = 11.08km +𝑅 = 10.983km +𝑅 = 11.114km +Table 2. CSI vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for +NS Masses 𝑀 ∼ 1.4𝑀⊙ +𝑅𝑝 Attractors Model +APR +SLy +WFF1 +𝛼 = 1 +𝑀 = 0.58 𝑀⊙ +𝑀 = 1.41 𝑀⊙ +𝑀 = 0.25 𝑀⊙ +𝛼 = 1 +𝑅 = 11.48km +𝑅 = 11.74km +𝑅 = 11.89km +𝛼 = 0.1 +𝑀 = 1.39 𝑀⊙ +𝑀 = 1.39 𝑀⊙ +𝑀 = 0.07 𝑀⊙ +𝛼 = 0.1 +𝑅 = 11.55km +𝑅 = 12.04km +𝑅 = 11.79km +𝛼 = 8 +𝑀 = 0.64 𝑀⊙ +𝑀 = 1.42 𝑀⊙ +𝑀 = 0.28 𝑀⊙ +𝛼 = 8 +𝑅 = 11.45km +𝑅 = 11.73km +𝑅 = 11.46km +Finally, the Einstein frame scalar potential is given in Eq. (3), which +we also quote it here for reading convenience, +𝑉(𝜑) = 𝑉0 𝑒−2 +√︃ +2 +3𝛼 𝜑 +� +𝑒 +√︃ +2 +3𝛼 𝜑 − 1 +� +𝑝 +𝑝−1 +, +(24) +and in Geometrized units, the constraint on 𝑉0 given in Eq. (15) +becomes, +𝑉0 ≃ 7.62 × 10−12 . +(25) +For the study of NS physics, we shall consider the following spheri- +cally symmetric metric, +𝑑𝑠2 = −𝑒𝜈(𝑟)𝑑𝑡2 + +𝑑𝑟2 +1 − 2𝑚(𝑟) +𝑟 ++ 𝑟2(𝑑𝜃2 + sin2 𝜃𝑑𝜙2) , +(26) +which describes a static NS, where the function 𝑚(𝑟) describes the +total gravitational mass of the NS and 𝑟 stands for the circumferential +radius. In the following, we shall calculate numerically the functions +𝜈(𝑟) and +1 +1− 2𝑚(𝑟) +𝑟 +following a simple procedure, in which the central +value of 𝜈(𝑟) and of the scalar field will be arbitrary and will be +optimally calculated numerically by using a double shooting method. +The double shooting aims to find the optimal values of the central +values of 𝜈(𝑟) and of the scalar field, which guarantee that the metric +at numerical infinity becomes identical to the Schwarzschild metric. +This procedure is different compared to standard General Relativity +(GR) NS, because in GR, the metric at the surface of the star abruptly +becomes the Schwarzschild metric. This is not true in the scalar- +tensor theories, because the scalar potential and the non-minimally +coupling function 𝐴(𝜑) have non-trivial effects on the NS beyond the +Figure 2. The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR +and SLy EoSs, for 𝛼 = 1 +surface of the star (scalarization). The Einstein frame TOV equations +take the following form, +𝑑𝑚 +𝑑𝑟 = 4𝜋𝑟2𝐴4(𝜑)𝜀 + 𝑟 +2 (𝑟 − 2𝑚(𝑟))𝜔2 + 4𝜋𝑟2𝑉(𝜑) , +(27) +𝑑𝜈 +𝑑𝑟 = 𝑟𝜔2+ +2 +𝑟(𝑟 − 2𝑚(𝑟)) +� +4𝜋𝐴4(𝜑)𝑟3𝑃−4𝜋𝑉(𝜑)𝑟3� ++ +2𝑚(𝑟) +𝑟(𝑟 − 2𝑚(𝑟)) , +(28) +𝑑𝜔 +𝑑𝑟 = 4𝜋𝑟 𝐴4(𝜑) +𝑟 − 2𝑚(𝑟) +� +𝛼(𝜑)(𝜖 − 3𝑃) + 𝑟𝜔(𝜖 − 𝑃) +� +− 2𝜔(𝑟 − 𝑚(𝑟)) +𝑟(𝑟 − 2𝑚(𝑟)) +(29) ++ +8𝜋𝜔𝑟2𝑉(𝜑) + 𝑟 𝑑𝑉 (𝜑) +𝑑𝜑 +𝑟 − 2𝑚(𝑟) +, +𝑑𝑃 +𝑑𝑟 = −(𝜖 + 𝑃) +� 1 +2 +𝑑𝜈 +𝑑𝑟 + 𝛼(𝜑)𝜔 +� +, +(30) +𝜔 = 𝑑𝜑 +𝑑𝑟 , +(31) +with 𝛼(𝜑) being defined in Eq. (22). Also note that the energy density +𝜖 and the pressure 𝑃 of the matter fluid are Jordan frame quantities. +We shall solve the TOV equations for both the interior and the exterior +of the NS, with the following set of initial conditions being used, +𝑃(0) = 𝑃𝑐 , 𝑚(0) = 0 , 𝜈(0) , = −𝜈𝑐 , 𝜑(0) = 𝜑𝑐 , 𝜔(0) = 0 . +(32) +Both 𝜈𝑐 and 𝜑𝑐 will be determined using a double shooting method, +and the numerical analysis shall be performed for three distinct piece- +wise polytropic EoS, with the central part being described by the +SLy, WFF1 or the APR EoS. For the calculation of the ADM mass +in the Jordan frame we shall use the following definition Odintsov & +Oikonomou (2021, 2022a); Oikonomou (2021), +𝑀 = 𝐴(𝜑(𝑟𝐸)) +� +𝑀𝐸 − +𝑟2 +𝐸 +2 𝛼(𝜑(𝑟𝐸)) 𝑑𝜑 +𝑑𝑟 +� +2 + 𝛼(𝜑(𝑟𝐸))𝑟𝐸 +𝑑𝜑 +𝑑𝑟 +� � +1 − 2𝑀𝐸 +𝑟𝐸 +�� +. +(33) +where 𝑟𝐸 denotes the Einstein frame circumferential radius of the +NS, and also we define 𝑑𝜑 +𝑑𝑟 = 𝑑𝜑 +𝑑𝑟 +���𝑟=𝑟𝐸 +. Finally, the circumferential +MNRAS 000, 1–8 (0000) + +MM -R Diagramm +25 +WFF1 EoS a=1 +APR EoS a=1 +2D +SLy EoSa=1 +15 +LD +0.5 +0.D +9 +1f +11 +12 +13 +R (krm)𝑅𝑝 Attractors Static Neutron Star Phenomenology +5 +Figure 3. The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR +and SLy EoSs, for 𝛼 = 8. +radii of the NS in the Jordan and Einstein frames are related as +𝑅 = 𝐴(𝜑(𝑅𝑠)) 𝑅𝑠. We shall measure the Jordan frame mass in solar +masses 𝑀⊙ and the Jordan frame radius in kilometers. +2.1 Results of the Numerical Analysis +Let us now present the results of our numerical analysis on the NS +phenomenology of the 𝑅𝑝 attractors. We considered three character- +istic cases of attractors, corresponding to three values of 𝛼, namely +𝛼 = 1, 𝛼 = 0.1 and 𝛼 = 8. All these values of 𝛼 produce a vi- +able inflationary phenomenology as was shown in Ref. Odintsov & +Oikonomou (2022b). Here we shall present the 𝑀 − 𝑅 graphs for the +𝑅𝑝 attractors for the three values of 𝛼. Accordingly the results will +be confronted with three distinct constraints on NS radii for specific +mass NS. Specifically we shall use the following constraints, devel- +oped in Refs. Altiparmak, Ecker & Rezzolla (2022), Raaijmakers et +al. (2021) and Bauswein, et al. (2017) to which we shall refer to as +CSI, CSII and CSIII respectively. The CSI indicates that the radius of +an 1.4𝑀⊙ mass NS should be 𝑅1.4𝑀⊙ = 12.42+0.52 +−0.99 and furthermore, +the radius of an 2𝑀⊙ mass NS should be 𝑅2𝑀⊙ = 12.11+1.11 +−1.23 km. Ac- +cordingly, CSII indicates that the radius of an 1.4𝑀⊙ mass NS should +be 𝑅1.4𝑀⊙ = 12.33+0.76 +−0.81 km. Lastly, CSIII indicates that the radius of +an 1.6𝑀⊙ mass NS should be larger than 𝑅1.6𝑀⊙ = 12.42+0.52 +−0.99 km +and the radius of a NS with maximum mass should be larger than +𝑅𝑀𝑚𝑎𝑥 > 10.68+0.15 +−0.04 km. The constraints CSI, CSII and CSIII are +pictorially represented in Fig. 11. Using a double shooting LSODA +python 3 numerical integration method Stergioulas (2019), and also +by setting the numerical infinity at 𝑟 ∼ 67.943 km, at this point we +shall present our results, which can be seen in the 𝑀 − 𝑅 plots and +the tables appearing in this work. Note that the numerical infinity +plays an important role for the double shooting method, in order for +the scalar field effects to be switched off at the numerical infinity. +To start with, in Figs. 2, 4 and 3 we present the 𝑀 − 𝑅 graphs of +the 𝑅𝑝 attractors for 𝛼 = 1, 𝛼 = 0.1 and 𝛼 = 8 NS respectively, for +1 This media was originally created by the European Southern Observatory +(ESO). I edited the figure for demonstrative purposes. Their website states: +”Unless specifically noted, the images, videos, and music distributed on the +public ESO website, along with the texts of press releases, announcements, +pictures of the week, blog posts and captions, are licensed under a Creative +Commons Attribution 4.0 International License, and may on a non-exclusive +basis be reproduced without fee provided the credit is clear and visible.” +Figure 4. The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR +and SLy EoSs, for 𝛼 = 0.1. +Figure 5. The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve), +𝛼 = 0.1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for +the WFF1 EoS. +the WFF1 EoS (red curve), the APR EoS (green curve) and the SLy +EoS (blue curve). In all the cases, the maximum masses of the NS +are larger compared to the GR case. Also it is notable that the 𝛼 = 1 +case is quite similar to the 𝛼 = 8 case, however strong differences are +observed for the 𝛼 = 0.1 case. Also in Figs. 5, 6 and 7 we present +for each EoS the 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red +curves), 𝛼 = 0.1 (green curves), 𝛼 = 8 (blue curves) and the GR +(magenta curves) for the WFF1 EoS (upper left plot) the SLy EoS +(upper right) and the APR EoS (bottom plot). Now let us present the +confrontation of the 𝑅𝑝 attractor NS with the constraints CSI, CSII +and CSIII. +The results of our analysis regarding the confrontation of the 𝑅𝑝 +inflationary attractors models with the observational constraints on +NS, namely CSI, CSII, AND CSIII are presented in Tables 1-5. +For the case with 𝛼 = 1, the SLy EoS is compatible with all the +constraints, with regard to the APR, it is not compatible with CSII, the +first constraint of CSI, but it is compatible with the second constraint +of CSII and the CSIII constraints. Also the WFF1 case is incompatible +with all the constraints. For the case with 𝛼 = 0.1, the SLy EoS is +compatible with all the constraints, and interestingly enough, for this +case the APR is also compatible with all the constraints. However, +in this case the WFF1 EoS satisfies the second constraint of CSI and +also satisfies all the constraints of CSIII. Finally, for the case with +𝛼 = 1, the SLy EoS is compatible with all the constraints, with regard +MNRAS 000, 1–8 (0000) + +MM -R Diagramm +25 +WFF1 EoS a=8 +APR EoS a=8 +2D +SLy EoS a=8 +15 +LD +0.5 +0.D +9 +1f +11 +12 +13 +R (krm)MM -R Diagramm +25 +*- WFF1 EoS a=0.1 +*- APR EoS a=0.1 +2D +SLy EoS a=0.1 +15 +LD +0.5 +0.0 +9 +1f +11 +12 +13 +R (krm)MM -R Diagramm +25 +*-WFF1EoSa=1 +WFF1 EoS a=0.1 +2D +WFF1 EoSa=8 +*- WFF1EoS GR +15 +LD +0.5 +0.D +9 +1f +11 +12 +13 +R (krm)6 +Oikonomou +Figure 6. The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve), +𝛼 = 0.1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for +the SLy EoS . +Table 3. CSIII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for +NS Masses 𝑀 ∼ 1.6𝑀⊙ +𝑅𝑝 Attractors Model +APR +SLy +WFF1 +𝛼 = 1 +𝑀 = 1.60 𝑀⊙ +𝑀 = 1.60 𝑀⊙ +𝑀 = 1.61 𝑀⊙ +𝛼 = 1 +𝑅 = 11.30km +𝑅 = 11.63km +𝑅 = 10.41km +𝛼 = 0.1 +𝑀 = 1.61 𝑀⊙ +𝑀 = 1.60 𝑀⊙ +𝑀 = 1.59 𝑀⊙ +𝛼 = 0.1 +𝑅 = 11.61km +𝑅 = 12.05km +𝑅 = 11.05km +𝛼 = 8 +𝑀 = 1.61 𝑀⊙ +𝑀 = 1.60 𝑀⊙ +𝑀 = 1.58 𝑀⊙ +𝛼 = 8 +𝑅 = 11.28km +𝑅 = 12.05km +𝑅 = 10.40km +to the APR, it is not compatible with CSII, and the first constraint of +CSI, but it is compatible with the second constraint of CSII and the +CSIII constraints. +Also the WFF1 case is incompatible with all the constraints, save +the first constraint of CSIII. Hence, the viable NS phenomenologies +that pass all the tests imposed by the constraints CSI, CSII and CSIII, +are provided by all the SLy cases for all the values of the parameter +𝛼, and also by the APR EoS, only when 𝛼 = 0.1. Thus apparently, +obtaining a viable NS phenomenology nowadays is not as easy it was +before the GW170817 event. Also regarding the 𝑅𝑝 attractors, these +can be discriminated in NS, for different values of 𝛼, especially for +0.1 < 𝛼 < 1. However, as 𝛼 grows larger than unity, it seems that +𝑅𝑝 attractors provide an almost identical NS phenomenology. This +is a notable feature for the class of 𝑅𝑝 attractors. Before closing, +we need to discuss an important issue, having to do with the NS +phenomenology of inflationary potentials, with regard to the tidal +deformability of NSs, the radial perturbations of static NSs and finally +the overall stability of NSs, by also taking into account the constraints +imposed by the GW170817 event. This issue however extends further +from the aims and scopes of this article, since a whole article could +be devoted to these issues, see for example Refs. Brown (2022) and +Yang et al. (2022), in which these issues are addressed in the context +of scalar-tensor gravity Brown (2022) and in unimodular gravity +Yang et al. (2022). +Figure 7. The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve), +𝛼 = 0.1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for +the APR EoS. +CONCLUDING REMARKS +In this article we studied the NS phenomenology of the 𝑅𝑝 infla- +tionary attractor scalar-tensor models in the Einstein frame. The 𝑅𝑝 +attractors constitute a class of models in the Einstein frame, which +originate from a large number of different models in the Jordan frame +These distinct Jordan frame models result to the same phenomenol- +ogy in the Einstein frame and this feature justifies the terminology +inflationary attractors. Our aim was to investigate whether these at- +tractor models can be distinguished when NSs are considered. As +we showed the NS phenomenology corresponding to different values +of the parameter 𝛼 which characterizes the attractors, is in general +different for 𝛼 < 1, however the models for 𝛼 > 1 show many +similarities and generate almost identical 𝑀 − 𝑅 diagrams. We also +confronted the NS phenomenology of the 𝑅𝑝 attractors to several +NS constraints, which we named CSI, CSII and CSIII. The con- +straint CSI was developed in Ref. Altiparmak, Ecker & Rezzolla +(2022) and indicates that the radius of an 1.4𝑀⊙ mass NS has to +be 𝑅1.4𝑀⊙ = 12.42+0.52 +−0.99 while the radius of an 2𝑀⊙ mass NS has +to be 𝑅2𝑀⊙ = 12.11+1.11 +−1.23 km. The constraint CSII was developed +in Ref. Raaijmakers et al. (2021) and indicates that the radius of +an 1.4𝑀⊙ mass NS has to be 𝑅1.4𝑀⊙ = 12.33+0.76 +−0.81 km and the +constraint CSIII was developed in Ref. Bauswein, et al. (2017) and +indicates that the radius of an 1.6𝑀⊙ mass NS has to be larger than +𝑅1.6𝑀⊙ = 12.42+0.52 +−0.99 km while the radius of the maximum mass NS +has to be larger than 𝑅𝑀𝑚𝑎𝑥 > 10.68+0.15 +−0.04 km. Our analysis indicated +that for 𝑅𝑝 attractors, for the case with 𝛼 = 1, only the SLy EoS is +compatible with all the constraints, while the APR is not compatible +with CSII, the first constraint of CSI, but it is compatible with the +second constraint of CSII and the CSIII constraints. Also the WFF1 +case is incompatible with all the constraints. +For the case with 𝛼 = 0.1, which is the most interesting case phe- +nomenologically, the SLy EoS is compatible with all the constraints, +and for this case the APR is also compatible with all the constraints. +However, in this case the WFF1 EoS satisfies the second constraint +of CSI and also satisfies all the constraints of CSIII. Finally, for +the case with 𝛼 = 1, only the SLy EoS is compatible with all the +constraints while the APR is not compatible with CSII, and the first +constraint of CSI, but it is compatible with the second constraint of +CSII and the CSIII constraints. Finally, the WFF1 case is incompat- +ible with all the constraints, save the first constraint of CSIII. Our +results indicate two main research lines, firstly that NS phenomenol- +MNRAS 000, 1–8 (0000) + +MM -R Diagramm +25 +SLy EoS a=1 +SLy EoS a=0.1 +2D +SLy EoS a=8 +SLy EoS GR +15 +LD +0.5 +0.D +9 +11 +12 +13 +R (kri)MM -R Diagramm +25 +APR EoS a=1 +APR EoS a=0.1 +2D +APR EoS a=8 +APR EoS GR +15 +LD +0.5 +0.D +9 +11 +12 +13 +R (kri)𝑅𝑝 Attractors Static Neutron Star Phenomenology +7 +Table 4. CSII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for +NS Masses 𝑀 ∼ 1.4𝑀⊙ +𝑅𝑝 Attractors Model +APR +SLy +WFF1 +𝛼 = 1 +𝑀 = 0.52 𝑀⊙ +𝑀 = 1.41 𝑀⊙ +𝑀 = 0.25 𝑀⊙ +𝛼 = 1 +𝑅 = 11.56km +𝑅 = 11.74km +𝑅 = 11.89km +𝛼 = 0.1 +𝑀 = 1.39 𝑀⊙ +𝑀 = 1.39 𝑀⊙ +𝑀 = 0.07 𝑀⊙ +𝛼 = 0.1 +𝑅 = 11.55km +𝑅 = 12.04km +𝑅 = 11.79km +𝛼 = 8 +𝑀 = 0.53 𝑀⊙ +𝑀 = 1.42 𝑀⊙ +𝑀 = 0.25 𝑀⊙ +𝛼 = 8 +𝑅 = 11.60km +𝑅 = 11.738km +𝑅 = 11.944km +Table 5. CSIII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for +Maximum NS Masses +𝑅𝑝 Attractors Model +APR +SLy +WFF1 +𝛼 = 1 +𝑀 = 2.41 𝑀⊙ +𝑀 = 2.24 𝑀⊙ +𝑀 = 2.33 𝑀⊙ +𝛼 = 1 +𝑅 = 9.91km +𝑅 = 9.99km +𝑅 = 9.30km +𝛼 = 0.1 +𝑀 = 2.41 𝑀⊙ +𝑀 = 2.27 𝑀⊙ +𝑀 = 2.32 𝑀⊙ +𝛼 = 0.1 +𝑅 = 10.40km +𝑅 = 10.09km +𝑅 = 11.06km +𝛼 = 8 +𝑀 = 2.41 𝑀⊙ +𝑀 = 2.27 𝑀⊙ +𝑀 = 2.34 𝑀⊙ +𝛼 = 8 +𝑅 = 9.91km +𝑅 = 10.72km +𝑅 = 9.28km +ogy for scalar-tensor theories is not easily rendered viable, since a +large number of astrophysical and cosmological constraints have to +be satisfied in order for the viability of the model to be guaranteed. +Thus a simple parameter assigning is not the correct way to study +NS nowadays, both cosmology and astrophysics constrain in a rigid +way NSs. Secondly, several inflationary attractors which are indistin- +guishable at the cosmological level, may be discriminated to some +extent when their NS phenomenology is considered. This research +line is not the general rule though, so work is in progress toward +comparing a large sample of cosmological attractors with respect to +their NS phenomenology. Finally, let us note that the scalar-tensor +inflationary framework we used in this work cannot be considered +more advantageous compared to other modified gravity theories, it is +one of the many possible modified gravity descriptions of the nature +of NSs. +ACKNOWLEDGMENTS +This work was supported by MINECO (Spain), project PID2019- +104397GB-I00 (S.D.O). This work by S.D.O was also partially +supported by the program Unidad de Excelencia Maria de Maeztu +CEX2020-001058-M, Spain. +Data availability. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Oikonomou1,2 1 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece 2 Institut für Theoretische Physik, Goethe Universität Frankfurt, Max-von-Laue-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1, 60438 Frankfurt am Main, Germany 31 January 2023 ABSTRACT In this work we study the neutron star phenomenology of 𝑅𝑝 attractor theories in the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The Einstein frame 𝑅𝑝 attractor theories have the attractor property that they originate from a large class of Jordan frame scalar theories with arbitrary non-minimal coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' These theories in the Einstein frame provide a viable class of inflationary models, and in this work we investigate their implications on static neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We numerically solve the Tolman-Oppenheimer-Volkoff equations in the Einstein frame, for three distinct equations of state, and we provide the mass-radius diagrams for several cases of interest of the 𝑅𝑝 attractor theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We confront the results with several timely constraints on the radii of specific mass neutron stars, and as we show, only a few cases corresponding to specific equations of state pass the stringent tests on neutron stars phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Key words: stars: neutron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Physical Data and Processes, cosmology: theory INTRODUCTION The direct gravitational wave observation GW170817 LIGO & Virgo Collaboration, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2017, 2020) initiated what is nowadays known as gravitational wave astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Neutron stars (NS) Haensel, Potekhin & Yakovlev (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kanakis- Pegios, Koliogiannis & Moustakidis (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Tsaloukidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2022), high energy physics Buschmann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Safdi, Sun & Chen (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Hook, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Edwards, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Nurmi, Schi- appacasse & Yanagida (2021), modified gravity Astashenok, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020, 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Capozziello, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Astashenok, Capozziello & Odintsov (2015, 2014, 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Arapoˇglu, Deliduman & Eksi (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Panotopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Lobato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Numajiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2022) and astrophysics Altiparmak, Ecker & Rezzolla (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Bauswein, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Vretinaris, Stergioulas & Bauswein (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Bauswein, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020a, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Most, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Rezzolla, Most & Weih (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Nathanail, Most & Rezzolla (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Köppel, Bovard & Rezzolla (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Raaijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Most, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ecker & Rezzolla (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Jiang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The perspective of modified gravity implications on NS has been for a long time in the mainstream of NS works, see for example Astashenok, Capozziello & Odintsov (2015, 2014) and also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Pani & Berti (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Staykov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Horbatsch, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Silva, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Doneva, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Xu, Gao & Shao (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Salgado, Sudarsky & Nucamendi (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Shibata, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Arapoğlu, Ekşi & Yükselci (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ramazanoğlu & Pretorius (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Motahar, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Chew, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Blázquez-Salcedo, Scen Khoo & Kunz (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Motahar, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Odintsov & Oikonomou (2021, 2022a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Oikonomou (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Pretel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Pretel & Duarte (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cuzinatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2016) for scalar-tensor descriptions of NS phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The main effect of modified gravity descriptions of NS is the significant elevation of the maximum NS masses, with modified gravity bring- ing this maximum mass near or inside the mass-gap region with 𝑀 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Regarding non-minimally coupled scalar field theo- ries, there exists a vast class of viable inflationary potentials which have the remarkable property of being attractors Kallosh, Linde & Roest (2014a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh & Linde (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ferrara, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh, Linde & Roest (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Linde (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cecotti & Kallosh (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Carrasco, Kallosh & Linde (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Carrasco, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh, Linde & Roest (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Roest & Scalisi (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh, Linde & Roest (2014b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ellis, Nanopoulos & Olive (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cai, Gong & Pi (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Yi & Gong (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Akrami, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Qum- mer, Jawad & Younas (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Fei, Yi & Yang (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kanfon, Mavoa & Houndjo (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Antoniadis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' García-García, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cedeño, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Karamitsos (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Canko, Gialamas & Kodaxis (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Miranda, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Karam, Pappas & Tam- vakis (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Nozari & Rashidi (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' García-García, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Rashidi & Nozari (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Gao, Gong & Fei (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Dimopoulos, Wood & Owen (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Miranda, Fabris & Piattella (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Karam, Pappas & Tamvakis (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Nozari & Rashidi (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Gao & Gong (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Geng, Lee & Wu (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Odintsov & Oikonomou (2020, 2016, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Järv, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The attractor terminology is justi- fied due to the fact that distinct non-minimally coupled scalar-tensor inflationary theories, lead to the same Einstein frame inflationary phenomenology, which is compatible with the latest Planck data Planck Collaboration (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The question always when studying these attractor models is whether these models can be distinguished in some way, phenomenologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' From an inflationary point of view, and regarding the large wavelength Cosmic Microwave Back- ground modes, a discrimination between these models is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' However, this discrimination is possible if NS are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Indeed, the phenomenologically indistinguishable attractor models can be discriminated in NS and vice versa, with the latter feature being phe- © 0000 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='12136v1 [gr-qc] 28 Jan 2023 2 Oikonomou nomenal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' That is, if some models are indistinguishable with respect to their NS phenomenology, they can be distinguished if their infla- tionary properties are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' To address these issues in a concrete way, in this work we shall study 𝑅𝑝 attractor theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The inflation- ary phenomenology of these theories is studied in the recent literature Odintsov & Oikonomou (2022b) see also Motohashi (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Renzi, Shokri & Melchiorri (2009) for subcases of the original 𝑅𝑝 at- tractors theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' For a spherically symmetric metric we derive and solve numerically the Einstein frame Tolman-Oppenheimer-Volkoff (TOV) equations, using an LSODA based double shooting python 3 numerical integration Stergioulas (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We derive the Jordan frame 𝑀 −𝑅 graphs for the 𝑅𝑝 attractors, for three different piecewise poly- tropic Read, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2009a,b) equations of state (EoS), WFF1 Wiringa, Fiks & Fabrocini (1988), the SLy Douchin & Haensel (2001), and the APR EoS Akmal, Pandharipande & Ravenhall (1998), using the Arnowitt-Deser-Misner (ADM) definition of Jordan frame masses of NS Arnowitt, Deser & Misner (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The NSs temperature is significantly lower than the Fermi energy of the constituent particles of NSs, thus NS matter can be in principle described by a single- parameter EoS that may describe perfectly cold matter at densities higher than the nuclear density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' However, a serious problem emerges, having to do with the uncertainty in the EoS, which is larger, and the pressure as a function of the baryonic mass density cannot be accurately defined and is uncertain to one order of magnitude at least above the nuclear density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Moreover, the exact nature of the phase of matter at the NSs core is highly uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Hence, a parameterized- type EoS at high densities is an optimal choice for an EoS, thus rendering the piecewise polytropic EoS a suitable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' In order to construct the piecewise polytropic EoS, astrophysical constraints are taken into account, both observational and theoretical, like the causality constraints, see Read, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2009a,b), to also confirm the causality fulfilment for all the piecewise polytropic EoS we shall use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' For the construction of the piecewise polytropic EoS one uses a low-density part with 𝜌 < 𝜌0, which is basically chosen to be a tabulated and well-known EoS for the crust, and furthermore, the piecewise polytropic EoS also has a large density part with 𝜌 ≫ 𝜌0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We finally confront the resulting NS phenomenologies with several recent constraints on the radii of specific mass NS Altiparmak, Ecker & Rezzolla (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Raaijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Bauswein, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2017) and as we show, only a few scenarios and EoS are compatible with the constraints on NS radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Obviously, the gravitational wave astron- omy era has changed the way of thinking on theoretical astrophysics, since several models of scalar-tensor gravity which in the recent past could be considered as viable, nowadays may no longer be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 1 INFLATIONARY PHENOMENOLOGY OF 𝑅𝑃 ATTRACTORS The full analysis of the generalized 𝑅𝑝 attractors is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Odintsov & Oikonomou (2022b), so we refer the reader for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Here we shall briefly discuss the inflationary phenomenological prop- erties of 𝑅𝑝 attractors in order to stress their importance among other cosmological attractors Kallosh, Linde & Roest (2014a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh & Linde (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ferrara, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh, Linde & Roest (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Linde (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cecotti & Kallosh (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Carrasco, Kallosh & Linde (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Carrasco, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh, Linde & Roest (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Roest & Scalisi (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kallosh, Linde & Roest (2014b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ellis, Nanopou- los & Olive (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cai, Gong & Pi (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Yi & Gong (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Akrami, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Qummer, Jawad & Younas (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Fei, Yi & Yang (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Kanfon, Mavoa & Houndjo (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Antoniadis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' García-García, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Cedeño, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Karamitsos (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Canko, Gialamas & Kodaxis (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Miranda, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Karam, Pappas & Tamvakis (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Nozari & Rashidi (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' García-García, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Rashidi & Nozari (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Gao, Gong & Fei (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Dimopoulos, Wood & Owen (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Miranda, Fabris & Piattella (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Karam, Pappas & Tamvakis (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Nozari & Rashidi (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Gao & Gong (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Geng, Lee & Wu (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Odintsov & Oikonomou (2020, 2016, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Järv, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑅𝑝 attractors constitute a class of their own among other attrac- tors, and all the 𝑅𝑝 attractors in the Einstein frame correspond to generalizations of the following Einstein frame potential, 𝑉(𝜑) = 𝑉0 𝑀4 𝑝𝑒−2 √︃ 2 3 𝜅 𝜑 � 𝑒 √︃ 2 3 𝜅 𝜑 − 1 � 𝑝 𝑝−1 , (1) where 𝑀𝑝 = 1 √ 8𝜋𝐺 is the reduced Planck mass and 𝐺 is Newton’s gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The inflationary properties of the above theory have been addressed in the recent literature, see for example Moto- hashi (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Renzi, Shokri & Melchiorri (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The scalar-tensor theory with the potential (1) corresponds to the Jordan frame 𝐹(𝑅) gravity, 𝐹(𝑅) = 𝑅 + 𝛽𝑅𝑝 , (2) with 𝛽 is a free parameter with its physical dimensions in natural units being [𝛽] = [𝑚]2−2𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑅𝑝 attractors have the following scalar potential in the Einstein frame, 𝑉(𝜑) = 𝑉0 𝑀4 𝑝𝑒−2 √︃ 2 3𝛼 𝜅 𝜑 � 𝑒 √︃ 2 3𝛼 𝜅 𝜑 − 1 � 𝑝 𝑝−1 , (3) where 𝑀𝑝 is the reduced Planck mass, and for 𝛼 = 1 we obtain the scalar theory with scalar potential (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Now the question is why these models are classified as attractor models, what justifies the terminology attractors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' It is the class of scalar-tensor Jordan frame theories which correspond to the Einstein frame potential (3) that justify the use of the terminology attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Basically, the potential (3) can be the Einstein frame potential for a large class of Jordan frame scalar-tensor theories, as we now evince.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝜙-Jordan frame action is, S𝐽 = ∫ 𝑑4𝑥 � Ω(𝜙) 2𝜅2 𝑅 − 𝜔(𝜙) 2 𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 − 𝑉𝐽 (𝜙) � , (4) with the scalar field describing a non-canonical scalar field in the Jordan frame, and the coupling function has the general form Ω(𝜙) = 1+𝜉 𝑓 (𝜙) with 𝜉 and 𝑓 (𝜙) being the arbitrary dimensionless coupling and an arbitrary dimensionless function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑅𝑝 attractors have the following 𝜙-Jordan frame scalar potential, 𝑉𝐽 (𝜙) = 𝑉0 (Ω(𝜙) − 1) 𝑝 𝑝−1 , (5) and more importantly, the kinetic term function 𝜔(𝜙) has the follow- ing form, 𝜔(𝜙) = 1 4𝜉 � 𝑑Ω(𝜙) 𝑑𝜙 �2 Ω(𝜙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (6) Hence the large class of the 𝑅𝑝-attractors correspond to the Jordan frame theories which are described by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Notice that the Jordan frame functions 𝑓 (𝜙) are arbitrary and we shall not need to specify these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' By performing the conformal transformation of the Jordan frame metric 𝑔𝜇𝜈, ˜𝑔𝜇𝜈 = Ω(𝜙)𝑔𝜇𝜈 , (7) MNRAS 000, 1–8 (0000) 𝑅𝑝 Attractors Static Neutron Star Phenomenology 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The constraints CSI, CSII and CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' This figure is inspired and based after editing on Credit: ESO/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='Calçada: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='org/ public/images/eso0831a/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' we get the Einstein frame action, S𝐸 = √︁ − ˜𝑔 � ˜𝑅 2𝜅2 − ˜𝑔𝜇𝜈𝜕𝜇𝜑𝜕𝜈𝜑 − 𝑉(𝜑) � , (8) with ˜𝑔𝜇𝜈 denoting the Einstein frame metric tensor, and the “tilde” indicates Einstein frame quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also the Einstein frame potential 𝑉(𝜙) and the Jordan frame potential 𝑉𝐽 (𝜙) are related as follows, 𝑉(𝜑) = Ω−2(𝜙)𝑉𝐽 (𝜙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (9) Notice that the general relation which connects the Jordan frame scalar field 𝜙 with the canonical Einstein frame scalar field 𝜑 is, � 𝑑𝜑 𝑑𝜙 �2 = 3 2 � 𝑑Ω(𝜙) 𝑑𝜙 �2 Ω(𝜙) + 𝜔(𝜙) Ω(𝜙) , (10) hence for the 𝑅𝑝 attractors, in which case the kinetic term function 𝜔(𝜙) is chosen to be that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (6), we finally have the important relation of the non-minimal scalar coupling function to gravity, Ω(𝜙) = 𝑒 √︃ 2 3𝛼 𝜑 , (11) with the parameter 𝛼 being defined to be, 𝛼 = 1 + 1 6𝜉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (12) Notice that by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (11) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (9) we obtain the gen- eralized 𝑅𝑝-attractor potential of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Furthermore, the impor- tant case with 𝛼 = 1 is realized when 𝜉 → ∞, or similarly when Ω(𝜙) ≪ 3 2 � 𝑑Ω(𝜙) 𝑑𝜙 �2 𝜔(𝜙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑅𝑝 attractors yield a viable inflationary phenomenology, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Odintsov & Oikonomou (2022b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' with the spectral index of the primordial scalar perturbations as a function of the canonical scalar field being,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 𝑛𝑠 = � � 3𝛼 + (3𝛼 − 2)𝑝2 + (8 − 6𝛼)𝑝 − 8 � 𝑒2 √︃ 2 3 √︃ 1 𝛼 𝜅 𝜑 (13) − 2(𝑝 − 1)(−3𝛼 + (3𝛼 − 2)𝑝 + 8)𝑒 √︃ 2 3 √︃ 1 𝛼 𝜅 𝜑 + (3𝛼 − 8)(𝑝 − 1)2� × 3𝛼(𝑝 − 1)2 � 𝑒 √︃ 2 3 √︃ 1 𝛼 𝜅 𝜑 − 1 �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' and the tensor-to-scalar ratio is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 𝑟 = 16 � (𝑝 − 2)𝑒 √︃ 2 3 √︃ 1 𝛼 𝜅 𝜑 − 2𝑝 + 2 �2 3𝛼(𝑝 − 1)2 � 𝑒 √︃ 2 3 √︃ 1 𝛼 𝜅 𝜑 − 1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (14) Also the free parameter 𝑉0 of the potential is constrained to have values 𝑉𝑠 ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='6 × 10−11 , (15) a results which originates from the constraints of the Planck data on the Einstein frame amplitude Δ2𝑠 of the scalar perturbations, Δ2 𝑠 = 1 24𝜋2 𝑉(𝜑 𝑓 ) 𝑀4𝑝 1 𝜖(𝜑 𝑓 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (16) For the purposes of this paper, we shall consider several limiting cases for the values of the parameter 𝛼, mainly the cases 𝛼 ≠ 1, and the case 𝛼 = 1, which corresponds to the strong 𝜉 coupling theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also in order to have a viable inflationary phenomenology, the parameter 𝑝 which is the exponent in the 𝑅𝑝 attractors potential, has to take values in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='91 ≤ 𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' It proves that this is irrelevant for NS studies, so we shall assume that 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='91 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' In the next section we shall specify the values of the various functions involved in the TOV equations of NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 2 NEUTRON STARS WITH 𝑅𝑃 ATTRACTORS For the purpose of studying NS in Einstein frame, we shall use the Geometrized physical units system 𝐺 = 𝑐 = 1, and we shall adopt the notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Pani & Berti (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The Jordan frame scalar-tensor theory has the following form, S = ∫ 𝑑4𝑥 √−𝑔 16𝜋 � Ω(𝜙)𝑅 − 1 2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 −𝑈(𝜙) � + 𝑆𝑚(𝜓𝑚, 𝑔𝜇𝜈) , (17) and by performing the following conformal transformation, ˜𝑔𝜇𝜈 = 𝐴−2𝑔𝜇𝜈 , 𝐴(𝜙) = Ω−1/2(𝜙) , (18) we obtain the Einstein frame action, S = ∫ 𝑑4𝑥 √︁ − ˜𝑔 � ˜𝑅 16𝜋 −1 2 ˜𝑔𝜇𝜈𝜕𝜇𝜑𝜕𝜈𝜑−𝑉(𝜑) 16𝜋 � +𝑆𝑚(𝜓𝑚, 𝐴2(𝜑)𝑔𝜇𝜈) , (19) with 𝜑 denoting the Einstein frame canonical scalar field as in the previous section, and 𝑉(𝜑) = 𝑈(𝜙) Ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (20) For the 𝑅𝑝 attractors with general 𝛼, the important function 𝐴(𝜑) has the following form, 𝐴(𝜑) = 𝑒− 1 2 √︃ 2 3𝛼 𝜑 , (21) therefore, the function 𝛼(𝜙) which is defined as follows, 𝛼(𝜑) = 𝑑 ln 𝐴(𝜑) 𝑑𝜑 , (22) takes the form, 𝑎(𝜑) = −1 2 √︂ 2 3𝛼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (23) MNRAS 000, 1–8 (0000) CS I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99 CS II R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4Mo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='81 CS III 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='034 Oikonomou Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' CSI vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for NS Masses 𝑀 ∼ 2𝑀⊙ 𝑅𝑝 Attractor Model APR SLy WFF1 𝛼 = 1 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='00 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='01 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='31 𝑀⊙ 𝛼 = 1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='10km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='17km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='06km 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='02 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='00 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='00 𝑀⊙ 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='52km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='818km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='012km 𝛼 = 8 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='00 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='09 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='32 𝑀⊙ 𝛼 = 8 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='08km 𝑅 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='983km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='114km Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' CSI vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for NS Masses 𝑀 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ 𝑅𝑝 Attractors Model APR SLy WFF1 𝛼 = 1 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='58 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='41 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='25 𝑀⊙ 𝛼 = 1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='48km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='74km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='89km 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='39 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='39 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='07 𝑀⊙ 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='55km 𝑅 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='04km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='79km 𝛼 = 8 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='64 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='42 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='28 𝑀⊙ 𝛼 = 8 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='45km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='73km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='46km Finally, the Einstein frame scalar potential is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (3), which we also quote it here for reading convenience, 𝑉(𝜑) = 𝑉0 𝑒−2 √︃ 2 3𝛼 𝜑 � 𝑒 √︃ 2 3𝛼 𝜑 − 1 � 𝑝 𝑝−1 , (24) and in Geometrized units, the constraint on 𝑉0 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (15) becomes, 𝑉0 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='62 × 10−12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (25) For the study of NS physics, we shall consider the following spheri- cally symmetric metric, 𝑑𝑠2 = −𝑒𝜈(𝑟)𝑑𝑡2 + 𝑑𝑟2 1 − 2𝑚(𝑟) 𝑟 + 𝑟2(𝑑𝜃2 + sin2 𝜃𝑑𝜙2) , (26) which describes a static NS, where the function 𝑚(𝑟) describes the total gravitational mass of the NS and 𝑟 stands for the circumferential radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' In the following, we shall calculate numerically the functions 𝜈(𝑟) and 1 1− 2𝑚(𝑟) 𝑟 following a simple procedure, in which the central value of 𝜈(𝑟) and of the scalar field will be arbitrary and will be optimally calculated numerically by using a double shooting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The double shooting aims to find the optimal values of the central values of 𝜈(𝑟) and of the scalar field, which guarantee that the metric at numerical infinity becomes identical to the Schwarzschild metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' This procedure is different compared to standard General Relativity (GR) NS, because in GR, the metric at the surface of the star abruptly becomes the Schwarzschild metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' This is not true in the scalar- tensor theories, because the scalar potential and the non-minimally coupling function 𝐴(𝜑) have non-trivial effects on the NS beyond the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR and SLy EoSs, for 𝛼 = 1 surface of the star (scalarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The Einstein frame TOV equations take the following form, 𝑑𝑚 𝑑𝑟 = 4𝜋𝑟2𝐴4(𝜑)𝜀 + 𝑟 2 (𝑟 − 2𝑚(𝑟))𝜔2 + 4𝜋𝑟2𝑉(𝜑) , (27) 𝑑𝜈 𝑑𝑟 = 𝑟𝜔2+ 2 𝑟(𝑟 − 2𝑚(𝑟)) � 4𝜋𝐴4(𝜑)𝑟3𝑃−4𝜋𝑉(𝜑)𝑟3� + 2𝑚(𝑟) 𝑟(𝑟 − 2𝑚(𝑟)) , (28) 𝑑𝜔 𝑑𝑟 = 4𝜋𝑟 𝐴4(𝜑) 𝑟 − 2𝑚(𝑟) � 𝛼(𝜑)(𝜖 − 3𝑃) + 𝑟𝜔(𝜖 − 𝑃) � − 2𝜔(𝑟 − 𝑚(𝑟)) 𝑟(𝑟 − 2𝑚(𝑟)) (29) + 8𝜋𝜔𝑟2𝑉(𝜑) + 𝑟 𝑑𝑉 (𝜑) 𝑑𝜑 𝑟 − 2𝑚(𝑟) , 𝑑𝑃 𝑑𝑟 = −(𝜖 + 𝑃) � 1 2 𝑑𝜈 𝑑𝑟 + 𝛼(𝜑)𝜔 � , (30) 𝜔 = 𝑑𝜑 𝑑𝑟 , (31) with 𝛼(𝜑) being defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also note that the energy density 𝜖 and the pressure 𝑃 of the matter fluid are Jordan frame quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We shall solve the TOV equations for both the interior and the exterior of the NS, with the following set of initial conditions being used, 𝑃(0) = 𝑃𝑐 , 𝑚(0) = 0 , 𝜈(0) , = −𝜈𝑐 , 𝜑(0) = 𝜑𝑐 , 𝜔(0) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (32) Both 𝜈𝑐 and 𝜑𝑐 will be determined using a double shooting method, and the numerical analysis shall be performed for three distinct piece- wise polytropic EoS, with the central part being described by the SLy, WFF1 or the APR EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' For the calculation of the ADM mass in the Jordan frame we shall use the following definition Odintsov & Oikonomou (2021, 2022a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Oikonomou (2021), 𝑀 = 𝐴(𝜑(𝑟𝐸)) � 𝑀𝐸 − 𝑟2 𝐸 2 𝛼(𝜑(𝑟𝐸)) 𝑑𝜑 𝑑𝑟 � 2 + 𝛼(𝜑(𝑟𝐸))𝑟𝐸 𝑑𝜑 𝑑𝑟 � � 1 − 2𝑀𝐸 𝑟𝐸 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (33) where 𝑟𝐸 denotes the Einstein frame circumferential radius of the NS, and also we define 𝑑𝜑 𝑑𝑟 = 𝑑𝜑 𝑑𝑟 ���𝑟=𝑟𝐸 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Finally, the circumferential MNRAS 000, 1–8 (0000) MM -R Diagramm 25 WFF1 EoS a=1 APR EoS a=1 2D SLy EoSa=1 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='D 9 1f 11 12 13 R (krm)𝑅𝑝 Attractors Static Neutron Star Phenomenology 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR and SLy EoSs, for 𝛼 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' radii of the NS in the Jordan and Einstein frames are related as 𝑅 = 𝐴(𝜑(𝑅𝑠)) 𝑅𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We shall measure the Jordan frame mass in solar masses 𝑀⊙ and the Jordan frame radius in kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 Results of the Numerical Analysis Let us now present the results of our numerical analysis on the NS phenomenology of the 𝑅𝑝 attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We considered three character- istic cases of attractors, corresponding to three values of 𝛼, namely 𝛼 = 1, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 and 𝛼 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' All these values of 𝛼 produce a vi- able inflationary phenomenology as was shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Odintsov & Oikonomou (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Here we shall present the 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractors for the three values of 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Accordingly the results will be confronted with three distinct constraints on NS radii for specific mass NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Specifically we shall use the following constraints, devel- oped in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Altiparmak, Ecker & Rezzolla (2022), Raaijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021) and Bauswein, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2017) to which we shall refer to as CSI, CSII and CSIII respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The CSI indicates that the radius of an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ mass NS should be 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99 and furthermore, the radius of an 2𝑀⊙ mass NS should be 𝑅2𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='11+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='23 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Ac- cordingly, CSII indicates that the radius of an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ mass NS should be 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='81 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Lastly, CSIII indicates that the radius of an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='6𝑀⊙ mass NS should be larger than 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='6𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99 km and the radius of a NS with maximum mass should be larger than 𝑅𝑀𝑚𝑎𝑥 > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='04 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The constraints CSI, CSII and CSIII are pictorially represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Using a double shooting LSODA python 3 numerical integration method Stergioulas (2019), and also by setting the numerical infinity at 𝑟 ∼ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='943 km, at this point we shall present our results, which can be seen in the 𝑀 − 𝑅 plots and the tables appearing in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Note that the numerical infinity plays an important role for the double shooting method, in order for the scalar field effects to be switched off at the numerical infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' To start with, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 2, 4 and 3 we present the 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 and 𝛼 = 8 NS respectively, for 1 This media was originally created by the European Southern Observatory (ESO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' I edited the figure for demonstrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Their website states: ”Unless specifically noted, the images, videos, and music distributed on the public ESO website, along with the texts of press releases, announcements, pictures of the week, blog posts and captions, are licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='0 International License, and may on a non-exclusive basis be reproduced without fee provided the credit is clear and visible.” Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR and SLy EoSs, for 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve), 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for the WFF1 EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' the WFF1 EoS (red curve), the APR EoS (green curve) and the SLy EoS (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' In all the cases, the maximum masses of the NS are larger compared to the GR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also it is notable that the 𝛼 = 1 case is quite similar to the 𝛼 = 8 case, however strong differences are observed for the 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' 5, 6 and 7 we present for each EoS the 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curves), 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 (green curves), 𝛼 = 8 (blue curves) and the GR (magenta curves) for the WFF1 EoS (upper left plot) the SLy EoS (upper right) and the APR EoS (bottom plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Now let us present the confrontation of the 𝑅𝑝 attractor NS with the constraints CSI, CSII and CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The results of our analysis regarding the confrontation of the 𝑅𝑝 inflationary attractors models with the observational constraints on NS, namely CSI, CSII, AND CSIII are presented in Tables 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' For the case with 𝛼 = 1, the SLy EoS is compatible with all the constraints, with regard to the APR, it is not compatible with CSII, the first constraint of CSI, but it is compatible with the second constraint of CSII and the CSIII constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also the WFF1 case is incompatible with all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' For the case with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1, the SLy EoS is compatible with all the constraints, and interestingly enough, for this case the APR is also compatible with all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' However, in this case the WFF1 EoS satisfies the second constraint of CSI and also satisfies all the constraints of CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Finally, for the case with 𝛼 = 1, the SLy EoS is compatible with all the constraints, with regard MNRAS 000, 1–8 (0000) MM -R Diagramm 25 WFF1 EoS a=8 APR EoS a=8 2D SLy EoS a=8 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='D 9 1f 11 12 13 R (krm)MM -R Diagramm 25 *- WFF1 EoS a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 *- APR EoS a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 2D SLy EoS a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='0 9 1f 11 12 13 R (krm)MM -R Diagramm 25 *-WFF1EoSa=1 WFF1 EoS a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 2D WFF1 EoSa=8 *- WFF1EoS GR 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='D 9 1f 11 12 13 R (krm)6 Oikonomou Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve), 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for the SLy EoS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' CSIII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for NS Masses 𝑀 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='6𝑀⊙ 𝑅𝑝 Attractors Model APR SLy WFF1 𝛼 = 1 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='60 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='60 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='61 𝑀⊙ 𝛼 = 1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='30km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='63km 𝑅 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='41km 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='61 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='60 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='59 𝑀⊙ 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='61km 𝑅 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='05km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='05km 𝛼 = 8 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='61 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='60 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='58 𝑀⊙ 𝛼 = 8 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='28km 𝑅 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='05km 𝑅 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='40km to the APR, it is not compatible with CSII, and the first constraint of CSI, but it is compatible with the second constraint of CSII and the CSIII constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also the WFF1 case is incompatible with all the constraints, save the first constraint of CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Hence, the viable NS phenomenologies that pass all the tests imposed by the constraints CSI, CSII and CSIII, are provided by all the SLy cases for all the values of the parameter 𝛼, and also by the APR EoS, only when 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Thus apparently, obtaining a viable NS phenomenology nowadays is not as easy it was before the GW170817 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also regarding the 𝑅𝑝 attractors, these can be discriminated in NS, for different values of 𝛼, especially for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 < 𝛼 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' However, as 𝛼 grows larger than unity, it seems that 𝑅𝑝 attractors provide an almost identical NS phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' This is a notable feature for the class of 𝑅𝑝 attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Before closing, we need to discuss an important issue, having to do with the NS phenomenology of inflationary potentials, with regard to the tidal deformability of NSs, the radial perturbations of static NSs and finally the overall stability of NSs, by also taking into account the constraints imposed by the GW170817 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' This issue however extends further from the aims and scopes of this article, since a whole article could be devoted to these issues, see for example Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Brown (2022) and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2022), in which these issues are addressed in the context of scalar-tensor gravity Brown (2022) and in unimodular gravity Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve), 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for the APR EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' CONCLUDING REMARKS In this article we studied the NS phenomenology of the 𝑅𝑝 infla- tionary attractor scalar-tensor models in the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The 𝑅𝑝 attractors constitute a class of models in the Einstein frame, which originate from a large number of different models in the Jordan frame These distinct Jordan frame models result to the same phenomenol- ogy in the Einstein frame and this feature justifies the terminology inflationary attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Our aim was to investigate whether these at- tractor models can be distinguished when NSs are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' As we showed the NS phenomenology corresponding to different values of the parameter 𝛼 which characterizes the attractors, is in general different for 𝛼 < 1, however the models for 𝛼 > 1 show many similarities and generate almost identical 𝑀 − 𝑅 diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' We also confronted the NS phenomenology of the 𝑅𝑝 attractors to several NS constraints, which we named CSI, CSII and CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The con- straint CSI was developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Altiparmak, Ecker & Rezzolla (2022) and indicates that the radius of an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ mass NS has to be 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99 while the radius of an 2𝑀⊙ mass NS has to be 𝑅2𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='11+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='23 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' The constraint CSII was developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Raaijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2021) and indicates that the radius of an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ mass NS has to be 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='81 km and the constraint CSIII was developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Bauswein, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' (2017) and indicates that the radius of an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='6𝑀⊙ mass NS has to be larger than 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='6𝑀⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99 km while the radius of the maximum mass NS has to be larger than 𝑅𝑀𝑚𝑎𝑥 > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='04 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Our analysis indicated that for 𝑅𝑝 attractors, for the case with 𝛼 = 1, only the SLy EoS is compatible with all the constraints, while the APR is not compatible with CSII, the first constraint of CSI, but it is compatible with the second constraint of CSII and the CSIII constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Also the WFF1 case is incompatible with all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' For the case with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1, which is the most interesting case phe- nomenologically, the SLy EoS is compatible with all the constraints, and for this case the APR is also compatible with all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' However, in this case the WFF1 EoS satisfies the second constraint of CSI and also satisfies all the constraints of CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Finally, for the case with 𝛼 = 1, only the SLy EoS is compatible with all the constraints while the APR is not compatible with CSII, and the first constraint of CSI, but it is compatible with the second constraint of CSII and the CSIII constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Finally, the WFF1 case is incompat- ible with all the constraints, save the first constraint of CSIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Our results indicate two main research lines, firstly that NS phenomenol- MNRAS 000, 1–8 (0000) MM -R Diagramm 25 SLy EoS a=1 SLy EoS a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 2D SLy EoS a=8 SLy EoS GR 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='D 9 11 12 13 R (kri)MM -R Diagramm 25 APR EoS a=1 APR EoS a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 2D APR EoS a=8 APR EoS GR 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='D 9 11 12 13 R (kri)𝑅𝑝 Attractors Static Neutron Star Phenomenology 7 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' CSII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for NS Masses 𝑀 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='4𝑀⊙ 𝑅𝑝 Attractors Model APR SLy WFF1 𝛼 = 1 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='52 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='41 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='25 𝑀⊙ 𝛼 = 1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='56km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='74km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='89km 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='39 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='39 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='07 𝑀⊙ 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='55km 𝑅 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='04km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='79km 𝛼 = 8 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='53 𝑀⊙ 𝑀 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='42 𝑀⊙ 𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='25 𝑀⊙ 𝛼 = 8 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='60km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='738km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='944km Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' CSIII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for Maximum NS Masses 𝑅𝑝 Attractors Model APR SLy WFF1 𝛼 = 1 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='41 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='24 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='33 𝑀⊙ 𝛼 = 1 𝑅 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='91km 𝑅 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='99km 𝑅 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='30km 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='41 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='27 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='32 𝑀⊙ 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='1 𝑅 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='40km 𝑅 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='09km 𝑅 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='06km 𝛼 = 8 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='41 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='27 𝑀⊙ 𝑀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='34 𝑀⊙ 𝛼 = 8 𝑅 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='91km 𝑅 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='72km 𝑅 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content='28km ogy for scalar-tensor theories is not easily rendered viable, since a large number of astrophysical and cosmological constraints have to be satisfied in order for the viability of the model to be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Thus a simple parameter assigning is not the correct way to study NS nowadays, both cosmology and astrophysics constrain in a rigid way NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Secondly, several inflationary attractors which are indistin- guishable at the cosmological level, may be discriminated to some extent when their NS phenomenology is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' This research line is not the general rule though, so work is in progress toward comparing a large sample of cosmological attractors with respect to their NS phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFLT4oBgHgl3EQfpi-7/content/2301.12136v1.pdf'} +page_content=' Finally, let us note that the scalar-tensor inflationary framework we used in this work cannot be considered more advantageous compared to other modified gravity theories, it is one of the many possible modified gravity descriptions of the nature of NSs.' 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new file mode 100644 index 0000000000000000000000000000000000000000..8658be1360f95ad7b2e1c16f01fa63c58a5695d0 --- /dev/null +++ b/2NAyT4oBgHgl3EQf1fm1/content/tmp_files/2301.00737v1.pdf.txt @@ -0,0 +1,968 @@ +1 +Rotational Abstractions for Verification of Quantum +Fourier Transform Circuits +1st Arun Govindankutty Department of Electrical and Computer Engineering +North Dakota State University +Fargo, USA +arun.g@ndsu.edu +2nd Sudarshan K. Srinivasan Department of Electrical and Computer Engineering +North Dakota State University +Fargo, USA +sudarshan.srinivasan@ndsu.edu +3rd Nimish Mathure Department of Electrical and Computer Engineering +North Dakota State University +Fargo, USA +nimish.mathure@ndsu.edu +Abstract—With the race to build large-scale quantum com- +puters and efforts to exploit quantum algorithms for efficient +problem solving in science and engineering disciplines, the +requirement to have efficient and scalable verification methods +are of vital importance. We propose a novel formal verification +method that is targeted at Quantum Fourier Transform (QFT) +circuits. QFT is a fundamental quantum algorithm that forms the +basis of many quantum computing applications. The verification +method employs abstractions of quantum gates used in QFT that +leads to a reduction of the verification problem from Hilbert +space to the quantifier free logic of bit-vectors. Very efficient +decision procedures are available to reason about bit-vectors. +Therefore, our method is able to scale up to the verification of +QFT circuits with 10,000 qubits and 50 million quantum gates, +providing a meteoric advance in the size of QFT circuits thus +far verified using formal verification methods. +Index +Terms—Formal +verification, +Quantum +algorithms, +Quantum computing, Quantum Fourier transform, Quantum +circuit verification. +1 +I. INTRODUCTION +The race to build large scale Quantum computers with +1,000 qubits and beyond is in full steam [1] [2]. The IBM +Condor quantum computer with 1,000 qubits is expected to +be released in 2023 [3]. After Condor, IBM plans to use +chip-to-chip couplers to build even larger quantum computing +systems [4], with a goal of building a system with 1 million +qubits [5]. Google’s road map is to built a quantum computer +with 1 million qubits as well in the near future [6]. There +are numerous other quantum computers being developed by +corporations such as Xanadu, Rigetti, IonQ, and D-Wave, to +name a few. The development of cryogenic control circuits +needed for quantum computing is also accelerated as demon- +strated by Intel (Horse Ridge chip) [7], which realizes quantum +computing and communication applications [8]. +1This paper is a preprint of a paper submitted to IET Quantum Computing. +If accepted, the copy of record will be available at the IET Digital Library. +The Quantum Algorithm Zoo website tracks algorithms +in this domain and currently lists 430 citations of various +Quantum algorithms [9]. +The 80/20 design rule is well know in computing, i.e., +20% of the design cycle time is spend in the actual design, +while 80% is spent in validation and verification. Without +verification technologies that can scale, the useful deployment +of these large-scale quantum systems will be significantly +hampered. It is imperative therefore to develop verification +methods for quantum circuits, which is the focus of this +work. Formal verification has become a standard in the +semiconductor industry with its ability to provide correctness +guarantees and flag hard-to-find corner case bugs. There are +various formal verification methods proposed for quantum +circuits [10]. +However, for example, the largest Quantum Fourier Trans- +form (QFT) circuit verified as reported in literature has only +31 qubits [11]. Scalable verification methods are thus the need +of the hour. +Contributions: One of the approaches to achieve scalability +in formal verification is to develop domain-specific methods. +In this work, we target one of the fundamental quantum +algorithms, the Quantum Fourier Transform (QFT). In com- +puting and engineering, transformations play a vital role in +problem solving and analysis. Quantum computing uses QFT +to tackle various problems. QFT is an integral part of numer- +ous quantum algorithms including Shor’s factoring algorithm, +quantum phase estimation algorithm, and computing discrete +logarithm to name a few [12] [13]. The real-world applications +where QFT is employed include portfolio optimization in +computational finance [14], Monte Carlo pricing of financial +derivatives [15], quantum meteorology for building interferom- +eters [16], materials examination and analysis [17], analysis of +image data [18] in medical applications, and risk analysis [19] +among others. +We have developed a formal verification method that can be +arXiv:2301.00737v1 [quant-ph] 2 Jan 2023 + +2 +used to efficiently verify Quantum Fourier Transform (QFT) +circuits for up to 10,000 qubits and 50 million gates. Our +specific contributions are as follows: +1) Abstractions of the Hadamard (H) gate and the control +rotation gate (Rn) that exploits the rotational impact of +these gates on the incoming qubit. +2) A correctness framework that exploits these abstractions +and allows the verification problem to be reduced from +Hilbert space (complex vector space) to the quantifier +free logic of bit-vectors (QF BV). +3) Theorems with proofs to show that the abstractions are +sound, i.e., if the abstract QFT circuit is verified to be +correct, then the correctness of the QFT circuit under +verification is guaranteed +While we have developed our approach with QFT as the +target, the key ideas used in the abstractions can be applied +to a much larger class of quantum circuits, which is what we +plan to do for future work. +The rest of the paper is organized as follows. Section II +covers background on quantum circuits and QFT circuits. +Section III overviews the related work on formal methods +for verification of quantum circuits. Section IV describes +the key contributions of the proposed work, including the +gate abstractions and the correctness framework. Section V +addresses the correctness of the abstractions and the overall +approach. Experimental results are provided and discussed +in Section VI. Conclusions and future work are outlined in +section VII. +II. BACKGROUND +In this section, we review background on qubits, quantum +gates, and QFT circuits. A detailed description of these topics +can be found in [12]. Information in the quantum computing +domain is represented by qubits. A qubit is the basic unit +of information analogous to a bit in classical computing. In +general, qubits are represented by a linear combination of +ortho-normal (orthogonal and normalized) vectors |0⟩ and |1⟩. +The vectors are linearly independent i.e., we cannot express +one as the linear combination of the other. The independent +vectors are shown below. +|0⟩ = +�1 +0 +� +, and |1⟩ = +�0 +1 +� +The above ortho-normal vectors can be used to represent any +vectors in the vector space by using vector addition and scaling +(linear combination), and thus they are called the basis vectors. +A standard representation of a qubit |q⟩ is shown below where, +α and β are complex numbers such that α2 + β2 = 1. +|q⟩ = α|0⟩ + β|1⟩ +Quantum gates are unitary operators that act on qubits and +produce a required output. A quantum algorithm is a step by +step process that utilizes quantum mechanical properties to +solve a particular problem. Quantum algorithms are run on +computation models for quantum computing and this work is +based on the quantum circuit model, which is the most widely +used method [20]. +QFT is analogous to the Discrete Fourier Transform (DFT) +in the classical domain and efficiently performs the quantum +mechanical model’s Fourier transform. The QFT operates on +the input qubit states (ortho-normal basis vectors |0⟩, ....., |N− +1⟩) and transforms them to the corresponding output states. +The transformation is shown below [12]. +|j⟩ → +1 +√ +N +N−1 +� +k=0 +e2πijk/N|k⟩ +In the above, |j⟩, N, i, and k represents the input qubit, +the number of QFT points, imaginary number (√−1), and the +iteration variable, respectively. Here N = 2n, where n is the +number of qubits in the QFT. +In the transformed domain, this resultant state (transformed +|j⟩) can be represented as a sum of individual components +whose frequencies are integer multiples of +2π +N . The same +equation can be re-organized to obtain the equivalent trans- +formation happening in each qubit independently, which we +exploit in this work. +Implementation of QFT as a circuit can be achieved by a +series of cascaded Hadamard (H) gates and controlled rotation +(Rn) gates. The H gates and Rn gates are defined below. +H = +1 +√ +2 +�1 +1 +1 +eπi +� += +1 +√ +2 +�1 +1 +1 +−1 +� +Rn = +�1 +0 +0 +e2πi/2n +� +The H gate introduces equal superposition of the input basis +vectors for the qubit. The Rn gates are responsible for the +frequency harmonics. QFT circuits are constructed by first +applying the H gate to all qubits. Qubit 1 of a QFT circuit +with m qubits should have gates R2, ..., Rm acting on it, with +control inputs qubit 2, ..., qubit m taken before the H gate is +applied to the control qubits, respectively. Qubit 2 should have +gates R2, ..., Rm−1 acting on it with control inputs qubit 3, ..., +qubit m taken before the H gate is applied, respectively, and +so on. Figure 1(a) shows the transformations happening while +QFT is performed on a 3 qubit system. +III. RELATED WORK +Formal verification of quantum algorithms and circuits has +been an active area of research. In this section, we overview +these related works and how they contrast with our approach. +The main takeaway is that the approaches have not demon- +strated the efficiency and scalability that we have been able to +achieve. In this sense, our approach is a meteoric advance in +the size of quantum circuits thus far verified. +Yamashita and Markov [22] have proposed an equivalence +checking approach for quantum circuits. In equivalence check- +ing, the circuit to be verified is compared with a reference + +3 +Fig. 1. (a) 3-qubit QFT circuit [21]. (b) Abstract Hadamard gate. (c) Abstract rotation gate. (d) 3-qubit QFT abstract circuit representation. +circuit. There are two prominent contrasts with our approach. +The first contrast is related to equivalence checking in general, +where a golden (already verified, trusted) circuit is required as +the reference circuit. For example, to verify a QFT circuit with +10,000 qubits and 50 million gates, a trusted QFT circuit of +the same size is required. Therefore, to enable equivalence +checking, methods that can verify functional correctness of +a given circuit is mandatory. This is the gap that we address. +Equivalence checking is useful in synthesis optimizations. Our +approach is property based and does not require a reference +circuit of the same size for verification. If a QFT circuit +with 10,000 qubits and 50 million gates satisfies our proposed +correctness property, it is guaranteed to be correct. The second +contrast is that if they are not able to reduce the problem to +a boolean space, then a hybrid approach is used [23], where +the verification problem is solved in the Hilbert space. We use +rotational abstractions to reduce the problem fully to a Boolean +space, solvers for which are orders of magnitude more efficient +and scalable. We also exploit the fact that our approach is +domain-specific to QFT circuit verification to enable this. The +largest circuits they verified have 5,000 gates and requires +about 59 seconds. In contrast, we are able to verify circuits +with 8,000 gates in 0.04 seconds, 5 million gates in about 60 +seconds, and 50 million gates in 2,380 seconds. +Amy [11] use complex path-sums to model quantum gates +for verification. They perform reductions on the resulting +circuit, which are implemented using rewrite rules. The re- +ductions are performed using the Haskell theorem prover. +The rewrite rules are guaranteed to reduce the circuit to +a normal form, which is then used to check correctness. +They verify a 16-qubit and a 31-qubit QFT, which required +1.250 seconds and 16.020 seconds for circuits without errors, +respectively. In contrast, our approach required 0.02 seconds +and 0.03 seconds for 16-qubit and 32-qubit QFT circuits, +respectively. They employ a dyadic arithmetic technique, the +current implementation of which causes an integer overflow +for QFT circuits larger than 31 qubits. Therefore, with this +current implementation, they are unable to handle QFT circuits +larger than 31 qubits. We are able to handle upto 10,000 qubits. +Liu et al. [24] formalize quantum hoare logic in the Is- +abelle/HOL theorem prover and use it to prove the correctness +of Grover’s search algorithm for infinite size input. They report +that the proof required 5 person months of effort. They do not +describe how this proof can be used to verify a given quantum +circuit that implements Grover’s algorithm. In contrast, our +approach is fully automated for verification of any QFT circuit. +They have not addressed QFT verification. +Feng et al. [25] have developed a model checking algorithm +that can check the Quantum CTL (QCTL) properties on +quantum Markov chains. The method is used to check the +correctness of the BB84 protocol when n=1, the corresponding +circuit for which has 8 qubits and 24 quantum gates. They have +not addressed QFT verification either. +IV. ROTATIONAL ABSTRACTIONS +There are three key ideas in developing the abstractions for +the Hadamard (H) gate and the controlled rotation (Rn) gate. +The first key idea is with regard to the basis vectors. If +a QFT circuit works correctly when the input qubits are the +basis vectors |0⟩ or |1⟩, then the circuit is guaranteed to work +correctly for any qubit inputs [26]. Therefore, for verification +purposes, we only consider the cases where the input qubits +are |0⟩ or |1⟩. +The second key idea is with regard to quantum gates and +is as follows. If the input qubits are limited to basis vectors, +then both the H gate and the Rn gate can be modelled as gates +causing rotation on the basis vectors. The H gate has only one +input. We call this the control input qc as shown in Figure 1(b), + +H +R2 +R3 +H +R2 +H +HA +HA +R2A +ReA +HA +R2A +R.A +HA4 +because if the input is |1⟩, then the H gate function can be +represented as a rotation on |1⟩. If this control input is |0⟩, then +no rotation is performed. The Rn gate has two inputs (control +and data) and one output, we call the control input qc, the data +input qd, and the output qo (as shown in Figure 1(c)). If qc is +|1⟩, then Rn performs a rotation on qd. Otherwise, if qc is |0⟩, +then no rotation is performed. +The third key idea is with regard to the amount of rotation +performed by the quantum gates on data input qubits and the +resulting output qubit states, and is as follows. The H gate +induces a π (2π/2) rotation on |1⟩ and does not rotate |0⟩. +The Rn gate induces a 2π/2n (π/2n−1 ) rotation on |1⟩ and +does not rotate |0⟩. For examle, R4 induces a rotation of π/8. +Thus, the rotation performed by the gates on |1⟩ are negative +powers of 2 with reference to 2π . +The QFT circuit structure is such that the control inputs to +the quantum gates are always initial qubit states and are used +only to make the decision, whether to rotate or not. +Thus, we can abstract the basis vector input values |0⟩ and +|1⟩ using Boolean values 0 and 1. +The qubits once transformed by these rotations are input +to the next quantum gate and finally the output state of the +circuit. +If the 2πi term is factored out of the exponent, the final +output state of each qubit (after transformation) can be ab- +stractly represented using fractional bit-vectors that essentially +capture the amount of rotation on |1⟩. The fractional bit-vector +⟨.b1b2b3⟩ corresponds to rotation value 2π ∗ (b1 ∗ 2−1 + b2 ∗ +2−2+b3∗2−3). For example, the bit-vector ⟨.101⟩ corresponds +to rotation value of 2π(1/2+0+1/8). Abstractions of the H gate +and the Rn gate can be obtained by defining their rotational +impact on the fractional bit-vectors, and an abstracted QFT +circuit can be obtained by using these abstracted gates. In a +QFT circuit with m qubits, the smallest amount of rotation +will be 2π/2m. Therefore, the fractional bit-vectors used to +represent qubits in the abstracted QFT circuit will have to +have m bits. +The abstract H gate is defined below and has one input qubit +qc, which is Boolean type. Output qubit qo is a bit-vector of +size equal to m, the number of qubits. +Definition 1. (Abstract Hadamard Gate) If +qc=1, then +qo +← ⟨.100...0⟩m, +else +qo ← ⟨.000...0⟩m. +The abstract Rn gate is defined below and has two qubit +inputs qc and qd. The control input qc is type Boolean, the +data input qd and the output qubit qo are both fractional bit- +vectors of size m, the number of qubits. +Definition 2. +(Abstract Rn Gate) If +qc=1, +then qo ← +qd +m ⟨.00..01m−n0...0⟩m, else +qo ← qd. +In the above, +m represents fixed-point modulo addition +w.r.t m. The abstracted QFT circuit is obtained by replacing +the H gates and Rn gates of the original circuit with the +abstracted gates. Input qubits are declared as type Boolean +and all other qubits are declared as type bit-vector of size m. +The abstracted QFT circuit with 3 qubits is shown in Figure +1(d). When the abstract H gate is applied, the qubits at the +output of the H gates of the QFT circuit in Figure 1(d) will +have the following values: +q1 +1 ← ⟨.b100⟩ +q1 +2 ← ⟨.b200⟩ +q1 +3 ← ⟨.b300⟩ +The QFT correctness property is given next. Let QFT- +Absi(b1, b2, ..., bm) denote the output state of the ith qubit +of an abstracted version of a QFT circuit, where m is the +number of qubits and b1, b2, ..., bm are Boolean variables. +Property 1. +(QFT Correctness Property) A QFT circuit is +functionally correct if, for all 1 ≤ i ≤ m, i is an integer, +QFT-Absi(b1, b2, ..., bm) = ⟨.bibi+1...bm0...0⟩m. +Based on the correctness property above, for the QFT +circuit from Figure 1(a) to be correct, the state of qubits at +the output should be as follows: +q3 +1 = ⟨.b1b2b3⟩ +q3 +2 = ⟨.b2b30⟩ +q3 +3 = ⟨.b300⟩ +The abstracted gates, abstracted QFT circuit, and Property +1 are expressible in the Quantifier Free logic of Bit Vectors +(QF BV). A number of SMT solvers exist that can very +efficiently check properties in this logic. Therefore, verification +of a given QFT circuit can be performed by encoding the +abstracted circuit and correctness property in this logic (using +the SMT LIB language). An SMT solver will check the +property automatically and indicate if the property is satisfied +or not. If the property is satisfied, then the QFT circuit is +guaranteed to be correct (as will be established in the next +section). If the property is not satisfied, the tool will generate +a counter example, which can be used to trace the error(s) in +the circuit. +V. ABSTRACTION CORRECTNESS +Fig. 2. QFT circuit showing error scenarios. +In this section, we provide a proof of correctness of our ver- +ification approach. The overall approach is that we enumerate +through all possible classes of errors in QFT circuits and show +how the verification approach will flag each error class. The +error classes are depicted in Figure 2. We call bit-vector values +as data values as well. + +H +R3 +R3 +R2 +R2 +H5 +TABLE I +VERIFICATION RESULTS +QFT Benchmark +Correct Circuit +Incorrect Gate Error +Incorrect Control Error +No Error +Error Depth +Error Depth +Gate-2 +Gate-n +Gate-2 +Gate-n +Verification Stats. +Verification Stats. +Verification Stats. +Verification Stats. +Verification Stats. +Qubits(n) +Gates +M(MB) +T(s) +M(MB) +T(s) +M(MB) +T(s) +M(MB) +T(s) +M(MB) +T(s) +16 +136 +19.0 +0.02 +27.2 +0.04 +27.3 +0.02 +19.0 +0.01 +27.2 +0.02 +32 +528 +19.0 +0.03 +27.2 +0.02 +27.5 +0.02 +19.0 +0.02 +19.0 +0.01 +64 +2,080 +19.0 +0.03 +27.3 +0.03 +27.6 +0.02 +19.1 +0.02 +19.1 +0.03 +128 +8,256 +19.3 +0.04 +27.4 +0.07 +27.9 +0.04 +19.3 +0.03 +19.3 +0.02 +256 +32,896 +20.1 +0.19 +27.7 +0.06 +28.6 +0.06 +20.0 +0.08 +20.0 +0.08 +512 +131,328 +22.1 +0.26 +28.3 +0.29 +29.8 +0.2 +22.2 +0.29 +22.2 +0.2 +1,024 +524,800 +29.1 +1.37 +29.5 +0.77 +32.2 +0.92 +29.5 +1.46 +29.5 +1.29 +2,048 +2,098,176 +56.1 +9.85 +56.9 +5.52 +73.7 +5.87 +56.9 +9.66 +56.9 +9.47 +4,096 +8,390,656 +169.3 +95.75 +169.6 +51.78 +203.3 +53.57 +169.6 +73.65 +169.6 +79.68 +8,192 +33,558,528 +592.1 +1,109.0 +593.6 +640.53 +596.2 +643.9 +593.6 +641.03 +593.6 +639.57 +10,000 +50,005,000 +888.7 +2,379.88 +890.7 +1,523.99 +894.5 +1,568.79 +890.6 +1,571.29 +890.6 +1,524.65 +Lemma 1. If a QFT circuit has an error, where an incorrect +input is fed to an H gate, verification of the abstracted version +of the QFT circuit will either generate a type error or will not +satisfy Property 1. +If the input to the abstract H gate is a bit-vector input, this +will be flagged as a type error as the H gate expects a Boolean +input. If Boolean input qubit bj is expected whereas bk is fed +for qubit qj, then the LHS of Property 1 for qj will be ⟨.bk...⟩ +and RHS will be ⟨.bj...⟩. Therefore, Property 1 will not be +satisfied. +Lemma 2. If a QFT circuit has an error, where an incorrect +input is fed to an Rn gate, verification of the abstracted version +of the QFT circuit will either generate a type error or will not +satisfy Property 1. +If a control value is fed to the data input of an Rn gate +or if a data value is fed to the control input of an Rn gate, +a type error will be generated. If bj is expected whereas bk +is fed for the control input of an Rn gate acting on qubit qj, +then the LHS of Property 1 for qj will be ⟨....bk...⟩ and RHS +will be ⟨....bj...⟩. Therefore, Property 1 will not be satisfied. +If an incorrect data value is fed to an Rn gate, this will result +in a missing Rn gate on a qubit and this case is dealt with +subsequently. +The error above is shown in Figure 2. R3 gate with input q2 +1 +should have b3 as its control input. Instead b2 is erroneously +fed as the control input. +Lemma 3. If a QFT circuit has an error, where an H gate +is missing on a qubit or there is more than one H gate acting +on a qubit, verification of the abstracted version of the QFT +circuit will generate a type error. +In the abstracted version of a QFT circuit, the input of an +H gate is a control value and the output is a data value. Thus, +if there is more than one H gate acting on a qubit, the H gates +after the first one will receive data inputs and this will result +in a type error. If there are no H gates acting on a qubit, the +subsequent Rn gates will not get a data value at its data input +and this will again result in a type error. +An example of a missing H gate error is shown in Figure +2. The H gate on q2 is missing. +Lemma 4. If a QFT circuit has an error where an incorrect +set of Rn gates are acting on a qubit, i.e., required Rn gates are +missing or additional Rn gates are present or both, verification +of the abstract version of the QFT circuit will not satisfy +Property 1. +Qubit 1 of a QFT circuit with m qubits should have gates +R2, ..., Rm acting on it. Qubit 2 should have gates R2, ..., +Rm−1 acting on it and so on. Thus, there is only one Rn gate +of a certain n value required to act on each qubit. If a required +Rn gate is missing, then its rotational impact on the fractional +bit-vector value abstracting the qubit will not be observed in +Property 1. If a qubit has additional erroneous Rn gates acting +on it, then the required rotation of the qubit will be incorrect +and this will be reflected on the final fractional bit-vector value +of the qubit. In both the above cases, Property 1 will not be +satisfied. Note that an Rn gate can be replaced with two Rn−1 +gates, with the same control inputs. For example, R2 can be +substituted with two R3 gates. If the total rotational impact of +a sequence of Rn gates is what is expected, even though it +does not conform with the Rn gate sequence described above, +Property 1 will be satisfied because the fractional bit-vector +abstraction accurately captures the rotations. +An example of an incorrect Rn gate is shown in Figure 2, +where the gate on q1 +1 should be R2 instead of R3. +Lemma 5. If a QFT circuit has a combination of errors from +the error classes described in Lemmas 1-4, verification of the +abstracted version of the QFT circuit will generate a type error +or will not satisfy Property 1. +As can be seen from Lemmas 1-4, the effect that flags each +error class is disjoint, i.e., there is no overlap in these effects +for type errors or Property 1. Thus a combination of errors +will also be flagged as a type error or will not satisfy Property +1. +Theorem 1. +(QFT-Rotational Abstraction Correctness) If a +QFT circuit has an error, verification of the abstracted version +of the QFT circuit will generate a type error or will not satisfy +Property 1. +A QFT circuit has only two types of gates, the H gate and +the Rn gate. Based on this, there are only four classes of + +6 +errors possible: Incorrect input to a H gate, incorrect input to +an Rn gate, missing or additional H gates in the circuit, and +incorrect set of Rn gates acting on a qubit. The fifth case of an +erroneous QFT circuit is any combination of the above. From +Lemmas 1-5, we see that in all the above cases, verification of +the abstracted version of the QFT circuit will generate a type +error or will not satisfy Property 1. +VI. RESULTS AND DISCUSSIONS +Table I gives the verification results. The verification bench- +marks were generated by varying the number of qubits in +the QFT circuit from 16 qubits to 10,000 qubits. The table +gives the number of quantum gates in each of the QFT +benchmark circuits as well (column 2: Gates). The verification +experiments were conducted on an Intel(R) Core(TM) i9 - +12900K CPU @ 3.2 GHz with 32 GB RAM and Ubuntu 64- +bit operating system. The z3 version 4.8.12 SMT solver [27] +was used to check Property 1 for all benchmarks. +In the table, ”T(s)” indicates verification time in seconds, +which is the z3 run time. ”M(MB)” gives the z3 run time +memory consumption in megabytes. ”Correct Circuit” gives +the verification statistics for the QFT circuits without errors. +For these circuits Property 1 is proved to be satisfied. Property +1 allows for each qubit output to be verified independently. +Therefore, the verification of all the qubit output in the circuit +were done in parallel and the memory and time reported +correspond to the worst case. +”Incorrect Gate Error” are circuits with gates errors and is +described as follows. The Gate-2 error here indicates that the +R3 gate is incorrectly acting on qubit 1 instead of R2. The +Gate-n error here indicates that the Rn−1 gate is incorrectly +acting on qubit 1 instead of Rn. ”Incorrect Control Error” +are circuits with incorrect control input to an Rn gate. The +Gate-2 error here indicates that the R2 gate in qubit 1 is +incorrectly controlled by qubit 3 instead of qubit 2. The Gate- +n error here indicates that the Rn gate in qubit 1 is incorrectly +controlled by qubit n-1 instead of qubit n. For the circuits with +errors, verification of Property 1 generates a counterexample. +The time and memory reported corresponds to the verification +of the first qubit output that caused a counterexample to be +generated. +Figures 3 and 4 plot the verification time and memory from +Table I versus the number of quantum gates, respectively. In +these graphs, both the x-axis and y-axis use a log scale. As +can be seen from these graphs, with increase in the number +of gates, both memory and verification time increase linearly +for both correct circuits and circuits with errors. The most +complex circuit with 10,000 qubits and 50 million gates is +verified in only about 37 minutes. This demonstrates the high +efficiency and scalability of our approach. The time taken to +verify circuits with errors is less than that of correct circuits. +However, there is not an order-of-magnitude reduction that is +often observed in formal verification. +Figure 5 shows both verification time and memory as the +position of the gate error is moved from qubit 1 to qubit +10,000 on the QFT circuit with 10,000 qubits. The x-scale +increases linearly, whereas the y-scale is logarithmic. The +graph indicates the variation of time and memory with the +vertical location of errors. We see that as the error moves +from qubit 1 to qubit 10,000, both time and memory reduce +exponentially. +Fig. 3. +Execution time requirement capture for QFT verification versus +quantum gate count. Correct circuit, control input error and value error at +qubit positions 2 and 10000 captured for elucidation. +Fig. 4. Execution memory requirement capture for QFT verification versus +quantum gate count. Correct circuit, control input error and value error at +qubit positions 2 and 10000 captured for elucidation. +VII. CONCLUSIONS AND FUTURE WORK +Our proposed approach for verification of Quantum Fourier +Transform (QFT) circuits achieves a meteoric advance in the +efficiency and scalability of quantum circuits thus far verified. +We have been able to verify a QFT circuit with 10,000 +qubits and over 50 million gates in only about 37 minutes. +We exploit the fact that our approach is domain specific +to QFT verification. This is a common theme to achieve +scalability in formal verification. For example, there are a +large number of formal verification techniques dedicated to +the verification of multipliers. We also exploit the idea that +the rotations performed by the gates are negative powers of 2 + +Execution time capture +103 +Correct Circuit +Incorrect Control Error at gate 2 +101. +Incorrect Control Erro at gate n +10-1 +102 +103 +104 +105 +106 +107 +Correct Circuit +102 +Incorrect Gate Error at gate 2 +Incorrect Gate Error at gate n +100 +102 +103 +104 +105 +106 +107 +Number of quantum gates in QFT circuitExecution memory capture +103 +Correct Circuit +Incorrect Control Error at gate 2 +Incorrect Control Erro at gate n +102 +102 +103 +104 +105 +106 +107 +103 +Correct Circuit +Incorrect Gate Error at gate 2 +Incorrect Gate Error at gate n +102 +102 +103 +104 +105 +106 +107 +Number of quantum gates in QFT circuit7 +Fig. 5. +Resource utilization (time and memory) capture versus qubit count +for erroneous QFT circuit. +and can therefore be encoded as fractional bit-vectors, thus +reducing the verification obligations from Hilbert space to +Boolean space. For future work, our goal is to extend these +ideas to other quantum algorithms to advance efficiency and +scalability of formal verification so as to cope with the size +and complexity of quantum hardware roadmaps of the near +future. +REFERENCES +[1] D. Gottesman and I. L. Chuang, “Demonstrating the viability of +universal quantum computation using teleportation and single-qubit +operations,” Nature, vol. 402, pp. 390–393, 11 1999. +[2] F. A. et.al., “Quantum supremacy using a programmable superconduct- +ing processor,” Nature, vol. 574, pp. 505–510, 10 2019. +[3] G. Jay, F. Ismael, and W. Karl. (2021, 2) Ibm’s roadmap for +building an open quantum software ecosystem. [Online]. Available: +https://research.ibm.com/blog/quantum-development-roadmap +[4] K. +Krewell +and +T. +Research. +(2022, +6) +The +next +generation +of +ibm +quantum +computers. +[On- +line]. 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Available: https://www.sciencedirect.com/ +science/article/pii/S0022000013000780 +[26] F. X. Lin, “Shor’s algorithm and the quantum fourier transform,” McGill +University, 2014. +[27] L. De Moura and N. Bjørner, “Z3: An efficient smt solver,” in +Proceedings of the Theory and Practice of Software, 14th International +Conference on Tools and Algorithms for the Construction and Analysis +of Systems, ser. TACAS’08/ETAPS’08. +Berlin, Heidelberg: Springer- +Verlag, 2008, p. 337–340. + +Resource capture for erroneous QFT circuit +103 +102 +101 +Time +100 +Memory +0 +2000 +4000 +6000 +8000 +10000 +Qubit line where error is located \ No newline at end of file diff --git a/2NAyT4oBgHgl3EQf1fm1/content/tmp_files/load_file.txt b/2NAyT4oBgHgl3EQf1fm1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae12c41b1e6b60e2986a0a3a3089696e3ae8ab98 --- /dev/null +++ b/2NAyT4oBgHgl3EQf1fm1/content/tmp_files/load_file.txt @@ -0,0 +1,736 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf,len=735 +page_content='1 Rotational Abstractions for Verification of Quantum Fourier Transform Circuits 1st Arun Govindankutty Department of Electrical and Computer Engineering North Dakota State University Fargo, USA arun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='g@ndsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='edu 2nd Sudarshan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Srinivasan Department of Electrical and Computer Engineering North Dakota State University Fargo, USA sudarshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='srinivasan@ndsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='edu 3rd Nimish Mathure Department of Electrical and Computer Engineering North Dakota State University Fargo, USA nimish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='mathure@ndsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='edu Abstract—With the race to build large-scale quantum com- puters and efforts to exploit quantum algorithms for efficient problem solving in science and engineering disciplines, the requirement to have efficient and scalable verification methods are of vital importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We propose a novel formal verification method that is targeted at Quantum Fourier Transform (QFT) circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' QFT is a fundamental quantum algorithm that forms the basis of many quantum computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The verification method employs abstractions of quantum gates used in QFT that leads to a reduction of the verification problem from Hilbert space to the quantifier free logic of bit-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Very efficient decision procedures are available to reason about bit-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, our method is able to scale up to the verification of QFT circuits with 10,000 qubits and 50 million quantum gates, providing a meteoric advance in the size of QFT circuits thus far verified using formal verification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Index Terms—Formal verification, Quantum algorithms, Quantum computing, Quantum Fourier transform, Quantum circuit verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' INTRODUCTION The race to build large scale Quantum computers with 1,000 qubits and beyond is in full steam [1] [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The IBM Condor quantum computer with 1,000 qubits is expected to be released in 2023 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' After Condor, IBM plans to use chip-to-chip couplers to build even larger quantum computing systems [4], with a goal of building a system with 1 million qubits [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Google’s road map is to built a quantum computer with 1 million qubits as well in the near future [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' There are numerous other quantum computers being developed by corporations such as Xanadu, Rigetti, IonQ, and D-Wave, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The development of cryogenic control circuits needed for quantum computing is also accelerated as demon- strated by Intel (Horse Ridge chip) [7], which realizes quantum computing and communication applications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 1This paper is a preprint of a paper submitted to IET Quantum Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If accepted, the copy of record will be available at the IET Digital Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Quantum Algorithm Zoo website tracks algorithms in this domain and currently lists 430 citations of various Quantum algorithms [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The 80/20 design rule is well know in computing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', 20% of the design cycle time is spend in the actual design, while 80% is spent in validation and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Without verification technologies that can scale, the useful deployment of these large-scale quantum systems will be significantly hampered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' It is imperative therefore to develop verification methods for quantum circuits, which is the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Formal verification has become a standard in the semiconductor industry with its ability to provide correctness guarantees and flag hard-to-find corner case bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' There are various formal verification methods proposed for quantum circuits [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' However, for example, the largest Quantum Fourier Trans- form (QFT) circuit verified as reported in literature has only 31 qubits [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Scalable verification methods are thus the need of the hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Contributions: One of the approaches to achieve scalability in formal verification is to develop domain-specific methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In this work, we target one of the fundamental quantum algorithms, the Quantum Fourier Transform (QFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In com- puting and engineering, transformations play a vital role in problem solving and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Quantum computing uses QFT to tackle various problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' QFT is an integral part of numer- ous quantum algorithms including Shor’s factoring algorithm, quantum phase estimation algorithm, and computing discrete logarithm to name a few [12] [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The real-world applications where QFT is employed include portfolio optimization in computational finance [14], Monte Carlo pricing of financial derivatives [15], quantum meteorology for building interferom- eters [16], materials examination and analysis [17], analysis of image data [18] in medical applications, and risk analysis [19] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We have developed a formal verification method that can be arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='00737v1 [quant-ph] 2 Jan 2023 2 used to efficiently verify Quantum Fourier Transform (QFT) circuits for up to 10,000 qubits and 50 million gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Our specific contributions are as follows: 1) Abstractions of the Hadamard (H) gate and the control rotation gate (Rn) that exploits the rotational impact of these gates on the incoming qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 2) A correctness framework that exploits these abstractions and allows the verification problem to be reduced from Hilbert space (complex vector space) to the quantifier free logic of bit-vectors (QF BV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 3) Theorems with proofs to show that the abstractions are sound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', if the abstract QFT circuit is verified to be correct, then the correctness of the QFT circuit under verification is guaranteed While we have developed our approach with QFT as the target, the key ideas used in the abstractions can be applied to a much larger class of quantum circuits, which is what we plan to do for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Section II covers background on quantum circuits and QFT circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Section III overviews the related work on formal methods for verification of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Section IV describes the key contributions of the proposed work, including the gate abstractions and the correctness framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Section V addresses the correctness of the abstractions and the overall approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Experimental results are provided and discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Conclusions and future work are outlined in section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' BACKGROUND In this section, we review background on qubits, quantum gates, and QFT circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' A detailed description of these topics can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Information in the quantum computing domain is represented by qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' A qubit is the basic unit of information analogous to a bit in classical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In general, qubits are represented by a linear combination of ortho-normal (orthogonal and normalized) vectors |0⟩ and |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The vectors are linearly independent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', we cannot express one as the linear combination of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The independent vectors are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' |0⟩ = �1 0 � , and |1⟩ = �0 1 � The above ortho-normal vectors can be used to represent any vectors in the vector space by using vector addition and scaling (linear combination), and thus they are called the basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' A standard representation of a qubit |q⟩ is shown below where, α and β are complex numbers such that α2 + β2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' |q⟩ = α|0⟩ + β|1⟩ Quantum gates are unitary operators that act on qubits and produce a required output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' A quantum algorithm is a step by step process that utilizes quantum mechanical properties to solve a particular problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Quantum algorithms are run on computation models for quantum computing and this work is based on the quantum circuit model, which is the most widely used method [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' QFT is analogous to the Discrete Fourier Transform (DFT) in the classical domain and efficiently performs the quantum mechanical model’s Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The QFT operates on the input qubit states (ortho-normal basis vectors |0⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', |N− 1⟩) and transforms them to the corresponding output states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The transformation is shown below [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' |j⟩ → 1 √ N N−1 � k=0 e2πijk/N|k⟩ In the above, |j⟩, N, i, and k represents the input qubit, the number of QFT points, imaginary number (√−1), and the iteration variable, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Here N = 2n, where n is the number of qubits in the QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In the transformed domain, this resultant state (transformed |j⟩) can be represented as a sum of individual components whose frequencies are integer multiples of 2π N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The same equation can be re-organized to obtain the equivalent trans- formation happening in each qubit independently, which we exploit in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Implementation of QFT as a circuit can be achieved by a series of cascaded Hadamard (H) gates and controlled rotation (Rn) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The H gates and Rn gates are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' H = 1 √ 2 �1 1 1 eπi � = 1 √ 2 �1 1 1 −1 � Rn = �1 0 0 e2πi/2n � The H gate introduces equal superposition of the input basis vectors for the qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Rn gates are responsible for the frequency harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' QFT circuits are constructed by first applying the H gate to all qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Qubit 1 of a QFT circuit with m qubits should have gates R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', Rm acting on it, with control inputs qubit 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', qubit m taken before the H gate is applied to the control qubits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Qubit 2 should have gates R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', Rm−1 acting on it with control inputs qubit 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', qubit m taken before the H gate is applied, respectively, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Figure 1(a) shows the transformations happening while QFT is performed on a 3 qubit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' RELATED WORK Formal verification of quantum algorithms and circuits has been an active area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In this section, we overview these related works and how they contrast with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The main takeaway is that the approaches have not demon- strated the efficiency and scalability that we have been able to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In this sense, our approach is a meteoric advance in the size of quantum circuits thus far verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Yamashita and Markov [22] have proposed an equivalence checking approach for quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In equivalence check- ing, the circuit to be verified is compared with a reference 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (a) 3-qubit QFT circuit [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (b) Abstract Hadamard gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (c) Abstract rotation gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (d) 3-qubit QFT abstract circuit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' There are two prominent contrasts with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The first contrast is related to equivalence checking in general, where a golden (already verified, trusted) circuit is required as the reference circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For example, to verify a QFT circuit with 10,000 qubits and 50 million gates, a trusted QFT circuit of the same size is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, to enable equivalence checking, methods that can verify functional correctness of a given circuit is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' This is the gap that we address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Equivalence checking is useful in synthesis optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Our approach is property based and does not require a reference circuit of the same size for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit with 10,000 qubits and 50 million gates satisfies our proposed correctness property, it is guaranteed to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The second contrast is that if they are not able to reduce the problem to a boolean space, then a hybrid approach is used [23], where the verification problem is solved in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We use rotational abstractions to reduce the problem fully to a Boolean space, solvers for which are orders of magnitude more efficient and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We also exploit the fact that our approach is domain-specific to QFT circuit verification to enable this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The largest circuits they verified have 5,000 gates and requires about 59 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In contrast, we are able to verify circuits with 8,000 gates in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='04 seconds, 5 million gates in about 60 seconds, and 50 million gates in 2,380 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Amy [11] use complex path-sums to model quantum gates for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They perform reductions on the resulting circuit, which are implemented using rewrite rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The re- ductions are performed using the Haskell theorem prover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The rewrite rules are guaranteed to reduce the circuit to a normal form, which is then used to check correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They verify a 16-qubit and a 31-qubit QFT, which required 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='250 seconds and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='020 seconds for circuits without errors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In contrast, our approach required 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='02 seconds and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='03 seconds for 16-qubit and 32-qubit QFT circuits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They employ a dyadic arithmetic technique, the current implementation of which causes an integer overflow for QFT circuits larger than 31 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, with this current implementation, they are unable to handle QFT circuits larger than 31 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We are able to handle upto 10,000 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' [24] formalize quantum hoare logic in the Is- abelle/HOL theorem prover and use it to prove the correctness of Grover’s search algorithm for infinite size input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They report that the proof required 5 person months of effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They do not describe how this proof can be used to verify a given quantum circuit that implements Grover’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In contrast, our approach is fully automated for verification of any QFT circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They have not addressed QFT verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' [25] have developed a model checking algorithm that can check the Quantum CTL (QCTL) properties on quantum Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The method is used to check the correctness of the BB84 protocol when n=1, the corresponding circuit for which has 8 qubits and 24 quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' They have not addressed QFT verification either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ROTATIONAL ABSTRACTIONS There are three key ideas in developing the abstractions for the Hadamard (H) gate and the controlled rotation (Rn) gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The first key idea is with regard to the basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit works correctly when the input qubits are the basis vectors |0⟩ or |1⟩, then the circuit is guaranteed to work correctly for any qubit inputs [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, for verification purposes, we only consider the cases where the input qubits are |0⟩ or |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The second key idea is with regard to quantum gates and is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If the input qubits are limited to basis vectors, then both the H gate and the Rn gate can be modelled as gates causing rotation on the basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The H gate has only one input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We call this the control input qc as shown in Figure 1(b), H R2 R3 H R2 H HA HA R2A ReA HA R2A R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='A HA4 because if the input is |1⟩, then the H gate function can be represented as a rotation on |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If this control input is |0⟩, then no rotation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Rn gate has two inputs (control and data) and one output, we call the control input qc, the data input qd, and the output qo (as shown in Figure 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If qc is |1⟩, then Rn performs a rotation on qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Otherwise, if qc is |0⟩, then no rotation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The third key idea is with regard to the amount of rotation performed by the quantum gates on data input qubits and the resulting output qubit states, and is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The H gate induces a π (2π/2) rotation on |1⟩ and does not rotate |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Rn gate induces a 2π/2n (π/2n−1 ) rotation on |1⟩ and does not rotate |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For examle, R4 induces a rotation of π/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Thus, the rotation performed by the gates on |1⟩ are negative powers of 2 with reference to 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The QFT circuit structure is such that the control inputs to the quantum gates are always initial qubit states and are used only to make the decision, whether to rotate or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Thus, we can abstract the basis vector input values |0⟩ and |1⟩ using Boolean values 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The qubits once transformed by these rotations are input to the next quantum gate and finally the output state of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If the 2πi term is factored out of the exponent, the final output state of each qubit (after transformation) can be ab- stractly represented using fractional bit-vectors that essentially capture the amount of rotation on |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The fractional bit-vector ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b1b2b3⟩ corresponds to rotation value 2π ∗ (b1 ∗ 2−1 + b2 ∗ 2−2+b3∗2−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For example, the bit-vector ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='101⟩ corresponds to rotation value of 2π(1/2+0+1/8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Abstractions of the H gate and the Rn gate can be obtained by defining their rotational impact on the fractional bit-vectors, and an abstracted QFT circuit can be obtained by using these abstracted gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In a QFT circuit with m qubits, the smallest amount of rotation will be 2π/2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, the fractional bit-vectors used to represent qubits in the abstracted QFT circuit will have to have m bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The abstract H gate is defined below and has one input qubit qc, which is Boolean type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Output qubit qo is a bit-vector of size equal to m, the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (Abstract Hadamard Gate) If qc=1, then qo ← ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='0⟩m, else qo ← ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='0⟩m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The abstract Rn gate is defined below and has two qubit inputs qc and qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The control input qc is type Boolean, the data input qd and the output qubit qo are both fractional bit- vectors of size m, the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (Abstract Rn Gate) If qc=1, then qo ← qd +m ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='.01m−n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='0⟩m, else qo ← qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In the above, +m represents fixed-point modulo addition w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='t m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The abstracted QFT circuit is obtained by replacing the H gates and Rn gates of the original circuit with the abstracted gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Input qubits are declared as type Boolean and all other qubits are declared as type bit-vector of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The abstracted QFT circuit with 3 qubits is shown in Figure 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' When the abstract H gate is applied, the qubits at the output of the H gates of the QFT circuit in Figure 1(d) will have the following values: q1 1 ← ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b100⟩ q1 2 ← ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b200⟩ q1 3 ← ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b300⟩ The QFT correctness property is given next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Let QFT- Absi(b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', bm) denote the output state of the ith qubit of an abstracted version of a QFT circuit, where m is the number of qubits and b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', bm are Boolean variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (QFT Correctness Property) A QFT circuit is functionally correct if, for all 1 ≤ i ≤ m, i is an integer, QFT-Absi(b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', bm) = ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='bibi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='bm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='0⟩m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Based on the correctness property above, for the QFT circuit from Figure 1(a) to be correct, the state of qubits at the output should be as follows: q3 1 = ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b1b2b3⟩ q3 2 = ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b2b30⟩ q3 3 = ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='b300⟩ The abstracted gates, abstracted QFT circuit, and Property 1 are expressible in the Quantifier Free logic of Bit Vectors (QF BV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' A number of SMT solvers exist that can very efficiently check properties in this logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, verification of a given QFT circuit can be performed by encoding the abstracted circuit and correctness property in this logic (using the SMT LIB language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' An SMT solver will check the property automatically and indicate if the property is satisfied or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If the property is satisfied, then the QFT circuit is guaranteed to be correct (as will be established in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If the property is not satisfied, the tool will generate a counter example, which can be used to trace the error(s) in the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ABSTRACTION CORRECTNESS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' QFT circuit showing error scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In this section, we provide a proof of correctness of our ver- ification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The overall approach is that we enumerate through all possible classes of errors in QFT circuits and show how the verification approach will flag each error class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The error classes are depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We call bit-vector values as data values as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' H R3 R3 R2 R2 H5 TABLE I VERIFICATION RESULTS QFT Benchmark Correct Circuit Incorrect Gate Error Incorrect Control Error No Error Error Depth Error Depth Gate-2 Gate-n Gate-2 Gate-n Verification Stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Verification Stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Verification Stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Verification Stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Verification Stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Qubits(n) Gates M(MB) T(s) M(MB) T(s) M(MB) T(s) M(MB) T(s) M(MB) T(s) 16 136 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='02 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='2 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='88 890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='7 1,523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='99 894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='5 1,568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='79 890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='6 1,571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='29 890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='6 1,524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='65 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit has an error, where an incorrect input is fed to an H gate, verification of the abstracted version of the QFT circuit will either generate a type error or will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If the input to the abstract H gate is a bit-vector input, this will be flagged as a type error as the H gate expects a Boolean input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If Boolean input qubit bj is expected whereas bk is fed for qubit qj, then the LHS of Property 1 for qj will be ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='⟩ and RHS will be ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, Property 1 will not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit has an error, where an incorrect input is fed to an Rn gate, verification of the abstracted version of the QFT circuit will either generate a type error or will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a control value is fed to the data input of an Rn gate or if a data value is fed to the control input of an Rn gate, a type error will be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If bj is expected whereas bk is fed for the control input of an Rn gate acting on qubit qj, then the LHS of Property 1 for qj will be ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='.bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='⟩ and RHS will be ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='.bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, Property 1 will not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If an incorrect data value is fed to an Rn gate, this will result in a missing Rn gate on a qubit and this case is dealt with subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The error above is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' R3 gate with input q2 1 should have b3 as its control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Instead b2 is erroneously fed as the control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit has an error, where an H gate is missing on a qubit or there is more than one H gate acting on a qubit, verification of the abstracted version of the QFT circuit will generate a type error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In the abstracted version of a QFT circuit, the input of an H gate is a control value and the output is a data value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Thus, if there is more than one H gate acting on a qubit, the H gates after the first one will receive data inputs and this will result in a type error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If there are no H gates acting on a qubit, the subsequent Rn gates will not get a data value at its data input and this will again result in a type error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' An example of a missing H gate error is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The H gate on q2 is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit has an error where an incorrect set of Rn gates are acting on a qubit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', required Rn gates are missing or additional Rn gates are present or both, verification of the abstract version of the QFT circuit will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Qubit 1 of a QFT circuit with m qubits should have gates R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', Rm acting on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Qubit 2 should have gates R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', Rm−1 acting on it and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Thus, there is only one Rn gate of a certain n value required to act on each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a required Rn gate is missing, then its rotational impact on the fractional bit-vector value abstracting the qubit will not be observed in Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a qubit has additional erroneous Rn gates acting on it, then the required rotation of the qubit will be incorrect and this will be reflected on the final fractional bit-vector value of the qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In both the above cases, Property 1 will not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Note that an Rn gate can be replaced with two Rn−1 gates, with the same control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For example, R2 can be substituted with two R3 gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If the total rotational impact of a sequence of Rn gates is what is expected, even though it does not conform with the Rn gate sequence described above, Property 1 will be satisfied because the fractional bit-vector abstraction accurately captures the rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' An example of an incorrect Rn gate is shown in Figure 2, where the gate on q1 1 should be R2 instead of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' If a QFT circuit has a combination of errors from the error classes described in Lemmas 1-4, verification of the abstracted version of the QFT circuit will generate a type error or will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' As can be seen from Lemmas 1-4, the effect that flags each error class is disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=', there is no overlap in these effects for type errors or Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Thus a combination of errors will also be flagged as a type error or will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' (QFT-Rotational Abstraction Correctness) If a QFT circuit has an error, verification of the abstracted version of the QFT circuit will generate a type error or will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' A QFT circuit has only two types of gates, the H gate and the Rn gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Based on this, there are only four classes of 6 errors possible: Incorrect input to a H gate, incorrect input to an Rn gate, missing or additional H gates in the circuit, and incorrect set of Rn gates acting on a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The fifth case of an erroneous QFT circuit is any combination of the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' From Lemmas 1-5, we see that in all the above cases, verification of the abstracted version of the QFT circuit will generate a type error or will not satisfy Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS Table I gives the verification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The verification bench- marks were generated by varying the number of qubits in the QFT circuit from 16 qubits to 10,000 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The table gives the number of quantum gates in each of the QFT benchmark circuits as well (column 2: Gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The verification experiments were conducted on an Intel(R) Core(TM) i9 - 12900K CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='2 GHz with 32 GB RAM and Ubuntu 64- bit operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The z3 version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='12 SMT solver [27] was used to check Property 1 for all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In the table, ”T(s)” indicates verification time in seconds, which is the z3 run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ”M(MB)” gives the z3 run time memory consumption in megabytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ”Correct Circuit” gives the verification statistics for the QFT circuits without errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For these circuits Property 1 is proved to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Property 1 allows for each qubit output to be verified independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Therefore, the verification of all the qubit output in the circuit were done in parallel and the memory and time reported correspond to the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ”Incorrect Gate Error” are circuits with gates errors and is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Gate-2 error here indicates that the R3 gate is incorrectly acting on qubit 1 instead of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Gate-n error here indicates that the Rn−1 gate is incorrectly acting on qubit 1 instead of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ”Incorrect Control Error” are circuits with incorrect control input to an Rn gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Gate-2 error here indicates that the R2 gate in qubit 1 is incorrectly controlled by qubit 3 instead of qubit 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The Gate- n error here indicates that the Rn gate in qubit 1 is incorrectly controlled by qubit n-1 instead of qubit n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For the circuits with errors, verification of Property 1 generates a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The time and memory reported corresponds to the verification of the first qubit output that caused a counterexample to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Figures 3 and 4 plot the verification time and memory from Table I versus the number of quantum gates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' In these graphs, both the x-axis and y-axis use a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' As can be seen from these graphs, with increase in the number of gates, both memory and verification time increase linearly for both correct circuits and circuits with errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The most complex circuit with 10,000 qubits and 50 million gates is verified in only about 37 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' This demonstrates the high efficiency and scalability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The time taken to verify circuits with errors is less than that of correct circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' However, there is not an order-of-magnitude reduction that is often observed in formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Figure 5 shows both verification time and memory as the position of the gate error is moved from qubit 1 to qubit 10,000 on the QFT circuit with 10,000 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The x-scale increases linearly, whereas the y-scale is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' The graph indicates the variation of time and memory with the vertical location of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We see that as the error moves from qubit 1 to qubit 10,000, both time and memory reduce exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Execution time requirement capture for QFT verification versus quantum gate count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Correct circuit, control input error and value error at qubit positions 2 and 10000 captured for elucidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Execution memory requirement capture for QFT verification versus quantum gate count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Correct circuit, control input error and value error at qubit positions 2 and 10000 captured for elucidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK Our proposed approach for verification of Quantum Fourier Transform (QFT) circuits achieves a meteoric advance in the efficiency and scalability of quantum circuits thus far verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We have been able to verify a QFT circuit with 10,000 qubits and over 50 million gates in only about 37 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We exploit the fact that our approach is domain specific to QFT verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' This is a common theme to achieve scalability in formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For example, there are a large number of formal verification techniques dedicated to the verification of multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' We also exploit the idea that the rotations performed by the gates are negative powers of 2 Execution time capture 103 Correct Circuit Incorrect Control Error at gate 2 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Control Erro at gate n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Correct Circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Gate Error at gate 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Gate Error at gate n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Number of quantum gates in QFT circuitExecution memory capture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Correct Circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Control Error at gate 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Control Erro at gate n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Correct Circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Gate Error at gate 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Incorrect Gate Error at gate n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Number of quantum gates in QFT circuit7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Resource utilization (time and memory) capture versus qubit count for erroneous QFT circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' and can therefore be encoded as fractional bit-vectors, thus reducing the verification obligations from Hilbert space to Boolean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' For future work, our goal is to extend these ideas to other quantum algorithms to advance efficiency and scalability of formal verification so as to cope with the size and complexity of quantum hardware roadmaps of the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Gottesman and I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} +page_content=' Resource capture for erroneous QFT circuit 103 102 101 Time 100 Memory 0 2000 4000 6000 8000 10000 Qubit line where error is located' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQf1fm1/content/2301.00737v1.pdf'} diff --git a/2dAzT4oBgHgl3EQf8_7Q/content/tmp_files/2301.01914v1.pdf.txt b/2dAzT4oBgHgl3EQf8_7Q/content/tmp_files/2301.01914v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5764e230da2c123578dc7d762937f393059b4de4 --- /dev/null +++ b/2dAzT4oBgHgl3EQf8_7Q/content/tmp_files/2301.01914v1.pdf.txt @@ -0,0 +1,720 @@ + +Accuracy and Fidelity Comparison of Luna and +DALL-E 2 Diffusion-Based Image Generation +Systems +Michael Cahyadi +School of Computer Science +Bina Nusantara University +Jakarta, Indonesia +michael.cahyadi001@binus.ac.id +Muhammad Rafi +School of Computer Science +Bina Nusantara University +Jakarta, Indonesia +muhammad.rafi007@binus.ac.id + + + + + +William Shan +School of Computer Science +Bina Nusantara University +Jakarta, Indonesia +william.sitanggang@binus.ac.id + + +Abstract — We qualitatively examine the accuracy and +fideltiy between two diffusion-based image generation systems, +namely DALL-E 2 and Luna, which have massive differences in +training datasets, algorithmic approaches, prompt resolvement, +and output upscaling. In our research we conclude that DALL- +E 2 significantly edges Luna in both alignment and fidelity +comparisons +I. +INTRODUCTION +Image generation systems is one of the many avenues +artificial intelligence research projects have been pursuing +ways to improve generative methods. Image generation +systems have immense potential to compliment and extend +human creativity[1], but on the other hand there are issues +with the field such as potential abuse to spread misinformation +and harrassment[2], bias against certain cultural groups[3], +and harmful associations against marginalized societies [4]. +The field of image generation using artificial intelligence +technologies has made great advancements in the last few +years, with recent models capable of generating images with +near human-like characteristics. Variations of Generative +Adversarial Networks (GAN)[5] were among the first models +to generate high-quality images, but recently there has been +more focus by researchers and the public[6] on diffusion- +based models trained using massive datasets. +The open nature of research information regarding +diffusion-based image generation models have also led to an +increase of image generation systems made by individuals, +which may not have sufficient guardrails against abuse[7]. +Image generation systems also use CLIP latents to +associate certain human concepts and understand them in +creative contexts[8]. With artificial intelligence systems +continuing to get better in non-analytical areas such as artistic +creativity that mimic closely human cognitive architectures, +researchers might soon get closer into the realm of General +Artificial Intelligence (GAI)[9]. +As artificial intelligence becomes a more pervaise tool in +day-to-day workflows, there needs to be an evaluation +regarding the quality of outputs generated by image +generation systems. Accurately judging the alignment and +perceived fidelity of generated outputs from these image +generation systems can help researchers and developers build +better systems that are aware of the pitfalls of current systems +in the market. +While there exists many diffusion-based image generation +systems, both open and closed sourced, we decided to search +for two systems who adopt resolvement approaches that lead +in their industry in terms of accuracy and widescale +implementation in the image generation technology +community. Latent diffusion models and CLIP-guided[10] +diffusion models represent the forefront methodologies for +image-generation technologies with alignment and fidelity +results surpassing previous GAN-based systems. +This paper also ultimately aims to examine the difference +in accuracy between images generated by diffusion-based +systems that are made by a large company using a large +training data set and a system created by an individual with +more limited training resources and less guardrails towards +abuse. To that effect, we consider the following two image +generation models for comparing our results: +1. +DALL-E 2[8] is an image generation system created +by OpenAI which can generate high-resolution images that +combine various concepts and art styles. The project was built +in PyTorch using ViT-H/16 text encounters with the training +data of 650M images scraped from the internet and aligned by +CLIP[10]. +2. +Luna is an image generation system built by Arfy +Slowly, a Senior Software Engineer at Google Research. The +project was built in Tensorflow and published on GitHub as +an open-source project. The system uses a latent diffusion +model[11] to condition the model on text prompts, however +the training dataset used is unknown. + +II. +RELATED WORKS +There have been many papers that try to compare the +performance of two image generation systems, whether +quantitatively or qualitatively. The metrics that are used to +verify the accuracy of image generation systems mostly rely +on image fidelity and its benchmark against real-world +equivalents. +While quantitative methods have been laid out to gauge +the accuracy of image generation systems such as the Fréchet +Inception Distance by Heusel et al.[12], the metric is used to +compare GAN performance at image generation using real- +life samples, which differs from our attempts to gauge the +Henry Lucky +School of Computer Science +Bina Nusantara University +Jakarta, Indonesia +henry.lucky@binus.ac.id +Jurike Moniaga +School of Computer Science +Bina Nusantara University +Jakarta, Indonesia +jurike@binus.edu + +accuracy of image generation systems at prompt resolving +novel concepts that have little to no real-life examples. +Existing evaluations of diffusion-based image generation +models are mostly based on the accuracy of the image +generated in specific fields such as in artificially generated +faces[13]. But no paper has qualitatively evaluated the +inherent accuracy between the prompt given to the model to +the generated image. +Qualititative methods of performance analysis are usually +done by human surveyors such as in research by Saharia et +al.[14]. This is due to the subjective nature of art[15], unlike +measurable things such as image fidelity, that cannot be +measured with common metric calculations. +In Saharia et al., the method used in evaluating image +accuracy consists of 2 questions given to human raters +inquriing about the fidelity and alignment (the accuracy +between human interpretation of a given concept and the +output given by an artificial intelligence[16]) of the system’s +output. +III. +METHODOLOGY +A. Prompt Creation +The prompts list below are modified prompts from +Google’s Drawbench Benchmark[14], whic covers a variety +of concepts, art styles, and common pitfalls of image +generation systems to generate images that can be a point of +evaluation for the alignment and fidelity of the image +generation systems. We have also listed the reasoning towards +why we choose each prompt. +The prompts detailed above have been screened by +running them through a Google image search and seeing how +easily images for these concepts could be retrieved; from this +process, we eliminated two prompts and modified another. +Number +Contents +Prompt +Explanation +1 +A +photorealistic +image of a machine +resembling +a +human being and +able to replicate +certain +human +movements +and +functions +automatically. +Prompt is used to evaluate +the ability of the image +generation system to build +photorealistic images that +don’t cross the uncanny +valley[17]. +2 + +A half-robot and +man +entity +with +chainsaws for their +head and hands in +the +style +of +Japanese anime. + +Prompt is used to evaluate +the +bias +in +machine +learning algorithms that are +trained with data from +westernized-culture[18]. +3 +Rbefraigerator. +The prompt is used as a +way to determine the image +generation system’s ability +to +navigate +word +misspellings[19]. +4 +A car on top of a +spoon. +The prompt is used to +examine the ability of +image generation systems +to generate images novel in +concept[19]. +Number +Contents +Prompt +Explanation +5 +Two bicycles and +one +car +on +an +empty grass field. +The prompt is used to +examine the ability of +image generation systems +to generate images with +accurate +positional +information[19]. +6 +In late afternoon in +January in Jakarta, +a man stands in the +shadow of a tree. +The prompt is used to +examine the ability of +image generation systems +to +accurately +create +shadows that correspond +with +differing +lighting +conditions[19] +7 +A Sumatran tiger +under the sea. +The prompt is used to +examine the ability of the +image generation system to +generate images that have +conflicting concepts[19]. +8 +Art +nouveau +stained +glass +window +art +depicting +Woody +from Toy Story. +The prompt is used to +examine the ability of the +image generation system to +generate images with pop +culture +products +in +a +medium +not +usually +associated +with +the +product[20]. + +B. Image Generation +Below each set of 4 images per-prompt on each model, we +will outline general observations from the researchers as with +the cited reasons of why each model behave in such a way. +The image results are compiled for further analysis in the +paper in methodologies outlined in later subsections. +The researchers will generate every 32 images from each +system, noting that DALL-E 2 generates four images per run. +For DALL-E 2, access was provided to the system during +September 2022 after a request for beta-testing research was +approved by the company. DALL-E 2 was accessed from the +OpenAI Beta website, with 8 credits dispensed every month +for non-commercial research use only. DALL-E 2 outputs 4 +photos in one-prompt execution. DALL-E 2 outputs +1024x1024 pixel images and Luna outputs 512x512 pixel +images. +For Luna, we use the provided Colab notebook by Google +to run the system. Considerations we’re made to run Luna +using on-premises hardware, however due to Tensorflow’s +requirement of NVIDIA CUDA cores we decided to opt for +cloud solutions instead due to faster compute times and as a +better benchmark against DALL-E 2 which is a cloud-based +system hosted in Microsoft Azure. Luna outputs 4 photos in +one-prompt execution. +C. Analysis +As laid out in the related works section, due to art being +inherently subjective in nature[15], normal metrics cannot be + +applied when analyzing the inherent accuracy of prompt +creations from image generation systems. +The methodologies to evaluate these images are based on +Drawbench by Saharia et al. [14] who used human raters to +judge prompt accuracy of Imagen, Google’s in-house +proprietary image generation system, against existing +competitors such as OpenAI DALL-E 2[8] and GLIDE[21]. +For the benchmark analysis, we conduct an independent +human evaluation run for each category. For each prompt, the +rater is shown two sets of images with one from DALL-E 2, +and second from Luna. Each set contains eight non-cherry- +picked generations from the corresponding model. The human +rater will be asked two questions. +1. Which set of images better represents the text +caption: [Text Caption]? Question subjectively evaluates +image-text alignment. +2. Which set of images is of higher quality? Question +subjectively evaluates image fidelity. +For each question, the rater is asked to select from two +choices: +1. I prefer set A. +2. I prefer set B. +The paper aggregates the scores from different raters and then +score it in a percentage value which will be presented in the +form of a candle graph. The authors did not perform any post +filtering of the data to identify unreliable raters, both for +expedience of the analysis process and because the task was +straightforward to explain and execute. + +IV. +RESULTS +After carefully compiling the results of the survey from +human raters over a span of two weeks. We analyze the +results according to generally acceptable benchmarks for +alignment and fidelity scores. + + + + + + + + + + + +Fig. 1. Alignment and Fidelity comparison between DALL-E 2 and Luna +using methodologies outlined in Saharia et al. plotted into a candle graph: User +preference rates for prompt alignment and image fidelity. + +Results show that when the output images are given to +human raters and evaluated using methods outlined in Saharia +et al, the results show that DALL-E 2 on average received a +higher image-text alignment (62.1%) and image fidelity +(83.4%) rating than Luna. +FID scores can be a more objective measurement of +fidelity of machine-generated images, but previous research +has shown that FID scores are not reflective of perceptual +quality[22]. + + + + + +Fig. 2. MS-COCO 256 × 256 FID-30K for DALL-E 2[8] and Luna (which +is based on stable diffusion, marked as LDM-KL[11]). Lower score is better. +While quantitative measurements are outside the scope of +this paper, FID scores cited from research papers of the +respective models show that DALL-E 2 outperforms other +methods on MS-COCO 256 x 256 with zero-shot FID-30K +with a score of 10.39, significantly outperforming systems +based on Latent Diffusion Models (LDM-KL) such as Luna. +The results line up with human raters’ indication of individual +samples fidelity ratings. + +Fig. 3. Selected image samples from the resolvement process of Prompt +4 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 +each generated per system. +It’s observed that both prompt systems have difficulties in +prompt resolvement of novel concepts, such as a car on a +spoon. While Luna seems to struggle with the concept, +DALL-E 2 interpret it as a request for a toy car, and not a real +car. + +Fig. 4. Selected image samples from the resolvement process of Prompt +5 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 +each generated per system. +It's also observed that Luna has difficulty assigning the +correct number of items in an image given a prompt that +contains numerical amounting values. While the issue is also +present in DALL-E 2, prior research has proven that the +system can atleast count to four objects[19]. +100% +50% +0% +DALLE-2 +Luna +Alignment +Fidelity +NParams +Model FID-30K +DALL-E 2 + 10.39 650M + Luna (LDM-KL) 12.63 n/a + +Researchers behind DALL-E 2 has also disclosed issues +regarding compositionality[8], which is the ability to +comprehend the merging of multiple object properties such as +shape and positioning within the image. Which is why the +placement of the objects inside of the picture generated might +look too symetrical. + +Fig. 5. Selected image samples from the resolvement process of Prompt +3 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 +each generated per system. +Resolvement of misspelled prompts[19] has also proved a +challenge for LDM-based systems such as Luna with DALL- +E 2 accurately representing the misspelled prompt as a +“refrigerator” and Luna failing to generate a comprehensible +image. This is theorized to be the result of significantly better +prompt alignment within DALL-E 2’s generation system that +enables it to edge out Luna in this prompt. +Resolvement of misspelled prompts[19] has also proved a +challenge for LDM-based systems such as Luna with DALL- +E 2 accurately representing the misspelled prompt as a +“refrigerator” and Luna failing to generate a comprehensible +image. This is theorized to be the result of significantly better +prompt alignment within DALL-E 2’s generation system that +enables it to edge out Luna in this prompt. + +Fig. 6. Selected image samples from the resolvement process of Prompt +2 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 +each generated per system. +Resolvement of prompts with non-westernized artstyles +both failed to generate anything resembling the inputted +prompt. Machine learning systems have consistently hit +difficulties in detecting and generating styles that are +uncommon outside of western culture such as the artstyle of +Japanese anime[23] This is possibly the result of bias within +large compiled datasets that’s mainly trained on webscrapes +of mostly western-aligned content[24]. This challenge will +also present itself more in bigger datasets, which will +complicate efforts to effectively scale computer vision and +generative datasets without significant alignment. +The possibility of training differences affecting the +performance of image generation systems are also observed to +be correlated. Comparing FID-30K scores and observing the +interception distance of FID-2K scores between the two +systems have yielded interesting technical observations. The +figures show that Luna experiences a distinct lowered amount +of TPU (Tensor Processing Units) training days compared to +DALL-E 2, which can negatively impact the alignment quality +and perceived fidelity of the image as less itterations are +performed within a specific timeframe. Luna as an +individually built system also possibly suffered from time +limitations during training. + + + + + + + + + + + + + +Fig. 6. +Comparison of TPU training time needed to achieve a 20 basis +point FID-2K rating between regular U-Nets (Luna or LDM-KL) vs efficient +U-Nets (DALLE-2). + +The differences mainly come down to complexity, which +might have caused worse FID-2K scores due the amount of +time used to train LDM-KL based systems compared to +OpenAI’s approach with DALL-E 2[25]. Time differences +may be attributed to the difference in libraries used, as Luna +uses Tensorflow and DALL-E 2 uses PyTorch, the latter of +which has been shown to be more performant than the former +resulting in faster compute times[26]. + +V. +CONCLUSIONS +The round of experimentation showcases the effectiveness +of frozen large pretrained language models as text encoders +for the text-to-image generation, but differences exist between +the capabilities of large models such as DALL-E 2 and smaller +scale models such as Luna. +Dramatically increasing the size of these language models +have significantly more impact than scaling the U-Net size on +overall performance on alignment and fidelity. This +encourages future research directions on exploring even +bigger language models as text encoders, both by companies +and individuals. +But increasing datasets has also several kinks other than +purely technical complications as there are ethical challenges +relating to large datasets used for the image generation +systems, particularly regarding subject data awareness and +consent[27], [28] and some datasents even reflect stereotypes, +offensive viewpoints, and derogatory associations of various +marginalized identity groups[24]. +While Luna was edged out in both alignment and fidelity +measurements both in qualitative benchmarks through human +raters and quantitative benchmarks through zero-shot FID-2K +and FID-30K scores, it has reached a remarkable level of +accuracy for a system that is built by an individual and trained +using a limited dataset. +We also find considerable performance penalties incurred +by Luna’s use of Tensorflow compared to DALL-E 2’s use of +FID-2K +Training Days +DALLE-2 equiv. +Luna equiv. + +DcoeceBrrees +RERD +Fceeecfor +TRBBEER +DBB40 +30 +20 +0 +1 +2 +3 +4 +5 +6 +7PyTorch which resulted in a slower comparative TPU training +days compared to the latter, which affects training accuracy. +We ultimately conclude that while differences exist +between large systems made by corporations and smaller +individual made systems, the advent of diffusion-based image +generation systems have lowered the barrier to enter the image +generation field significantly. The advancement in research of +generative AI technologies need to be paired with safeguards +and acknowledgement of ethical concerns, working towards a +safer implementation of systems. + +ACKNOWLEDGMENT +We give thanks to Arfy Slowy from the Google Brain +Research Team in Singapore and Imre Bard from the OpenAI +Alignment Research team for helping early discussions, and +providing +many +helpful comments and suggestions +throughout the project. We thank you the team at Kaggle and +OpenAI for the free tiers given for testing and exploratory +research purposes. Special thanks to Agneta Viola for +reviewing grammatical and linguistical errors. We thank +Herendra Kurniawan for their consistent and critical help with +TPU resource allocation and Kaggle notebook initialization. + +REFERENCES +[1] +R. T. Hughes, L. Zhu, and T. 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Panda, “Performance +Characterization of DNN Training Using +Tensorflow and PyTorch on Modern Clusters,” in +2019 IEEE International Conference on Cluster +Computing (CLUSTER), 2019, pp. 1–11. +[27] +C. Dulhanty, “Issues in Computer Vision Data +Collection: Bias, Consent, and Label Taxonomy,” +2020. Accessed: Dec. 01, 2022. [Online]. Available: +https://uwspace.uwaterloo.ca/handle/10012/16414 +[28] +A. Paullada, I. D. Raji, E. M. Bender, E. Denton, +and A. Hanna, “Data and its (dis)contents: A survey +of dataset development and use in machine learning +research,” Patterns, vol. 2, no. 11, p. 100336, Nov. +2021, doi: 10.1016/J.PATTER.2021.100336. + + + diff --git a/2dAzT4oBgHgl3EQf8_7Q/content/tmp_files/load_file.txt b/2dAzT4oBgHgl3EQf8_7Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d55755c35492c5f1e5c90bca7ecb535062944992 --- /dev/null +++ b/2dAzT4oBgHgl3EQf8_7Q/content/tmp_files/load_file.txt @@ -0,0 +1,342 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf,len=341 +page_content='Accuracy and Fidelity Comparison of Luna and DALL-E 2 Diffusion-Based Image Generation Systems Michael Cahyadi School of Computer Science Bina Nusantara University Jakarta, Indonesia michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='cahyadi001@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='id Muhammad Rafi School of Computer Science Bina Nusantara University Jakarta, Indonesia muhammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='rafi007@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='id William Shan School of Computer Science Bina Nusantara University Jakarta, Indonesia william.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='sitanggang@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='id Abstract — We qualitatively examine the accuracy and fideltiy between two diffusion-based image generation systems, namely DALL-E 2 and Luna, which have massive differences in training datasets, algorithmic approaches, prompt resolvement, and output upscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' In our research we conclude that DALL- E 2 significantly edges Luna in both alignment and fidelity comparisons I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' INTRODUCTION Image generation systems is one of the many avenues artificial intelligence research projects have been pursuing ways to improve generative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Image generation systems have immense potential to compliment and extend human creativity[1], but on the other hand there are issues with the field such as potential abuse to spread misinformation and harrassment[2], bias against certain cultural groups[3], and harmful associations against marginalized societies [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The field of image generation using artificial intelligence technologies has made great advancements in the last few years, with recent models capable of generating images with near human-like characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Variations of Generative Adversarial Networks (GAN)[5] were among the first models to generate high-quality images, but recently there has been more focus by researchers and the public[6] on diffusion- based models trained using massive datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The open nature of research information regarding diffusion-based image generation models have also led to an increase of image generation systems made by individuals, which may not have sufficient guardrails against abuse[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Image generation systems also use CLIP latents to associate certain human concepts and understand them in creative contexts[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' With artificial intelligence systems continuing to get better in non-analytical areas such as artistic creativity that mimic closely human cognitive architectures, researchers might soon get closer into the realm of General Artificial Intelligence (GAI)[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' As artificial intelligence becomes a more pervaise tool in day-to-day workflows, there needs to be an evaluation regarding the quality of outputs generated by image generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Accurately judging the alignment and perceived fidelity of generated outputs from these image generation systems can help researchers and developers build better systems that are aware of the pitfalls of current systems in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' While there exists many diffusion-based image generation systems, both open and closed sourced, we decided to search for two systems who adopt resolvement approaches that lead in their industry in terms of accuracy and widescale implementation in the image generation technology community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Latent diffusion models and CLIP-guided[10] diffusion models represent the forefront methodologies for image-generation technologies with alignment and fidelity results surpassing previous GAN-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' This paper also ultimately aims to examine the difference in accuracy between images generated by diffusion-based systems that are made by a large company using a large training data set and a system created by an individual with more limited training resources and less guardrails towards abuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' To that effect, we consider the following two image generation models for comparing our results: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' DALL-E 2[8] is an image generation system created by OpenAI which can generate high-resolution images that combine various concepts and art styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The project was built in PyTorch using ViT-H/16 text encounters with the training data of 650M images scraped from the internet and aligned by CLIP[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Luna is an image generation system built by Arfy Slowly, a Senior Software Engineer at Google Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The project was built in Tensorflow and published on GitHub as an open-source project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The system uses a latent diffusion model[11] to condition the model on text prompts, however the training dataset used is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' RELATED WORKS There have been many papers that try to compare the performance of two image generation systems, whether quantitatively or qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The metrics that are used to verify the accuracy of image generation systems mostly rely on image fidelity and its benchmark against real-world equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' While quantitative methods have been laid out to gauge the accuracy of image generation systems such as the Fréchet Inception Distance by Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' [12], the metric is used to compare GAN performance at image generation using real- life samples, which differs from our attempts to gauge the Henry Lucky School of Computer Science Bina Nusantara University Jakarta, Indonesia henry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='lucky@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='id Jurike Moniaga School of Computer Science Bina Nusantara University Jakarta, Indonesia jurike@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='edu accuracy of image generation systems at prompt resolving novel concepts that have little to no real-life examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Existing evaluations of diffusion-based image generation models are mostly based on the accuracy of the image generated in specific fields such as in artificially generated faces[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' But no paper has qualitatively evaluated the inherent accuracy between the prompt given to the model to the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Qualititative methods of performance analysis are usually done by human surveyors such as in research by Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' This is due to the subjective nature of art[15], unlike measurable things such as image fidelity, that cannot be measured with common metric calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' In Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=', the method used in evaluating image accuracy consists of 2 questions given to human raters inquriing about the fidelity and alignment (the accuracy between human interpretation of a given concept and the output given by an artificial intelligence[16]) of the system’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Prompt Creation The prompts list below are modified prompts from Google’s Drawbench Benchmark[14], whic covers a variety of concepts, art styles, and common pitfalls of image generation systems to generate images that can be a point of evaluation for the alignment and fidelity of the image generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' We have also listed the reasoning towards why we choose each prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompts detailed above have been screened by running them through a Google image search and seeing how easily images for these concepts could be retrieved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' from this process, we eliminated two prompts and modified another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Number Contents Prompt Explanation 1 A photorealistic image of a machine resembling a human being and able to replicate certain human movements and functions automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Prompt is used to evaluate the ability of the image generation system to build photorealistic images that don’t cross the uncanny valley[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 2 A half-robot and man entity with chainsaws for their head and hands in the style of Japanese anime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Prompt is used to evaluate the bias in machine learning algorithms that are trained with data from westernized-culture[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 3 Rbefraigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompt is used as a way to determine the image generation system’s ability to navigate word misspellings[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 4 A car on top of a spoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompt is used to examine the ability of image generation systems to generate images novel in concept[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Number Contents Prompt Explanation 5 Two bicycles and one car on an empty grass field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompt is used to examine the ability of image generation systems to generate images with accurate positional information[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 6 In late afternoon in January in Jakarta, a man stands in the shadow of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompt is used to examine the ability of image generation systems to accurately create shadows that correspond with differing lighting conditions[19] 7 A Sumatran tiger under the sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompt is used to examine the ability of the image generation system to generate images that have conflicting concepts[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 8 Art nouveau stained glass window art depicting Woody from Toy Story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The prompt is used to examine the ability of the image generation system to generate images with pop culture products in a medium not usually associated with the product[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Image Generation Below each set of 4 images per-prompt on each model, we will outline general observations from the researchers as with the cited reasons of why each model behave in such a way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The image results are compiled for further analysis in the paper in methodologies outlined in later subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The researchers will generate every 32 images from each system, noting that DALL-E 2 generates four images per run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' For DALL-E 2, access was provided to the system during September 2022 after a request for beta-testing research was approved by the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' DALL-E 2 was accessed from the OpenAI Beta website, with 8 credits dispensed every month for non-commercial research use only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' DALL-E 2 outputs 4 photos in one-prompt execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' DALL-E 2 outputs 1024x1024 pixel images and Luna outputs 512x512 pixel images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' For Luna, we use the provided Colab notebook by Google to run the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Considerations we’re made to run Luna using on-premises hardware, however due to Tensorflow’s requirement of NVIDIA CUDA cores we decided to opt for cloud solutions instead due to faster compute times and as a better benchmark against DALL-E 2 which is a cloud-based system hosted in Microsoft Azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Luna outputs 4 photos in one-prompt execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Analysis As laid out in the related works section, due to art being inherently subjective in nature[15], normal metrics cannot be applied when analyzing the inherent accuracy of prompt creations from image generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The methodologies to evaluate these images are based on Drawbench by Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' [14] who used human raters to judge prompt accuracy of Imagen, Google’s in-house proprietary image generation system, against existing competitors such as OpenAI DALL-E 2[8] and GLIDE[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' For the benchmark analysis, we conduct an independent human evaluation run for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' For each prompt, the rater is shown two sets of images with one from DALL-E 2, and second from Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Each set contains eight non-cherry- picked generations from the corresponding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The human rater will be asked two questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Which set of images better represents the text caption: [Text Caption]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Question subjectively evaluates image-text alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Which set of images is of higher quality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Question subjectively evaluates image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' For each question, the rater is asked to select from two choices: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' I prefer set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' I prefer set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The paper aggregates the scores from different raters and then score it in a percentage value which will be presented in the form of a candle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The authors did not perform any post filtering of the data to identify unreliable raters, both for expedience of the analysis process and because the task was straightforward to explain and execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' RESULTS After carefully compiling the results of the survey from human raters over a span of two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' We analyze the results according to generally acceptable benchmarks for alignment and fidelity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Alignment and Fidelity comparison between DALL-E 2 and Luna using methodologies outlined in Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' plotted into a candle graph: User preference rates for prompt alignment and image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Results show that when the output images are given to human raters and evaluated using methods outlined in Saharia et al, the results show that DALL-E 2 on average received a higher image-text alignment (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='1%) and image fidelity (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='4%) rating than Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' FID scores can be a more objective measurement of fidelity of machine-generated images, but previous research has shown that FID scores are not reflective of perceptual quality[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' MS-COCO 256 × 256 FID-30K for DALL-E 2[8] and Luna (which is based on stable diffusion, marked as LDM-KL[11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Lower score is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' While quantitative measurements are outside the scope of this paper, FID scores cited from research papers of the respective models show that DALL-E 2 outperforms other methods on MS-COCO 256 x 256 with zero-shot FID-30K with a score of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='39, significantly outperforming systems based on Latent Diffusion Models (LDM-KL) such as Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The results line up with human raters’ indication of individual samples fidelity ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Selected image samples from the resolvement process of Prompt 4 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Images we’re picked from a set of 4 each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' It’s observed that both prompt systems have difficulties in prompt resolvement of novel concepts, such as a car on a spoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' While Luna seems to struggle with the concept, DALL-E 2 interpret it as a request for a toy car, and not a real car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Selected image samples from the resolvement process of Prompt 5 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Images we’re picked from a set of 4 each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=" It's also observed that Luna has difficulty assigning the correct number of items in an image given a prompt that contains numerical amounting values." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' While the issue is also present in DALL-E 2, prior research has proven that the system can atleast count to four objects[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 100% 50% 0% DALLE-2 Luna Alignment Fidelity NParams Model FID-30K DALL-E 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='39 650M Luna (LDM-KL) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content='63 n/a Researchers behind DALL-E 2 has also disclosed issues regarding compositionality[8], which is the ability to comprehend the merging of multiple object properties such as shape and positioning within the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Which is why the placement of the objects inside of the picture generated might look too symetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Selected image samples from the resolvement process of Prompt 3 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Images we’re picked from a set of 4 each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Resolvement of misspelled prompts[19] has also proved a challenge for LDM-based systems such as Luna with DALL- E 2 accurately representing the misspelled prompt as a “refrigerator” and Luna failing to generate a comprehensible image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' This is theorized to be the result of significantly better prompt alignment within DALL-E 2’s generation system that enables it to edge out Luna in this prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Resolvement of misspelled prompts[19] has also proved a challenge for LDM-based systems such as Luna with DALL- E 2 accurately representing the misspelled prompt as a “refrigerator” and Luna failing to generate a comprehensible image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' This is theorized to be the result of significantly better prompt alignment within DALL-E 2’s generation system that enables it to edge out Luna in this prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Selected image samples from the resolvement process of Prompt 2 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Images we’re picked from a set of 4 each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Resolvement of prompts with non-westernized artstyles both failed to generate anything resembling the inputted prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Machine learning systems have consistently hit difficulties in detecting and generating styles that are uncommon outside of western culture such as the artstyle of Japanese anime[23] This is possibly the result of bias within large compiled datasets that’s mainly trained on webscrapes of mostly western-aligned content[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' This challenge will also present itself more in bigger datasets, which will complicate efforts to effectively scale computer vision and generative datasets without significant alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The possibility of training differences affecting the performance of image generation systems are also observed to be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Comparing FID-30K scores and observing the interception distance of FID-2K scores between the two systems have yielded interesting technical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The figures show that Luna experiences a distinct lowered amount of TPU (Tensor Processing Units) training days compared to DALL-E 2, which can negatively impact the alignment quality and perceived fidelity of the image as less itterations are performed within a specific timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Luna as an individually built system also possibly suffered from time limitations during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Comparison of TPU training time needed to achieve a 20 basis point FID-2K rating between regular U-Nets (Luna or LDM-KL) vs efficient U-Nets (DALLE-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The differences mainly come down to complexity, which might have caused worse FID-2K scores due the amount of time used to train LDM-KL based systems compared to OpenAI’s approach with DALL-E 2[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Time differences may be attributed to the difference in libraries used, as Luna uses Tensorflow and DALL-E 2 uses PyTorch, the latter of which has been shown to be more performant than the former resulting in faster compute times[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' CONCLUSIONS The round of experimentation showcases the effectiveness of frozen large pretrained language models as text encoders for the text-to-image generation, but differences exist between the capabilities of large models such as DALL-E 2 and smaller scale models such as Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Dramatically increasing the size of these language models have significantly more impact than scaling the U-Net size on overall performance on alignment and fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' This encourages future research directions on exploring even bigger language models as text encoders, both by companies and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' But increasing datasets has also several kinks other than purely technical complications as there are ethical challenges relating to large datasets used for the image generation systems, particularly regarding subject data awareness and consent[27], [28] and some datasents even reflect stereotypes, offensive viewpoints, and derogatory associations of various marginalized identity groups[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' While Luna was edged out in both alignment and fidelity measurements both in qualitative benchmarks through human raters and quantitative benchmarks through zero-shot FID-2K and FID-30K scores, it has reached a remarkable level of accuracy for a system that is built by an individual and trained using a limited dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' We also find considerable performance penalties incurred by Luna’s use of Tensorflow compared to DALL-E 2’s use of FID-2K Training Days DALLE-2 equiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Luna equiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' DcoeceBrrees RERD Fceeecfor TRBBEER DBB40 30 20 0 1 2 3 4 5 6 7PyTorch which resulted in a slower comparative TPU training days compared to the latter, which affects training accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' We ultimately conclude that while differences exist between large systems made by corporations and smaller individual made systems, the advent of diffusion-based image generation systems have lowered the barrier to enter the image generation field significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' The advancement in research of generative AI technologies need to be paired with safeguards and acknowledgement of ethical concerns, working towards a safer implementation of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' ACKNOWLEDGMENT We give thanks to Arfy Slowy from the Google Brain Research Team in Singapore and Imre Bard from the OpenAI Alignment Research team for helping early discussions, and providing many helpful comments and suggestions throughout the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' We thank you the team at Kaggle and OpenAI for the free tiers given for testing and exploratory research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' Special thanks to Agneta Viola for reviewing grammatical and linguistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' We thank Herendra Kurniawan for their consistent and critical help with TPU resource allocation and Kaggle notebook initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'} +page_content=' REFERENCES [1] R.' 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Technology, +Department of Electrical Engineering and Information Science, +Ruhr University Bochum, 44780 Bochum, Germany +2Theoretical Electrical Engineering, Department of Electrical and Information Engineering, +Kiel University, Kaiserstraße 2, 24143 Kiel, Germany +3Kiel Nano, Surface and Interface Science KiNSIS, +Kiel University, Christian-Albrechts-Platz 4, 24118 Kiel, Germany +(Dated: January 10, 2023) +Abstract +Understanding and modeling plasma-surface interactions frame a multi-scale as well as multi- +physics problem. Scale-bridging machine learning surface surrogate models have been demonstrated +to perceive the fundamental atomic fidelity for the physical vapor deposition of pure metals. How- +ever, the immense computational cost of the data-generating simulations render a practical appli- +cation with predictions on relevant timescales impracticable. This issue is resolved in this work +for the sputter deposition of AlN in Ar/N2 discharges by developing a scheme that populates +the parameter spaces effectively. Hybrid reactive molecular dynamics / time-stamped force-bias +Monte Carlo simulations of randomized plasma-surface interactions / diffusion processes are used +to setup a physics-separating artificial neural network. The application of this generic machine +learning model to a specific experimental reference case study enables the systematic analysis of +the particle flux emission as well as underlying system state (e.g., composition, mass density, stress, +point defect structure) evolution within process times of up to 45 minutes. +∗ tobias.gergs@rub.de +† thomas.mussenbrock@rub.de +‡ jt@tf.uni-kiel.de +1 +arXiv:2301.03524v1 [cond-mat.mtrl-sci] 9 Jan 2023 + +I. +INTRODUCTION +In most technological applications of plasmas (e.g., thin film sputter deposition, catalysis) +surfaces and, hence, plasma-surface interactions (e.g., growth, sputtering, surface chemical +reactions) are involved [1–4]. Analyzing, understanding, and modeling the last is considered +to be essential for a knowledge-driven process design. However, the physics of these two +states of matter (i.e., plasma, solid-state) demand for descriptions on length as well as time +scales that differ in orders of magnitudes (see Figure 1) [5–8]. +Common scale bridging solutions include event dependent coefficients, lookup-tables, and +analytic formulas (e.g., Berg-model [9, 10], Sigmund–Thompson theory [11–13]). However, +they altogether lack a fundamental atomic fidelity. +An issue that has been addressed by applying machine learning (ML) models. +They +have been shown to be capable of describing physical processes relevant to plasma science +with high accuracy while mitigating statistical noise, generalizing successfully [5, 14–21]. +In particular, a series of ML plasma-surface interaction (PSI) surrogate models have been +proposed for the sputter deposition of Ti1−xAlx thin films. First, a multi-layer-perceptron +(MLP) was trained to predict the Ar+ ion bombardment induced sputtering of a Ti0.5Al0.5 +composite target [5]. Second, a more advanced artificial neural network (ANN) combining a +dedicated mapper network with the decoder of a β-variational autoencoder (β-VAE [22–26]) +was established for Ti1−xAlx composite targets [20]. Therein, the stoichiometry has been +introduced as a basic surface state descriptor. Both studies are based on transport of ions +in matter (TRIM) simulation data. Further, a physics-separating artificial neural network +Figure 1. Schematic of the physical time and length scales for thin film sputter depositions. +2 + +Heavy particle +dynamics +mm +Electron +dynamics +Nanostructured thin +film deposition +Surface processes(PSNN) was proposed to describe the PSIs at the substrate as well as target in a generalized +manner for Al and Ar as material system and working gas, respectively. The PSNN consists +of two conditional variational autoencoders (CVAEs [21, 26, 27]). One describes the PSIs +(e.g., sputtering, ion bombardment induced damage formation). The other one describes +the conversion of the defect structure (i.e., ring statistical connectivity profile [28–30]) to the +surface state (i.e., stoichiometry, mass density, biaxial stress, tensile stress). It was demon- +strated that both (i.e., defect structure, surface state) are sufficient for a complete system +description that may evolve in time. However, being based on molecular dynamics (MD) +simulations for data generation, the latter was limited to the impingement of two consecutive +particle doses (in total: 2.42 × 1015 particles/cm2) due to the immense computational cost. +Hence, the input parameter space (i.e., particle flux composition, ion energy, surface state) +was found to be sampled insufficiently to setup a long-term evolution ML PSI surrogate +model for the sputter deposition of metal thin films. +In this work, the concept of a ML surface surrogate model is advanced by – among other +aspects – proposing a randomized data generating scheme which enables PSNNs to predict +the reactive sputter deposition of AlN thin films in Ar/N2 discharges for up to hours. The +considered process is relevant for the preparation of hard coatings, protective wear (e.g., +transition metal aluminium nitride, transition metal aluminium oxynitride), and energy +harvesting (scavenging) [31–35]. This manuscript is structured as follows: The considered +scenario is presented in Section II. In Section III, applied methods and parameters are +described. The results are presented and discussed in Section IV. Finally, conclusions are +drawn in Section V. +II. +SETUP +The general scenario of an Ar/N2 plasma discharge interacting with AlN surfaces is con- +sidered. While the gas discharge and sputtered particle transport dynamics are considered +predetermined, the focus is on the substrate side AlN thin film deposition. The target side +sputtering of AlN is not of main concern, but is included up to the maximum considered ion +energy (i.e., 300 eV). The key aspect for robust and reliable data-driven ML model develop- +ment is to efficiently populate the parameter space relevant for representing the dynamics +of PSI and diffusion. This is achieved by random sampling of a given number of initial +3 + +Figure 2. Illustration of the PSI setup. The atom configuration is rendered with the Open Visu- +alization Tool (OVITO) [36]. Al and N atoms are colored gray and light blue, respectively. +AlN bulk systems, which are subsequently subject to a series of diffusion process and PSI +simulations (e.g., ion bombardment). The corresponding evolution is recorded and used for +ML. A brief description of the procedure is as follows: +System state +A bulk wurtzite AlN supercell is considered with a point defect structure +that includes up to 5 % Ar, 10 % Al, and 10 % N interstitials as well as 20 % Al and 20 +% N vacancies. The defect structure is assumed to define the system sufficiently [21, 37]. +Complementing properties are determined after the atom configuration is relaxed. +The +system is characterized by the mass density ρ, lattice constant a, heat of formation ∆Hf, +bulk modulus B0, its derivative B′ +0, and 12 point defect populations ρvAl, ρAlN, ρAli, ρvN, +ρNAl, ρ(N-N)Al, ρNi, ρ(N-N)N, ρ(N-N)i, ρArAl, ρArN, ρAri. The Kr¨oger-Vink notation is used for +the defect types (subscripts) [38]. The defect populations define the total number of atoms +in the system: +ntot = (1 + ρvAl + ρvN − ρAli − ρNi − ρAri − 2ρ(N-N)i − ρ(N-N)N − ρ(N-N)Al)−1nideal +tot +(1) +nideal +tot +refers to the total number of atoms in the ideal AlN supercell (8 atoms per unit cell). +The point defect structure defines the Al, N, and Ar concentrations cAl, cN, and cAr, which +4 + +Fout +Tinare denoted as the composition: +cAl = 0.5nideal +tot +ntot +− ρvAl + ρAli + ρAlN − ρNAl − ρ(N-N)Al − ρArAl +(2a) +cN = 0.5nideal +tot +ntot +− ρvN + ρNi + 2ρ(N-N)i + ρ(N-N)N + 2ρ(N-N)Al − ρAlN − ρArN +(2b) +cAr = ρAri + ρArAl + ρArN +(2c) +The first terms on the right hand side of Eqs. (2a) and (2b) refer to the ideal configuration, +as 0.5nideal +tot +is the number of Al or N atoms when point defects are absent. The mass density +is determined by the lattice constants, the total number of atoms ntot, and the composition: +ρ = mAlcAl + mNcN + mArcAr +√ +3nuca2c +ntot +(3) +mAl, mN, and mAr are the masses of Al, N, and Ar atoms, respectively. nuc is the number +of unit cells (detailed later) and the lattice constant c = 1.6a is kept constant (anisotropic +deformations are suppresesed). +Plasma-Surface Interaction and Diffusion +For each initialized system, seven diffusion +and PSI simulations are performed alternately (detailed later). +First, the effect of bulk +diffusion processes on the system state is studied. For this a temperature T is imposed. +Second, an AlN surface is obtained by cleaving the bulk system either in [100] or [002] +direction. Third, the effect of individual particles s (i.e., Al, N, N2, Ar) bombarding the +AlN surface with specified kinetic energies Ekin is investigated. The contribution from the +plasma onto the surface is characterized by the particle fluxes Γin +s , the kinetic energy of the +particles Ekin, and the species s. The emitted fluxes are denoted by Γout +s . +The first and the last are used to setup two individual machine learning regression models +(i.e., PSI-CVAE, Diffusion-CVAE) that eventually are used to form a PSNN. +III. +METHODS +First, the data generating hybrid reactive molecular dynamics (RMD) / time-stamped +force-bias Monte Carlo (tfMC) simulations are described. +Second, the data processing, +training workflow and included metric are introduced. Third, the structure and information +flow of the PSNN is outlined. Fourth, physics-constraints and their implementation are +introduced. Fifth, the hyperparameter (HP) optimization is descried. Sixth, the production +run is presented. +5 + +Figure 3. +Schematic of the workflow and information flow for the data generating hybrid +RMD/tfMC simulations. +A. +Hybrid reactive molecular dynamics / time-stamped force-bias Monte Carlo +RMD, tfMC, and hybrid RMD/tfMC simulations are performed with the open-source +Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [39–43]. The in- +teractions of AlN complexes are described by the third-generation charge-optimized many- +body (COMB3) potential that is tapered with the Ziegler-Biersack-Littmark (ZBL) potential +(COMB3/ZBL potential) to account for high-energy collisions by including screened nuclear +repulsions [44–46]. The COMB3 formalism is outlined in [44]. The COMB3 AlN parame- +terization and combination with the ZBL potential is described in [46]. Its predecessor was +setup for nanostructures as well as heterogeneous interfaces and revisited to describe plasma- +surface interactions more accurately (e.g., ion bombardment induced damage production) +[46, 47]. The atomic charges are equilibrated by applying the charge transfer equilibration +(QTE+) method to account for meaningful charge exchange during PSIs (e.g., ion bombard- +ment, sputtering) [48]. In the following, charge equilibration refers to the application of the +QTE+ method with a timestep of 10−2 fs. The exponents of the 1s Slater type orbitals used +for the overlap integral computations are 0.668 ˚A−1 and 1.239 ˚A−1 for Al and N, respectively +[48]. +a. +System state initialization +It has been argued and demonstrated that the defect +structure is sufficient to describe a system [21, 37]. Hence, the initial atom configuration +is constructed by specifying the point defect structure. +The Ar and Al (N) interstitial +population ρAri and ρAli are sampled from a normal distribution N(0, σ) with the standard +6 + +System state Ss, +System state Ss +Bulk cleavage +PSI +PSI: +Diffusion: +Diffusion +Molecular dynamics +Monte Carlo +data +data +Bulk reinforcement +System state Ss. +System state S. +flux Fout +temperature Tdeviations 3σ = 5 % and 3σ = 10 %, respectively. The N interstitial populations account +for single as well as split interstitials (N-N) [49]. They are distinguished from each other +at the end of the surface state initialization. The Al and N vacancy population ρvAl and +ρvN are sampled from a normal distribution N(0, σ) with a standard deviation 3σ = 20 %. +Initially, no anti-sites (i.e., NAl, (N-N)Al, AlN, ArAl, ArN) are defined. +The surface orientation (i.e., AlN(002), AlN(100)) is determined by a coin flip. In either +case, a bulk supercell consisting of 8 × 5 × 7 orthorhombic unit cells is constructed with +the lattice constants a=3.136 ˚A and c = 1.6a. The total number of atoms in the ideal AlN +supercell (8 atoms per unit cell) is nideal +tot += 2240. The targeted total number of atoms ntot +is calculated as a function of the point defect population following Eq. (1). The absolute +number of point defects is obtained by multiplying the total number of atoms with the +individual point defect population. +First, Al and N vacancies are created by removing the required number of Al and N +atoms from the system. Second, interstitials are taken care of by randomly inserting new +atoms (i.e., Al, N, Ar) into the simulation domain. The Ar atoms’ coordinate in surface +normal direction is constrained to fall in between 5 ˚A above and below the lower and upper +boundary of the simulation domain, respectively. If the new atoms overlap with each other +or old atoms, they are deleted. This second step is repeated until the correct number of Al, +N and Ar atoms are generated. +The atom configuration is then relaxed. Minor discontinuities of the COMB3 interaction +potential hinder the successful application of a single conjugate gradient descent algorithm. +This issue is addressed by performing multiple energy minimizations (relaxations) as de- +picted in Figure 4. +The alternation between charge equilibration (i.e., applying QTE+) +and relaxation is meant to increase the computational efficiency. It is easier to relax an +expanding than shrinking atom configuration. Hence, the system is compressed when the +instantaneous pressure falls below -1 MPa. +The resultant point defect structure is determined by comparing the position of each +atom mapped into the unit cell with the Al as well as the N atom sites of the ideal AlN(002) +or AlN(100) structures (periodic images are taken into account). The distance tolerance is +defined by the halved Al-N bond length 1.9/2 ˚A. Nitrogen split interstitials (N-N)i, (N-N)N +or anti-sites (N-N)Al are identified by searching for interatomic distances between N atoms +that fall below 1.5 ˚A (the N-N bond length equals 1.3 ˚A [49]). The number of Al and N +7 + +Figure 4. Workflow of the bulk relaxation. Relaxation: Application of the conjugate gradient +descent algorithm implemented in LAMMPS. The tolerance for the residual force on any atom +is 1 eV/˚A. Charge equilibration: Performing one time step while the QTE+ method is applied. +∆U, ∆V and p refer to the change of the potential energy, volume, and instantaneous pressure +value, respectively. Compression: Application of the strain −10−6 along each direction. Volume +relaxation: In addition to the atom site relaxation, the simulation box dimensions are adjusted +isotropically to remove the residual stress from the system. +vacancies are computed at last to fulfill the particle balances: +nvAl = 0.5nideal +tot +− ntot,Al + nAli + nAlN − nNAl − n(N−N)Al − nArAl +(4a) +nvN = 0.5nideal +tot +− ntot,N + nNi + n(N−N)N + 2n(N−N)i + nNAl + 2n(N−N)Al − nAlN − nArN (4b) +The symbol n describes the absolute number of point defects, while the indexes denote +the particular point defect type. When the provided distance tolerance results in negative +8 + +(start) +interstitial relaxation + charge equilibration +relaxation +AU < 0.1 eV +no +yes +charge equilibration +OU +end +△U > 0.1 eV +yes +compression +-1 MPa +no +yes +charge equilibration +volume relaxation +charge equilibration +relaxationnumbers for vacancies, Frenkel pairs (i.e., vacancies plus interstitials) are added to even +out this diagnostic artifact. However, this procedure is applied rarely and is only meant to +guarantee physically meaningful results (i.e., non-negative numbers of vacancies). +At last, the minimum of the potential energy and corresponding lattice constant is ob- +tained by fitting the third-order Birch-Murnaghan equation of state (EOS) to the p-V/ntot +and U/ntot-V/ntot-curve of the just relaxed structure [50, 51]. The system dimensions are +scaled isotropically to evaluate ten strains distributed equidistantly in between −10−2 and +10−2. The system is compressed before it is expanded. The atom sites are relaxed for each +probed atom configuration. +b. +Diffusion +The tfMC method is applied for the simulation of the diffusion processes +[39–41]. The maximal displacement length of the lightest atom (i.e., N) is ∆ = 0.19˚A, that +is approximately 10 % of the typical nearest neighbor distance for AlN. The temperature is +sampled from a uniform distribution in between 300 and 1000 K. Simultaneously, the QTE+ +method is applied with a time step of 10−2 fs, tolerance of 1 V, and damping constant of +0.45 [48]. The simulation is run for 104 steps. The resultant atom configuration is relaxed +and evaluated as detailed previously (the interstitial relaxation is skipped). +c. +Plasma-surface interaction +The surface is established by cleaving the bulk system +either in [100] or [002] direction and elongating the simulation domain in surface normal +direction by additional 35 ˚A. This value is defined to be the sum of 11 ˚A and 24 ˚A, that +are attributed to the COMB3 cutoff radii and serve as a buffer for recognizing reflected or +sputtered particles, respectively. The atom sites are adjusted by alternating between charge +equilibration (i.e., QTE+) for fixed atom configuraiton and relaxation (i.e., performing a +conjugate gradient descent minimization until the residual force perceived by each atom +falls below 1 eV/˚A) for a fixed charge distribution until the potential energy is changed by +less than 0.1 eV per iteration. The surface slab is displaced randomly in both surface parallel +directions to include different impingement sites (the impinging particles are centered above +the surface slab). +Following this procedure, the system can be subdivided into four regions as depicted +in Figure 5: i) Excluded atoms whose surface normal coordinate falls below a threshold +zth = hz − 15 ˚A − (hz − 15 ˚A)Ekin/Ekin,max. zth is decreased linearly for increasing kinetic +energies of impinging particles Ekin, ranging from 0 eV to 300 eV (Ekin,max = 300 eV). hz is +the surface slab height. Excluded atoms are not allowed to interact with any other atom, +9 + +Figure 5. Illustration of the PSI setup. The atom configuration is rendered with OVITO [36]. Al +and N atoms are colored gray and light blue, respectively. The regions containing i) excluded, ii) +immobile, iii) temperature-controlled, and iv) all remaining atoms are colored transparent gray, +light blue, red, and not at all respectively. +effectively reducing the surface slab thickness to reduce the computational cost. These atoms +are also excluded from the charge equilibration. The interactions of reflected or sputtered +particles with other atoms are excluded too, when their surface normal coordinate exceeds +hz + 12 ˚A. However, they are kept in the simulation domain to evaluate them later. The +members of this group are updated dynamically (i.e., every 2000 steps). ii) Immobile atoms +whose surface normal coordinate falls below a threshold zth + 5 ˚A. These are not evolved in +time to anchor the surface slab in the simulation domain. iii) Mobile atoms whose distance +to the left or right periodic boundary in the surface parallel directions falls below 5 ˚A are +coupled to a Langevin thermostat with a damping constant of 100 fs to gradually remove +their kinetic energy, targeting 0 K. Ar atoms are always excluded. iv) All remaining atoms. +10 + +4 +i) +24 A +12 A +5A +5A +iv) +15 A +iii) +iii) +5 A +i) +ZthAn impinging particle is created 11 ˚A above the surface and centered laterally. Its species +(i.e., Al, N, N2, Ar) is determined randomly, whereas projectiles are assumed to be charge +neutral prior interaction. The likelihood for each candidate is distributed equally among +them. +Its kinetic energy ranging from 0 eV to 300 eV is found by squaring a sample +from a uniform distribution U(0 +√ +eV, +√ +300 eV). The atoms are assumed to hit the surface +perpendicularly (the surface parallel components of its velocity vector equals zero). The time +step equals 0.25 fs and is eventually lowered to secure that the maximum displacement and +change in kinetic energy of any atom does not exceed 0.1 ˚A and 0.01 eV, respectively. The +simulation is run for 1 ps repeatedly until the temperature of the mobile atoms falls below 100 +K. Impinging atoms that bypass the lower simulation domain boundary due to channeling +are reinserted at a random position in surface normal direction (lateral coordinates are +maintained) within the surface slab (overlapping with another atom by less than 0.5 ˚A leads +to a repetition). +To again transfer from a surface to a bulk configuration, the atom sites are relaxed +as outlined previously. +The random shifts in surface parallel directions outlined in the +beginning of this section are reversed. The change of the surface normal coordinate of the +uppermost temperature controlled atom ∆zup is used as a reference to invert the particle +impingement induced thermal expansion of the mobile surface slab. The surface normal +coordinates z of all mobile atoms are updated by z → z − ∆zup(z − zth − 5 ˚A)/(zup − zth − +5 ˚A) (assuming a linear expansion). All atoms that exceed the original surface slab height +prior to the particle impingement are removed from the system. This includes reflected +particles, sputtered particles, and in general atoms atop the surface (e.g., adatoms). The +last are assumed to contribute to the film growth of following layers, but do not effect the +subsurface region. Hence, they are neglected when making a prediction for the bulk system +by reestablishing a periodic boundary in surface normal direction. This procedure has been +validated by comparing lattice constants and stresses obtained with density functional theory +based molecular dynamics thermal spike simulations to experimentally measured reference +values for metal aluminium nitrides [37, 52, 53]. The resultant atom configuration is relaxed +and further evaluated as detailed previously (the interstitial relaxation is omitted). +11 + +B. +Data preparation, training and metrics +a. +Data set splitting +The data sets consisting of 6496 diffusion processes and 4470 PSIs +are shuffled and split for the HP optimization to train, validate and test the ML model with +80 %, 10 %, and 10 % of the available data, respectively. The size of the training, validation +and test set are referred to by ntrain +data , nval +data, and ntest +data, respectively. +b. +Data normalization +Min-max normalization is utilized, whereas the minimum and +maximum values are taken from the training set to avoid data leakage. +c. +Data augmentation +The normalized data is augmented to virtually extend the train- +ing database and, therewith, setup a more robust ML model [54–56]. In this work, a gener- +alized version of the constrained mixup augmentation is utilized. Input and output samples +are determined by ˆxij = λxi + (1 − λ)xj and ˆyij = λyi + (1 − λ)yj, respectively [57]. Hence, +a hypothesis of linear superposition is provided to the network. Its validity is reflected by +the probability distribution function (i.e., Beta(α)) of the λ value. α approaching zero, one, +or infinity resembles a coinflip, uniform distribution, or 0.5, respectively. In this work, sam- +ples i and j are only mixed up when the length of the vector pointing from one to another +falls below rc, that is +��nx +k=1(xi − xj)2 < rc. nx is the number of input parameters. rc +is considered a HP. The augmented data set size equals the original training data set size +(ntrain +data,aug = ntrain +data ). The training data is augmented anew once per epoch. +d. +Training procedure and metrics +Backpropagation of the mean absolute errors +(MAEs) is used to update the internal degrees of freedom of the ANNs (e.g., weights) +once per batch. +The stochastic gradient descent algorithm adaptive moment estimation +(Adam) is applied [58]. The applied batch size nbatch is defined to match the set up and +the ideal batch size nbatch,ideal included as HP as close as possible, but required to fulfill +|nbatch − ntrain +data,aug%nbatch| ≤ nbatch,ideal. Hence, all data samples contribute almost equally to +the learning progress. +The learning rate rl is initialized with rl-0 and kept constant for a simulated annealing +phase, that is outlined later. Afterwards, it is divided by ten whenever the validation MAE +falls below its previous minimum value over the course of nl-patience epochs. Early stopping +stops the training when there is no further reduction of the validation MAE after 2.5nl-patience +epochs. +12 + +Figure 6. Schematic of the CVAE structure. Input variables enter from the left and predictions +are extracted on the right of the graph. The coordinates zls of the latent space are indicated by +the center white box. The figure is taken from [21]. +e. +Hyperparameter study +10-fold Monte Carlo cross validation (MCCV) is utilized to +determine a more accurate final validation MAE. Training, validation, and test data sets are +randomly selected according to the given split. In 10-fold MCCV, an ensemble of 10 ANNs +is trained with 10 different random splits. It used in the selection of the best HPs using an +evolution strategy (described later). The coefficient of determination R2 is introduced as an +additional, secondary metric. The test set is meant to provide an unbiased measure for the +ML model’s performance. The last is determined even more thoroughly by applying a 100- +fold MCCV for the eventually selected set of HPs to compute the final training, validation +and test errors. +f. +Production run +The test set is not required for the production run. +Thus, it is +combined with the training set, that is 90 % of the data. An ensemble of ten ML models is +set up and trained to reduce the bias introduced to the model by splitting the data into the +two subsets [59]. +C. +Physics-separating artificial neural network methodology +a. +Conditional variational autoencoders +The proposed PSNN combines two regression +ML models (i.e., PSI-CVAE, Diffusion-CVAE), that are implemented as conditional vari- +ational autoencoders (CVAEs) [21, 26, 27]. +Their network architecture and information +13 + +Input +Input +Mis (ylac) +Decoder +Output y +Encoder +S +is (ylc) +2 +Latent +Train +space +Input y +phase? +True +False +N(0, I) +N(0, I)flow are shown schematically in Figure 6. CVAEs resemble β-variational autoencoders (β- +VAEs [22–26]), whose encoder and decoder are conditioned on the regression input variables. +These are set up symmetrically. The number of hidden layer nhl and nodes per layer nnpl +are considered as HPs. The activation functions for any hidden and output layer are set as +rectified linear unit (ReLU) and linear, respectively. The encoder projects information of the +regression output variables yi to an nls dimensional latent space representation. Similarity +to a standard normal distribution in latent space is enforced by introducing an additional +Kullback-Leibler (KL) divergence. [22, 23]. The HP β is used to scale the KL loss. It is +additionally scaled with a simulated annealing factor that is increased logarithmically from +10−3 to 1 per batch over the course of nSA (also a HP) epochs. The decoder is conditioned +on the regression input and the latent space (a standard normal distribution after successful +training) tries to reconstruct the regression output. The decoder resembles the regression +model to be utilized for prediction (after training is completed). CVAEs are described in +detail in [26, 27]. The outputs of the CVAEs are passed through physics-constraint enforcing +custom layers. +b. +Physics-constraints +The physics-constraints enforced by the last output layer sim- +plify the regression problem to be solved by the individual CVAEs and are described in the +following. First, the suppression of extrapolation is outlined. Second, the particle conversa- +tion for prediction on bulk diffusion processes are introduced. Note that predicted quantities +are denoted by primes (e.g., y′). +1. Extrapolation Suppression Constrained predictions were utilized in previous works +to secure physically plausible predictions (e.g., an Ar concentration in the range of 0 % to +100 %) [21]. This procedure is developed further and generalized in this work. In general ML +models are well suited for interpolation but often fail to extrapolate beyond known input +data. +Hence, predictions below (beyond) the minimal (maximal) training reference ymin +(ymax) are suppressed by folding them back three times to facilitate a more stable system +state evolution and guarantee positive quantities when required (e.g., mass density, sputter +yield): +y′ → 2ymin − y′ +if y′ ≤ ymin +(5a) +y′ → 2ymax − y′ +if y′≥ ymax +(5b) +2. Particle conservation (diffusion) The absolute number of Ar, Al, and N atoms must be +14 + +conserved during bulk diffusion processes, which are modeled by the Diffusion-CVAE. This +also demands a balance of the individual point defect populations. Using three corresponding +constraints (e.g., based on Eqs. (2c)-(2a)) to determine them reduces the number of the ML +model’s output descriptors, but may eventually contradict the extrapolation suppression +constraint introduced in the preceding paragraph. For example, the conservation of Ar atoms +prior and post diffusion (prediction) could be realized by determining the Ar population +occupying Al lattice sites ρ′ +ArAl = ntot/n′ +tot(ρAri +ρArAl +ρArN)−ρ′ +Ari −ρ′ +ArN and using Eq. (1). +However, some predictions may require n′ +ArAl to be negative (i.e., nAri + nArAl + nArN < +n′ +Ari + n′ +ArN), even though the number of Ar atoms occupying Al lattice sites cannot be +negative. +Enforcing the constraint outlined in the preceding paragraph may resolve the +issue, but being evaluated sequentially again may lead to a violation of particle conversation +during diffusion processes. +Thus, a more careful point defect balancing is required and +introduced in the following. +All Ar point defect population predictions (i.e., ρ′ +Ari + ρ′ +ArAl + ρ′ +ArN) are multiplied with a +correction factor fAr: +fAr = ntot +n′ +tot +ρAri + ρArAl + ρArN +ρ′ +Ari + ρ′ +ArAl + ρ′ +ArN + 10−7 +(6) +The deviation of predicted Al and N atoms prior/post diffusion is defined by ∆nAl and ∆nN, +respectively: +∆nAl =ntot(ρAli + ρAlN − ρvAl − ρNAl − ρ(N-N)Al − ρArAl) +− n′ +tot(ρ′ +Ali + ρ′ +AlN − ρ′ +vAl − ρ′ +NAl − ρ′ +(N-N)Al − fArρ′ +ArAl) +(7a) +∆nN =ntot(ρNi + 2ρ(N-N)i + ρ(N-N)N − ρvN + ρNAl + 2ρ(N-N)Al − ρAlN − ρArN) +− n′ +tot(ρ′ +Ni + 2ρ′ +(N-N)i + ρ′ +(N-N)N − ρ′ +vN + ρ′ +NAl + 2ρ′ +(N-N)Al − ρ′ +AlN − fArρ′ +ArN) +(7b) +with ntot as a function of the point defect population following Eq. (1). All defect populations +15 + +but anti-sites are compensated for the particle balancing using these deviations: +ρ′ +vAl → n′ +totρ′ +vAl − ∆nAl +ntot +if ∆nAl< 0 +(8a) +ρ′ +Ali → n′ +totρ′ +Ali + ∆nAl +ntot +if ∆nAl> 0 +(8b) +ρ′ +vN → n′ +totρ′ +vN − ∆nN +ntot +if ∆nN < 0 +(8c) +ρ′ +(N-N)N → +n′ +totρ′ +(N-N)N + +ρ′ +(N-N)N +ρ′ +Ni+ρ′ +(N-N)N ∆nN +ntot +if ∆nN > 0 +(8d) +ρ′ +Ni → +n′ +totρ′ +Ni + +ρ′ +Ni +ρ′ +Ni+ρ′ +(N-N)N ∆nN +ntot +if ∆nN > 0 +(8e) +All point defect populations, which have not been altered up this point, are scaled with the +quotient n′ +tot/ntot to account for the changed total number of atoms, ensuring consistent +predictions. +c. +Physics-separating artificial neural network +Each CVAE (i.e., PSI-CVAE, Diffusion- +CVAE) describes one physical process, separating one from another. The (trained) decoders +are combined to form a PSNN that allows for an evolution in time by passing the surface +state Ss from one surrogate model to another. The information flow of the PSNN is depicted +in Figure 7. It resembles closely the information flow inherent to the physical simulations +(Fig. 3. The input to the PSI-Decoder is a single particle sampled from the particle flux of +the plasma, characterized by the particles’ kinetic energy Ekin, species s, and surface state Ss. +It predicts the updated surface state S′ +s and emitted flux for each species Γout ′ +s +. The former is +fed together with the temperature T to the Diffusion-Decoder, which predicts a new surface +state S′′ +s . It is passed on to the PSI-Decoder, establishing an recurrent link within the PSNN. +Note that the direct correspondence of the physical simulations and the separated PSI- +Decoder and Diffusion-Decoder structure allows for an efficient parameter space exploration, +as outlined in Section III A. However, relying on single PSIs, the predictions after training +may be subject to vastly different plasma conditions and are not limited to the specific flux +ratios or ion energy distributions used for setting up the data set. +16 + +Figure 7. Schematic of the PSNN structure and information flow. Plasma dynamics can be imposed +by hand, simulation or experiment. +D. +Hyperparameter study +The HP of each CVAE (i.e., PSI-CVAE, Diffusion-CVAE) are optimized by applying +an individual anisotropic self-adaptive evolution strategy with intermediate recombination +(µ/µI, λ)-σSA-ESs. µ, λ, and σ refer to the number of parents, populations size, and step +sizes (mutation strengths), respectively. +Generalized and topic-wise related descriptions +of this method can be found in [21, 60–62]. The HPs considered in this work and their +initialization ranges are listed in Table I. +The evolution strategies are inialized as (7/7I, 70)-σSA-ESs. +The population sizes λ +are reduced by one per generation over the course of 63 generations and, afterwards, kept +constant. The numbers of parents are determined by µ = λ/7 (integer values are enforced) +[63]. The ESs are conducted for 200 generations and, hence, end as (1, 7)-σSA-ESs. +E. +Production run: Reference experiment +First, the experimental scenario considered for production as well as validation is outlined. +Second, the fluxes onto the AlN surfaces required for the ML simulation are calculated and +used to introduce an estimated process time. +a. +Reference experiment +Ries et al. +used a large-area multi-frequency capacitively +coupled plasma (MFCCP) to sputter deposit AlN with Ar and N2 as working gases (Ar/N2 +gas inlet ratio equal 8/1) [64]. The electrical asymmetry effect was taken advantage of to +17 + +System state Ss: +System state S" +Plasma dynamics +PSI-Decoder +Diffusion-Decoder +System state $ +flux Fouti +temperature TTable I. The HPs to be optimized, their initialization range, and final values for the PSI-CVAE as +well as Diffusion-CVAE. +HP +Init. range PSI-CVAE Diffusion-CVAE +rc +[0.0,1.0] +0.50 +0.44 +α +[10−5,1.0] +0.47 ·10−2 +0.17 +rl-0 +[10−3,10−2] 9.14 ·10−3 +1.08 ·10−3 +nl-patience +[4,7] +9 +7 +λL2 +[0,10−4] +2.23 ·10−7 +1.22 ·10−7 +nhl +[1,5] +1 +3 +nnpl +[8,128] +107 +155 +nls +[1,6] +1 +5 +β +[10−1,10] +56.69 +0.14 +nSA +[1,102] +102 +102 +nbatch,ideal +[16,64] +37 +53 +decouple the ion flux from the ion energy. The former was kept approximately constant and +the latter was controlled by applying voltage waveform tailoring (i.e., adjusting the relative +phase shift between the two excitation frequencies). +Four cases with mean ion energies +Eion of 47 eV, 53 eV, 57 eV, and 81 eV were considered. The predominant AlN surface +orientation was found to vary as function of the mean ion energy: AlN(002) for Eion =47 +eV and Eion =53 eV as well as AlN(100) for Eion =57 eV and Eion =81 eV [64]. +In this work, the species most relevant for PSI (i.e., Al, N+, N+ +2 , Ar+) are sampled from +the experimentally determined fluxes impinging onto the substrate (cf. next subsection). The +kinetic energy Ekin of ions and Al neutrals are sampled from measured ion energy distribution +functions (IEDFs) (depicted in Figure 11 of [64]) and from Monte Carlo transport simulations +(Al in pure Ar, but assumed invariant), respectively [64–66]. A threshold for the IEDFs +is imposed to avoid sampling from noise. Monte Carlo accept-reject sampling (rejection +sampling) is used to determine the individual particle energies. +The evolution and response of both monocrystalline systems (i.e., AlN(002), AlN(100)) +predicted by the PSNN is to be studied as a function of the mentioned four ion energy +18 + +distribution functions (IEDFs). Each case starts with ideal, defect free AlN and is run until +a steady-state is reached, that is approximately 45 minutes (experimental process time). +All cases are re-run 100 times to evaluate their statistics accurately. The final results are +averaged over the last minute. Intrinsic stresses are determined as a function of the predicted +lattice constants by utilizing the third-order Birch-Murnaghan EOS of the ideal, defect free +AlN reference system as proposed and validated in [37]. The final stresses and compositions +predicted by the PSNN are compared to experimentally measured reference values [64]. +Spurious Fe and O concentrations observed in the experiment are substituted with Al and +N concentrations to define a comparable reference for the simulation. +b. +Flux and process time estimations +The process time tp for npi particle impingements +may be estimated by the sum over all reciprocal impingement rates, tp = �npi +i=1 1/(ΓiA′ +RMD). +Γin +i +and A′ +RMD denote the experimental particle flux onto the AlN surface and predicted +RMD AlN surface area, respectively. +A′ +RMD is computed as a function of the predicted +lattice constant a′ and imposed surface orientation. +The ion flux onto the target and substrate is assumed to be approximately equal for +the given geometry and approximated by Γin +ion = hnevB, with assumed edge-to-center ration +h = 0.61, electron density ne = 5·1015 m−3, and Bohm velocity vB = 3.21·103 m/s (for Ar+ +ions and a given electron temperature of kBTe = 3 eV) [7, 67]. +The flux of Al neutrals onto the substrate is calculated from Γin +Al = ctYAr+(352 eV) Γin +ion = +3.47·1018 m−2s−1 with the collisional transport coefficient ct = 0.6 obtained from Monte Carlo +transport simulations (Al in pure Ar, but assumed invariant) as well as an Ar sputtering +yield YAr+(352 eV) = 0.579 (clean Al target) [64, 65]. The Al flux from the target is obtained +by multiplying the ion flux Γin +ion with the Ar+ sputtering yield YAr+. +The considered ion fluxes (i.e., Γin +N+, Γin +N+ +2 , Γin +Ar+) are determined by assuming that the +composition of the ion fluxes onto the substrate Γin +ion resemble the volumetric composition. +The total gas density is given by ng,tot = p/(kBTg) with the Boltzmann constant kb, and gas +temperature Tg = 650 K [67]. The species specific gas densities are calculated as relative +fractions assuming ng,N/ng,N2 = 1/9 and (ng,N + ng,N2)/ng,Ar = 1/8 [64]. Hence, the working +gas approximately consists of 1.16 % N, 10.47 % N2, and 88.37 % Ar. The ion (Bohm) flux +onto the substrate is split up accordingly: ΓAl+ = 3.11·1014 m−2s−1, ΓN+ = 1.14·1017 m−2s−1, +ΓN+ +2 = 1.03 · 1018 m−2s−1, and ΓAr+ = 8.65 · 1018 m−2s−1. The contribution due to Al+ is +neglected due to their rare occurrence. +19 + +IV. +RESULTS +A. +Hyperparameter study +Following the outlined evolution strategy with MCCV, an optimum set of HPs is de- +termined and listed in Table I. As apparent, data augmentation by means of constrained +mixup augmentation is beneficial for the Diffusion-CVAE (α = 0.17). This means that the +hypothesis of linear superposition is accepted to some extend for the diffusion processes but +declined for the PSIs (α = 0.47·10−2). Kernel regularization is found to be disadvantageous +for either ML model (λL2 ≈ 10−7). The network structure of the Diffusion-CVAE (i.e., 3 +hidden layer with 155 nodes per layer) allows for higher order of complexity than the PSI- +CVAE’s one (i.e., 1 hidden layer with 107 nodes per layer). It is also interesting to note that +the optimum number of simulated annealing epochs is 100 for both ML models, which is +the imposed upper boundary for this HP for the evolution strategies. Hence, the simulated +annealing step is assumed to be of great use for the training procedure. +In addition to the MAE, the performance of the PSI-CVAE and Diffusion-CVAE with +their final set of HPs listed in Table I can be assessed by the coefficient of determination +R2. It is calculated on the training, validation, and test set to equal 0.87, 0.86, and 0.87 for +the PSI-CVAE as well as 0.94, 0.93, and 0.93 for the Diffusion-CVAE, respectively. These +values ≳ 0.9 signify an accurate model approach (R2 = 1 signifies fully explained variance +in the data). The negligible difference between the three subsets indicates that the ML +models learned successfully to generalize on the training data. This finding is analyzed more +thoroughly in the following by comparing the unnormalized mean absolute errors (MAEs) of +each system property (e.g., mass density). It is important to note though that the reference +data does not resemble any kind of ground truth but contains statistical fluctuations (e.g., +a single ion hitting the surface on a different surface sites is likely to inflict different kinds +of defect structures) which intrinsically provide limits for the MAEs. +The MAE of all considered defect populations are shown in Figure 8. The PSI-CVAE and +Diffusion-CVAE is found to predict the defect structure accurately with errors that are of +the order/below 0.1 %. The error of the PSI-CVAE’s predictions on the training, validation, +and test set are barely distinguishable from each other, resembling excellent generalization. +The Diffusion-CVAE is found to perform best on the training set, showing minor signs of +20 + +Figure 8. MAE of the unnormalized predictions on the point defect populations and data sets. +Point defect types are listed on the x-axis. +Table II. MAE of the unnormalized predictions on all data sets for the PSI-PSNN. +Property +Train. set Val. set Test set +a (˚A) +0.002 +0.002 +0.002 +∆Ef (eV) +0.005 +0.005 +0.005 +B (GPa) +3.029 +3.067 +3.055 +B′ (GPa) +0.913 +0.922 +0.931 +Γout +Al /Γin +s (.) +0.015 +0.016 +0.015 +Γout +N /Γin +s (.) +0.089 +0.089 +0.087 +Γout +Ar /Γin +s (.) +0.218 +0.219 +0.219 +Γout +N2 /Γin +s (.) +0.139 +0.139 +0.140 +overfitting. However, the difference between the validation and test set is negligible. +The high accuracy prediction of the PSI-CVAE and the Diffusion-CVAE on the lattice +constant a, the formation energy ∆Ef, the bulk modulus B, and its derivative B′ are pre- +sented in Table II and Table III, respectively. The almost interchangeable performance on +the training, validation, and test set shows again that the models successfully learned to +21 + +Training set +Validation set +Test set +X +0.125 +PSI-CVAE +Diffusion-CVAE +0.100 +0.075 +0.050 +MA +X +0.025 +X +0.000Table III. MAE of the unnormalized predictions on all data sets for the Diffusion-PSNN. +Property Train. set Val. set Test set +a (˚A) +0.001 +0.001 +0.001 +∆Ef (eV) +0.007 +0.008 +0.008 +B (GPa) +2.905 +3.094 +3.124 +B′ (GPa) +0.918 +0.963 +0.973 +generalize on the provided data. However, the MAEs of the emitted Al, N, Ar and N2 flux +per incident flux, as listed in Table II, are relatively large when compared to typical sputter +yields as well as reflection ratios in the considered regime of kinetic energies (i.e., Ekin in +[0 eV, 300 eV]). It is argued that these larger errors do not signify bad performance, but +are rather a consequence of the data assembly for the PSIs. One PSI data sample contains +the information on a single PSI, which leads to the emission of, for example, none, one, or +maybe two particles. This will be perceived as noise to the ML model, which consequently +learns to predict the mean number of emitted particles per PSI for a given surface state. +This inherently leads to relatively large MAEs but ultimately is exactly what the PSI-CVAE +is meant to learn. +B. +Production run +The production run resembles the reference experiment of AlN thin-film deposition for +four discharge conditions as previously discussed. In the following, they are investigated for +two surface orientations (100) and (002). Initially the emitted particle fluxes are discussed: +Particles are emitted from the surface due to reflection of the incident particle or sputtering +of surface atoms. It is observed that most fluxes reach a steady-state after a few seconds. +Minor changes on the minute time-scale are observed only for three cases (i.e., Eion =47 eV: +(002), Eion =53 eV: (002), Eion =57 eV: (100)) due to a change of the Al sticking probability +of approximately 0.5 %. This transient variation is a side-effect of slowly evolving system +states, described in detail later. All Al sticking coefficients are in between 98-99 %. +The emitted per incident particle fluxes averaged over the last, 45th minute are shown +22 + +Figure 9. The emission of all film forming species per incident fluxes are presented for all considered +IEDFs as well as surface orientations. Circle and error bars represent mean values and root-mean- +squared deviations, respectively. +in Figure 9 for all film forming flux combinations (i.e., the emission of Ar is omitted). No +significant difference between the two surface orientations is recognizable, which is attributed +to the considered ion energy regime of 30 to 100 eV. Higher ion energies are expected to +present surface orientation dependent sputtering yields. +The impingement of N+ and Ar+ ions leads to an almost similar removal of Al atoms, +whereas Ar+ ions achieve a slightly increased Al sputtering yield (i.e., Γout +Al /Γin +N+ < Γout +Al /Γin +Ar+). +This is attributed to elastic collisions of bombarding N+ ions with N surface atoms, distribut- +ing the momentum more rapidly and evenly among them than Al atom. The displacement +of N atoms in the subsurface regions leads to the temporary formation of (N-N)N close to +the surface, where they eventually leave as N2. Higher ion energies lead to deeper collision +cascades spawned with higher momenta. The proportionality with the mean ion energy +indicates that for neither IEDF a relevant proportion of N+ ions directly form temporary +(N-N)N at the surface (and desorb as N2). +Bombarding N+ +2 ions are split apart when they hit the surface and, thus, inhibit a reduced +individual momentum compared to the initially shared one. This favors an even stronger +distribution of the momenta in the surface slab and, thus, lessens the likelihood of sputtering +23 + +0.3 +Fout: (002) +(002) +Tout. +out. +(002) +Al +N +N2 +47 eV +0.2 +53 eV +0.1 +57 eV +81 eV +0.0 +0.3 +Tout. +out. +:(100) +(100) +(100) +Al +N2 +47 eV +0.2 +53 eV +0.1 +57 eV +81 ev +0.0Figure 10. Transient evolution of the most relevant point defect populations for all considered +IEDFs as well as surface orientations. Error bars and the height of transparent region resemble +the mean plus / minus the RMSD. +Al atoms in the considered ion energy regime. Moreover, for smaller ion energies a shallower +subsurface region is affected, which enables incident N+ +2 ions to directly form temporary +(N-N)N at the surface before leaving as N2. This is reflected by the decreased flux ratio +Γout +N2 /Γin +N+ +2 for increased ion energies. +The transient evolutions of the most relevant point defect populations are shown in Fig- +24 + +2.4 +Eion +pvn: (002) +P(N-N)n: (002) +47eV +.8 + 53 eV +1.2 +57eV +81eV +0.0 +2.4 +Eion +pvA1: (002) +PAl: (002) +(%) +1.8 +47 eV +53 eV +1.2 +5 57 eV +0.6 +81 eV +0.0 +2.4 +Eion +Pvn: (100) +P(N-N)N: (100) +47 eV +1.8 + 53 eV +1.2 +57eV +0.6 +81 eV +0.0 +2.4 +Eion +pvAl: (100) +pAl:: (100) +(%) +1.8 +47eV +53 eV +fect +1.2 +57 eV +0.6 +_81eV +0.0 +0 +10 +20 +30 +40 +0 +10 +20 +30 +40 +t (min) +t (min)ure 10 for all considered IEDFs and surface orientations. The deposition onto AlN(002) with +Eion =47 eV takes up to 30 minutes to reach a steady-state. The ongoing ion bombardment +spawns collision cascades in the subsurface region, which once they have worn off may leave +vacancies and interstitials behind. Sputtering events or the desorption of N2 remove atoms +from the surface and, thus, facilitate the accumulation of vacancies. The Al and N vacancy +populations are approximately equal. The Al interstitial population is greater than the N +split interstitial population, and both exceed the corresponding vacancy populations. This +is due forward sputtering (peening) of surface atoms as well as incorporation of energetic +particles (i.e., N+, N2, small proportion of Al), which eventually either reside as interstitials +or recombine with vacancies. IEDFs with slightly higher mean ion energies (i.e., 53 eV, 57 +eV) converge to a similar point defect structure with marginally increased N split and Al +interstitial populations, but require significantly less time for equilibration. These require +a few minutes and seconds for Eion =53 eV and Eion =57 eV, respectively. Therefore it is +assumed that for Eion =47 eV scarcely sampled ions with relatively high kinetic energies +push the systems to their final state. The likelihood for encountering such ions is naturally +increased when increasing the mean ion energy. +This effect is enhanced by a change of +the IEDF shapes (i.e., narrow unimodal → narrow bimodal → broad unimodal). A more +significantly increased mean ion energy of 81 eV leads to the evolution to a different sys- +tem state with less Al and N vacancies (ρvAl ≈ ρvN) and more interstitials (ρAli > ρ(N-N)N). +The evolution of the vacancy populations inhibits intermediate maxima after a few seconds. +Subsequently, vacancies are removed due to recombination as described before and reach a +steady-state after 10 seconds. The evolution of the point defect structures are depicted in +Figure 10 for up to 45 minutes (and are available in the appendix for up to 100 seconds). +The deposition onto AlN(100) leads to similar system dynamics for Eion =47 eV and +Eion =53 eV. The Al and N vacancy populations are approximately equal too. But a greater +number of N split and smaller number of Al interstitials are observed. Scarce Al atoms +hitting the surface with relative high kinetic energies of up to 30 eV provide an insufficient +momentum when penetrating the AlN(100) surfaces to be persistently incorporated, i.e. +they end up atop the surface. Increasing the mean ion energy to 57 eV leads to a system +evolution that requires up to 10 minutes to reach a steady-state that differs significantly +from the previous one. The equilibration on the minute-time scale is again attributed to the +contribution of only a small proportion of the incident ions with sufficient kinetic energies, +25 + +Figure 11. Transient evolution of the mass density for all considered IEDFs as well as surface +orientations. Error bars and the height of transparent region resemble the mean plus / minus the +root-mean-squared deviations. +which are pushing the systems to their final state (cf. Eion =53 eV: (002)). The (N-N)N +populations remain unchanged. The Al and N vacancy populations are doubled. Hence, +the probability for the recombination of surface near Al vacancies and incident Al atoms is +increased too. The final Al interstitial population are therefore even more than doubled. The +point defect structure is characterized predominantly by Al and N Frenkel pairs (vacancies +plus interstitials). The evolution to this new system state is caused by a change of the +IEDF shapes. +The IEDF with Eion =53 eV (narrow bimodal IEDF) and Eion =57 eV +(broad unimodal IEDF) reaches up to 60 eV and 75 eV, respectively. The cases with the +highest mean ion energies of 83 eV converge to a similar system state with slightly increased +interstitial populations, but it takes only a few seconds. +The evolution of the mass densities are presented in Figure 11. The equilibration time of +the individual cases shows a consistent behavior. However, it is interesting to note that all +cases converge to a similar mass density for the surface orientation (002). The accumulation +of interstitials is balanced out by a corresponding volumetric expansion. In case of AlN(100), +two final point defect structures were discussed in the preceding paragraph. +These two +system states are reflected by two distinctly separate mass densities. Higher ion energies +26 + +3.175 +(002) +Eion +47 +eV +3.150 +53 eV +3.125 +57 eV +3.100 +81 eV +3.075 +3.175 +Eion +(100) +47 +eV +3.150 +53 eV +3.125 +57 +eV +3.100 +81 +eV +3.075 +0 +10 +20 +30 +40 +t (min)Figure 12. The final composition (i.e., Al and Ar concentration cAl and cAr, respectively) for all +considered IEDFs as well as surface orientations averaged over the last minute are compared to +experimental reference values [64]. Circles and error bars represent mean values and root-mean- +squared deviations. +lead to a great number of Al as well as N Frenkel pairs, which do not alter the mass of the +atomic system but cause stress and correspondingly a volumetric relaxation. Hence, smaller +mass densities are observed. +The composition of the deposited AlN(002) and AlN(100) thin films averaged over the +last minute are shown in Figure 12 in comparison to experimental reference values [64]. A +good agreement with the experiment is achieved when predicting stoichiometric AlN thin +films even though the Ar concentration of 2.5 ± 0.1 % for Eion = 47 eV and Eion = 53 eV is +not reproduced. +The stresses predicted by the PSNN and measured in the experiment are presented in +Figure 13 (a). An increasingly compressive stress is observed for greater mean ion energies in +either case due to the enhanced ion bombardment induced point defect formation. Vacancies +and interstitials cause tensile and compressive stresses, respectively. The interplay of all +point defects define the film stresses in the ML simulation. However, the contributions due +to Al interstitials dominate the stress formation due to their larger size and high formation +energies [68]. This finding is illustrated by a similar dependence of the stresses and the +negated Al interstitial populations (multiplied by -1) on the mean ion energy Eion, as shown +in Figures 13. The preferential surface orientation was found to change from (002) to (100) +in the experiment when increasing the mean ion energies from 47-53 eV to 57-81 eV [64]. +27 + +45 +CAl +30 +c +Experiment +CAr +15 +Simulation: (002) +Simulation: (100) +48 +56 +64 +72 +80 +Eion (eV)Figure 13. The (a) final stress and (b) negated Al interstitial population for all considered IEDFs +as well as surface orientations averaged over the last minute are compared to experimental reference +values [64]. Circles and error bars represent mean values and root-mean-squared deviations. +By comparison with the ML prediction for the (002) surface orientation, it can be inferred +that the predicted stresses for the two IEDFs with smaller mean ion energies (i.e., 47 eV, 53 +eV) are overestimated. However, from comparison with the prediction for the (100) surface +orientation, the two IEDFs with greater mean ion energies (i.e., 57 eV, 81 eV) are in excellent +agreement with the experiment. The change of the predominant surface orientation (i.e., +(002)→(100)) observed in the experiment may be attributed to the reduced compressive +stresses predicted to reach up to -12 GPa for (002), compared to -8 GPa for (100). +28 + +0 +a) +Experiment +Simulation: (002) +-3 +(GPa) +Simulation: (100) +stress α +-12 +48 +56 +64 +72 +80 +Eion (eV) +0.0 +(b) +Simulation: (002) +Simulation: (100) +-0.6 +.1.2 +-pAli +-1.8 +2.4 +48 +56 +64 +72 +80 +Eion (eV)V. +CONCLUSION +This work is meant to further advance the development of data-driven plasma-surface +interaction models with atomic fidelity [21]. Reactive processes (i.e., sputtering and depo- +sition of AlN in an Ar/N2 discharges) are taken into account. A data-generating scheme +is proposed that overcomes the burden of computationally too demanding simulations (i.e., +hybrid RMD/tfMC) and, hence, undersampled parameter spaces. The latter are effectively +populated by evolving randomly sampled system states Ss by means of random PSIs (i.e., +species s, kinetic energy Ekin) and diffusion processes (i.e., temperature T). The effect of a +single PSI on the deposited film is estimated by cleaving and reinforcing the corresponding +bulk structure in surface normal direction [37]. A PSNN is used to separate the PSIs from +the diffusion processes, which allows for a more efficient data-generation and enforcement of +physics-constraints (e.g., particle conservation during bulk diffusion). +The trained PSNN model is applied to an experimental reference sputter deposition of +AlN by taking the corresponding particle fluxes and IEDFs with mean ion energies in the +range of 47-81 eV into account [64]. Ar+ ions are found to remove more Al than N atoms +from the surface. +The inverse is observed for N+ ions, which spawn collision cascades +that distribute their momenta more rapidly with the N surface atoms. This facilitates the +temporary formation of (N-N)N at the very surface that eventually leave as N2. N+ +2 ions +are split up when they hit the surface and, thus, spawn two collision cascades with reduced +individual momenta compared to the initially shared one. +A diminishing amount of Al +atoms is sputtered and a shallower subsurface region is effected. The latter allows for the +direct formation of (N-N)N at the surface and subsequent emission as N2. Atomic nitrogen is +rarely sputtered by either ion species. Higher mean ion energies decrease the outgoing flux +of N2 due N+ +2 ion bombardment but increase the formation of persistent, deeper (N-N)N. +The predicted film depositions take either a few seconds or up to 30 minutes to reach their +respective steady-state. Long equilibration times are observed when rare ions whose kinetic +energy originates from the high energy tail of the IEDF push the systems to their final states. +The latter is found to be dependent on the imposed surface orientation. In particular, a +greater Al interstitials population is predicted for AlN(002) than for AlN(100). This point +defect type predominantly determines the compressive stress evolution in the deposited AlN +thin films. +The stresses predicted by the PSNN are quantitatively and qualitatively in +29 + +good agreement with the experimental reference values in spite of neglecting for instance +thermal stresses or point defect annihilation at grain boundaries. The ML model predicts +stoichiometric AlN that is observed in the experiment too. +In summary, 200 million plasma-surface interactions and diffusion processes were pre- +dicted with high physical fidelity (hybrid RMD/tfMC). This enabled the evolution of 800 +AlN systems (100 × four IEDFs × two surface orientations) in time for up to 45 minutes. +It took about 34 hours to perform all machine learning predictions with a single GPU. +Hence, predictions can be readily extended to cover up the total experimental deposition +time of up to hours when required. In contrast, conducting the same case study with hybrid +RMD/tfMC simulations is unattainable as it would take more than approximately 8 million +CPU years. +ACKNOWLEDGEMENT +Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +– Project-ID 138690629 – TRR 87 and – Project-ID 434434223 – SFB 1461. The authors +thank Dr.-Ing. S. Ries from Ruhr University Bochum, S. Karimi Aghda, M. Sc. from RWTH +Aachen University, and L. Vialetto, Ph.D. from Kiel University for fruitful discussions. +DATA AVAILABILITY +The data that support the findings of this study are available from the corresponding +author upon reasonable request. +ORCID +T. Gergs: https://orcid.org/0000-0001-5041-2941 +T. Mussenbrock: https://orcid.org/0000-0001-6445-4990 +J. Trieschmann: https://orcid.org/0000-0001-9136-8019 +30 + +APPENDIX +Figure 14. +Schematic of the CVAE network structure. +The shape of the data is provided in +parenthesis. Machine learning operations are indicated by colored arrows. The inputs and outputs +for the PSI-CVAE are given by x = {Ekin, s, Ss} and y = {Γout +s +, Ss}, respectively. The inputs +and outputs for the Diffusion-CVAE are given by x = {T, Ss} and y = {Ss}, respectively. +31 + +Concat +Dense +Calc +EnforceConstraints +2 +μls,r +Train +phase? +True +False +Ols,r +N(0, 1) +N(0,1) +m +m +nhl = 2 +nhl = 2 +(nis) +(SIu)Figure 15. Transient evolution of the most relevant point defect populations for all considered +IEDFs as well as surface orientations. Error bars and the height of transparent region resemble +the mean plus / minus the root-mean-squared deviations. +32 + +2.4 +Eion +pvn: (002) +P(N-N)n: (002) +47eV +1.8 + 53 eV +1.2 +57 eV +81eV +0.0 +2.4 +Eion +pvA1: (002) +PAl: (002) +(%) +1.8 +47 eV +53 eV +1.2 +57eV +0.6 +81 eV +0.0 +2.4 +Eion +Pvn: (100) +P(N-N)N: (100) +47 eV + 53 eV +1.2 +57eV +0.6 +81 eV +0.0 +2.4 +Eion +pvAr: (100) +PAl;: (100) +1.8 +47 eV +53 eV +fect +1.2 +557 eV +0.6 +81eV +0.0 +0 +25 +50 +75 +100 +0 +25 +50 +75 +100 +t (s) +(s) [1] P. J. Kelly and R. D. Arnell, Vacuum 56, 159 (2000). +[2] J. T. Gudmundsson, Plasma Sources Science and Technology 29, 113001 (2020), publisher: +IOP Publishing. +[3] A. Baptista, F. Silva, J. Porteiro, J. M´ıguez, and G. Pinto, Coatings 8, 402 (2018), number: +11 Publisher: Multidisciplinary Digital Publishing Institute. +[4] S. M. Rossnagel, J. J. 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Van de Walle, Physical Review B 65, 155212 (2002), publisher: American +Physical Society. +37 + diff --git a/5dE1T4oBgHgl3EQf6gWk/content/tmp_files/load_file.txt b/5dE1T4oBgHgl3EQf6gWk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..63e219185db0d6b3e5678a4aa95477e3cc4a48ad --- /dev/null +++ b/5dE1T4oBgHgl3EQf6gWk/content/tmp_files/load_file.txt @@ -0,0 +1,1470 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf,len=1469 +page_content='Physics-separating artificial neural networks for predicting sputtering and thin film deposition of AlN in Ar/N2 discharges on experimental timescales Tobias Gergs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ∗ Thomas Mussenbrock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' † and Jan Trieschmann2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ‡ 1Chair of Applied Electrodynamics and Plasma Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Department of Electrical Engineering and Information Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Ruhr University Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 44780 Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Germany 2Theoretical Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Department of Electrical and Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Kiel University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Kaiserstraße 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 24143 Kiel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Germany 3Kiel Nano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Surface and Interface Science KiNSIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Kiel University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Christian-Albrechts-Platz 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 24118 Kiel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Germany (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 2023) Abstract Understanding and modeling plasma-surface interactions frame a multi-scale as well as multi- physics problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Scale-bridging machine learning surface surrogate models have been demonstrated to perceive the fundamental atomic fidelity for the physical vapor deposition of pure metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' How- ever, the immense computational cost of the data-generating simulations render a practical appli- cation with predictions on relevant timescales impracticable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This issue is resolved in this work for the sputter deposition of AlN in Ar/N2 discharges by developing a scheme that populates the parameter spaces effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hybrid reactive molecular dynamics / time-stamped force-bias Monte Carlo simulations of randomized plasma-surface interactions / diffusion processes are used to setup a physics-separating artificial neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The application of this generic machine learning model to a specific experimental reference case study enables the systematic analysis of the particle flux emission as well as underlying system state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', composition, mass density, stress, point defect structure) evolution within process times of up to 45 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ∗ tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='gergs@rub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='de † thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='mussenbrock@rub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='de ‡ jt@tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='uni-kiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='03524v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='mtrl-sci] 9 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' INTRODUCTION In most technological applications of plasmas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', thin film sputter deposition, catalysis) surfaces and, hence, plasma-surface interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', growth, sputtering, surface chemical reactions) are involved [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Analyzing, understanding, and modeling the last is considered to be essential for a knowledge-driven process design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, the physics of these two states of matter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', plasma, solid-state) demand for descriptions on length as well as time scales that differ in orders of magnitudes (see Figure 1) [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Common scale bridging solutions include event dependent coefficients, lookup-tables, and analytic formulas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Berg-model [9, 10], Sigmund–Thompson theory [11–13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, they altogether lack a fundamental atomic fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' An issue that has been addressed by applying machine learning (ML) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' They have been shown to be capable of describing physical processes relevant to plasma science with high accuracy while mitigating statistical noise, generalizing successfully [5, 14–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In particular, a series of ML plasma-surface interaction (PSI) surrogate models have been proposed for the sputter deposition of Ti1−xAlx thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' First, a multi-layer-perceptron (MLP) was trained to predict the Ar+ ion bombardment induced sputtering of a Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5 composite target [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Second, a more advanced artificial neural network (ANN) combining a dedicated mapper network with the decoder of a β-variational autoencoder (β-VAE [22–26]) was established for Ti1−xAlx composite targets [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Therein, the stoichiometry has been introduced as a basic surface state descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Both studies are based on transport of ions in matter (TRIM) simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Further, a physics-separating artificial neural network Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Schematic of the physical time and length scales for thin film sputter depositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 2 Heavy particle dynamics mm Electron dynamics Nanostructured thin film deposition Surface processes(PSNN) was proposed to describe the PSIs at the substrate as well as target in a generalized manner for Al and Ar as material system and working gas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The PSNN consists of two conditional variational autoencoders (CVAEs [21, 26, 27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' One describes the PSIs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', sputtering, ion bombardment induced damage formation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The other one describes the conversion of the defect structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', ring statistical connectivity profile [28–30]) to the surface state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', stoichiometry, mass density, biaxial stress, tensile stress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It was demon- strated that both (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', defect structure, surface state) are sufficient for a complete system description that may evolve in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, being based on molecular dynamics (MD) simulations for data generation, the latter was limited to the impingement of two consecutive particle doses (in total: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='42 × 1015 particles/cm2) due to the immense computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, the input parameter space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', particle flux composition, ion energy, surface state) was found to be sampled insufficiently to setup a long-term evolution ML PSI surrogate model for the sputter deposition of metal thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In this work, the concept of a ML surface surrogate model is advanced by – among other aspects – proposing a randomized data generating scheme which enables PSNNs to predict the reactive sputter deposition of AlN thin films in Ar/N2 discharges for up to hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The considered process is relevant for the preparation of hard coatings, protective wear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', transition metal aluminium nitride, transition metal aluminium oxynitride), and energy harvesting (scavenging) [31–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This manuscript is structured as follows: The considered scenario is presented in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In Section III, applied methods and parameters are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The results are presented and discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Finally, conclusions are drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' SETUP The general scenario of an Ar/N2 plasma discharge interacting with AlN surfaces is con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' While the gas discharge and sputtered particle transport dynamics are considered predetermined, the focus is on the substrate side AlN thin film deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The target side sputtering of AlN is not of main concern, but is included up to the maximum considered ion energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', 300 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The key aspect for robust and reliable data-driven ML model develop- ment is to efficiently populate the parameter space relevant for representing the dynamics of PSI and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This is achieved by random sampling of a given number of initial 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Illustration of the PSI setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atom configuration is rendered with the Open Visu- alization Tool (OVITO) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Al and N atoms are colored gray and light blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' AlN bulk systems, which are subsequently subject to a series of diffusion process and PSI simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', ion bombardment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The corresponding evolution is recorded and used for ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A brief description of the procedure is as follows: System state A bulk wurtzite AlN supercell is considered with a point defect structure that includes up to 5 % Ar, 10 % Al, and 10 % N interstitials as well as 20 % Al and 20 % N vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The defect structure is assumed to define the system sufficiently [21, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Complementing properties are determined after the atom configuration is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The system is characterized by the mass density ρ, lattice constant a, heat of formation ∆Hf, bulk modulus B0, its derivative B′ 0, and 12 point defect populations ρvAl, ρAlN, ρAli, ρvN, ρNAl, ρ(N-N)Al, ρNi, ρ(N-N)N, ρ(N-N)i, ρArAl, ρArN, ρAri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Kr¨oger-Vink notation is used for the defect types (subscripts) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The defect populations define the total number of atoms in the system: ntot = (1 + ρvAl + ρvN − ρAli − ρNi − ρAri − 2ρ(N-N)i − ρ(N-N)N − ρ(N-N)Al)−1nideal tot (1) nideal tot refers to the total number of atoms in the ideal AlN supercell (8 atoms per unit cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The point defect structure defines the Al, N, and Ar concentrations cAl, cN, and cAr, which 4 Fout Tinare denoted as the composition: cAl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5nideal tot ntot − ρvAl + ρAli + ρAlN − ρNAl − ρ(N-N)Al − ρArAl (2a) cN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5nideal tot ntot − ρvN + ρNi + 2ρ(N-N)i + ρ(N-N)N + 2ρ(N-N)Al − ρAlN − ρArN (2b) cAr = ρAri + ρArAl + ρArN (2c) The first terms on the right hand side of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' (2a) and (2b) refer to the ideal configuration, as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5nideal tot is the number of Al or N atoms when point defects are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The mass density is determined by the lattice constants, the total number of atoms ntot, and the composition: ρ = mAlcAl + mNcN + mArcAr √ 3nuca2c ntot (3) mAl, mN, and mAr are the masses of Al, N, and Ar atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' nuc is the number of unit cells (detailed later) and the lattice constant c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6a is kept constant (anisotropic deformations are suppresesed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Plasma-Surface Interaction and Diffusion For each initialized system, seven diffusion and PSI simulations are performed alternately (detailed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' First, the effect of bulk diffusion processes on the system state is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' For this a temperature T is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Second, an AlN surface is obtained by cleaving the bulk system either in [100] or [002] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Third, the effect of individual particles s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Al, N, N2, Ar) bombarding the AlN surface with specified kinetic energies Ekin is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The contribution from the plasma onto the surface is characterized by the particle fluxes Γin s , the kinetic energy of the particles Ekin, and the species s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The emitted fluxes are denoted by Γout s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The first and the last are used to setup two individual machine learning regression models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', PSI-CVAE, Diffusion-CVAE) that eventually are used to form a PSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' METHODS First, the data generating hybrid reactive molecular dynamics (RMD) / time-stamped force-bias Monte Carlo (tfMC) simulations are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Second, the data processing, training workflow and included metric are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Third, the structure and information flow of the PSNN is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Fourth, physics-constraints and their implementation are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Fifth, the hyperparameter (HP) optimization is descried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Sixth, the production run is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Schematic of the workflow and information flow for the data generating hybrid RMD/tfMC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hybrid reactive molecular dynamics / time-stamped force-bias Monte Carlo RMD, tfMC, and hybrid RMD/tfMC simulations are performed with the open-source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The in- teractions of AlN complexes are described by the third-generation charge-optimized many- body (COMB3) potential that is tapered with the Ziegler-Biersack-Littmark (ZBL) potential (COMB3/ZBL potential) to account for high-energy collisions by including screened nuclear repulsions [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The COMB3 formalism is outlined in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The COMB3 AlN parame- terization and combination with the ZBL potential is described in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Its predecessor was setup for nanostructures as well as heterogeneous interfaces and revisited to describe plasma- surface interactions more accurately (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', ion bombardment induced damage production) [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atomic charges are equilibrated by applying the charge transfer equilibration (QTE+) method to account for meaningful charge exchange during PSIs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', ion bombard- ment, sputtering) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In the following, charge equilibration refers to the application of the QTE+ method with a timestep of 10−2 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The exponents of the 1s Slater type orbitals used for the overlap integral computations are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='668 ˚A−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='239 ˚A−1 for Al and N, respectively [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' System state initialization It has been argued and demonstrated that the defect structure is sufficient to describe a system [21, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, the initial atom configuration is constructed by specifying the point defect structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Ar and Al (N) interstitial population ρAri and ρAli are sampled from a normal distribution N(0, σ) with the standard 6 System state Ss, System state Ss Bulk cleavage PSI PSI: Diffusion: Diffusion Molecular dynamics Monte Carlo data data Bulk reinforcement System state Ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' System state S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' flux Fout temperature Tdeviations 3σ = 5 % and 3σ = 10 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The N interstitial populations account for single as well as split interstitials (N-N) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' They are distinguished from each other at the end of the surface state initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Al and N vacancy population ρvAl and ρvN are sampled from a normal distribution N(0, σ) with a standard deviation 3σ = 20 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Initially, no anti-sites (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', NAl, (N-N)Al, AlN, ArAl, ArN) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The surface orientation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', AlN(002), AlN(100)) is determined by a coin flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In either case, a bulk supercell consisting of 8 × 5 × 7 orthorhombic unit cells is constructed with the lattice constants a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='136 ˚A and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The total number of atoms in the ideal AlN supercell (8 atoms per unit cell) is nideal tot = 2240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The targeted total number of atoms ntot is calculated as a function of the point defect population following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The absolute number of point defects is obtained by multiplying the total number of atoms with the individual point defect population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' First, Al and N vacancies are created by removing the required number of Al and N atoms from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Second, interstitials are taken care of by randomly inserting new atoms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Al, N, Ar) into the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Ar atoms’ coordinate in surface normal direction is constrained to fall in between 5 ˚A above and below the lower and upper boundary of the simulation domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' If the new atoms overlap with each other or old atoms, they are deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This second step is repeated until the correct number of Al, N and Ar atoms are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atom configuration is then relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Minor discontinuities of the COMB3 interaction potential hinder the successful application of a single conjugate gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This issue is addressed by performing multiple energy minimizations (relaxations) as de- picted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The alternation between charge equilibration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', applying QTE+) and relaxation is meant to increase the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is easier to relax an expanding than shrinking atom configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, the system is compressed when the instantaneous pressure falls below -1 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The resultant point defect structure is determined by comparing the position of each atom mapped into the unit cell with the Al as well as the N atom sites of the ideal AlN(002) or AlN(100) structures (periodic images are taken into account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The distance tolerance is defined by the halved Al-N bond length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='9/2 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Nitrogen split interstitials (N-N)i, (N-N)N or anti-sites (N-N)Al are identified by searching for interatomic distances between N atoms that fall below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5 ˚A (the N-N bond length equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='3 ˚A [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The number of Al and N 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Workflow of the bulk relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Relaxation: Application of the conjugate gradient descent algorithm implemented in LAMMPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The tolerance for the residual force on any atom is 1 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Charge equilibration: Performing one time step while the QTE+ method is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ∆U, ∆V and p refer to the change of the potential energy, volume, and instantaneous pressure value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Compression: Application of the strain −10−6 along each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Volume relaxation: In addition to the atom site relaxation, the simulation box dimensions are adjusted isotropically to remove the residual stress from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' vacancies are computed at last to fulfill the particle balances: nvAl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5nideal tot − ntot,Al + nAli + nAlN − nNAl − n(N−N)Al − nArAl (4a) nvN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5nideal tot − ntot,N + nNi + n(N−N)N + 2n(N−N)i + nNAl + 2n(N−N)Al − nAlN − nArN (4b) The symbol n describes the absolute number of point defects, while the indexes denote the particular point defect type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' When the provided distance tolerance results in negative 8 (start) interstitial relaxation charge equilibration relaxation AU < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 eV no yes charge equilibration OU end △U > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 eV yes compression 1 MPa no yes charge equilibration volume relaxation charge equilibration relaxationnumbers for vacancies, Frenkel pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', vacancies plus interstitials) are added to even out this diagnostic artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, this procedure is applied rarely and is only meant to guarantee physically meaningful results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', non-negative numbers of vacancies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' At last, the minimum of the potential energy and corresponding lattice constant is ob- tained by fitting the third-order Birch-Murnaghan equation of state (EOS) to the p-V/ntot and U/ntot-V/ntot-curve of the just relaxed structure [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The system dimensions are scaled isotropically to evaluate ten strains distributed equidistantly in between −10−2 and 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The system is compressed before it is expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atom sites are relaxed for each probed atom configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Diffusion The tfMC method is applied for the simulation of the diffusion processes [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The maximal displacement length of the lightest atom (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', N) is ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='19˚A, that is approximately 10 % of the typical nearest neighbor distance for AlN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The temperature is sampled from a uniform distribution in between 300 and 1000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Simultaneously, the QTE+ method is applied with a time step of 10−2 fs, tolerance of 1 V, and damping constant of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='45 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The simulation is run for 104 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The resultant atom configuration is relaxed and evaluated as detailed previously (the interstitial relaxation is skipped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Plasma-surface interaction The surface is established by cleaving the bulk system either in [100] or [002] direction and elongating the simulation domain in surface normal direction by additional 35 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This value is defined to be the sum of 11 ˚A and 24 ˚A, that are attributed to the COMB3 cutoff radii and serve as a buffer for recognizing reflected or sputtered particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atom sites are adjusted by alternating between charge equilibration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', QTE+) for fixed atom configuraiton and relaxation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', performing a conjugate gradient descent minimization until the residual force perceived by each atom falls below 1 eV/˚A) for a fixed charge distribution until the potential energy is changed by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 eV per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The surface slab is displaced randomly in both surface parallel directions to include different impingement sites (the impinging particles are centered above the surface slab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Following this procedure, the system can be subdivided into four regions as depicted in Figure 5: i) Excluded atoms whose surface normal coordinate falls below a threshold zth = hz − 15 ˚A − (hz − 15 ˚A)Ekin/Ekin,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' zth is decreased linearly for increasing kinetic energies of impinging particles Ekin, ranging from 0 eV to 300 eV (Ekin,max = 300 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' hz is the surface slab height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Excluded atoms are not allowed to interact with any other atom, 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Illustration of the PSI setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atom configuration is rendered with OVITO [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Al and N atoms are colored gray and light blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The regions containing i) excluded, ii) immobile, iii) temperature-controlled, and iv) all remaining atoms are colored transparent gray, light blue, red, and not at all respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' effectively reducing the surface slab thickness to reduce the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' These atoms are also excluded from the charge equilibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The interactions of reflected or sputtered particles with other atoms are excluded too, when their surface normal coordinate exceeds hz + 12 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, they are kept in the simulation domain to evaluate them later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The members of this group are updated dynamically (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', every 2000 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ii) Immobile atoms whose surface normal coordinate falls below a threshold zth + 5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' These are not evolved in time to anchor the surface slab in the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' iii) Mobile atoms whose distance to the left or right periodic boundary in the surface parallel directions falls below 5 ˚A are coupled to a Langevin thermostat with a damping constant of 100 fs to gradually remove their kinetic energy, targeting 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Ar atoms are always excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' iv) All remaining atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 10 4 i) 24 A 12 A 5A 5A iv) 15 A iii) iii) 5 A i) ZthAn impinging particle is created 11 ˚A above the surface and centered laterally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Its species (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Al, N, N2, Ar) is determined randomly, whereas projectiles are assumed to be charge neutral prior interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The likelihood for each candidate is distributed equally among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Its kinetic energy ranging from 0 eV to 300 eV is found by squaring a sample from a uniform distribution U(0 √ eV, √ 300 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The atoms are assumed to hit the surface perpendicularly (the surface parallel components of its velocity vector equals zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The time step equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='25 fs and is eventually lowered to secure that the maximum displacement and change in kinetic energy of any atom does not exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 ˚A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='01 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The simulation is run for 1 ps repeatedly until the temperature of the mobile atoms falls below 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Impinging atoms that bypass the lower simulation domain boundary due to channeling are reinserted at a random position in surface normal direction (lateral coordinates are maintained) within the surface slab (overlapping with another atom by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5 ˚A leads to a repetition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' To again transfer from a surface to a bulk configuration, the atom sites are relaxed as outlined previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The random shifts in surface parallel directions outlined in the beginning of this section are reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The change of the surface normal coordinate of the uppermost temperature controlled atom ∆zup is used as a reference to invert the particle impingement induced thermal expansion of the mobile surface slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The surface normal coordinates z of all mobile atoms are updated by z → z − ∆zup(z − zth − 5 ˚A)/(zup − zth − 5 ˚A) (assuming a linear expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' All atoms that exceed the original surface slab height prior to the particle impingement are removed from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This includes reflected particles, sputtered particles, and in general atoms atop the surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', adatoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The last are assumed to contribute to the film growth of following layers, but do not effect the subsurface region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, they are neglected when making a prediction for the bulk system by reestablishing a periodic boundary in surface normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This procedure has been validated by comparing lattice constants and stresses obtained with density functional theory based molecular dynamics thermal spike simulations to experimentally measured reference values for metal aluminium nitrides [37, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The resultant atom configuration is relaxed and further evaluated as detailed previously (the interstitial relaxation is omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 11 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Data preparation, training and metrics a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Data set splitting The data sets consisting of 6496 diffusion processes and 4470 PSIs are shuffled and split for the HP optimization to train, validate and test the ML model with 80 %, 10 %, and 10 % of the available data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The size of the training, validation and test set are referred to by ntrain data , nval data, and ntest data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Data normalization Min-max normalization is utilized, whereas the minimum and maximum values are taken from the training set to avoid data leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Data augmentation The normalized data is augmented to virtually extend the train- ing database and, therewith, setup a more robust ML model [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In this work, a gener- alized version of the constrained mixup augmentation is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Input and output samples are determined by ˆxij = λxi + (1 − λ)xj and ˆyij = λyi + (1 − λ)yj, respectively [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, a hypothesis of linear superposition is provided to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Its validity is reflected by the probability distribution function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Beta(α)) of the λ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' α approaching zero, one, or infinity resembles a coinflip, uniform distribution, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In this work, sam- ples i and j are only mixed up when the length of the vector pointing from one to another falls below rc, that is ��nx k=1(xi − xj)2 < rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' nx is the number of input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' rc is considered a HP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The augmented data set size equals the original training data set size (ntrain data,aug = ntrain data ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The training data is augmented anew once per epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Training procedure and metrics Backpropagation of the mean absolute errors (MAEs) is used to update the internal degrees of freedom of the ANNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', weights) once per batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The stochastic gradient descent algorithm adaptive moment estimation (Adam) is applied [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The applied batch size nbatch is defined to match the set up and the ideal batch size nbatch,ideal included as HP as close as possible, but required to fulfill |nbatch − ntrain data,aug%nbatch| ≤ nbatch,ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, all data samples contribute almost equally to the learning progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The learning rate rl is initialized with rl-0 and kept constant for a simulated annealing phase, that is outlined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Afterwards, it is divided by ten whenever the validation MAE falls below its previous minimum value over the course of nl-patience epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Early stopping stops the training when there is no further reduction of the validation MAE after 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5nl-patience epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 12 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Schematic of the CVAE structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Input variables enter from the left and predictions are extracted on the right of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The coordinates zls of the latent space are indicated by the center white box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The figure is taken from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hyperparameter study 10-fold Monte Carlo cross validation (MCCV) is utilized to determine a more accurate final validation MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Training, validation, and test data sets are randomly selected according to the given split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In 10-fold MCCV, an ensemble of 10 ANNs is trained with 10 different random splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It used in the selection of the best HPs using an evolution strategy (described later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The coefficient of determination R2 is introduced as an additional, secondary metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The test set is meant to provide an unbiased measure for the ML model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The last is determined even more thoroughly by applying a 100- fold MCCV for the eventually selected set of HPs to compute the final training, validation and test errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Production run The test set is not required for the production run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Thus, it is combined with the training set, that is 90 % of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' An ensemble of ten ML models is set up and trained to reduce the bias introduced to the model by splitting the data into the two subsets [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Physics-separating artificial neural network methodology a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Conditional variational autoencoders The proposed PSNN combines two regression ML models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', PSI-CVAE, Diffusion-CVAE), that are implemented as conditional vari- ational autoencoders (CVAEs) [21, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Their network architecture and information 13 Input Input Mis (ylac) Decoder Output y Encoder S is (ylc) 2 Latent Train space Input y phase?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' True False N(0, I) N(0, I)flow are shown schematically in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' CVAEs resemble β-variational autoencoders (β- VAEs [22–26]), whose encoder and decoder are conditioned on the regression input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' These are set up symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The number of hidden layer nhl and nodes per layer nnpl are considered as HPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The activation functions for any hidden and output layer are set as rectified linear unit (ReLU) and linear, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The encoder projects information of the regression output variables yi to an nls dimensional latent space representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Similarity to a standard normal distribution in latent space is enforced by introducing an additional Kullback-Leibler (KL) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The HP β is used to scale the KL loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is additionally scaled with a simulated annealing factor that is increased logarithmically from 10−3 to 1 per batch over the course of nSA (also a HP) epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The decoder is conditioned on the regression input and the latent space (a standard normal distribution after successful training) tries to reconstruct the regression output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The decoder resembles the regression model to be utilized for prediction (after training is completed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' CVAEs are described in detail in [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The outputs of the CVAEs are passed through physics-constraint enforcing custom layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Physics-constraints The physics-constraints enforced by the last output layer sim- plify the regression problem to be solved by the individual CVAEs and are described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' First, the suppression of extrapolation is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Second, the particle conversa- tion for prediction on bulk diffusion processes are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Note that predicted quantities are denoted by primes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Extrapolation Suppression Constrained predictions were utilized in previous works to secure physically plausible predictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', an Ar concentration in the range of 0 % to 100 %) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This procedure is developed further and generalized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In general ML models are well suited for interpolation but often fail to extrapolate beyond known input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, predictions below (beyond) the minimal (maximal) training reference ymin (ymax) are suppressed by folding them back three times to facilitate a more stable system state evolution and guarantee positive quantities when required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', mass density, sputter yield): y′ → 2ymin − y′ if y′ ≤ ymin (5a) y′ → 2ymax − y′ if y′≥ ymax (5b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Particle conservation (diffusion) The absolute number of Ar, Al, and N atoms must be 14 conserved during bulk diffusion processes, which are modeled by the Diffusion-CVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This also demands a balance of the individual point defect populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Using three corresponding constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' (2c)-(2a)) to determine them reduces the number of the ML model’s output descriptors, but may eventually contradict the extrapolation suppression constraint introduced in the preceding paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' For example, the conservation of Ar atoms prior and post diffusion (prediction) could be realized by determining the Ar population occupying Al lattice sites ρ′ ArAl = ntot/n′ tot(ρAri +ρArAl +ρArN)−ρ′ Ari −ρ′ ArN and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, some predictions may require n′ ArAl to be negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', nAri + nArAl + nArN < n′ Ari + n′ ArN), even though the number of Ar atoms occupying Al lattice sites cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Enforcing the constraint outlined in the preceding paragraph may resolve the issue, but being evaluated sequentially again may lead to a violation of particle conversation during diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Thus, a more careful point defect balancing is required and introduced in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' All Ar point defect population predictions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ρ′ Ari + ρ′ ArAl + ρ′ ArN) are multiplied with a correction factor fAr: fAr = ntot n′ tot ρAri + ρArAl + ρArN ρ′ Ari + ρ′ ArAl + ρ′ ArN + 10−7 (6) The deviation of predicted Al and N atoms prior/post diffusion is defined by ∆nAl and ∆nN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' respectively: ∆nAl =ntot(ρAli + ρAlN − ρvAl − ρNAl − ρ(N-N)Al − ρArAl) − n′ tot(ρ′ Ali + ρ′ AlN − ρ′ vAl − ρ′ NAl − ρ′ (N-N)Al − fArρ′ ArAl) (7a) ∆nN =ntot(ρNi + 2ρ(N-N)i + ρ(N-N)N − ρvN + ρNAl + 2ρ(N-N)Al − ρAlN − ρArN) − n′ tot(ρ′ Ni + 2ρ′ (N-N)i + ρ′ (N-N)N − ρ′ vN + ρ′ NAl + 2ρ′ (N-N)Al − ρ′ AlN − fArρ′ ArN) (7b) with ntot as a function of the point defect population following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' All defect populations 15 but anti-sites are compensated for the particle balancing using these deviations: ρ′ vAl → n′ totρ′ vAl − ∆nAl ntot if ∆nAl< 0 (8a) ρ′ Ali → n′ totρ′ Ali + ∆nAl ntot if ∆nAl> 0 (8b) ρ′ vN → n′ totρ′ vN − ∆nN ntot if ∆nN < 0 (8c) ρ′ (N-N)N → n′ totρ′ (N-N)N + ρ′ (N-N)N ρ′ Ni+ρ′ (N-N)N ∆nN ntot if ∆nN > 0 (8d) ρ′ Ni → n′ totρ′ Ni + ρ′ Ni ρ′ Ni+ρ′ (N-N)N ∆nN ntot if ∆nN > 0 (8e) All point defect populations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' which have not been altered up this point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' are scaled with the quotient n′ tot/ntot to account for the changed total number of atoms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ensuring consistent predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Physics-separating artificial neural network Each CVAE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', PSI-CVAE, Diffusion- CVAE) describes one physical process, separating one from another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The (trained) decoders are combined to form a PSNN that allows for an evolution in time by passing the surface state Ss from one surrogate model to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The information flow of the PSNN is depicted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It resembles closely the information flow inherent to the physical simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The input to the PSI-Decoder is a single particle sampled from the particle flux of the plasma, characterized by the particles’ kinetic energy Ekin, species s, and surface state Ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It predicts the updated surface state S′ s and emitted flux for each species Γout ′ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The former is fed together with the temperature T to the Diffusion-Decoder, which predicts a new surface state S′′ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is passed on to the PSI-Decoder, establishing an recurrent link within the PSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Note that the direct correspondence of the physical simulations and the separated PSI- Decoder and Diffusion-Decoder structure allows for an efficient parameter space exploration, as outlined in Section III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, relying on single PSIs, the predictions after training may be subject to vastly different plasma conditions and are not limited to the specific flux ratios or ion energy distributions used for setting up the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 16 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Schematic of the PSNN structure and information flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Plasma dynamics can be imposed by hand, simulation or experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hyperparameter study The HP of each CVAE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', PSI-CVAE, Diffusion-CVAE) are optimized by applying an individual anisotropic self-adaptive evolution strategy with intermediate recombination (µ/µI, λ)-σSA-ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' µ, λ, and σ refer to the number of parents, populations size, and step sizes (mutation strengths), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Generalized and topic-wise related descriptions of this method can be found in [21, 60–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The HPs considered in this work and their initialization ranges are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The evolution strategies are inialized as (7/7I, 70)-σSA-ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The population sizes λ are reduced by one per generation over the course of 63 generations and, afterwards, kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The numbers of parents are determined by µ = λ/7 (integer values are enforced) [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The ESs are conducted for 200 generations and, hence, end as (1, 7)-σSA-ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Production run: Reference experiment First, the experimental scenario considered for production as well as validation is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Second, the fluxes onto the AlN surfaces required for the ML simulation are calculated and used to introduce an estimated process time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Reference experiment Ries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' used a large-area multi-frequency capacitively coupled plasma (MFCCP) to sputter deposit AlN with Ar and N2 as working gases (Ar/N2 gas inlet ratio equal 8/1) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The electrical asymmetry effect was taken advantage of to 17 System state Ss: System state S" Plasma dynamics PSI-Decoder Diffusion-Decoder System state $ flux Fouti temperature TTable I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The HPs to be optimized, their initialization range, and final values for the PSI-CVAE as well as Diffusion-CVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' HP Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' range PSI-CVAE Diffusion-CVAE rc [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='44 α [10−5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='47 ·10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='17 rl-0 [10−3,10−2] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='14 ·10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='08 ·10−3 nl-patience [4,7] 9 7 λL2 [0,10−4] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='23 ·10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='22 ·10−7 nhl [1,5] 1 3 nnpl [8,128] 107 155 nls [1,6] 1 5 β [10−1,10] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='14 nSA [1,102] 102 102 nbatch,ideal [16,64] 37 53 decouple the ion flux from the ion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The former was kept approximately constant and the latter was controlled by applying voltage waveform tailoring (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', adjusting the relative phase shift between the two excitation frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Four cases with mean ion energies Eion of 47 eV, 53 eV, 57 eV, and 81 eV were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The predominant AlN surface orientation was found to vary as function of the mean ion energy: AlN(002) for Eion =47 eV and Eion =53 eV as well as AlN(100) for Eion =57 eV and Eion =81 eV [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In this work, the species most relevant for PSI (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Al, N+, N+ 2 , Ar+) are sampled from the experimentally determined fluxes impinging onto the substrate (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' next subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The kinetic energy Ekin of ions and Al neutrals are sampled from measured ion energy distribution functions (IEDFs) (depicted in Figure 11 of [64]) and from Monte Carlo transport simulations (Al in pure Ar, but assumed invariant), respectively [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A threshold for the IEDFs is imposed to avoid sampling from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Monte Carlo accept-reject sampling (rejection sampling) is used to determine the individual particle energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The evolution and response of both monocrystalline systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', AlN(002), AlN(100)) predicted by the PSNN is to be studied as a function of the mentioned four ion energy 18 distribution functions (IEDFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Each case starts with ideal, defect free AlN and is run until a steady-state is reached, that is approximately 45 minutes (experimental process time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' All cases are re-run 100 times to evaluate their statistics accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The final results are averaged over the last minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Intrinsic stresses are determined as a function of the predicted lattice constants by utilizing the third-order Birch-Murnaghan EOS of the ideal, defect free AlN reference system as proposed and validated in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The final stresses and compositions predicted by the PSNN are compared to experimentally measured reference values [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Spurious Fe and O concentrations observed in the experiment are substituted with Al and N concentrations to define a comparable reference for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Flux and process time estimations The process time tp for npi particle impingements may be estimated by the sum over all reciprocal impingement rates, tp = �npi i=1 1/(ΓiA′ RMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Γin i and A′ RMD denote the experimental particle flux onto the AlN surface and predicted RMD AlN surface area, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A′ RMD is computed as a function of the predicted lattice constant a′ and imposed surface orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The ion flux onto the target and substrate is assumed to be approximately equal for the given geometry and approximated by Γin ion = hnevB, with assumed edge-to-center ration h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='61, electron density ne = 5·1015 m−3, and Bohm velocity vB = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='21·103 m/s (for Ar+ ions and a given electron temperature of kBTe = 3 eV) [7, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The flux of Al neutrals onto the substrate is calculated from Γin Al = ctYAr+(352 eV) Γin ion = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='47·1018 m−2s−1 with the collisional transport coefficient ct = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 obtained from Monte Carlo transport simulations (Al in pure Ar, but assumed invariant) as well as an Ar sputtering yield YAr+(352 eV) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='579 (clean Al target) [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Al flux from the target is obtained by multiplying the ion flux Γin ion with the Ar+ sputtering yield YAr+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The considered ion fluxes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Γin N+, Γin N+ 2 , Γin Ar+) are determined by assuming that the composition of the ion fluxes onto the substrate Γin ion resemble the volumetric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The total gas density is given by ng,tot = p/(kBTg) with the Boltzmann constant kb, and gas temperature Tg = 650 K [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The species specific gas densities are calculated as relative fractions assuming ng,N/ng,N2 = 1/9 and (ng,N + ng,N2)/ng,Ar = 1/8 [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, the working gas approximately consists of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='16 % N, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='47 % N2, and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='37 % Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The ion (Bohm) flux onto the substrate is split up accordingly: ΓAl+ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='11·1014 m−2s−1, ΓN+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='14·1017 m−2s−1, ΓN+ 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='03 · 1018 m−2s−1, and ΓAr+ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='65 · 1018 m−2s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The contribution due to Al+ is neglected due to their rare occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 19 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hyperparameter study Following the outlined evolution strategy with MCCV, an optimum set of HPs is de- termined and listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' As apparent, data augmentation by means of constrained mixup augmentation is beneficial for the Diffusion-CVAE (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This means that the hypothesis of linear superposition is accepted to some extend for the diffusion processes but declined for the PSIs (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='47·10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Kernel regularization is found to be disadvantageous for either ML model (λL2 ≈ 10−7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The network structure of the Diffusion-CVAE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', 3 hidden layer with 155 nodes per layer) allows for higher order of complexity than the PSI- CVAE’s one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', 1 hidden layer with 107 nodes per layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is also interesting to note that the optimum number of simulated annealing epochs is 100 for both ML models, which is the imposed upper boundary for this HP for the evolution strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, the simulated annealing step is assumed to be of great use for the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In addition to the MAE, the performance of the PSI-CVAE and Diffusion-CVAE with their final set of HPs listed in Table I can be assessed by the coefficient of determination R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is calculated on the training, validation, and test set to equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='87, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='86, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='87 for the PSI-CVAE as well as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='94, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='93, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='93 for the Diffusion-CVAE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' These values ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='9 signify an accurate model approach (R2 = 1 signifies fully explained variance in the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The negligible difference between the three subsets indicates that the ML models learned successfully to generalize on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This finding is analyzed more thoroughly in the following by comparing the unnormalized mean absolute errors (MAEs) of each system property (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', mass density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is important to note though that the reference data does not resemble any kind of ground truth but contains statistical fluctuations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', a single ion hitting the surface on a different surface sites is likely to inflict different kinds of defect structures) which intrinsically provide limits for the MAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The MAE of all considered defect populations are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The PSI-CVAE and Diffusion-CVAE is found to predict the defect structure accurately with errors that are of the order/below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The error of the PSI-CVAE’s predictions on the training, validation, and test set are barely distinguishable from each other, resembling excellent generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Diffusion-CVAE is found to perform best on the training set, showing minor signs of 20 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' MAE of the unnormalized predictions on the point defect populations and data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Point defect types are listed on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' MAE of the unnormalized predictions on all data sets for the PSI-PSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Property Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' set Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' set Test set a (˚A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='002 ∆Ef (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='005 B (GPa) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='029 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='067 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='055 B′ (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='931 Γout Al /Γin s (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='015 Γout N /Γin s (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='087 Γout Ar /Γin s (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='219 Γout N2 /Γin s (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='140 overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, the difference between the validation and test set is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The high accuracy prediction of the PSI-CVAE and the Diffusion-CVAE on the lattice constant a, the formation energy ∆Ef, the bulk modulus B, and its derivative B′ are pre- sented in Table II and Table III, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The almost interchangeable performance on the training, validation, and test set shows again that the models successfully learned to 21 Training set Validation set Test set X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='125 PSI-CVAE Diffusion-CVAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='050 MA X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='025 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='000Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' MAE of the unnormalized predictions on all data sets for the Diffusion-PSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Property Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' set Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' set Test set a (˚A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='001 ∆Ef (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='008 B (GPa) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='905 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='094 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='124 B′ (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='973 generalize on the provided data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, the MAEs of the emitted Al, N, Ar and N2 flux per incident flux, as listed in Table II, are relatively large when compared to typical sputter yields as well as reflection ratios in the considered regime of kinetic energies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Ekin in [0 eV, 300 eV]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is argued that these larger errors do not signify bad performance, but are rather a consequence of the data assembly for the PSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' One PSI data sample contains the information on a single PSI, which leads to the emission of, for example, none, one, or maybe two particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This will be perceived as noise to the ML model, which consequently learns to predict the mean number of emitted particles per PSI for a given surface state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This inherently leads to relatively large MAEs but ultimately is exactly what the PSI-CVAE is meant to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Production run The production run resembles the reference experiment of AlN thin-film deposition for four discharge conditions as previously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In the following, they are investigated for two surface orientations (100) and (002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Initially the emitted particle fluxes are discussed: Particles are emitted from the surface due to reflection of the incident particle or sputtering of surface atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It is observed that most fluxes reach a steady-state after a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Minor changes on the minute time-scale are observed only for three cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Eion =47 eV: (002), Eion =53 eV: (002), Eion =57 eV: (100)) due to a change of the Al sticking probability of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This transient variation is a side-effect of slowly evolving system states, described in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' All Al sticking coefficients are in between 98-99 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The emitted per incident particle fluxes averaged over the last, 45th minute are shown 22 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The emission of all film forming species per incident fluxes are presented for all considered IEDFs as well as surface orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Circle and error bars represent mean values and root-mean- squared deviations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' in Figure 9 for all film forming flux combinations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', the emission of Ar is omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' No significant difference between the two surface orientations is recognizable, which is attributed to the considered ion energy regime of 30 to 100 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Higher ion energies are expected to present surface orientation dependent sputtering yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The impingement of N+ and Ar+ ions leads to an almost similar removal of Al atoms, whereas Ar+ ions achieve a slightly increased Al sputtering yield (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Γout Al /Γin N+ < Γout Al /Γin Ar+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This is attributed to elastic collisions of bombarding N+ ions with N surface atoms, distribut- ing the momentum more rapidly and evenly among them than Al atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The displacement of N atoms in the subsurface regions leads to the temporary formation of (N-N)N close to the surface, where they eventually leave as N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Higher ion energies lead to deeper collision cascades spawned with higher momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The proportionality with the mean ion energy indicates that for neither IEDF a relevant proportion of N+ ions directly form temporary (N-N)N at the surface (and desorb as N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Bombarding N+ 2 ions are split apart when they hit the surface and, thus, inhibit a reduced individual momentum compared to the initially shared one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This favors an even stronger distribution of the momenta in the surface slab and, thus, lessens the likelihood of sputtering 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='3 Fout: (002) (002) Tout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' (002) Al N N2 47 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 53 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 57 eV 81 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='3 Tout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' :(100) (100) (100) Al N2 47 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 53 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 57 eV 81 ev 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Transient evolution of the most relevant point defect populations for all considered IEDFs as well as surface orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Error bars and the height of transparent region resemble the mean plus / minus the RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Al atoms in the considered ion energy regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Moreover, for smaller ion energies a shallower subsurface region is affected, which enables incident N+ 2 ions to directly form temporary (N-N)N at the surface before leaving as N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This is reflected by the decreased flux ratio Γout N2 /Γin N+ 2 for increased ion energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The transient evolutions of the most relevant point defect populations are shown in Fig- 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion pvn: (002) P(N-N)n: (002) 47eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 57eV 81eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion pvA1: (002) PAl: (002) (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 47 eV 53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 5 57 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 81 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion Pvn: (100) P(N-N)N: (100) 47 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 57eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 81 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion pvAl: (100) pAl:: (100) (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 47eV 53 eV fect 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 57 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 _81eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 0 10 20 30 40 0 10 20 30 40 t (min) t (min)ure 10 for all considered IEDFs and surface orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The deposition onto AlN(002) with Eion =47 eV takes up to 30 minutes to reach a steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The ongoing ion bombardment spawns collision cascades in the subsurface region, which once they have worn off may leave vacancies and interstitials behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Sputtering events or the desorption of N2 remove atoms from the surface and, thus, facilitate the accumulation of vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Al and N vacancy populations are approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Al interstitial population is greater than the N split interstitial population, and both exceed the corresponding vacancy populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This is due forward sputtering (peening) of surface atoms as well as incorporation of energetic particles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', N+, N2, small proportion of Al), which eventually either reside as interstitials or recombine with vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' IEDFs with slightly higher mean ion energies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', 53 eV, 57 eV) converge to a similar point defect structure with marginally increased N split and Al interstitial populations, but require significantly less time for equilibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' These require a few minutes and seconds for Eion =53 eV and Eion =57 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Therefore it is assumed that for Eion =47 eV scarcely sampled ions with relatively high kinetic energies push the systems to their final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The likelihood for encountering such ions is naturally increased when increasing the mean ion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This effect is enhanced by a change of the IEDF shapes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', narrow unimodal → narrow bimodal → broad unimodal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A more significantly increased mean ion energy of 81 eV leads to the evolution to a different sys- tem state with less Al and N vacancies (ρvAl ≈ ρvN) and more interstitials (ρAli > ρ(N-N)N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The evolution of the vacancy populations inhibits intermediate maxima after a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Subsequently, vacancies are removed due to recombination as described before and reach a steady-state after 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The evolution of the point defect structures are depicted in Figure 10 for up to 45 minutes (and are available in the appendix for up to 100 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The deposition onto AlN(100) leads to similar system dynamics for Eion =47 eV and Eion =53 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Al and N vacancy populations are approximately equal too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' But a greater number of N split and smaller number of Al interstitials are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Scarce Al atoms hitting the surface with relative high kinetic energies of up to 30 eV provide an insufficient momentum when penetrating the AlN(100) surfaces to be persistently incorporated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' they end up atop the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Increasing the mean ion energy to 57 eV leads to a system evolution that requires up to 10 minutes to reach a steady-state that differs significantly from the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The equilibration on the minute-time scale is again attributed to the contribution of only a small proportion of the incident ions with sufficient kinetic energies, 25 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Transient evolution of the mass density for all considered IEDFs as well as surface orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Error bars and the height of transparent region resemble the mean plus / minus the root-mean-squared deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' which are pushing the systems to their final state (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Eion =53 eV: (002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The (N-N)N populations remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The Al and N vacancy populations are doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, the probability for the recombination of surface near Al vacancies and incident Al atoms is increased too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The final Al interstitial population are therefore even more than doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The point defect structure is characterized predominantly by Al and N Frenkel pairs (vacancies plus interstitials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The evolution to this new system state is caused by a change of the IEDF shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The IEDF with Eion =53 eV (narrow bimodal IEDF) and Eion =57 eV (broad unimodal IEDF) reaches up to 60 eV and 75 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The cases with the highest mean ion energies of 83 eV converge to a similar system state with slightly increased interstitial populations, but it takes only a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The evolution of the mass densities are presented in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The equilibration time of the individual cases shows a consistent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, it is interesting to note that all cases converge to a similar mass density for the surface orientation (002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The accumulation of interstitials is balanced out by a corresponding volumetric expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In case of AlN(100), two final point defect structures were discussed in the preceding paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' These two system states are reflected by two distinctly separate mass densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Higher ion energies 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='175 (002) Eion 47 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='150 53 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='125 57 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='100 81 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='075 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='175 Eion (100) 47 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='150 53 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='125 57 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='100 81 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='075 0 10 20 30 40 t (min)Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The final composition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', Al and Ar concentration cAl and cAr, respectively) for all considered IEDFs as well as surface orientations averaged over the last minute are compared to experimental reference values [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Circles and error bars represent mean values and root-mean- squared deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' lead to a great number of Al as well as N Frenkel pairs, which do not alter the mass of the atomic system but cause stress and correspondingly a volumetric relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, smaller mass densities are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The composition of the deposited AlN(002) and AlN(100) thin films averaged over the last minute are shown in Figure 12 in comparison to experimental reference values [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A good agreement with the experiment is achieved when predicting stoichiometric AlN thin films even though the Ar concentration of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1 % for Eion = 47 eV and Eion = 53 eV is not reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The stresses predicted by the PSNN and measured in the experiment are presented in Figure 13 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' An increasingly compressive stress is observed for greater mean ion energies in either case due to the enhanced ion bombardment induced point defect formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Vacancies and interstitials cause tensile and compressive stresses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The interplay of all point defects define the film stresses in the ML simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, the contributions due to Al interstitials dominate the stress formation due to their larger size and high formation energies [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This finding is illustrated by a similar dependence of the stresses and the negated Al interstitial populations (multiplied by -1) on the mean ion energy Eion, as shown in Figures 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The preferential surface orientation was found to change from (002) to (100) in the experiment when increasing the mean ion energies from 47-53 eV to 57-81 eV [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 27 45 CAl 30 c Experiment CAr 15 Simulation: (002) Simulation: (100) 48 56 64 72 80 Eion (eV)Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The (a) final stress and (b) negated Al interstitial population for all considered IEDFs as well as surface orientations averaged over the last minute are compared to experimental reference values [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Circles and error bars represent mean values and root-mean-squared deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' By comparison with the ML prediction for the (002) surface orientation, it can be inferred that the predicted stresses for the two IEDFs with smaller mean ion energies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', 47 eV, 53 eV) are overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' However, from comparison with the prediction for the (100) surface orientation, the two IEDFs with greater mean ion energies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', 57 eV, 81 eV) are in excellent agreement with the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The change of the predominant surface orientation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', (002)→(100)) observed in the experiment may be attributed to the reduced compressive stresses predicted to reach up to -12 GPa for (002), compared to -8 GPa for (100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 28 0 a) Experiment Simulation: (002) 3 (GPa) Simulation: (100) stress α 12 48 56 64 72 80 Eion (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 (b) Simulation: (002) Simulation: (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 pAli 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 48 56 64 72 80 Eion (eV)V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' CONCLUSION This work is meant to further advance the development of data-driven plasma-surface interaction models with atomic fidelity [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Reactive processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', sputtering and depo- sition of AlN in an Ar/N2 discharges) are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A data-generating scheme is proposed that overcomes the burden of computationally too demanding simulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', hybrid RMD/tfMC) and, hence, undersampled parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The latter are effectively populated by evolving randomly sampled system states Ss by means of random PSIs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', species s, kinetic energy Ekin) and diffusion processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', temperature T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The effect of a single PSI on the deposited film is estimated by cleaving and reinforcing the corresponding bulk structure in surface normal direction [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A PSNN is used to separate the PSIs from the diffusion processes, which allows for a more efficient data-generation and enforcement of physics-constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=', particle conservation during bulk diffusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The trained PSNN model is applied to an experimental reference sputter deposition of AlN by taking the corresponding particle fluxes and IEDFs with mean ion energies in the range of 47-81 eV into account [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Ar+ ions are found to remove more Al than N atoms from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The inverse is observed for N+ ions, which spawn collision cascades that distribute their momenta more rapidly with the N surface atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This facilitates the temporary formation of (N-N)N at the very surface that eventually leave as N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' N+ 2 ions are split up when they hit the surface and, thus, spawn two collision cascades with reduced individual momenta compared to the initially shared one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' A diminishing amount of Al atoms is sputtered and a shallower subsurface region is effected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The latter allows for the direct formation of (N-N)N at the surface and subsequent emission as N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Atomic nitrogen is rarely sputtered by either ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Higher mean ion energies decrease the outgoing flux of N2 due N+ 2 ion bombardment but increase the formation of persistent, deeper (N-N)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The predicted film depositions take either a few seconds or up to 30 minutes to reach their respective steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Long equilibration times are observed when rare ions whose kinetic energy originates from the high energy tail of the IEDF push the systems to their final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The latter is found to be dependent on the imposed surface orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In particular, a greater Al interstitials population is predicted for AlN(002) than for AlN(100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This point defect type predominantly determines the compressive stress evolution in the deposited AlN thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The stresses predicted by the PSNN are quantitatively and qualitatively in 29 good agreement with the experimental reference values in spite of neglecting for instance thermal stresses or point defect annihilation at grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The ML model predicts stoichiometric AlN that is observed in the experiment too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In summary, 200 million plasma-surface interactions and diffusion processes were pre- dicted with high physical fidelity (hybrid RMD/tfMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' This enabled the evolution of 800 AlN systems (100 × four IEDFs × two surface orientations) in time for up to 45 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' It took about 34 hours to perform all machine learning predictions with a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hence, predictions can be readily extended to cover up the total experimental deposition time of up to hours when required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' In contrast, conducting the same case study with hybrid RMD/tfMC simulations is unattainable as it would take more than approximately 8 million CPU years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ACKNOWLEDGEMENT Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138690629 – TRR 87 and – Project-ID 434434223 – SFB 1461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='-Ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Ries from Ruhr University Bochum, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Karimi Aghda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' from RWTH Aachen University, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Vialetto, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' from Kiel University for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' ORCID T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Gergs: https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='org/0000-0001-5041-2941 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Mussenbrock: https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='org/0000-0001-6445-4990 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Trieschmann: https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='org/0000-0001-9136-8019 30 APPENDIX Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Schematic of the CVAE network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The shape of the data is provided in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Machine learning operations are indicated by colored arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The inputs and outputs for the PSI-CVAE are given by x = {Ekin, s, Ss} and y = {Γout s , Ss}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' The inputs and outputs for the Diffusion-CVAE are given by x = {T, Ss} and y = {Ss}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 31 Concat Dense Calc EnforceConstraints 2 μls,r Train phase?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' True False Ols,r N(0, 1) N(0,1) m m nhl = 2 nhl = 2 (nis) (SIu)Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Transient evolution of the most relevant point defect populations for all considered IEDFs as well as surface orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Error bars and the height of transparent region resemble the mean plus / minus the root-mean-squared deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion pvn: (002) P(N-N)n: (002) 47eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 57 eV 81eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion pvA1: (002) PAl: (002) (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 47 eV 53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 57eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 81 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion Pvn: (100) P(N-N)N: (100) 47 eV 53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 57eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 81 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='4 Eion pvAr: (100) PAl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=': (100) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='8 47 eV 53 eV fect 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='2 557 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='6 81eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content='0 0 25 50 75 100 0 25 50 75 100 t (s) (s) [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' J.' metadata={'source': 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Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Donk´o, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Graves, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hamaguchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hegemann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Hori, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Walt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' Sanden, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQf6gWk/content/2301.03524v1.pdf'} +page_content=' v.' metadata={'source': 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[math.OC] 27 Jan 2023 +A continuity result for the adjusted normal cone operator +Marco Castellania, Massimiliano Giulia +aDepartment of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via +Vetoio, L’Aquila, Italy +Abstract +The concept of adjusted sublevel set for a quasiconvex function was introduced in [5] and the local +existence of a norm-to-weak∗ upper semicontinuous base-valued submap of the normal operator +associated to the adjusted sublevel set was proved. When the space is finite dimensional, a globally +defined upper semicontinuous base-valued submap is obtained taking the intersection of the unit +sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique +does not work in the infinite dimensional case. +We propose a partition of unity technique to +overcome this problem in Banach spaces. Application is given to a quasiconvex quasioptimization +problem through the use of a new existence result for generalized quasivariational inequalities which +is based on the Schauder fixed point theorem. +Keywords: +Cone upper semicontinuity, Normal operator, Quasivariational inequality, +Quasioptimization +1. Introduction +The notion of upper semicontinuity seems to be unappropriate for cone-valued maps and hence +modified definitions were been introduced and studied [5, 8, 12]. The normal cone operator to the +adjusted sublevel sets of a quasiconvex function f defined on a Banach space was introduced in [5] +and it was proved to be both quasimonotone and cone upper semicontinuous. In particular, the +authors showed that the normal cone operator admits a locally defined base-valued submap being +norm-to-weak∗ upper semicontinuous. In [4], when the space is Euclidian, the authors obtained a +globally defined upper semicontinuous base-valued submap taking the convex hull of the normalized +Email addresses: marco.castellani@univaq.it (Marco Castellani), massimiliano.giuli@univaq.it +(Massimiliano Giuli) +Preprint submitted to Journal of LATEX Templates +January 31, 2023 + +normal operator which is the intersection of the unit sphere, which is compact, with the normal +operator, which is closed. Since the unit sphere is not compact in the infinite dimensional case, this +approach is unsuccessful in a Banach space X. +The first aim of this paper is to overcome this problem by using a partition of unity tech- +nique. +Theorem 3 states the existence of a norm-to-weak∗ upper semicontinuous submap A : +X \ arg min f ⇒ X∗ such that each A(x) is a nonempty weak∗ compact convex set not containing +the origin which generates the normal cone to the adjusted sublevel set at x. Subsequently, we +establish an existence result (Theorem 5) for a generalized quasivariational inequality which im- +proves the famous Tan’s result [14]. Finally, combining both results, we present an application to +quasioptimization problems. +In the last part of this introduction, we present some preliminary notions and results. Let X be +a real Banach space with norm ∥ · ∥, X∗ its topological dual with norm ∥ · ∥∗, and ⟨·, ·⟩ the duality +pairing between X∗ and X. From now on, unless otherwise indicated, the spaces X and X∗ will be +equipped by the strong (norm) topology s and the weak∗ topology w∗, respectively. The closed unit +balls in X and X∗ are denoted by B and B∗, respectively. Given a nonempty set A ⊆ X and x ∈ X, +dist(x, A) = inf{∥y−x∥ : y ∈ A} is the distance of x from A and B(A, r) = {x ∈ X : dist(x, A) ≤ r} +is the neighbourhood of A with radius r ≥ 0. +A subset K of X∗ is a cone if for each x∗ ∈ K and scalar t > 0, the product tx∗ ∈ K (note +that some authors define cone with the scalar t ranging over all non-negative scalars). Clearly the +empty set is a cone. +Let K ⊆ X∗ be a cone. A convex subset A of K is called a base if K = {tx∗ : t ≥ 0, x∗ ∈ A} +and 0 ̸∈ w∗- cl A, where w∗- cl denotes the closure with respect to the weak∗ topology. Clearly the +empty set is a base of the empty cone. Vice versa, if K admits a nonempty base then K is a convex +cone such that {0} ⊊ K. In particular, if the base is compact then K is closed. +The domain and the graph of a set-valued map Φ : X ⇒ X∗ are denoted by dom Φ and gph Φ, +respectively. The map Φ is norm-to-weak∗ upper semicontinuous at x ∈ X if for every open set Ω +such that Φ(x) ⊆ Ω, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω, for all x′ ∈ Ux. +The map Φ is norm-to-weak∗ closed at x ∈ dom f if for each x∗ ∈ Φ(x) and for each net {(xα, x∗ +α)} +with x∗ +α ∈ Φ(xα) which converges to (x, x∗) in the s × w∗ topology, we have that x∗ ∈ Φ(x). The +map is norm-to-weak∗ closed if its graph is closed with respect to the topology s × w∗. The Closed +Graph Theorem states that a closed-valued map Φ with values in a compact set is norm-to-weak∗ +2 + +upper semicontinuous if and only if it is norm-to-weak∗ closed. +When we are dealing with a cone-valued map Φ, the concept of norm-to-weak∗ upper semicon- +tinuity is not appropriate to picture the behaviour of Φ and it is convenient to slightly alter the +definition. The cone-valued map Φ is called +• norm-to-weak∗ cone upper semicontinuous at x ∈ X if for every open cone Ω such that +Φ(x) ⊆ Ω ∪ {0}, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω ∪ {0}, for all +x′ ∈ Ux; +• norm-to-weak∗ base upper semicontinuous at x ∈ X if there exist a neighbourhood Ux of x +and a set-valued map A : Ux ⇒ X∗ such that A(x′) is a base of Φ(x′) for each x′ ∈ Ux and A +is norm-to-weak∗ upper semicontinuous at x. +Some remarks are needed. If Φ is norm-to-weak∗ base upper semicontinuous at x ∈ X then there +exists a neighbourhood Ux of x such that Φ(x′) ̸= {0} for each x′ ∈ Ux. Instead, if Φ is norm-to- +weak∗ cone upper semicontinuous at x ̸∈ dom Φ then there exists a neighbourhood Ux of x such +that Φ(x′) ⊆ {0} for each x′ ∈ Ux. Therefore, if Φ is norm-to-weak∗ cone upper semicontinuous +and Φ(x) admits a base for each x ∈ X then dom Φ is closed. Moreover the norm-to-weak∗ base +upper semicontinuity of Φ at x implies the norm-to-weak∗ cone upper semicontinuity at the same +point. The reverse implication holds if Φ(x) admits a base and Φ(x′) ̸= {0} for all x′ in a suitable +neighbourhood of x [5]. +The norm-to-weak∗ cone upper semicontinuity of a map implies its norm-to-weak∗ closedness if +the map admits a compact base at every point [7, Proposition 2.3]. The same proof works for local +closedness. +Theorem 1. Let Φ : X ⇒ X∗ be a cone-valued map which is norm-to-weak∗ cone upper semicon- +tinuous at x ∈ dom Φ. If Φ(x) has a compact base then Φ is norm-to-weak∗ closed at x. +2. The result +Let f : X → R∪{+∞} be an extended-valued function. Define for any λ ∈ R∪{+∞} the sublevel +and the strict sublevel set of f at level λ by Sλ = {x ∈ X : f(x) ≤ λ} and S< +λ = {x ∈ X : f(x) < λ}, +respectively. Clearly S∞ = X and S< +∞ = dom f. The function f is quasiconvex if Sλ is convex for +all λ ∈ R. Now, we recall the notion of adjusted level set introduced in [5]. +3 + +Definition 2.1. Let f : X → R ∪ {+∞} and x ∈ X. The adjusted sublevel set of f at x is +Sa +f (x) = + + + +Sf(x) +if x ∈ arg min f +Sf(x) ∩ B(S< +f(x), ρx) +if x /∈ arg min f +where ρx = dist(x, S< +f(x)). +Note that S< +f(x) ⊆ Sa +f (x) ⊆ Sf(x) for all x ∈ X; moreover the convexity of the adjusted sublevel +sets characterizes the quasiconvexity of the function. +Theorem 2 (Proposition 2.4 in [5]). The extended-valued function f is quasiconvex if and only if +Sa +f (x) is convex, for every x ∈ X. +To any function f we associate the set-valued map N a : X ⇒ X∗ defined by +N a(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ Sa +f (x)} +In [5, Proposition 3.5] the authors showed that N a is norm-to-weak∗ base upper semicontinuous +under regularity assumptions on f. Combining Theorem 1 and Proposition 3.5 in [5], the following +result can be easily deduced. +Corollary 2.1. Let f be quasiconvex and lower semicontinuous at x ∈ dom f \ arg min f. If there +exists λ < f(x) such that int Sλ ̸= ∅ then N a is closed at x. +Such a result has been proved in [4] and, with weaker assumptions but in a finite dimensional +case, in [1]. Taking advantage of Corollary 2.1, the authors deduce [4, Proposition 4.4] the upper +semicontinuity of the normalized map N a ∩ S : Rn \ arg min f ⇒ B, being the unit sphere S in +Rn compact. Moreover, the assumptions in [4, Proposition 4.4] guarantee that the convex hull of +N a ∩ S is an upper semicontinuous base-valued submap of N a. Our aim is to extend their result to +the infinite dimensional case. Since the sphere is not weak∗ compact in the dual of a Banach space, +the previous technique does not work. +Theorem 3. Let f : X → R ∪ {+∞} be proper, quasiconvex and lower semicontinuous. Assume +that for each x ∈ X \ arg min f there exists λ < f(x) such that int Sλ ̸= ∅. Then there exists a +norm-to-weak∗ upper semicontinuous set-valued map A : X \ arg min f ⇒ B∗ such that A(x) is a +compact base of N a(x), for all x. +4 + +Proof. For the first step of the proof, we argue as in [5, Lemma 3.6]. Let z ∈ X \ arg min f +be fixed. +Choose z0 ∈ X and λ ∈ R such that λ < f(z) and z0 ∈ int S< +λ . +Since f is lower +semicontinuous, there exists ε > 0 such that +z0 + 2εB ⊆ S< +λ ⊆ S< +f(x), +∀x ∈ z + εB +Thus, for every x ∈ z + εB and for every +x∗ ∈ N <(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ S< +f(x)} +we obtain the following: +⟨x∗, z0 + 2εu − x⟩ ≤ 0, +∀u ∈ B +It follows that +2ε∥x∗∥∗ = 2ε sup +u∈B +⟨x∗, u⟩ +≤ +⟨x∗, x − z0⟩ += +⟨x∗, z − z0⟩ + ⟨x∗, x − z⟩ +≤ +⟨x∗, z − z0⟩ + ε∥x∗∥∗ +Thus, +⟨x∗, z − z0⟩ ≥ ε∥x∗∥∗, +∀x ∈ z + εB, x∗ ∈ N <(x) +Set Hz = {x∗ ∈ X∗ : ⟨x∗, z − z0⟩ = ε}. Obviously, for every x ∈ z + εB we have N <(x) ∩ Hz ⊆ B∗ +and, since N a(x) ⊆ N <(x), the set N a(x) ∩ Hz ⊆ B∗ is a compact base for the cone N a(x). Now, +following the proof of [5, Proposition 3.5], we get the norm-to-weak∗ upper semicontinuity of the +set-valued map Az : z + εB ⇒ X∗ defined by Fz(x) = N a(x) ∩ Hz, for all x ∈ z + εB. +The last step of the proof consists in finding the selection A as convex combination of the local +maps Az through a partition of unity technique. +Since X \ arg min f is paracompact, there exists a locally finite open covering U = {Ui : i ∈ I} +where every Ui ∈ U is a subset of some ball z + εB: let us denote by Ai the map Az corresponding +to the ball z + εB. Moreover, there is a partition of unity {λi : i ∈ I} subordinate to U such that +each λi : X \ arg min f → [0, 1] is continuous, the finite sum � +i∈I λi(y) = 1 for any y and λi(y) = 0 +for each y ̸∈ Ui. For every x ∈ X \ arg min f, let I(x) = {i ∈ I : λi(x) > 0}, which is nonempty and +finite, and define the map A : X \ arg min f ⇒ X∗ as follows +A(x) = +� +i∈I(x) +λi(x)Ai(x) +5 + +Clearly A(x) is a compact base of N a(x), for all x. Moreover, since the values of A are all contained +in the compact ball B∗, the norm-to-weak∗ upper semicontinuity of A is equivalent to prove that +the graph of A is closed with respect to the s × w∗ topology. Assume that the net {xα} converges +to x. Since all the λi are continuous, it is not restrictive to assume that I(x) ⊆ I(xα) for all α and +we get: +A(xα) = +� +i∈I(x) +λi(xα)Ai(xα) + +� +i∈I(xα)\I(x) +λi(xα)Ai(xα) +Moreover, from the continuity of the functions λi, we deduce +lim +α +� +i∈I(xα)\I(x) +λi(xα) = 1 − lim +α +� +i∈I(x) +λi(xα) = 0 +(1) +Now, let {x∗ +α} be a net which weakly∗ converges to x∗ and such that x∗ +α ∈ A(xα) for any α. Then, +there exist x∗ +i,α ∈ Ai(xα) for every i ∈ I(xα) such that +x∗ +α = +� +i∈I(x) +λi(xα)x∗ +i,α + +� +i∈I(xα)\I(x) +λi(xα)x∗ +i,α +(2) +The second addend of (2) weakly∗ converges to zero since, thanks to (1), it converges to zero in +norm +������ +� +i∈I(xα)\I(x) +λi(xα)x∗ +i,α +������ +∗ +≤ +� +i∈I(xα)\I(x) +λi(xα)∥x∗ +i,α∥∗ ≤ +� +i∈I(xα)\I(x) +λi(xα) +On the other hand, without loss of generality, we may assume that {x∗ +i,α} weakly∗ converges to +some x∗ +i , for every i ∈ I(x). Since Ai has closed graph, we obtain x∗ +i ∈ Ai(x) and x∗ ∈ A(x) follows +from (2) taking the weak∗ limit. +✷ +3. An application +In this section, our aim is to consider a special optimization problem, called quasioptimization +problem, and to provide an existence result for this problem through the study of an associated +generalized quasivariational inequality where Theorem 3 plays a key role. We start establishing a +new existence result for a generalized quasivariational inequality without requiring any assumption +of monotonicity. +Let C be a nonempty subset of X and T : C ⇒ X∗ and K : C ⇒ C be two set-valued maps; +the generalized quasivariational inequality GQV I(T, K) consists in finding +x ∈ K(x) such that ∃x∗ ∈ T (x) with ⟨x∗, y − x⟩ ≥ 0, +∀y ∈ K(x) +6 + +One of the most classic existence results for GQV I(T, K) in the infinite dimensional setting is due +to Tan and it was originally stated for locally convex topological vector spaces. We recall that the +set-valued map K : C ⇒ C is said to be lower semicontinuous if for every open set Ω the lower +inverse image {x ∈ C : K(x) ∩ Ω ̸= ∅} is open in C. Moreover K is called compact if K(C) is +contained in a compact subset of C. +Theorem 4 (Theorem 1 in [14]). Let C be compact and convex and K be closed and lower semi- +continuous with nonempty convex values. Assume that T is norm-to-norm upper semicontinuous +with nonempty norm compact convex values, then GQV I(T, K) has a solution. +The existence of solutions for GQV I(T, K) can be obtained with a weaker continuity assumption +on T than in Theorem 4 if the space X is normed. To this purpose, we need to recall the notion +of inside point of a convex set that appeared in 1956 in a paper by Michael [13]. The convex set +S ⊆ C is a face of C if x1, x2 ∈ C, t ∈ (0, 1) and tx1 + (1 − t)x2 ∈ S imply x1, x2 ∈ S. Let FC be +the (possibly empty) collection of all proper closed faces of cl C, which is the closure of C +Definition 3.1. A point x ∈ C is an inside point if it is not in any proper closed face of cl C. +Denote by +I(C) = C \ +� +S∈FC +S +the set of the inside points of C. +A comparison with other notions of relative interior is given in [9, 10]. Thanks to this concept +of interior point, we can define the following family of convex sets +D(X) = {C ⊆ X : C is convex and I(cl C) ⊆ C} +It was proved [13] that D(X) contains all the convex sets which are either closed, or with nonempty +interior, or finite dimensional. In particular, when X is finite dimensional the class D(X) coincides +with the family of all convex sets. Now we are in position to state and prove our existence result. +Let us denote by fix K the set of the fixed points of K. +Theorem 5. Let C be convex and K be a compact and lower semicontinuous set-valued map with +nonempty values in D(X), and fix K closed. Assume that T is norm-to-weak∗ upper semicontinuous +with nonempty weak∗ compact convex values, then GQV I(T, K) has a solution. +7 + +Proof. Notice that K admits a continuous selection thanks to [10, Theorem 3.2]. Hence the +Schauder fixed point theorem as formulated in [11, Proposition 6.3.2] guarantees fix K ̸= ∅. +Let us consider the set-valued map F : fix K ⇒ X defined as +F(x) = +� +x∗∈T (x) +{y ∈ X : ⟨x∗, y − x⟩ < 0} = +� +y ∈ X : +max +x∗∈T (x)⟨x∗, y − x⟩ < 0 +� +Clearly, F has convex values. To prove that F has open graph in fix K × X, it is sufficient to show +that the function m : fix K × X → R defined as +m(x, y) = +max +x∗∈T (x)⟨x∗, y − x⟩ +is upper semicontinuous. +First, fix K is compact since closed subset of the compact set which +contains K(C). From [2, Lemma 17.8], the subset T (fix K) is weak∗ compact; hence, it is norm +bounded. Thanks to [2, Corollary 6.40] the duality pairing ⟨·, ·⟩ restricted to T (fix K)× X is jointly +continuous, where X has its norm topology and X∗ has its weak∗ topology; hence, [2, Lemma 17.30] +guarantees the upper semicontinuity of m. +By contradiction, assume that F(x) ∩ K(x) ̸= ∅ for all x ∈ fix K. Fix (x0, y0) ∈ gph K and +define the map K0 : C ⇒ C as +K0(x) = + + + +K(x) +if x ̸= x0 +{y0} +if x = x0 +K0 is compact and lower semicontinuous, and K0(x) ∈ D(X) for every x ∈ C. From [10, Theorem +3.2] the map K0 admits a continuous selection, hence K is locally selectionable (see Definition +1.10.1 in [3]). From [3, Proposition 1.10.4] we deduce that also F ∩ K is locally selectionable and +[3, Proposition 1.10.2] guarantees that F ∩ K has a continuous selection f : fix K → C. Therefore, +the set-valued map Υ : C ⇒ C defined as +Υ(x) = + + + +K(x) +if x /∈ fix K +{f(x)} +if x ∈ fix K +is lower semicontinuous [10, Lemma 2.3] with values in the class D(X). Hence [10, Theorem 3.2] +guarantees that f can be extended to a continuous selection ϕ for Υ. The Schauder fixed point +theorem guarantees that ϕ has a fixed point, that is, there exists x ∈ C such that x = ϕ(x) ∈ Υ(x). +Clearly x ∈ fix K and this implies x = f(x) ∈ F(x) which is absurd. +Therefore, there exists +8 + +x ∈ fix K such that F(x) ∩ K(x) = ∅, that is, +min +y∈K(x) max +x∗∈T (x)⟨x∗, y − x⟩ ≥ 0 +Invoking the Sion’s minimax theorem we deduce that +max +x∗∈T (x) min +y∈K(x)⟨x∗, y − x⟩ ≥ 0 +which means that x solves the generalized quasivariational inequality. +✷ +Remark 3.1. Let us compare our result with Theorem 4 due to Tan. The first difference is about +the setting: Tan’s result works in a locally convex topological vector space instead Theorem 5 is +stated in a Banach space. Nevertheless, the other assumptions of Theorem 5 are rather weaker +than the ones in Theorem 4. Maybe, the most significant improvement consists in requiring the +norm-to-weak∗ upper semicontinuity of T instead of the stronger norm-to-norm upper semiconti- +nuity. Moreover, the values of T are assumed weakly∗ compact instead of norm compact. Also the +assumptions on K are weaker. In Theorem 4 the map K is closed, which implies the closedness of +K(x), for all x. Conversely, in Theorem 5 we require only the closedness of fix K, that is necessary +for the closedness of K, and K(x) may not be closed but belonging to the class D(X) only. Lastly, +we do not assume the compactness of C, not even its closedness, but only the fact that K(C) is +contained in a compact set. +Taking advantage of Theorem 5 and the good properties of the normal operator Na, our last +aim is to obtain an existence result for a quasioptimization problem through the study of a suitable +associated generalized quasivariational inequality. +A quasioptimization problem is an optimization problem in which the constraint set is subject +to modifications depending on the considered point. Given C ⊆ X nonempty, K : C ⇒ C and +f : C → R, a quasioptimization problem consists in finding +x ∈ K(x) such that f(x) ≤ f(y), +∀y ∈ K(x) +Clearly, if K(x) = C for all x ∈ C, quasioptimization problem reduces to a classical optimization +problem. +Theorem 6. Let C be convex and K be a compact and lower semicontinuous set-valued map with +nonempty values in D(X), and fix K closed. Assume that f is continuous and quasiconvex, then +the quasioptimization problem has a solution. +9 + +Proof. Let T : C ⇒ X∗ be defined as +T (x) = + + + +B∗ +if x ∈ arg min f +A(x) +if x /∈ arg min f +where A is the norm-to-weak∗ upper semicontinuous set-valued map obtained in Theorem 3. Since +arg min f is closed and A(x) ⊆ B∗, then T is norm-to-weak∗ upper semicontinuous. In this way, +thanks to Theorem 5, it follows that GQV I(T, K) has a solution x ∈ C. Clearly, if x ∈ arg min f, +then f(x) ≤ f(y) for all y ∈ K(x). Instead, if x /∈ arg min f, then it results that +x∗ ∈ T (x) = A(x) ⊆ N a(x) \ {0} +Hence, x is a solution to the generalized variational inequality associated to the operator N a \ {0} +and the feasible set K(x). Thanks to [6, Proposition 3.2], the thesis follows. +✷ +Theorem 6 extends Proposition 4.5 in [4] which is stated in a finite dimensional space and +requires also the compactness of C and the closedness of K. +References +[1] S. Al-Homidan, N. Hadjisavvas, L. Shaalan, Transformation of quasiconvex functions to elim- +inate local minima, J. Optim. Theory Appl. 177 (2018) 93–105. +[2] C.D. Aliprantis, K.C. Border, Infinite dimensional analysis. A hitchhikers guide, Springer- +Verlag, third ed., Berlin, 2006. +[3] J.P. Aubin, A. Cellina, Differential inclusions. Set-valued maps and viability theory, Springer- +Verlag, Berlin, 1984. +[4] D. Aussel, J. Cotrina, Quasimonotone quasivariational inequalities: existence results and ap- +plications, J. Optim. Theory Appl. 158 (2013) 637–652. +[5] D. Aussel, N. Hadjisavvas, Adjusted sublevel sets, normal operator, and quasi-convex program- +ming, SIAM J. Optim. 16 (2005) 358–367. +[6] D. Aussel, J.J. Ye, Quasiconvex programming with locally starshaped constraint region and +applications to quasiconvex MPEC, Optimization 55 (2006) 433–457. +10 + +[7] M. Bianchi, N. Hadjisavvas, R. Pini, Continuity and maximal quasimonotonicity of normal +cone operators, Stud. Univ. Babeş-Bolyai Math. 67 (2022) 31–45. +[8] J. Borde, J.-P Crouzeix, Continuity properties of the normal cone to the level sets of a quasi- +convex function, J. Optim. Theory Appl. 66 (1990) 415–429. +[9] M. Castellani, M. Giuli, An existence result for quasiequilibrium problems in separable Banach +spaces, J. Math. Anal. Appl. 425 (2015) 85–95. +[10] M. Castellani, M. Giuli, Existence of quasiequilibria in metric vector spaces, J. Math. Anal. +Appl. 484 (2020) 123751. +[11] A. Granas, J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003. +[12] D.T. Luc, J.-P. Penot, Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 +(2001) 4095–4121. +[13] E. Michael, Continuous selections. I, Ann. of Math. 63 (1956) 361–382. +[14] N.X. Tan, Quasi-variational inequalities in topological linear locally convex Hausdorff spaces, +Math. Nachr. 122 (1985) 231–245. +11 + diff --git a/5dFKT4oBgHgl3EQf-S4-/content/tmp_files/load_file.txt b/5dFKT4oBgHgl3EQf-S4-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a8fd0967393ce65aea804647a5976383ccb67e1 --- /dev/null +++ b/5dFKT4oBgHgl3EQf-S4-/content/tmp_files/load_file.txt @@ -0,0 +1,277 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf,len=276 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='11957v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='OC] 27 Jan 2023 A continuity result for the adjusted normal cone operator Marco Castellania, Massimiliano Giulia aDepartment of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via Vetoio, L’Aquila, Italy Abstract The concept of adjusted sublevel set for a quasiconvex function was introduced in [5] and the local existence of a norm-to-weak∗ upper semicontinuous base-valued submap of the normal operator associated to the adjusted sublevel set was proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' When the space is finite dimensional, a globally defined upper semicontinuous base-valued submap is obtained taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Unfortunately, this technique does not work in the infinite dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' We propose a partition of unity technique to overcome this problem in Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Keywords: Cone upper semicontinuity, Normal operator, Quasivariational inequality, Quasioptimization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Introduction The notion of upper semicontinuity seems to be unappropriate for cone-valued maps and hence modified definitions were been introduced and studied [5, 8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The normal cone operator to the adjusted sublevel sets of a quasiconvex function f defined on a Banach space was introduced in [5] and it was proved to be both quasimonotone and cone upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In particular, the authors showed that the normal cone operator admits a locally defined base-valued submap being norm-to-weak∗ upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In [4], when the space is Euclidian, the authors obtained a globally defined upper semicontinuous base-valued submap taking the convex hull of the normalized Email addresses: marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='castellani@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='it (Marco Castellani), massimiliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='giuli@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='it (Massimiliano Giuli) Preprint submitted to Journal of LATEX Templates January 31, 2023 normal operator which is the intersection of the unit sphere, which is compact, with the normal operator, which is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since the unit sphere is not compact in the infinite dimensional case, this approach is unsuccessful in a Banach space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The first aim of this paper is to overcome this problem by using a partition of unity tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 3 states the existence of a norm-to-weak∗ upper semicontinuous submap A : X \\ arg min f ⇒ X∗ such that each A(x) is a nonempty weak∗ compact convex set not containing the origin which generates the normal cone to the adjusted sublevel set at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Subsequently, we establish an existence result (Theorem 5) for a generalized quasivariational inequality which im- proves the famous Tan’s result [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Finally, combining both results, we present an application to quasioptimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In the last part of this introduction, we present some preliminary notions and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let X be a real Banach space with norm ∥ · ∥, X∗ its topological dual with norm ∥ · ∥∗, and ⟨·, ·⟩ the duality pairing between X∗ and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' From now on, unless otherwise indicated, the spaces X and X∗ will be equipped by the strong (norm) topology s and the weak∗ topology w∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The closed unit balls in X and X∗ are denoted by B and B∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Given a nonempty set A ⊆ X and x ∈ X, dist(x, A) = inf{∥y−x∥ : y ∈ A} is the distance of x from A and B(A, r) = {x ∈ X : dist(x, A) ≤ r} is the neighbourhood of A with radius r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' A subset K of X∗ is a cone if for each x∗ ∈ K and scalar t > 0, the product tx∗ ∈ K (note that some authors define cone with the scalar t ranging over all non-negative scalars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Clearly the empty set is a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let K ⊆ X∗ be a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' A convex subset A of K is called a base if K = {tx∗ : t ≥ 0, x∗ ∈ A} and 0 ̸∈ w∗- cl A, where w∗- cl denotes the closure with respect to the weak∗ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Clearly the empty set is a base of the empty cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Vice versa, if K admits a nonempty base then K is a convex cone such that {0} ⊊ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In particular, if the base is compact then K is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The domain and the graph of a set-valued map Φ : X ⇒ X∗ are denoted by dom Φ and gph Φ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The map Φ is norm-to-weak∗ upper semicontinuous at x ∈ X if for every open set Ω such that Φ(x) ⊆ Ω, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω, for all x′ ∈ Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The map Φ is norm-to-weak∗ closed at x ∈ dom f if for each x∗ ∈ Φ(x) and for each net {(xα, x∗ α)} with x∗ α ∈ Φ(xα) which converges to (x, x∗) in the s × w∗ topology, we have that x∗ ∈ Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The map is norm-to-weak∗ closed if its graph is closed with respect to the topology s × w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The Closed Graph Theorem states that a closed-valued map Φ with values in a compact set is norm-to-weak∗ 2 upper semicontinuous if and only if it is norm-to-weak∗ closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' When we are dealing with a cone-valued map Φ, the concept of norm-to-weak∗ upper semicon- tinuity is not appropriate to picture the behaviour of Φ and it is convenient to slightly alter the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The cone-valued map Φ is called norm-to-weak∗ cone upper semicontinuous at x ∈ X if for every open cone Ω such that Φ(x) ⊆ Ω ∪ {0}, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω ∪ {0}, for all x′ ∈ Ux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' norm-to-weak∗ base upper semicontinuous at x ∈ X if there exist a neighbourhood Ux of x and a set-valued map A : Ux ⇒ X∗ such that A(x′) is a base of Φ(x′) for each x′ ∈ Ux and A is norm-to-weak∗ upper semicontinuous at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Some remarks are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' If Φ is norm-to-weak∗ base upper semicontinuous at x ∈ X then there exists a neighbourhood Ux of x such that Φ(x′) ̸= {0} for each x′ ∈ Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Instead, if Φ is norm-to- weak∗ cone upper semicontinuous at x ̸∈ dom Φ then there exists a neighbourhood Ux of x such that Φ(x′) ⊆ {0} for each x′ ∈ Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Therefore, if Φ is norm-to-weak∗ cone upper semicontinuous and Φ(x) admits a base for each x ∈ X then dom Φ is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Moreover the norm-to-weak∗ base upper semicontinuity of Φ at x implies the norm-to-weak∗ cone upper semicontinuity at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The reverse implication holds if Φ(x) admits a base and Φ(x′) ̸= {0} for all x′ in a suitable neighbourhood of x [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The norm-to-weak∗ cone upper semicontinuity of a map implies its norm-to-weak∗ closedness if the map admits a compact base at every point [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The same proof works for local closedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let Φ : X ⇒ X∗ be a cone-valued map which is norm-to-weak∗ cone upper semicon- tinuous at x ∈ dom Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' If Φ(x) has a compact base then Φ is norm-to-weak∗ closed at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The result Let f : X → R∪{+∞} be an extended-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Define for any λ ∈ R∪{+∞} the sublevel and the strict sublevel set of f at level λ by Sλ = {x ∈ X : f(x) ≤ λ} and S< λ = {x ∈ X : f(x) < λ}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Clearly S∞ = X and S< ∞ = dom f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The function f is quasiconvex if Sλ is convex for all λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Now, we recall the notion of adjusted level set introduced in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let f : X → R ∪ {+∞} and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The adjusted sublevel set of f at x is Sa f (x) = \uf8f1 \uf8f2 \uf8f3 Sf(x) if x ∈ arg min f Sf(x) ∩ B(S< f(x), ρx) if x /∈ arg min f where ρx = dist(x, S< f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Note that S< f(x) ⊆ Sa f (x) ⊆ Sf(x) for all x ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' moreover the convexity of the adjusted sublevel sets characterizes the quasiconvexity of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 2 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='4 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The extended-valued function f is quasiconvex if and only if Sa f (x) is convex, for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' To any function f we associate the set-valued map N a : X ⇒ X∗ defined by N a(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ Sa f (x)} In [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='5] the authors showed that N a is norm-to-weak∗ base upper semicontinuous under regularity assumptions on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Combining Theorem 1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='5 in [5], the following result can be easily deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let f be quasiconvex and lower semicontinuous at x ∈ dom f \\ arg min f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' If there exists λ < f(x) such that int Sλ ̸= ∅ then N a is closed at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Such a result has been proved in [4] and, with weaker assumptions but in a finite dimensional case, in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Taking advantage of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='1, the authors deduce [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='4] the upper semicontinuity of the normalized map N a ∩ S : Rn \\ arg min f ⇒ B, being the unit sphere S in Rn compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Moreover, the assumptions in [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='4] guarantee that the convex hull of N a ∩ S is an upper semicontinuous base-valued submap of N a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Our aim is to extend their result to the infinite dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since the sphere is not weak∗ compact in the dual of a Banach space, the previous technique does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let f : X → R ∪ {+∞} be proper, quasiconvex and lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Assume that for each x ∈ X \\ arg min f there exists λ < f(x) such that int Sλ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Then there exists a norm-to-weak∗ upper semicontinuous set-valued map A : X \\ arg min f ⇒ B∗ such that A(x) is a compact base of N a(x), for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' For the first step of the proof, we argue as in [5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let z ∈ X \\ arg min f be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Choose z0 ∈ X and λ ∈ R such that λ < f(z) and z0 ∈ int S< λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since f is lower semicontinuous, there exists ε > 0 such that z0 + 2εB ⊆ S< λ ⊆ S< f(x), ∀x ∈ z + εB Thus, for every x ∈ z + εB and for every x∗ ∈ N <(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ S< f(x)} we obtain the following: ⟨x∗, z0 + 2εu − x⟩ ≤ 0, ∀u ∈ B It follows that 2ε∥x∗∥∗ = 2ε sup u∈B ⟨x∗, u⟩ ≤ ⟨x∗, x − z0⟩ = ⟨x∗, z − z0⟩ + ⟨x∗, x − z⟩ ≤ ⟨x∗, z − z0⟩ + ε∥x∗∥∗ Thus, ⟨x∗, z − z0⟩ ≥ ε∥x∗∥∗, ∀x ∈ z + εB, x∗ ∈ N <(x) Set Hz = {x∗ ∈ X∗ : ⟨x∗, z − z0⟩ = ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Obviously, for every x ∈ z + εB we have N <(x) ∩ Hz ⊆ B∗ and, since N a(x) ⊆ N <(x), the set N a(x) ∩ Hz ⊆ B∗ is a compact base for the cone N a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Now, following the proof of [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='5], we get the norm-to-weak∗ upper semicontinuity of the set-valued map Az : z + εB ⇒ X∗ defined by Fz(x) = N a(x) ∩ Hz, for all x ∈ z + εB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The last step of the proof consists in finding the selection A as convex combination of the local maps Az through a partition of unity technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since X \\ arg min f is paracompact, there exists a locally finite open covering U = {Ui : i ∈ I} where every Ui ∈ U is a subset of some ball z + εB: let us denote by Ai the map Az corresponding to the ball z + εB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Moreover, there is a partition of unity {λi : i ∈ I} subordinate to U such that each λi : X \\ arg min f → [0, 1] is continuous, the finite sum � i∈I λi(y) = 1 for any y and λi(y) = 0 for each y ̸∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' For every x ∈ X \\ arg min f, let I(x) = {i ∈ I : λi(x) > 0}, which is nonempty and finite, and define the map A : X \\ arg min f ⇒ X∗ as follows A(x) = � i∈I(x) λi(x)Ai(x) 5 Clearly A(x) is a compact base of N a(x), for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Moreover, since the values of A are all contained in the compact ball B∗, the norm-to-weak∗ upper semicontinuity of A is equivalent to prove that the graph of A is closed with respect to the s × w∗ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Assume that the net {xα} converges to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since all the λi are continuous, it is not restrictive to assume that I(x) ⊆ I(xα) for all α and we get: A(xα) = � i∈I(x) λi(xα)Ai(xα) + � i∈I(xα)\\I(x) λi(xα)Ai(xα) Moreover, from the continuity of the functions λi, we deduce lim α � i∈I(xα)\\I(x) λi(xα) = 1 − lim α � i∈I(x) λi(xα) = 0 (1) Now, let {x∗ α} be a net which weakly∗ converges to x∗ and such that x∗ α ∈ A(xα) for any α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Then, there exist x∗ i,α ∈ Ai(xα) for every i ∈ I(xα) such that x∗ α = � i∈I(x) λi(xα)x∗ i,α + � i∈I(xα)\\I(x) λi(xα)x∗ i,α (2) The second addend of (2) weakly∗ converges to zero since, thanks to (1), it converges to zero in norm ������ � i∈I(xα)\\I(x) λi(xα)x∗ i,α ������ ∗ ≤ � i∈I(xα)\\I(x) λi(xα)∥x∗ i,α∥∗ ≤ � i∈I(xα)\\I(x) λi(xα) On the other hand, without loss of generality, we may assume that {x∗ i,α} weakly∗ converges to some x∗ i , for every i ∈ I(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since Ai has closed graph, we obtain x∗ i ∈ Ai(x) and x∗ ∈ A(x) follows from (2) taking the weak∗ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' ✷ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' An application In this section, our aim is to consider a special optimization problem, called quasioptimization problem, and to provide an existence result for this problem through the study of an associated generalized quasivariational inequality where Theorem 3 plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' We start establishing a new existence result for a generalized quasivariational inequality without requiring any assumption of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let C be a nonempty subset of X and T : C ⇒ X∗ and K : C ⇒ C be two set-valued maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' the generalized quasivariational inequality GQV I(T, K) consists in finding x ∈ K(x) such that ∃x∗ ∈ T (x) with ⟨x∗, y − x⟩ ≥ 0, ∀y ∈ K(x) 6 One of the most classic existence results for GQV I(T, K) in the infinite dimensional setting is due to Tan and it was originally stated for locally convex topological vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' We recall that the set-valued map K : C ⇒ C is said to be lower semicontinuous if for every open set Ω the lower inverse image {x ∈ C : K(x) ∩ Ω ̸= ∅} is open in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Moreover K is called compact if K(C) is contained in a compact subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 4 (Theorem 1 in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let C be compact and convex and K be closed and lower semi- continuous with nonempty convex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Assume that T is norm-to-norm upper semicontinuous with nonempty norm compact convex values, then GQV I(T, K) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The existence of solutions for GQV I(T, K) can be obtained with a weaker continuity assumption on T than in Theorem 4 if the space X is normed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' To this purpose, we need to recall the notion of inside point of a convex set that appeared in 1956 in a paper by Michael [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The convex set S ⊆ C is a face of C if x1, x2 ∈ C, t ∈ (0, 1) and tx1 + (1 − t)x2 ∈ S imply x1, x2 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let FC be the (possibly empty) collection of all proper closed faces of cl C, which is the closure of C Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' A point x ∈ C is an inside point if it is not in any proper closed face of cl C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Denote by I(C) = C \\ � S∈FC S the set of the inside points of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' A comparison with other notions of relative interior is given in [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Thanks to this concept of interior point, we can define the following family of convex sets D(X) = {C ⊆ X : C is convex and I(cl C) ⊆ C} It was proved [13] that D(X) contains all the convex sets which are either closed, or with nonempty interior, or finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In particular, when X is finite dimensional the class D(X) coincides with the family of all convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Now we are in position to state and prove our existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let us denote by fix K the set of the fixed points of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let C be convex and K be a compact and lower semicontinuous set-valued map with nonempty values in D(X), and fix K closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Assume that T is norm-to-weak∗ upper semicontinuous with nonempty weak∗ compact convex values, then GQV I(T, K) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Notice that K admits a continuous selection thanks to [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Hence the Schauder fixed point theorem as formulated in [11, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='2] guarantees fix K ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let us consider the set-valued map F : fix K ⇒ X defined as F(x) = � x∗∈T (x) {y ∈ X : ⟨x∗, y − x⟩ < 0} = � y ∈ X : max x∗∈T (x)⟨x∗, y − x⟩ < 0 � Clearly, F has convex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' To prove that F has open graph in fix K × X, it is sufficient to show that the function m : fix K × X → R defined as m(x, y) = max x∗∈T (x)⟨x∗, y − x⟩ is upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' First, fix K is compact since closed subset of the compact set which contains K(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' From [2, Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='8], the subset T (fix K) is weak∗ compact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' hence, it is norm bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Thanks to [2, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='40] the duality pairing ⟨·, ·⟩ restricted to T (fix K)× X is jointly continuous, where X has its norm topology and X∗ has its weak∗ topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' hence, [2, Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='30] guarantees the upper semicontinuity of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' By contradiction, assume that F(x) ∩ K(x) ̸= ∅ for all x ∈ fix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Fix (x0, y0) ∈ gph K and define the map K0 : C ⇒ C as K0(x) = \uf8f1 \uf8f2 \uf8f3 K(x) if x ̸= x0 {y0} if x = x0 K0 is compact and lower semicontinuous, and K0(x) ∈ D(X) for every x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' From [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='2] the map K0 admits a continuous selection, hence K is locally selectionable (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='1 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' From [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='4] we deduce that also F ∩ K is locally selectionable and [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='2] guarantees that F ∩ K has a continuous selection f : fix K → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Therefore, the set-valued map Υ : C ⇒ C defined as Υ(x) = \uf8f1 \uf8f2 \uf8f3 K(x) if x /∈ fix K {f(x)} if x ∈ fix K is lower semicontinuous [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='3] with values in the class D(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Hence [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='2] guarantees that f can be extended to a continuous selection ϕ for Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The Schauder fixed point theorem guarantees that ϕ has a fixed point, that is, there exists x ∈ C such that x = ϕ(x) ∈ Υ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Clearly x ∈ fix K and this implies x = f(x) ∈ F(x) which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Therefore, there exists 8 x ∈ fix K such that F(x) ∩ K(x) = ∅, that is, min y∈K(x) max x∗∈T (x)⟨x∗, y − x⟩ ≥ 0 Invoking the Sion’s minimax theorem we deduce that max x∗∈T (x) min y∈K(x)⟨x∗, y − x⟩ ≥ 0 which means that x solves the generalized quasivariational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' ✷ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let us compare our result with Theorem 4 due to Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' The first difference is about the setting: Tan’s result works in a locally convex topological vector space instead Theorem 5 is stated in a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Nevertheless, the other assumptions of Theorem 5 are rather weaker than the ones in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Maybe, the most significant improvement consists in requiring the norm-to-weak∗ upper semicontinuity of T instead of the stronger norm-to-norm upper semiconti- nuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Moreover, the values of T are assumed weakly∗ compact instead of norm compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Also the assumptions on K are weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In Theorem 4 the map K is closed, which implies the closedness of K(x), for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Conversely, in Theorem 5 we require only the closedness of fix K, that is necessary for the closedness of K, and K(x) may not be closed but belonging to the class D(X) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Lastly, we do not assume the compactness of C, not even its closedness, but only the fact that K(C) is contained in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Taking advantage of Theorem 5 and the good properties of the normal operator Na, our last aim is to obtain an existence result for a quasioptimization problem through the study of a suitable associated generalized quasivariational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' A quasioptimization problem is an optimization problem in which the constraint set is subject to modifications depending on the considered point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Given C ⊆ X nonempty, K : C ⇒ C and f : C → R, a quasioptimization problem consists in finding x ∈ K(x) such that f(x) ≤ f(y), ∀y ∈ K(x) Clearly, if K(x) = C for all x ∈ C, quasioptimization problem reduces to a classical optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let C be convex and K be a compact and lower semicontinuous set-valued map with nonempty values in D(X), and fix K closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Assume that f is continuous and quasiconvex, then the quasioptimization problem has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Let T : C ⇒ X∗ be defined as T (x) = \uf8f1 \uf8f2 \uf8f3 B∗ if x ∈ arg min f A(x) if x /∈ arg min f where A is the norm-to-weak∗ upper semicontinuous set-valued map obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Since arg min f is closed and A(x) ⊆ B∗, then T is norm-to-weak∗ upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' In this way, thanks to Theorem 5, it follows that GQV I(T, K) has a solution x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Clearly, if x ∈ arg min f, then f(x) ≤ f(y) for all y ∈ K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Instead, if x /∈ arg min f, then it results that x∗ ∈ T (x) = A(x) ⊆ N a(x) \\ {0} Hence, x is a solution to the generalized variational inequality associated to the operator N a \\ {0} and the feasible set K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' Thanks to [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='2], the thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' ✷ Theorem 6 extends Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content='5 in [4] which is stated in a finite dimensional space and requires also the compactness of C and the closedness of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'} 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0000000000000000000000000000000000000000..94a5e9444f6c016fd429ff5d52177931b19c63e1 --- /dev/null +++ b/7dE4T4oBgHgl3EQfCQuc/content/tmp_files/2301.04859v1.pdf.txt @@ -0,0 +1,682 @@ +JONES-WENZL IDEMPOTENTS IN THE TWISTED +I-BUNDLE OF THE M¨OBIUS BAND +DIONNE IBARRA +Abstract. The Jones-Wenzl idempotent plays a vital role in quan- +tum invariants of 3-manifolds and the colored Jones polynomial; +it also serves as a useful tool for simplifying computations and +proving theorems in knot theory. The relative Kauffman bracket +skein module (RKBSM) for surface I-bundles and manifolds with +marked boundaries have a well understood algebraic structure due +to the work of J. H. Przytycki and T. T. Q. Lˆe. +It has been +well documented that the RKBSM of the I-bundle of the annulus +and the twisted I-bundle of the M¨obius band have a distinct alge- +braic structures even though the manifolds are homeomorphic. In +this paper we will give various results on Jones-Wenzl idempotents +in the twisted I-bundle of the M¨obius band when it is partially +closed through the crosscap of the M¨obius band. In doing so we +will uncover properties that differ from properties of Jones-Wenzl +idempotents in Ann × I. +Contents +1. +Introduction +1 +1.1. +Acknowledgements +2 +2. +Introduction to Jones-Wenzl idempotents +2 +3. +Crossingless connection in the M¨obius band +9 +4. +Jones-Wenzl idempotents in the M¨obius band +10 +References +15 +1. Introduction +The Jones-Wenzl idempotent, discovered by V. F. R. Jones in [Jon], +is an idempotent element in the Temperley-Lieb algebra. Originally, +it was described as a certain symmetrizer using the Artin braid group +Date: January 13, 2023. +2020 Mathematics Subject Classification. Primary: 57K10. Secondary: 57K31. +Key words and phrases. Jones-Wenzl idempotents, M¨obius band, twisted I- +bundles, Kauffman bracket skein module, relative Kauffman bracket skein module. +1 +arXiv:2301.04859v1 [math.GT] 12 Jan 2023 + +2 +DIONNE IBARRA +and the projection to the Temperley-Lieb algebra. In the late 1980’s, +H. Wenzl in [Wen] discovered a recursive formula to the Jones-Wenzl +idempotent. +This formula is now widely used as the definition, see +[Lic2]. +The Jones-Wenzl idempotent has played a significant role in defining +quanum invariants of knots and 3-manifolds. For example, W. B. R. +Lickorish’s Kauffman bracket skein theoretic approach to the Witten- +Reshetikhin-Turaev 3-manifold invariants in [Lic1] uses a linear combi- +nation of the trace (closure) of the idempotent elements along a framed +knot or link. Similarly, the colored Jones polynomial quantum knot in- +variant is defined by taking the trace of the nth Jones-Wenzl idempotent +along a 0-framed knot in S3, see [Le, PBIMW]. These idempotent ele- +ments are also used to decorate the edges of a tetrahedra to obtain the +quantum 6j-symbols that are used in the definition of the Turaev-Viro +quantum 3-manifold invariants, see [TV]. +The Jones-Wenzl idempotent has been a vital tool for simplifying +computations and proving theorems in knot theory. An example of this +is seen in X. Cai’s proof of a closed formula for the Gram determinant +of type A in [Cai] and a closed formula for its generalization in [BIMP]. +In fact, this paper was conceived by needing properties of the Jones- +Wenzl idempotents when it is closed in the twisted I-bundle of the +M¨obius band in hopes to take a similar approach to [Cai] and [BIMP] +to prove a closed formula for the Gram determinant of type Mb. +In Section 2 we introduce the original definition of Jones-Wenzl idem- +potents and also the RKBSM of the twisted I-bundle of the M¨obius +band, then in Section 3 we give an illustration of the two different +models of the M¨obius band as well as the antipodal properties of the +crosscap. In Section 4 we prove many corollaries to the trace of Jones- +Wenzl idempotents intersecting or surrounding the crosscap, then we +end with a formula for when n − 1 curves from fn are closed around +the crosscap and the last arc is closed through the crosscap. +1.1. Acknowledgements. This work was supported by the Australian +Research Council grant DP210103136. +2. Introduction to Jones-Wenzl idempotents +The first formal definition of the Temperley-Lieb algebra, denoted +by TLn, was given by R. J. Baxter in [Bax] while describing the work +of physicists N. Temperley and E. Lieb in [TL]. Jones independently +introduced TLn in [Jon] while working on von Neumann algebras. + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +3 +Definition 2.1. Let R be a commutative ring with unity and d ∈ R. +Let n ∈ N be fixed, then the nth Temperley-Lieb algebra, TLn, +is defined to be the unital associative algebra over R with generators +e1, . . . , en−1, identity element 1n, and relations +(1) eiejei = ei for |i − j| = 1, +(2) eiej = ejei for |i − j| > 1, +(3) e2 +i = dei. +L. H. Kauffman in [Kau], motivated by utilizing the Kauffman bracket, +considered the Temperley-Lieb algebra over R = Z[A±1], where A is an +indeterminate and d = −A2 − A−2. He then constructed a graphical +interpretation using tangles. +We will consider an n-tangle to be a rectangular shaped disk with +n marked boundary points on the left (input points) and n marked +boundary points on the right (output points). Kauffman’s graphical +interpretation of the Temperley-Lieb algebra is obtained from the basis +of crossingless tangles where the identity element corresponds to an n- +tangle with n parallel arcs in which each ith input point is connected +to the ith output point, and each ei corresponds to an n-tangle that +has one input and one output cap on the ith and i + 1th position as +illustrated in Figure 1. +For simplicity we will label an arc by n to +denote n parallel arcs as shown in Figure 1a. +n +(a) Identity element. +n − i − 1 +i − 1 +(b) ei. +Figure 1. The graphical interpretation of TLn. +Definition 2.2. The n-tangle algebra is an R-module with basis +elements consisting of n-tangles where multiplication of two n-tangles +is defined by identifying the right side of the first n-tangle to the left +side of the second n-tangle while respecting the boundary points and +by letting any resulting trivial curve be denoted by d, see Figure 2 for +an illustrative example. Kauffman’s diagrammatic interpretation of the +Temperley-Lieb algebra, also known as the diagrammatic algebra, +is a subalgebra of the n-tangle algebra. It is generated by tangles with +no crossings where homotopically trivial curves are denoted by d ∈ R. + +4 +DIONNE IBARRA +e3e3 = += d += de3. +Figure 2. An illustration of multiplication. +Theorem 2.3. [Kau] The diagrammatic algebra is isomorphic to TLn +and can be thought of as a diagrammatic interpretation of it. +We will give Jones’ constructive definition of the Jones-Wenzl idem- +potent by using the relative Kauffman bracket skein module (RKBSM) +and the Artin braid group before introducing Wenzl’s recursive formula. +In doing so, we will first introduce the RKBSM and emphasize that the +RKBSM of the twisted I-bundle of the M¨obius band and the RKBSM +of Ann × I are different modules even though the two manifolds are +homeomorphic. This will give us motivation to study the Jones-Wenzl +idempotent in the twisted I-bundle of the M¨obius band. Furthermore, +the corollaries and proposition in the last section will show that there +are distinct differences when simple closed curves intersect the crosscap. +Definition 2.4. Let M be an oriented 3-manifold and {xi}2n +i=1 be the +set of 2n framed points on ∂M. Let I = [−1, 1], and let Lfr(2n) be the +set of all relative framed links (which consists of all framed links in M +and all framed arcs, I×I, where I×∂I is connected to framed points on +the boundary of M) up to ambient isotopy while keeping the boundary +fixed in such a way that L ∩ ∂M = {xi}2n +1 . Let R be a commutative +ring with unity, A ∈ R be invertible, and let Ssub +2,∞(2n) be the submodule +of RLfr(2n) that is generated by the Kauffman bracket skein relations: +(i) L+ − AL0 − A−1L∞, and +(ii) L ⊔ ⃝ +⃝ +⃝ + (A2 + A−2)L, +where ⃝ +⃝ +⃝ denotes the framed unknot and the skein triple (L+, L0, L∞) +denotes three framed links in M that are identical except in a small +3-ball in M where the difference is shown in Figure 3. +Then, the relative Kauffman bracket skein module (RKBSM) +of M is the quotient: +S2,∞(M, {xi}2n +1 ; R, A) = RLfr(2n)/Ssub +2,∞(2n). +Theorem 2.5. [Prz] Let F be a surface with ∂F ̸= ∅. If F is orientable +then let M = F × I , otherwise let M = F ˆ×I. Let all {xi}2n +1 be marked + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +5 +(a) L+. +(b) L0. +(c) L∞. +Figure 3. The skein triple. +points that lie on ∂F × {0}. Then S2,∞(M, {xi}2n +1 ; R, A) is a free R- +module whose basis is composed of relative links in F without trivial +components. When n = 0, the empty link is also a generator. +J. H. Przytycki’s corollary to Theorem 2.5 explicitly details the dif- +ferences between the RKBSM of Ann × I and the RKBSM of Mbˆ×I +even though both manifolds are homeomorphic to the solid torus. +Corollary 2.6. [Prz] +(1) S2,∞(Ann×I, {xi}2n +1 ; R, A) where {xi}2n +1 are located in the outer +boundary component of the annulus is a free R[x]-module with +Dn = +�2n +n +� +basis elements, where x denotes the homotopically +nontrivial curve in the annulus and d = −A2 − A−2 denotes +the homotopically trivial curve in the annulus. The basis is the +set of all crossingless connections in the annulus with no trivial +components or boundary parallel curves. +(2) S2,∞(Mbˆ×I, {xi}2n +1 ; R, A) is a free R-module. The standard ba- +sis contains an infinite number of elements of the form bzi, bxzi +for i ≥ 0, where x denotes the simple closed curve that intersects +the M¨obius band once, z denotes the boundary parallel curve of +the M¨obius band, and b is an element in the set of crossingless +connections in the M¨obius band with no trivial components or +boundary parallel curves for which the arcs do not intersect the +crosscap. +The rest of the elements in the standard basis are +from a finite number of crossingless connections consisting of +a collection of n − k arcs for 0 ≤ k < n that non-trivially in- +tersects the crosscap. Among the finite collection there are +�2n +k +� +crossingless connections that intersect the crosscap n−k times. +Definition 2.7. The Artin braid group is defined by the following +group presentation: +Bn =< σ1, . . . , σn−1; σiσj = σjσi for |i−j| > 1, σi±1σiσi±1 = σiσi±1σi > . +The Artin braid group can be interpreted using n-tangles where el- +ements of Bn are positive braids. More precisely, an element in Bn + +6 +DIONNE IBARRA +can be represented as an n-tangle with positive crossings such that the +boundary of each arc is attached to one input and one output point +and when read from left to right each generator σi corresponds to the +positive crossing of the ith and i + 1th arcs. That is, the ith generator +element σi is a positive transposition of the ith and i + 1th arcs. +Furthermore, there exists an epimorphism p : Bn → Sn from the +Artin braid group to the permutation group that uniquely interprets a +braid word. Let p be defined by sending generators of Bn, σi, to the +transpositions in Sn; si = (i, i+1) for 1 ≤ i ≤ n−1. For a permutation +π ∈ Sn, let bπ denote the unique minimal positive braid word such that +p(bπ) = π. +Definition 2.8. Let Z[A±1] denote the ring of Laurent polynomials +in the variable A and Q(A) denote the field of rational functions in +the variable A; whose elements are functions of the form P/Q where +P, Q ∈ Z[A±1]. We define an unnormalized A-symmetrizer, Fn ∈ +Z[A±1]Bn, by the following +Fn = +� +π∈Sn +(A3)|π|bπ, +and the normalized symmetrizer, also known as the A-symmetrizer +and denoted by fn ∈ Q(A)Bn, by the formula +fn = +1 +[n]A4!Fn, +where |π| denotes the minimal length of the permutation π written as +elementary transposition generators, [n]A4 = 1+A4+A8+· · ·+A4(n−1) = +A4n−1 +A4−1 and [n]A4! = +n� +i=1 +[i]A4. +Fn evaluated in S2,∞(D2×I, {xi}2n +1 ; R, A) is an element in the Temperley- +Lieb algebra and the normalization is chosen so that fn is an idempotent +element in TLn. The most recognized name for this A-symmetrizer is +the Jones-Wenzl idempotent. We will denote this element as a +square with n strands entering and n strands exiting, as shown in Fig- +ure 4 and fn will denote the Jones-Wenzl idempotent. +Wenzl’s recursive formula uses the Chebyshev polynomial of the sec- +ond kind. +Definition 2.9. The nth Chebyshev polynomial of the first kind +is defined recursively by the initial conditions T0(d) = 2, T1(d) = d and +Equation 2.1. +(2.1) +Tn(d) = dTn−1(d) − Tn−2(d). + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +7 +n +Figure 4. The Jones-Wenzl idempotent. +The nth Chebyshev polynomial of the second kind is defined +recursively by the initial conditions S0(d) = 1, S1(d) = d and the same +recursive relation as the first kind, Sn(d) = dSn−1(d) − Sn−2(d). +When we substitute d = −A2 − A−2, the Chebyshev polynomial of +the first kind has the following closed formula +Tn(d) = (−1)n(A2n + A−2n), +and the Chebyshev polynomial of the second kind has the following +closed formula,denoted by ∆n, +∆n = (−1)nA2n+2 − A−2n−2 +A2 − A−2 += (−1)nA−2n[n + 1]A4. +Theorem 2.10. [Wen] A recursion formula for the nth Jones-Wenzl +idempotent, fn, is described in Equation 2.2: +(2.2) +fn = +n − 1 +− ∆n−2 +∆n−1 +n − 1 +n − 1 +n − 2 +. +The following lemma can be obtained from Wenzl’s recursive formula +as discussed in [Lic2] or from the constructive definition of the Jones- +Wenzl idempotent as detailed in [PBIMW]. +Lemma 2.11. [Lic2] +(a) (fn − 1) is an element of the algebra generated by {ei}n−1 +i=1 . +(b) eifn = fnei = 0 for 1 ≤ i ≤ n − 1. +(c) fnfn = fn. +A direct application of the next corollary will be given in Section 4. + +8 +DIONNE IBARRA +Corollary 2.12. [Lic2] Let tr 1(fn) be obtained from fn by closing the +top string in fn (see Figure 5). Then +tr 1(fn) = +∆n +∆n−1 +fn−1. +n − 1 += +∆n +∆n−1 +n − 1 +Figure 5. Illustration of tr 1(fn). +When defining the colored Jones polynomial, many authors tend to +use the term “decorating a knot by the Chebyshev polynomial” when +describing taking the trace of fn along a framed knot. This is because +the trace of fn along the standard annulus S1×I where S1 is the trivial +knot is equal to the Chebyshev polynomial of the second kind as stated +in the next corollary. +Corollary 2.13. [Lic2] Let Sn(z) denote the nth Chebyshev polynomial +of the second kind and z denote the homotopically non-trivial curve in +the annulus. Then +tr Ann(fn) = Sn(z). +Lemma 2.14. [Lic2] +b +a += ς +b +, +where ς = +(−1)a(A2(b+1)(a+1)−A−2(b+1)(a+1)) +A2(b+1)−A−2(b+1) +. +The following result is a well known corollary to Lemma 2.14, we will +see similar corollaries in Section 4 for elements in the twisted I-bundle +of the M¨obius band. +Corollary 2.15. [Lic2] +(2.3) +m +k += (−A2(k+1) − A−2(k+1))m∆k = ((−1)kTk+1)m∆k, +where d = −A2 − A−2 and Tn(d) = (−1)n(A2n + A−2n) is the nth +Chebyshev polynomial of the first kind. + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +9 +3. Crossingless connection in the M¨obius band +Throughout this paper we will use the crosscap model of the M¨obius +band where the boundary will be given in a rectangular form when +marked points are included, as shown in Figure 6b, otherwise it will +be displayed as a smooth circle as shown in Figure 7b. +The three +homotopically distinct arcs fixed on the boundary of the M¨obius band +are given in Figure 6. In order to relate the arcs from the first and +second model, a convention was chosen on the two distinct arcs fixed +on the boundary that do not intersect the crosscap. +(a) Formed from [0, 1] × [0, 1] by +identifying {0} × [0, 1] with {1} × +[0, 1] as shown by the arrows. +(b) Consists of a crosscap and +highlights the boundary of the +M¨obius band. +Figure 6. Two models of the M¨obius band with 3 ho- +motopically distinct arcs fixed on the boundary. +d +z +x +(a) First model. +d +z +x +(b) Second model. +Figure 7. Two models of the M¨obius band with 3 ho- +motopically distinct simple closed curves in M¨obius band +denoted by d, x, and z, respectively. +Figure 7 pictorially describes the three homotopically distinct simple +closed curves in the M¨obius band. If a simple closed curve intersects +the crosscap more than once then the number of intersection points can +be reduced by two at a time. The following example will illustrate the +process of removing two intersection points from the crosscap. Similar + +10 +DIONNE IBARRA +moves can be applied to arcs attached to the boundary that intersect +the crosscap more than once. +Example 3.1. We will illustrate, in the first model then the second +model, the removal of two intersection points of the crosscap from a +simple closed curve. In the two examples, the curve will be multicolored +in order to show which portion of the curve passes through the crosscap. +Suppose we have a simple closed curve that is homotopically trivial +and intersects the crosscap twice then, as shown in Equation 3.1, we +may use a sequence of isotopy moves to remove the two intersection +points. +(3.1) +∼ +∼ +. +In Equation 3.2, we will illustrate the removal of the same intersec- +tion points presented in the second model. +(3.2) +∼ +. +Now, suppose we have a homotopically non-trivial curve that inter- +sects the crosscap twice, for example the curve illustrated in Equation +3.3. Then, we may remove the two intersection points by using one +isotopy move as given below. +(3.3) +∼ +. +Equation 3.4 gives an illustration of this move in the second model. +(3.4) +∼ +. +4. Jones-Wenzl idempotents in the M¨obius band +In this section we will introduce various properties associated to the +Jones-Wenzl idempotents in the twisted I-bundle of the M¨obius band. + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +11 +n +Figure 8. Illustration of the unique element in Mbn +with n arcs intersecting the crosscap, denoted by 1Mbn. +Lemma 4.1. Let (Mbn)k denote the set of elements of Mbn that in- +tersect the crosscap k times, and let 1Mbn denote the unique element in +Mbn that intersect the crosscap n times, then +1Mbnei = en−i1Mbn ∈ (Mbn)n−2. +Proof. This is a direct result from the antipodal property of the cross- +cap that is explained in Section 3 and illustrated in Equations 4.1 and +4.2. +(4.1) +1Mbnei = +n − i − 1 +i − 1 +n − i − 1 +i − 1 += +n − i − 1 +i − 1 +n − i − 1 +i − 1 +∈ (Mbn)n−2. +(4.2) +en−i1Mbn = +n − i − 1 +i − 1 +n − i − 1 +i − 1 += +n − i − 1 +i − 1 +n − i − 1 +i − 1 +∈ (Mbn)n−2. +□ +Corollary 4.2. Sliding fn through the crosscap is achieved by the fol- +lowing equations. +n +n += +n +n +(4.3) += +n +n . +(4.4) + +12 +DIONNE IBARRA +Proof. By Lemma 2.11(a) and Lemma 4.1, all of the ei’s coming from +the left fn from left hand side of Equation 4.3 can be pulled through +the crosscap. Furthermore, for each ei this action results in a turn +back on the second fn. Therefore, we obtain our desired result after +applying Lemma 2.11(b). Equation 4.4 is obtained similarly. +□ +We will now present the three direct corollaries to Lemma 2.14 that +are obtained from closing fb through the crosscap. +Corollary 4.3. +n += x +∆1 +(−1)nTn+1(d)∆n, +where Tk(d) = (−1)k(A2k + A−2k) is the kth Chebyshev polynomial of +the first kind. +Proof. Let a = n and b = 1 in Lemma 2.14. Then by closing fb through +the crosscap we have +n += (−1)n(A4(n+1) − A−4(n+1)) +A4 − A−4 +. +After simplification we have our desired result, where x denotes the +simple closed curve that intersects the crosscap once. +□ +Corollary 4.4. +n +m += (−1)n(A2(n+1)(m+1) − A−2(n+1)(m+1)) +A2(n+1) − A−2(n+1) +m +. +Proof. This is obtained from directly applying Lemma 2.14 where a = +n, b = m, and fb is closed through the crosscap. +□ + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +13 +Corollary 4.5. +m +n += ((−1)nTn+1(d))m +n +. +Proof. We may remove one of the m meridional curves by applying +Lemma 2.14 where a = 1 and b = n and closing fb through the crosscap. +After simplification we have +m +n += ((−1)nTn+1(d)) +m − 1 +n +. +We obtain our desired result after repeating this argument m − 1 more +times. +□ +The following corollary is obtained from Corollary 2.13 by gluing a +crosscap to the inner boundary of the annulus. +Corollary 4.6. Let z denote the homotopically non-trivial curve in the +M¨obius band that does not intersect the crosscap, then +n += Sn(z). +Corollary 4.7. +m +n += +∆m+n +∆n +n +. +Proof. Apply Corollary 2.12 m times when closing n strands from fn+m +through the crosscap then closing the rest away from the crosscap. +□ +Let tr Mb1(fn) be obtained from closing one arc from fn through the +crosscap and closing the rest of the arcs in such a way that it surrounds +the crosscap, as shown in Figure 9. Then Lemma 2.12 can no longer + +14 +DIONNE IBARRA +be directly applied. Instead we start with applying Wenzl’s recursion +formula, Theorem 2.2, to obtain a recursive formula for tr Mb1(fn). +n − 1 +Figure 9. Illustration of tr Mb1(fn). +Lemma 4.8. +tr Mb1(fn) = xSn−1(z) − ∆n−2 +∆n−1 +tr Mb1(fn−1). +Proof. By applying Wenzl’s formula to the x-curve we have the follow- +ing recursive formula. +n − 1 += +n − 1 +− ∆n−2 +∆n−1 +n − 1 +. +By Corollary 4.6 and Lemma 2.11(c), +n − 1 += +xSn−1(z) − ∆n−2 +∆n−1 +n − 2 +. +□ +Proposition 4.9. +tr Mb1(fn) = +x +∆n−1 +n−1 +� +k=0 +(−1)n−1+kSk(z)∆k. + +JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND +15 +Proof. The base case is trivial, +tr Mb1(f1) = += x. +Suppose tr Mb1(fn−1) = +x +∆n−2 +�n−2 +k=0(−1)n−2+kSk(z)∆k. Then by Lemma +4.8, +tr Mb1(fn) += +xSn−1(z) − ∆n−2 +∆n−1 +tr Mb1(fn−1) += +xSn−1(z) − +x +∆n−1 +n−2 +� +k=0 +(−1)n−2+kSk(z)∆k += +x +∆n−1 +Sn−1(z)∆n−1 + +x +∆n−1 +n−2 +� +k=0 +(−1)n−1+kSk(z)∆k. +□ +References +[BIMP] +R. P. Bakshi, D. Ibarra, S. Mukherjee, J. H. Przytycki, A generalization +of the Gram determinant of type A, Topology Appl. 295 (2021), Paper +No. 107663, 15 pp. e-print: arXiv:1905.07834 [math.GT]. +[Bax] +R. J. Baxter, Exactly solved models in statistical mechanics. Academic +Press, Inc., London (1982). +[Cai] +X. Cai, A Gram determinant of Lickorish’s bilinear form. Math. Proc. +Cambridge Philos. Soc. 151 (2011), no. 1, 83–94. arXiv:1006.1297v3 +[math.GT]. +[Jon] +V. F. R. Jones, Index for subfactors. Invent. Math. 72, 1983, 1-25. +[Kau] +L. H. Kauffman, An invariant of regular isotopy. Trans. Amer. Math. +Soc. 318 (1990), no. 2, 417–471. +[Le] +T. T. Q. Lˆe, The colored Jones polynomial and the A-polynomial of +knots. Adv. Math. 207 (2006), no. 2, 782–804. arXiv:math/0407521 +[math.GT]. +[Lic1] +W. B. R. Lickorish, Invariants for 3-manifolds from the combinatorics +of the Jones polynomial, Pacific Journ. Math.,149(2), 1991, 337-347. +[Lic2] +W. B. R. Lickorish, An introduction to knot theory. Graduate Texts in +Mathematics, 175. Springer-Verlag, New York, 1997. +[Prz] +J. H. Przytycki, Fundamentals of Kauffman bracket skein modules. Kobe +Math. J., 16(1), 1999, 45-66. arXiv:math/9809113 [math.GT]. +[PBIMW] +J. H. Przytycki, R. P. Bakshi, D. Ibarra, G. Montoya-Vega, D. E. Weeks, +Lectures on Knot Theory: An Exploration of Contemporary Topics, +Springer Universitext (to appear). +[TL] +H. Temperley and E. Lieb, Relations Between the ‘Percolation’ and +‘Colouring’ Problem and Other Graph-Theoretic Problems Associated + +16 +DIONNE IBARRA +with Regular Plane Lattices: Some Exact Results for the ‘Percolation’ +Problem, Proceeds of the Royal Society of London 322 (1971), 251 - 280. +[TV] +V. G. Turaev, O. Ya. Viro, State sum invariants of 3-manifolds and +quantum 6j-symbols. Topology 31 (1992), no. 4, 865–902. +[Wen] +H. Wenzl, On sequences of projections, C.R. Math. Rep. Acad. Sci., IX, +1987, 5-9. +School of Mathematics, 9 Rainforest Walk, Floor 4, Monash Uni- +versity, VIC 3800, Australia +Email address: dionne.ibarra@monash.edu + diff --git a/7dE4T4oBgHgl3EQfCQuc/content/tmp_files/load_file.txt b/7dE4T4oBgHgl3EQfCQuc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fc0fad3caeb93aafd70428dffd327c34ff38237 --- /dev/null +++ b/7dE4T4oBgHgl3EQfCQuc/content/tmp_files/load_file.txt @@ -0,0 +1,427 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf,len=426 +page_content='JONES-WENZL IDEMPOTENTS IN THE TWISTED I-BUNDLE OF THE M¨OBIUS BAND DIONNE IBARRA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The Jones-Wenzl idempotent plays a vital role in quan- tum invariants of 3-manifolds and the colored Jones polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' it also serves as a useful tool for simplifying computations and proving theorems in knot theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The relative Kauffman bracket skein module (RKBSM) for surface I-bundles and manifolds with marked boundaries have a well understood algebraic structure due to the work of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Przytycki and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lˆe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' It has been well documented that the RKBSM of the I-bundle of the annulus and the twisted I-bundle of the M¨obius band have a distinct alge- braic structures even though the manifolds are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In this paper we will give various results on Jones-Wenzl idempotents in the twisted I-bundle of the M¨obius band when it is partially closed through the crosscap of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In doing so we will uncover properties that differ from properties of Jones-Wenzl idempotents in Ann × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Acknowledgements 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Introduction to Jones-Wenzl idempotents 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Crossingless connection in the M¨obius band 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Jones-Wenzl idempotents in the M¨obius band 10 References 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Introduction The Jones-Wenzl idempotent, discovered by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Jones in [Jon], is an idempotent element in the Temperley-Lieb algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Originally, it was described as a certain symmetrizer using the Artin braid group Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Primary: 57K10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Secondary: 57K31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Jones-Wenzl idempotents, M¨obius band, twisted I- bundles, Kauffman bracket skein module, relative Kauffman bracket skein module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='04859v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='GT] 12 Jan 2023 2 DIONNE IBARRA and the projection to the Temperley-Lieb algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In the late 1980’s, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Wenzl in [Wen] discovered a recursive formula to the Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' This formula is now widely used as the definition, see [Lic2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The Jones-Wenzl idempotent has played a significant role in defining quanum invariants of knots and 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' For example, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lickorish’s Kauffman bracket skein theoretic approach to the Witten- Reshetikhin-Turaev 3-manifold invariants in [Lic1] uses a linear combi- nation of the trace (closure) of the idempotent elements along a framed knot or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Similarly, the colored Jones polynomial quantum knot in- variant is defined by taking the trace of the nth Jones-Wenzl idempotent along a 0-framed knot in S3, see [Le, PBIMW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' These idempotent ele- ments are also used to decorate the edges of a tetrahedra to obtain the quantum 6j-symbols that are used in the definition of the Turaev-Viro quantum 3-manifold invariants, see [TV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The Jones-Wenzl idempotent has been a vital tool for simplifying computations and proving theorems in knot theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' An example of this is seen in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Cai’s proof of a closed formula for the Gram determinant of type A in [Cai] and a closed formula for its generalization in [BIMP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In fact, this paper was conceived by needing properties of the Jones- Wenzl idempotents when it is closed in the twisted I-bundle of the M¨obius band in hopes to take a similar approach to [Cai] and [BIMP] to prove a closed formula for the Gram determinant of type Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In Section 2 we introduce the original definition of Jones-Wenzl idem- potents and also the RKBSM of the twisted I-bundle of the M¨obius band, then in Section 3 we give an illustration of the two different models of the M¨obius band as well as the antipodal properties of the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In Section 4 we prove many corollaries to the trace of Jones- Wenzl idempotents intersecting or surrounding the crosscap, then we end with a formula for when n − 1 curves from fn are closed around the crosscap and the last arc is closed through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' This work was supported by the Australian Research Council grant DP210103136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Introduction to Jones-Wenzl idempotents The first formal definition of the Temperley-Lieb algebra, denoted by TLn, was given by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Baxter in [Bax] while describing the work of physicists N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Temperley and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lieb in [TL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Jones independently introduced TLn in [Jon] while working on von Neumann algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let R be a commutative ring with unity and d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let n ∈ N be fixed, then the nth Temperley-Lieb algebra, TLn, is defined to be the unital associative algebra over R with generators e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' , en−1, identity element 1n, and relations (1) eiejei = ei for |i − j| = 1, (2) eiej = ejei for |i − j| > 1, (3) e2 i = dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Kauffman in [Kau], motivated by utilizing the Kauffman bracket, considered the Temperley-Lieb algebra over R = Z[A±1], where A is an indeterminate and d = −A2 − A−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' He then constructed a graphical interpretation using tangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We will consider an n-tangle to be a rectangular shaped disk with n marked boundary points on the left (input points) and n marked boundary points on the right (output points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Kauffman’s graphical interpretation of the Temperley-Lieb algebra is obtained from the basis of crossingless tangles where the identity element corresponds to an n- tangle with n parallel arcs in which each ith input point is connected to the ith output point, and each ei corresponds to an n-tangle that has one input and one output cap on the ith and i + 1th position as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' For simplicity we will label an arc by n to denote n parallel arcs as shown in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n (a) Identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n − i − 1 i − 1 (b) ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The graphical interpretation of TLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The n-tangle algebra is an R-module with basis elements consisting of n-tangles where multiplication of two n-tangles is defined by identifying the right side of the first n-tangle to the left side of the second n-tangle while respecting the boundary points and by letting any resulting trivial curve be denoted by d, see Figure 2 for an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Kauffman’s diagrammatic interpretation of the Temperley-Lieb algebra, also known as the diagrammatic algebra, is a subalgebra of the n-tangle algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' It is generated by tangles with no crossings where homotopically trivial curves are denoted by d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 4 DIONNE IBARRA e3e3 = = d = de3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' An illustration of multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Kau] The diagrammatic algebra is isomorphic to TLn and can be thought of as a diagrammatic interpretation of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We will give Jones’ constructive definition of the Jones-Wenzl idem- potent by using the relative Kauffman bracket skein module (RKBSM) and the Artin braid group before introducing Wenzl’s recursive formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In doing so, we will first introduce the RKBSM and emphasize that the RKBSM of the twisted I-bundle of the M¨obius band and the RKBSM of Ann × I are different modules even though the two manifolds are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' This will give us motivation to study the Jones-Wenzl idempotent in the twisted I-bundle of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Furthermore, the corollaries and proposition in the last section will show that there are distinct differences when simple closed curves intersect the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let M be an oriented 3-manifold and {xi}2n i=1 be the set of 2n framed points on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let I = [−1, 1], and let Lfr(2n) be the set of all relative framed links (which consists of all framed links in M and all framed arcs, I×I, where I×∂I is connected to framed points on the boundary of M) up to ambient isotopy while keeping the boundary fixed in such a way that L ∩ ∂M = {xi}2n 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let R be a commutative ring with unity, A ∈ R be invertible, and let Ssub 2,∞(2n) be the submodule of RLfr(2n) that is generated by the Kauffman bracket skein relations: (i) L+ − AL0 − A−1L∞, and (ii) L ⊔ ⃝ ⃝ ⃝ + (A2 + A−2)L, where ⃝ ⃝ ⃝ denotes the framed unknot and the skein triple (L+, L0, L∞) denotes three framed links in M that are identical except in a small 3-ball in M where the difference is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then, the relative Kauffman bracket skein module (RKBSM) of M is the quotient: S2,∞(M, {xi}2n 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R, A) = RLfr(2n)/Ssub 2,∞(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Prz] Let F be a surface with ∂F ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' If F is orientable then let M = F × I , otherwise let M = F ˆ×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let all {xi}2n 1 be marked JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 5 (a) L+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (b) L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (c) L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The skein triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' points that lie on ∂F × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then S2,∞(M, {xi}2n 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R, A) is a free R- module whose basis is composed of relative links in F without trivial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' When n = 0, the empty link is also a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Przytycki’s corollary to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='5 explicitly details the dif- ferences between the RKBSM of Ann × I and the RKBSM of Mbˆ×I even though both manifolds are homeomorphic to the solid torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Prz] (1) S2,∞(Ann×I, {xi}2n 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R, A) where {xi}2n 1 are located in the outer boundary component of the annulus is a free R[x]-module with Dn = �2n n � basis elements, where x denotes the homotopically nontrivial curve in the annulus and d = −A2 − A−2 denotes the homotopically trivial curve in the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The basis is the set of all crossingless connections in the annulus with no trivial components or boundary parallel curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (2) S2,∞(Mbˆ×I, {xi}2n 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R, A) is a free R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The standard ba- sis contains an infinite number of elements of the form bzi, bxzi for i ≥ 0, where x denotes the simple closed curve that intersects the M¨obius band once, z denotes the boundary parallel curve of the M¨obius band, and b is an element in the set of crossingless connections in the M¨obius band with no trivial components or boundary parallel curves for which the arcs do not intersect the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The rest of the elements in the standard basis are from a finite number of crossingless connections consisting of a collection of n − k arcs for 0 ≤ k < n that non-trivially in- tersects the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Among the finite collection there are �2n k � crossingless connections that intersect the crosscap n−k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The Artin braid group is defined by the following group presentation: Bn =< σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' , σn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' σiσj = σjσi for |i−j| > 1, σi±1σiσi±1 = σiσi±1σi > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The Artin braid group can be interpreted using n-tangles where el- ements of Bn are positive braids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' More precisely, an element in Bn 6 DIONNE IBARRA can be represented as an n-tangle with positive crossings such that the boundary of each arc is attached to one input and one output point and when read from left to right each generator σi corresponds to the positive crossing of the ith and i + 1th arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' That is, the ith generator element σi is a positive transposition of the ith and i + 1th arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Furthermore, there exists an epimorphism p : Bn → Sn from the Artin braid group to the permutation group that uniquely interprets a braid word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let p be defined by sending generators of Bn, σi, to the transpositions in Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' si = (i, i+1) for 1 ≤ i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' For a permutation π ∈ Sn, let bπ denote the unique minimal positive braid word such that p(bπ) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let Z[A±1] denote the ring of Laurent polynomials in the variable A and Q(A) denote the field of rational functions in the variable A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' whose elements are functions of the form P/Q where P, Q ∈ Z[A±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We define an unnormalized A-symmetrizer, Fn ∈ Z[A±1]Bn, by the following Fn = � π∈Sn (A3)|π|bπ, and the normalized symmetrizer, also known as the A-symmetrizer and denoted by fn ∈ Q(A)Bn, by the formula fn = 1 [n]A4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='Fn, where |π| denotes the minimal length of the permutation π written as elementary transposition generators, [n]A4 = 1+A4+A8+· · ·+A4(n−1) = A4n−1 A4−1 and [n]A4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' = n� i=1 [i]A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Fn evaluated in S2,∞(D2×I, {xi}2n 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' R, A) is an element in the Temperley- Lieb algebra and the normalization is chosen so that fn is an idempotent element in TLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The most recognized name for this A-symmetrizer is the Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We will denote this element as a square with n strands entering and n strands exiting, as shown in Fig- ure 4 and fn will denote the Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Wenzl’s recursive formula uses the Chebyshev polynomial of the sec- ond kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The nth Chebyshev polynomial of the first kind is defined recursively by the initial conditions T0(d) = 2, T1(d) = d and Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1) Tn(d) = dTn−1(d) − Tn−2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 7 n Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The nth Chebyshev polynomial of the second kind is defined recursively by the initial conditions S0(d) = 1, S1(d) = d and the same recursive relation as the first kind, Sn(d) = dSn−1(d) − Sn−2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' When we substitute d = −A2 − A−2, the Chebyshev polynomial of the first kind has the following closed formula Tn(d) = (−1)n(A2n + A−2n), and the Chebyshev polynomial of the second kind has the following closed formula,denoted by ∆n, ∆n = (−1)nA2n+2 − A−2n−2 A2 − A−2 = (−1)nA−2n[n + 1]A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Wen] A recursion formula for the nth Jones-Wenzl idempotent, fn, is described in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2) fn = n − 1 − ∆n−2 ∆n−1 n − 1 n − 1 n − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The following lemma can be obtained from Wenzl’s recursive formula as discussed in [Lic2] or from the constructive definition of the Jones- Wenzl idempotent as detailed in [PBIMW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Lic2] (a) (fn − 1) is an element of the algebra generated by {ei}n−1 i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (b) eifn = fnei = 0 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (c) fnfn = fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' A direct application of the next corollary will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 8 DIONNE IBARRA Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Lic2] Let tr 1(fn) be obtained from fn by closing the top string in fn (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then tr 1(fn) = ∆n ∆n−1 fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n − 1 = ∆n ∆n−1 n − 1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Illustration of tr 1(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' When defining the colored Jones polynomial, many authors tend to use the term “decorating a knot by the Chebyshev polynomial” when describing taking the trace of fn along a framed knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' This is because the trace of fn along the standard annulus S1×I where S1 is the trivial knot is equal to the Chebyshev polynomial of the second kind as stated in the next corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Lic2] Let Sn(z) denote the nth Chebyshev polynomial of the second kind and z denote the homotopically non-trivial curve in the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then tr Ann(fn) = Sn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Lic2] b a = ς b , where ς = (−1)a(A2(b+1)(a+1)−A−2(b+1)(a+1)) A2(b+1)−A−2(b+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The following result is a well known corollary to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='14, we will see similar corollaries in Section 4 for elements in the twisted I-bundle of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' [Lic2] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3) m k = (−A2(k+1) − A−2(k+1))m∆k = ((−1)kTk+1)m∆k, where d = −A2 − A−2 and Tn(d) = (−1)n(A2n + A−2n) is the nth Chebyshev polynomial of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Crossingless connection in the M¨obius band Throughout this paper we will use the crosscap model of the M¨obius band where the boundary will be given in a rectangular form when marked points are included, as shown in Figure 6b, otherwise it will be displayed as a smooth circle as shown in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The three homotopically distinct arcs fixed on the boundary of the M¨obius band are given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In order to relate the arcs from the first and second model, a convention was chosen on the two distinct arcs fixed on the boundary that do not intersect the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (a) Formed from [0, 1] × [0, 1] by identifying {0} × [0, 1] with {1} × [0, 1] as shown by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (b) Consists of a crosscap and highlights the boundary of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Two models of the M¨obius band with 3 ho- motopically distinct arcs fixed on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' d z x (a) First model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' d z x (b) Second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Two models of the M¨obius band with 3 ho- motopically distinct simple closed curves in M¨obius band denoted by d, x, and z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Figure 7 pictorially describes the three homotopically distinct simple closed curves in the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' If a simple closed curve intersects the crosscap more than once then the number of intersection points can be reduced by two at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The following example will illustrate the process of removing two intersection points from the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Similar 10 DIONNE IBARRA moves can be applied to arcs attached to the boundary that intersect the crosscap more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We will illustrate, in the first model then the second model, the removal of two intersection points of the crosscap from a simple closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In the two examples, the curve will be multicolored in order to show which portion of the curve passes through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Suppose we have a simple closed curve that is homotopically trivial and intersects the crosscap twice then, as shown in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1, we may use a sequence of isotopy moves to remove the two intersection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1) ∼ ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' In Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2, we will illustrate the removal of the same intersec- tion points presented in the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2) ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Now, suppose we have a homotopically non-trivial curve that inter- sects the crosscap twice, for example the curve illustrated in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then, we may remove the two intersection points by using one isotopy move as given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3) ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='4 gives an illustration of this move in the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='4) ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Jones-Wenzl idempotents in the M¨obius band In this section we will introduce various properties associated to the Jones-Wenzl idempotents in the twisted I-bundle of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 11 n Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Illustration of the unique element in Mbn with n arcs intersecting the crosscap, denoted by 1Mbn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let (Mbn)k denote the set of elements of Mbn that in- tersect the crosscap k times, and let 1Mbn denote the unique element in Mbn that intersect the crosscap n times, then 1Mbnei = en−i1Mbn ∈ (Mbn)n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' This is a direct result from the antipodal property of the cross- cap that is explained in Section 3 and illustrated in Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1) 1Mbnei = n − i − 1 i − 1 n − i − 1 i − 1 = n − i − 1 i − 1 n − i − 1 i − 1 ∈ (Mbn)n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2) en−i1Mbn = n − i − 1 i − 1 n − i − 1 i − 1 = n − i − 1 i − 1 n − i − 1 i − 1 ∈ (Mbn)n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Sliding fn through the crosscap is achieved by the fol- lowing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n n = n n (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3) = n n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='4) 12 DIONNE IBARRA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='11(a) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='1, all of the ei’s coming from the left fn from left hand side of Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3 can be pulled through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Furthermore, for each ei this action results in a turn back on the second fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Therefore, we obtain our desired result after applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='4 is obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ We will now present the three direct corollaries to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='14 that are obtained from closing fb through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n = x ∆1 (−1)nTn+1(d)∆n, where Tk(d) = (−1)k(A2k + A−2k) is the kth Chebyshev polynomial of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let a = n and b = 1 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then by closing fb through the crosscap we have n = (−1)n(A4(n+1) − A−4(n+1)) A4 − A−4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' After simplification we have our desired result, where x denotes the simple closed curve that intersects the crosscap once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n m = (−1)n(A2(n+1)(m+1) − A−2(n+1)(m+1)) A2(n+1) − A−2(n+1) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' This is obtained from directly applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='14 where a = n, b = m, and fb is closed through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 13 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' m n = ((−1)nTn+1(d))m n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We may remove one of the m meridional curves by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='14 where a = 1 and b = n and closing fb through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' After simplification we have m n = ((−1)nTn+1(d)) m − 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' We obtain our desired result after repeating this argument m − 1 more times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ The following corollary is obtained from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='13 by gluing a crosscap to the inner boundary of the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Let z denote the homotopically non-trivial curve in the M¨obius band that does not intersect the crosscap, then n = Sn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' m n = ∆m+n ∆n n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Apply Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='12 m times when closing n strands from fn+m through the crosscap then closing the rest away from the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ Let tr Mb1(fn) be obtained from closing one arc from fn through the crosscap and closing the rest of the arcs in such a way that it surrounds the crosscap, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='12 can no longer 14 DIONNE IBARRA be directly applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Instead we start with applying Wenzl’s recursion formula, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='2, to obtain a recursive formula for tr Mb1(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n − 1 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Illustration of tr Mb1(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' tr Mb1(fn) = xSn−1(z) − ∆n−2 ∆n−1 tr Mb1(fn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' By applying Wenzl’s formula to the x-curve we have the follow- ing recursive formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' n − 1 = n − 1 − ∆n−2 ∆n−1 n − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='6 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='11(c), n − 1 = xSn−1(z) − ∆n−2 ∆n−1 n − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' tr Mb1(fn) = x ∆n−1 n−1 � k=0 (−1)n−1+kSk(z)∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' The base case is trivial, tr Mb1(f1) = = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} +page_content=' Suppose tr Mb1(fn−1) = x ∆n−2 �n−2 k=0(−1)n−2+kSk(z)∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'} diff --git a/7tAyT4oBgHgl3EQf2_kh/content/tmp_files/2301.00759v1.pdf.txt b/7tAyT4oBgHgl3EQf2_kh/content/tmp_files/2301.00759v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd4efacc1143c91148c50f4c2cb6e0d3eb4ff549 --- /dev/null +++ b/7tAyT4oBgHgl3EQf2_kh/content/tmp_files/2301.00759v1.pdf.txt @@ -0,0 +1,1607 @@ +Avalanche scaling in large neural +populations with distributed +coupling to multiple dynamical +latent variables +Mia Morrell1, Ilya Nemenman2, Audrey J. Sederberg3* +*For correspondence: +sede0018@umn.edu (AJS) +1Department of Physics, New York University; 2Department of Physics, Department of +Biology, Initiative in Theory and Modeling of Living Systems, Emory University; +3Department of Neuroscience, University of Minnesota Medical School +Abstract +Observations of power laws in neural activity data have raised the intriguing notion +that brains may operate in a critical state. One example of this critical state is “avalanche +criticality,” which has been observed in a range of systems, including cultured neurons, zebrafish, +and human EEG. More recently, power laws have also been observed in neural populations in the +mouse under a coarse-graining procedure, and they were explained as a consequence of the +neural activity being coupled to multiple latent dynamical variables. An intriguing possibility is +that avalanche criticality emerges due to a similar mechanism. Here, we determine the +conditions under which dynamical latent variables give rise to avalanche criticality. We find that a +single, quasi-static latent variable can generate critical avalanches, but that multiple latent +variables lead to critical behavior in a broader parameter range. We identify two regimes of +avalanches, both of which are critical, but differ in the amount of information carried about the +latent variable. Our results suggest that avalanche criticality arises in neural systems in which +there is an emergent dynamical variable or shared inputs creating an effective latent dynamical +variable, and when this variable can be inferred from the population activity. +Introduction +The neural criticality hypothesis – the idea that neural systems operate close to a phase transition, +perhaps for optimal information processing – is at the same time ambitious and banal. Measure- +ments from biological systems are limited in the range of spatial and temporal scales that can be +sampled, not only because of limits of recording techniques but also due to fundamentally non- +stationary behavior of most, if not all, biological systems. These limitations make proving that an +observation indicates critical behavior difficult. At the same time, the idea that brain networks are +critical echoes the anthropic principle: tuned another way, a network becomes quiescent or epilep- +tic, and in either case seems unlikely to support perception, thought, or flexible behavior. Further +muddying the water, researchers have reported multiple kinds of criticality in neural networks, in- +cluding through analysis of avalanches (Beggs and Plenz, 2003; Plenz et al., 2021; O’Byrne and Jerbi, +2022; Girardi-Schappo, 2021) and of coarse-grained activity (Meshulam et al., 2019), as well as of +correlations (Dahmen et al., 2019). How these flavors of critical behavior relate to each other or to +any functional network mechanism is not known. +The phenomenon that we will refer to as “avalanche criticality” appears to be remarkably widespread. +1 of 18 +arXiv:2301.00759v1 [q-bio.NC] 2 Jan 2023 + +It was first observed in cultured neurons in a dish (Beggs and Plenz, 2003) and later studied in ze- +brafish (Ponce-Alvarez et al., 2018), turtles (Shew et al., 2015), rodents (Ma et al., 2019), monkeys +(Petermann et al., 2009), and even humans (Poil et al., 2008). The standard analysis, described +thoroughly later, requires extracting power-law exponents from a fit to a distribution of avalanche +size and duration and assessing the relationship between exponents. There is debate over whether +these observations reflect true power laws, but within the resolution achievable from experiments, +neural avalanches exhibit power laws with exponent relationships predicted from theory devel- +oped in physical systems (Perković et al., 1995). +Avalanche criticality is not the only form of criticality observed in neural systems. Zipf’s law (fre- +quency of the network state being inversely proportional to its rank) appears in systems as diverse +as fly motion estimation and salamander retina (Mora and Bialek, 2010; Schwab et al., 2014; Aitchi- +son et al., 2016). More recently, Meshulam et al. (2019) measured various statistics of population +activity in a mouse hippocampus, including the eigenvalue spectrum of the covariance matrix and +the variance of activity. These were found to scale as populations were “coarse-grained” through +a procedure in which neural activities were iteratively combined based on similarity. Neither the +Zipf’s law nor the coarse-grained criticality can be explained by simple mechanistic models. +Even though these three forms of criticality are observed through different analyses, it is pos- +sible that they may originate from similar mechanisms. While avalanche power laws may result +from critical dynamics, they can also appear due to quasi-static latent variables, which can pro- +duce power laws, but not the relationships expected between the critical exponents (Priesemann +and Shriki, 2018). We have previously shown that a dynamical latent variable (DLV) model, based +on the coupling of neural populations to multiple dynamical latent variables, can reproduce scaling +under coarse-graining analysis within experimental uncertainty (Morrell et al., 2021). The Zipf’s law +has been explained by a similar mechanism (Schwab et al., 2014; Aitchison et al., 2016). However, +it is not known under what conditions, if any, the DLV model generates avalanche criticality. +In this paper, we systematically investigate avalanche statistics in the DLV model. We show that +a system coupled to multiple dynamical latent variables can generate avalanche criticality, and we +examine the requirements for the number and timescale of variables for this criticality to occur. +We find that avalanche criticality is observed over a wide range of parameters, some of which may +be optimal for information representation. Our results suggest that latent dynamical structure in +large-scale neural recordings may be responsible for the observation of signatures of criticality +across many systems. +Results +Critical exponents values and crackling noise +We begin by defining the metrics used to quantify avalanche statistics and briefly summarize ex- +perimental observations, which have been reviewed in detail elsewhere (Plenz et al., 2021; O’Byrne +and Jerbi, 2022; Girardi-Schappo, 2021). Activity is recorded across a set of neurons and binned +in time. Avalanches are then defined as contiguous time bins in which at least one neuron in the +population is active. The duration of an avalanche is the number of contiguous time bins and the +size is the summed activity during the avalanche. The distributions of avalanche size and duration +are fit to power laws (푃(푆) ∼ 푆−휏 for size 푆, and 푃(퐷) ∼ 퐷−훼 for duration 퐷) using standard methods +(Clauset et al., 2009). +Power laws can be indicative of criticality, but they can also result from non-critical mechanisms +(Touboul and Destexhe, 2017; Priesemann and Shriki, 2018). A more stringent test of criticality is +the “crackling” relationship (Perković et al., 1995; Touboul and Destexhe, 2017), which involves +fitting a third power-law relationship, ̄푆(퐷) ∼ 퐷훾fit, and comparing 훾fit to the predicted exponent +훾pred, derived from the size and duration exponents, 휏 and 훼: +훾fit +?= 훾pred ≡ 훼 − 1 +휏 − 1 . +(1) +2 of 18 + +Figure 1. Dynamical Latent Variable model produces avalanche criticality. A: Model structure. Latent +dynamical variables ℎ휇(푡) are broadly coupled to neurons 푠푖(푡) in the recorded population. B: Raster plot of a +sample of activity binned at 3-ms resolution across 128 neurons with five latent variables, each with +correlation timescale 휏퐹 = 15 s. C: Projection of activity into a simulated field of view for illustration. D-F: +Avalanche analysis in a network (parameters 푁퐹 = 5, 휏퐹 = 104, 휂 = 4 and 휖 = 12), showing size distribution (D), +duration distribution (E), and size with duration scaling (F). Lower cutoffs used in fitting are shown with +vertical lines and their values are indicated in the figures. There are 푁obs = 42725 avalanches of size 푆 ≥ 푆min in +this simulated dataset. Estimated values of the critical exponents are shown in the titles of the panels. +Previous work demonstrating approximate power laws in size and duration distributions through +the mechanism of a slowly changing latent variable did not generate crackling (Touboul and Des- +texhe, 2017; Priesemann and Shriki, 2018). +Measuring power-laws in empirical data is challenging: it generally requires setting a lower cut- +off in the size and duration, and the power-law behavior only has limited range due to the finite +size and duration of the recording itself. Nonetheless, there is some consensus (Shew et al., 2015; +Fontenele et al., 2019; Ma et al., 2019) that even if 휏 and 훼 vary over a wide range (1.5 to about 3) +across recordings, the values of 훾fit and 훾pred stay in a relatively narrow range, from about 1.1 to 1.3. +Avalanche scaling in the Dynamical Latent Variable (DLV) model +We studied a population of neurons that are coupled to dynamical latent variables but not coupled +to each other (Fig. 1A). We refer to this model as the Dynamical Latent Variable (DLV) model. The +latent variables determine the inputs to the simulated population of neurons. We are agnostic as +to the origin of these inputs: they may be externally driven from other brain areas, or they may +arise from recurrent dynamics locally. We have previously shown that the DLV model with at least +about five latent variables can produce power laws under the coarse-graining analysis (Morrell +et al., 2021). In this paper, we examine avalanche criticality in the same model. +Specifically, we model the neurons as binary units (푠푖) that are randomly (퐽푖휇 ∼ 푁(0, 1)) coupled +to dynamical variables ℎ휇(푡). The probability of any pattern {푠푖}, given the current state of the latent +3 of 18 + +B +h2(t) +100 +1.. +neurons +") +50 +hm(t) +S;(t) +.... +time (s) +100 +10° +103 +Probability Density +Probability Density +Average Size +10 +102 +10 +107 +10 +101 +100 +100 +102 +104 +106 +100 +102 +104 +100 +101 +102 +103 +Avalanche Size S +Avalanche Duration D +Duration Dvariables, is +푃({푠푖}|ℎ휇(푡)) = +1 +푍(ℎ휇(푡)) exp +( +−휂 +푁퐹 +∑ +휇=1 +푠푖퐽푖휇ℎ휇(푡) − 휖푠푖 +) +, +(2) +where the parameter 휂 controls the scaling of the variables and 휖 controls the overall activity level. +We modeled each latent variable as an Ornstein-Uhlenbeck process with the time scale 휏퐹 (see +Methods). Thus our model has four parameters: 휂 (input scaling), 휖 (activity threshold), 휏퐹 (dynami- +cal timescale), and 푁퐹 (number of neurons). +Distributions of avalanche size and avalanche duration within this model followed approximate +power laws (Fig. 1C; see Methods). In the example shown (푁퐹 = 5, 휏퐹 = 104, 휂 = 4 and 휖 = 12), we +found exponents 휏 = 1.89 ± 0.02 (size) and 훼 = 2.11 ± 0.02 (duration). Further, the average size of +avalanches with fixed duration scaled as 푆 ∼ 퐷훾, with the fitted 훾fit = 1.24 ± 0.02, in agreement with +the predicted value 훾pred = 1.24±0.02. Thus, our model could generate avalanche scaling, at least for +some parameter choices. In the following sections, we examine how avalanche scaling depends +on model parameters (푁퐹 , 휏퐹 , 휂 and 휖; see Table 2). We first focus on two sets of simulations: one +set with 푁퐹 = 1 latent variable, which does not generate scaling under coarse-graining (Morrell +et al., 2021), and one set with 푁퐹 = 5 latent variables, which can generate such scaling for some +values of parameters 휏퐹 , 휂, and 휖 (Morrell et al., 2021). +Avalanche scaling depends on the number of latent variables +We analyzed avalanches from one- and five-variable simulations, each with fixed latent dynamical +timescale (휏퐹 = 5 × 103 time steps; see Table 2 for parameters). In the following sections, time is +measured in simulation time steps, see Methods for converting time steps to seconds. We used es- +tablished methods for measuring empirical power laws (Clauset et al., 2009). The minimum cutoffs +for size (푆min) and duration (퐷min) are indicated by vertical lines in Fig. 2. For the population coupled +to a single latent variable, the avalanche size distribution was not well fit by a power law (Fig. 2A). +With a sufficiently high minimum cut-off (퐷min), the duration distribution was approximately power- +law (Fig. 2B). +We next assessed whether the simulation produced crackling. If so, the value 훾fit obtained by +fitting ̄푆(퐷) ∼ 퐷훾fit would be similar to 훾pred = 훼−1 +휏−1 . In many cases, such as the one-variable example +shown in Fig. 2C, the full range of avalanche durations were not fit by a single power law. There- +fore, we determined the largest range over which a power law was a good fit to the simulated +observations. In this case, slightly over two decades of apparent scaling were observed starting +from avalanches with minimum duration slightly less than 100 time steps (Fig. 2C), with a best-fit +value of 훾푓푖푡 ∈ [1.69, 1.74]. As we did not find a power-law in the size distribution, calculating 훾pred is +meaningless. If we do it anyway, we obtain 훾푝푟푒푑 = 0.83 ± 0.03 (yellow line in Fig. 2C), which clearly +deviates from the fitted value of 훾. Thus, for the single dynamical latent variable model (휏퐹 = 5000), +power-law fits are poor, and there is no crackling. +The five-variable model produces a different picture. We now find avalanches for which size and +duration distributions are much better fit by power-law models starting from very low minimum +cutoffs (Fig. 2D-E, Fig. 2-Supp. Fig. 2). Average size scaled with duration, again over more than +two decades, with 훾fit = 1.27 ± 0.03, which was in close agreement with 훾pred = 1.25 ± 0.02 (Fig. 2F). +Holding other parameters constant, we thus found that scaling relationships and crackling arise in +the multi-variable model but not the single-variable model. +Avalanche scaling depends on the time scale of latent variables +Based on simulations in the previous section, we surmised that the five-variable simulation gen- +erated scaling more readily due to creating an “effective” latent variable that had slower dynam- +ics than any individual latent variable. We reasoned that at any moment in time, the latent vari- +able state ℎ휇(푡) is a vector in the latent space. Because coupling to the latent variables is random +throughout the population, only the length (∼ +√ +푁퐹 ) and not the direction of this vector matters, +4 of 18 + +Figure 2. Multiple latent variables generate avalanche scaling at shorter timescales than a single latent +variable. Parameters used for simulations for this figure are found in Table 2. A-C: Scaling analysis for one +variable models where the dynamic timescale is equal to 5 × 103 time steps. A: Distribution of avalanche sizes. +MLE value of exponent for best-fit power law is 휏 = 1.98 (0.02 SE), with lower cutoff indicated by the vertical +line. B: Distribution of avalanche duration. MLE value of 훼 is 1.81 (0.02 SE). C: Average size plotted against +avalanche duration (blue points), with power-law fit (black line) and predicted relationship (yellow line) from +MLE values for exponents in A and B. Gray bar on the horizontal axis indicates range over which a power law +with 훾 = 1.72 fits the data (see Methods). D-F: Analysis of avalanches from a simulation of a population coupled +to five independent latent variables where the dynamic timescale is equal to 5 × 103 time steps. G: Exponents +휏 for avalanche size distributions across timescales for one-variable (blue) and five-variable (red) simulations. +Each circle is a simulation with independently drawn coupling parameters. Simulations had to show scaling +over at least two decades to be included in panels (G-J). H: Exponents 훼 for avalanche duration distributions +for simulations in G. I: Fitted values of 훾 for simulations in G. J: Difference between fitted and predicted 훾 +values. Five-variable simulations produce crackling over a wider range of timescales than single-variable +simulations. Figure 2–Figure supplement 1. Methods, power law distribution fits, one variable example. +Figure 2–Figure supplement 2. Methods, power law distribution fits, five variable example. +Figure 2–Figure supplement 3. Methods, gamma fit and range, one variable example. +Figure 2–Figure supplement 4. Methods, gamma fit and range, five variables example. +5 of 18 + +5 +average size +4 +2 +2 +601) +３ +4 +pdf +4 +¥2 +¥6 +-6 +S +D +min +8 +8 +avalanche size +avalanche duration +duration +5 +(10g10) +average size +4 +2 +¥３ +4 +pdf +4 +2 +¥6 +OZIs +6 +D +min +8 +8 +0 +avalanche size +avalanche duration +duration +2.2 +1.8 +CDAD +D +團 +④ +2.4 +D OD) +0 +2 +1.6 +0.5 +CD +CRDD +2.2 +KED +CTXD +( +.8 +1.4 +00 +0 +BXD +ED +1.8 +1.6 +.2 +0.5 +10000 +00 +1000 +1000 +10000 +10000 +10000 +30000 +30000 +30000 +100000 +100000 +100000 +100000 +dynamical timescale +dynamical timescale +dynamical timescale +dynamical timescale +TEand the timescale of changes in this length would be much slower than 휏퐹 , the timescale of each +of the components ℎ휇(푡). We therefore speculated that increasing the timescale of dynamics of the +latent variables should eventually lead to scaling and crackling in the single-variable model as well +as the five-variable one. To examine the dependence of avalanche scaling on this timescale, we +simulated one-variable and five-variable networks at fixed 휂 and 휖 coupled to latent variables with +the correlation time of their Ornstein-Uhlenbeck dynamics of 휏퐹 ∈ [103, 105] time steps, spanning +from a factor of 10 faster to a factor of 10 slower than the original 휏퐹 in Fig. 1. Simulations were +replicated five times at each combination of parameters by drawing new latent variable coupling +values (퐽푖휇), as well as new latent variable dynamics and instances of neural firing. For simulations +that passed the criteria to be fitted by power laws, we plot the fitted values of 휏 , 훼, 훾fit and 훾fit − 훾pred +(Fig. 2G-J). Missing points are those for which distributions did not pass the power law fit criteria. +In the single-variable model, best-fit exponents changed abruptly for latent variable timescale +around 휏퐹 = 104 (Fig. 2G, H), while in the five-variable model, exponents tended to increase grad- +ually (Fig. 2G, H, red). The discontinuity in the single-variable case reflected a change in the lower +cutoff values in the power-law fits: size and duration distributions generated with faster latent +dynamics could be fit reasonably well to a power law by using a high value of the lower cutoff +(Fig. 2-Supp. Fig. 3). For time scales greater than ∼ 104, the minimum cutoffs dropped, and the +single-variable model generated power-law distributed avalanches and crackling (Fig. 2J), similar +to the five-variable model. In summary, in the DLV model, introducing multiple variables gener- +ated scaling at faster timescales. However, by slowing the timescale of the latent dynamics, the +DLV model generated signatures of critical avalanche scaling for both multi- and single-variable +simulations. +Avalanche criticality, input scaling, and firing threshold +In the previous section, we found that a very slow single DLV model generated scaling. Thus, from +now on, we simplify the model in order to characterize avalanche statistics across values of input +scaling 휂 and firing threshold 휖. Specifically, we modeled a population of 푁 = 128 neurons coupled +to a single quasi-static latent variable. We simulated 103 segments of 104 steps each and drew a +new value of the latent variable (ℎ ∼ 푁(0, 1)) for each segment. Ten replicates of the simulation +were generated at each of the combinations of 휂 and 휖 (see Methods). +Almost independent of 휂 and 휖, we found quality power law fits and crackling. Fig. 3 shows +the average (across 푛 = 10 network realizations) of the exponents extracted from size (휏, Fig. 3A) +and duration (훼, Fig. 3C) distributions. At small firing threshold (휖 = 2), we do not observe scaling +because the system is always active, and all avalanches merge into one. At large firing threshold 휖 +and low input scaling 휂, we do not observe scaling because activity is so sparse that all avalanches +are small. At intermediate values of the parameters, the simulations generated plausible scaling +relationships in size and duration. The difference between 훾fit and 훾pred was typically less than 0.1 +(Fig. 4D-F) which was consistent with previously reported differences between fit and predicted +exponents (Ma et al., 2019). Thus, there appears to be no need for fine-tuning to generate apparent +scaling in this model, at least in the limit of (near) infinite observation time. Wherever 휂 and 휖 +generate avalanches, there are approximate power-law distributions and crackling. +To determine where avalanches occur, we derive the avalanche rate across values of the latent +variable ℎ. In the quasi-static model, the probability of an avalanche initiation is the probability of a +transition from the quiet to an active state. Because all neurons are conditionally independent, this +is 푃ava = 푃silence(1 − 푃silence). Then the expected number of avalanches ̂푁ava is obtained by integrating +푃ava over ℎ at each value of 휂 and 휖: +̂푁ava = ∫ 푃ava(휖, 휂, ℎ; 퐽푖, 푁)푝(ℎ)푑ℎ = ∫ +∏ +푖 +( +1 +1 + 푒휂퐽푖ℎ+휖 +) ( +1 − +∏ +푖 +( +1 +1 + 푒휂퐽푖ℎ+휖 +)) +푝(ℎ)푑ℎ, +(3) +where 푝(ℎ) is the standard normal distribution. This probability tracks the observed number of +avalanches across simulations, Fig. 4A. +6 of 18 + +Figure 3. Exponents across network simulations. Each parameter combination 휂, 휖 was simulated for ten +replicates, each time drawing randomly the couplings 퐽푖, the latent variable values, and the neural activities. +A: Average across replicates for the size exponent 휏. B: Scatter plot of 훼 vs. 휏 for each network replicate for +parameter combinations indicated in A. Linear relationships between 휏 and 훼, corresponding to the minimum +and maximum values of 훾fit from panel E, are shown to guide the eye. C: Same as A, for duration exponent 훼. +D: Predicted exponent, 훾pred, derived from A and C. E: Value of 훾fit from fit to ̄푆퐷 ∼ 퐷훾. F: Difference between +훾pred and 훾fit. +7 of 18 + +Average Exponents: Size +.8 +口 +0 +2.6 +las +口 +2.4 +2.8 +米 +8 +2.2 +* +(duration exponent) +2.6 +2 +4 +6 +10 +米 +米 +n (gain) +米 +柔 +2 +米 +Average Exponents: Duration +2.8 +2 +2.6 +米 +(sel +2.4 +8 +24680 +2.5 +3 +n (gain) +(sizeexponent) +Crackling Exponents +Predicted +Difference +2 +0.2 +2 +1.3 +0. +as +Dias +.2 +8 +1. +-0.1 +0.2 +10 +810 +4 +aairTo gain an intuition for the conditions under which avalanches occur, we show two slices of +the avalanche probability, at fixed 휂 (Fig. 4B) and at fixed 휖 (Fig. 4C). Black regions indicate where +avalanches are likely to occur. If, for a given value of 휖 and 휂, there is no overlap between high +avalanche probability regions and the distribution of ℎ, then there will be no avalanches. For large +휖, avalanches occur because neurons with large coupling to the latent variable (휂|퐽푖| >> 1, recall +퐽푖 ∼ 푁(0, 1)) are occasionally activated by a value of the latent variable ℎ that is sufficient to exceed +휖 (Fig. 4B). Thus, the scaling parameter 휂 controls the value of ℎ for which avalanches occur most +frequently (Fig. 4C). As 휖 decreases, avalanches occur for smaller and smaller ℎ until avalanches +primarily occur when ℎ = 0. +To calculate the probability of avalanches, we must integrate over all values of ℎ, but we can +gain a qualitative understanding of which avalanche regime the system is in by examining the +probability of avalanches at ℎ = 0. At ℎ = 0, the avalanche probability (see Methods) is +푃ava(휖, 휂, ℎ = 0; 퐽푖, 푁) = +( +1 +1 + 푒휖 +)푁 ( +1 − +( +1 +1 + 푒휖 +)푁) +, +(4) +which is maximized at 휖0 = − log(21∕푁 − 1), independent of 퐽푖 and 휂. The dependence on 푁 re- +flects that a larger threshold is required for larger networks: large networks (푁 → ∞) are unlikely +to achieve complete network silence, therefore preventing avalanches from occurring. Similarly, +small networks (푁 ∼ 1) are unlikely to fire consecutively and thus are unlikely to avalanche. +We plot 푃ava(휖, 휂; 퐽푖, 푁, ℎ = 0) as a function of 휖 in Fig. 4B. The peak at 휖0 divides the space into +two regions. For 휖 < 휖0, a power-law is only observed in the large-size avalanches, which are rare +(Fig. 4E, green). By contrast, when 휖 > 휖0, minimum size cutoffs are low (Fig. 4F, orange). Both +regions, 휖 < 휖0 and 휖 > 휖0, exhibit crackling noise scaling. If observation times are not sufficiently +long (estimated in Fig. 4-Supp. Fig. 1), then scaling will not be observed in the 휖 < 휖0 region, whose +scaling relations consist of rare events. Insufficient observation times may explain experiments +and simulations where avalanche scaling was not found. +Inferring the latent variable +Our analysis of 푃ava(휖, 휂, ℎ) at ℎ = 0 suggested that there are two types of avalanche regimes: one +with high activity and high minimum cutoffs in the power law fit (Type 1), and the other with lower +activity and size cutoffs (Type 2). Further, when 푃ava drops to zero, avalanches disappear because +the activity is too high or too low. We now examine how information about the value of the latent +variables represented in the network activity relates to the activity type. To delineate these types, +we calculated numerically 휖∗(휂), the value of 휖 for which the probability of avalanches is maximized, +and the contours of 푃ava (Fig. 5A). Curves for 휖∗(휂) and 휖0 and 푃ava = 10−3 are shown in Fig. 5A and B. +We expect that the more cells fire, the more information they would convey, until the firing +rate saturates, and inferring the value of the latent variable becomes impossible. Fig. 5B supports +the prediction: generally, information is higher in regions with more activity (lower 휖, higher 휂), but +only up to a limit: as 휖 → 0, information decreases. This decrease begins approximately where +the probability of avalanches drops to nearly zero (dashed black lines, Fig. 5B-E) because all of +the activity merges into a few very large avalanches. In other words, the Type-1 avalanche region +coincides with the highest information about the latent variable. +The critical brain hypothesis suggests that the brain operates in a critical state, and its func- +tional role may be in optimizing information processing (Beggs, 2008; Chialvo, 2010). Under this +hypothesis, we would expect the information conveyed by the network to be maximized in the +regions we observe avalanche criticality. However, we see that critical regions do not always have +optimal information transmission. In Fig. 5, the region that displays crackling noise is that where +avalanches exist (푃ava > 0.001), which corresponds to any 휂 value and 휖 ≳ 3. This avalanche re- +gion encompasses both networks with high information transmission and networks with low in- +formation transmission. In summary, observing avalanche criticality in a system does not imply a +high-information processing network state. However, the scaling can be seen at smaller cutoffs, +8 of 18 + +Figure 4. Avalanches in the DLV model with a single quasistatic variable. A: Number of avalanches in +simulations as a function of the calculated probability of avalanches at fixed 휂 across values of 휖 and latent +variable ℎ. Line indicates equality. B: Analytically calculated probability of avalanches with 휂 = 2 across values +of 휖 and ℎ. The latent variable ℎ is normally distributed with mean 0 and variance 1. Where the distribution of +ℎ overlaps with regions of high probability (black), avalanches occur. C: Analytically calculated probability of +avalanches at 휖 = 8 across values of 휂 and ℎ. Increasing 휂 narrows the range of ℎ that generates avalanches. D: +Analytically calculated probability of avalanches at ℎ = 0 for a populations of 128 neurons (black line) and for a +varying 휖. Size distributions corresponding to simulations marked by the green and orange crosses are in E, F. +E: Example of size distribution with 휖 < 휖0 (orange marker in D). Size cutoff is close to 100. F: Example of size +distribution with 휖 > 휖0 (green marker in D). Size cutoff is < 10. +Figure 4–Figure supplement 1. Estimated simulation time to observe avalanche criticality. +and hence with shorter recordings, in the high-information state. This parallels the discussion by +Schwab et al. (2014), who noticed that the Zipf’s law always emerges in neural populations driven +by quasi-stationary latent fields, but it emerges at smaller system sizes when the information about +the latent variable is high. +Discussion +Here we studied systems with distributed, random coupling to Dynamical Latent Variables (DLV) +and we found that avalanche criticality is nearly always observed, with no fine-tuning required. +Avalanche criticality was surprisingly robust to changes in input gain and firing rate threshold. Loss +of avalanche criticality could occur if the latent process was not well-sampled, either because the +simulation was not long enough or the dynamics of the latent variables were too fast. Finally, while +information about the latent variables in the network activity was higher where avalanches were +generated compared to when they were not, there was a range of information values across the +critical avalanche regime. Thus, avalanche criticality alone was not a predictor of optimal informa- +tion transmission. +Explaining experimental exponents +A wide range of critical exponents have been found in ex vivo and in vivo recordings from various +systems. For instance, the seminal work on avalanche statistics in cultured neuronal networks +by Beggs and Plenz (2003) found size and duration exponents of 1.5 and 2.0 respectively, along +with 훾 = 2, when time was discretized with a time bin equal to the average inter-event interval in +the system. These values are predicted by a theoretical model of a critical branching process. By +contrast, a survey of many in vivo and ex vivo recordings found power-law size distributions with +exponents ranging from 1 to 3 depending on the system (Fontenele et al., 2019). Separately, Ma +9 of 18 + +A +B +c +n= 2 +E=8 +0.25 +10 +0.25 +Avalanche count (Na) +2 +2 +8 +1.5 +(seIc +a +6 +1 +8 +m +4 +0.5 +2 +0 +14 +0 +2 +-5 +0 +5 +-5 +0 +5 +7 +P +(1-P +h +h +D +silence +silence +E +F +T = 2.24 (0.01 SE) +T = 2.00 (0.01 SE) +0.25 +0 +0.2 +-2 +-2 +0.15 +-4 +-4 +0.1 +-6 +-6 +P°0.05 +0 +-8 +-8 +10 +15 +0 +E (bias) +avalanche size +avalanche sizeFigure 5. Information in the neural activity about the latent variable is higher in the low-휖 avalanche region, +compared to high-휖 avalanche or high-rate avalanche-free activity. A: Probability of avalanche per time step +across values of 휂 and 휖. Solid magenta curve follows 휖∗(휂), the value of 휖 maximizing the probability of +avalanches at fixed 휂. Dashed magenta line indicates 휖0, calculated analytically, which matches 휖∗ at 휂 = 0. B: +Information about latent variable, calculated from maximum likelihood estimate of ℎ using population +activity. MLE approximation is invalid in the dark-blue region bounded by gray curve. Magenta line marks the +maximum values of 푃ava, reproduced from A. Dashed black curve indicates 푃ava = 0.001. The highest +information region falls between 휖∗(휂) and the contour for 푃ava = 0.001. C - E: Slices of B, showing 퐼MLE(휖) for +휂 = {2, 5, 9}. Magenta and dashed black lines again indicate 휖∗ and 푃ava = 0.001, respectively, as in B. Black +dashed line marks the approximate boundary between the high-activity/no avalanche and the high-cutoff +avalanche, and magenta line marks boundary between high-cutoff and low-cutoff avalanche regions. +10 of 18 + +0 +0.25 +B +0 +2.5 +2 +2 + cells (MLE) +0.2 +2 +0.001 +Probability of Avalanche +三 +4 +4 +0.02 +0.15 +1.5 +(bias) +6 +(bias) +9 +128 +8 +8 +0.1 +for N +0.08 +1 +0 +10 +0.04 +10 +D +=0.001 +ava +0.05 +12 +12 +0 +e(n) +14 +0 +14 +0 +0 +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +n (input scaling) +n (input scaling) +c +D +n=2 +n=5 +E +n=9 +2 +2.5 +128 +128 +128 +2 +1.5 += +z +z +h*), +h*), +1 +0.5 +≤ 0.5 +0 +0 +0 +0 +5 +10 +0 +5 +10 +0 +5 +10 +E( +(bias) +E (bias) +E (bias)et al. (2019) reported recordings in freely moving rats with size exponents ranging from 1.5 to 2.7. +In all of the these recordings, when the crackling relationship held, the reported value of 훾 was +near 1.2 (Fontenele et al., 2019; Ma et al., 2019). +Our DLV model, across the parameters we tested that produced exponents consistent with the +scaling relationship, generated 휏 values that ranged from 1.9 to about 2.5. Across those simulations, +we found values 훾 within a narrow band from 1.1 to 1.3 (see Fig. 2I, J and Fig. 3H). While the exponent +values our model produces are inconsistent with a critical branching process (훾 = 2), they match +very closely the ranges of exponents reported by Fontenele et al. (2019). +One possible resolution to the discrepancy in exponents derives from how the system is sub- +sampled in space or coarse-grained in time, both of which systematically change exponents 휏 and +훼 (Beggs and Plenz, 2003; Shew et al., 2015). Were we to change the time bin, our modeling results +would exhibit different exponent values. However, neither manipulations of the latent variable +timescale (휏퐹 or 푁퐹 ), nor of the overall activity level (휂, 휖) produced exponents close to 1.5 and 2.0, +despite maintaining the crackling relationship across many different choices of parameters. +A second possibility is that different experiments study similar, but distinct biological phenom- +ena. In other words, the underlying biology can differ between networks that were cultured in vitro +and those that were not, whether they are in vivo or ex vivo (i.e., brain slices). This could happen +if cultured networks develop connections between neurons such that they truly do produce dy- +namics that approximate a critical branching process, while brain networks that develop in a living +brain have different structure and resulting dynamics and can be better understood as a system +coupled to latent dynamical variables. This is especially true in sensory systems, where coupling +to (latent) external stimuli in a way that the neural activity can be used to infer the stimuli is the +reason for the networks’ existence (Schwab et al., 2014). +Relationship to past modeling work +Our model is not the first to produce approximate power-law size and duration distributions for +avalanches from a latent variable process (Touboul and Destexhe, 2017; Priesemann and Shriki, +2018). In particular, Priesemann and Shriki (2018) derived the conditions under which an inhomo- +geneous Poisson process could produce such approximate scaling. The basic idea is to generate a +weighted sum of exponentially distributed event sizes, each of which are generated from a homo- +geneous Poisson process. How each process is weighted in this sum determines the approximate +power-law exponent, allowing one to tune the system to obtain the critical values of 1.5 and 2. In- +terestingly, this model did not generate non-trivial scaling of size with duration (푆 ∼ 퐷훾). Instead, +they found 훾 = 1, not the predicted 훾 = 2. Our results differ significantly, in that 훾 was typically +between 1.1 and 1.3 and it was nearly always close to the prediction from 훼 and 휏. We speculate +that this is due to nonlinearity in the mapping from latent variable to spiking activity, as doubling +the latent field ℎ does not double the population activity, but doubling the rate of a homogeneous +Poisson process does double the expected spike count. As biological networks are likely to have +such nonlinearities in their responses to common inputs, this scenario may be more applicable to +certain kinds of recordings. +Summary +Latent variables – whether they are emergent from network dynamics (Clark et al., 2022; Seder- +berg and Nemenman, 2020) or derived from shared inputs – are ubiquitous in large-scale neural +population recordings. This fact is reflected most directly in the relatively low-dimensional struc- +ture in large-scale population recordings (Stringer et al., 2019; Pandarinath et al., 2018; Nieh et al., +2021). We previously used a model based on this observation to examine signatures of neural crit- +icality under a coarse-graining analysis and found that coarse-grained criticality is generated by +systems driven by many latent variables (Morrell et al., 2021). Here we showed that the same +model also generates avalanche criticality, and that when information about the latent variables +can be inferred from the network, avalanche criticality is also observed. Crucially, finding signa- +11 of 18 + +tures of avalanche criticality required long observation times, such that the latent variable was +well-sampled. Previous studies showed that Zipf’s law appears generically in systems coupled to a +latent variable that changes slowly relative to the sampling time, and that the Zipf’s behavior is eas- +ier to observe in the higher information regime (Schwab et al., 2014; Aitchison et al., 2016). How- +ever, this also suggests that observation of either scaling at modest data set sizes indeed points +to some fine-tuning — namely to the increase of the information in the individual neurons (and, +since neurons in these models are conditionally independent, also in the entire network) about +the value of the latent variables. In other words, one would expect a sensory part of the brain, if +adapted to the statistics of the external stimuli, to exhibit all of these critical signatures at relatively +modest data set sizes. In monocular deprivation experiments, when the activity in the visual cor- +tex is transiently not adapted to its inputs, scaling disappears, at least for recordings of a typical +duration, and is restored as the system adapts to the new stimulus (Ma et al., 2019). We speculate +that the observed recovery of criticality by Ma et al. (2019) could be driven by neurons adapting +to the reduced stimuli state, for instance, by adjusting 휂 (input scaling) and 휖 (firing rate threshold). +Taken together, these results suggest that critical behavior in neural systems – whether based on +the Zipf’s law, avalanches, or coarse-graining analysis – is expected whenever neural recordings ex- +hibit some latent structure in population dynamics and this latent structure can be inferred from +observations of the population activity. +Methods and Materials +Simulation of Dynamic Latent Variable (DLV) model +We study a model from Morrell et al. (2021), incorporating only latent variables (no place variables), +and assuming that every cell is coupled to every latent variable with some randomly drawn coupling +strength. +The probability of observing a certain population state {푠푖} given latent variables {ℎ휇(푡)} at time +푡 is +푃({푠푖}|{ℎ휇}) = +1 +푍({ℎ휇})푒퐻({푠푖},{ℎ휇}), +(5) +where 푍 is the normalization, and 퐻 is the “energy”: +퐻 = +푁,푁f +∑ +푖,푚=1 +휂ℎ휇(푡)퐽푖휇푠푖 + 휖푠푖. +(6) +The latent variables {ℎ휇(푡)} are Ornstein-Uhlenbeck processes with zero mean, unit variance, and +time constant 휏푚. Couplings 퐽푖휇 are drawn from the standard normal distribution. +The parameters 휂, 휖, and 휏푚 are constants, and we simulate 푁 = 1024 cells. For the infinite time +constant simulation, we reset ℎ푛 ∼  (0, 1) (for each of 푛 = 1..푁푛) and simulate for 10000 time steps, +then repeat for 1000 draws of ℎ푛. +Time step units +Most results were presented using arbitrary time units: all times (i.e., 휏퐹 and avalanche duration 퐷) +are measured in units of an unspecified time step. Specifying a time bin converts the probability +Table 1. Simulation parameters for Fig. 1. +Parameter +Description +Value +휖 +bias towards silence +휖 = 12 +휂 +variance multiplier +휂 = 4.0 +푁F +number of latent fields +푁F = 5 +휏퐹 +latent field time constant +휏 = 104 +푁 +number of cells +푁 = 1024 +12 of 18 + +of firing into actual firing rates, in spikes per second, and this choice determines which part of the +휂-휖 phase space is most relevant to a given experiment. +The time step is the temporal resolution at which activity is discretized, which varies from sev- +eral to hundreds of milliseconds across different experimental studies (Beggs and Plenz, 2003; +Fontenele et al., 2019; Ma et al., 2019). In physical units and assuming a bin size of 3 ms to 10 ms, +our choice of 휂 and 휖 in Fig. 2 would yield physiologically realistic firing rate ranges (Hengen et al., +2016), with high-firing neurons reaching averages rates of 20 − 50 spikes/second and median firing- +rate neurons around 1 − 2 spikes/second. The timescales of latent variables examined range from +about 3 seconds to 3000 seconds, assuming 3-ms bins. Simulations were carried out for the same +number of time steps (2 × 106), which would be approximately 1 to 2 “hours,” which is a reasonable +duration for in vivo neural recordings. Note that at large values of 휏퐹 , the latent variable space is +not well sampled during this time period. +Analysis of avalanche statistics +Setting the threshold for observing avalanches +In our model, we count avalanches as periods of continuous activity (in any subset of neurons) +that is book-ended by time bins with no activity in the entire simulated neural network. For real +neural populations of modest size, this method fails because there are no periods of quiescence. +The typical solution is to set a threshold, and to only count avalanches when the population activity +exceeds that threshold, with the hope that results are relatively robust to that choice. In our model, +this operation is equivalent to changing 휖, which shifts the probability of firing up or down by a +constant amount across all cells independent of inputs. Our results in Fig. 3 show that 훼 and 휏 +decrease as the threshold for detection is increased (equivalent to large |휖|), but that the scaling +relationship is maintained. The model predicts that 훾pred − 훾fit would initially increase slightly with +the detection threshold before decreasing back to near zero. +Following the algorithm laid out in Clauset et al. (2009), we fit power laws to the size and dura- +tion distributions from simulations generating avalanches. We use least-squares fitting to estimate +훾fit, the scaling exponent for size with duration, assessing the consistency of the fit across decades. +Reading power laws from data +We want, from each simulation, a quantification of the quality of scaling (how many decades, min- +imally) and an estimate of the scaling exponents (휏 for the size distribution, 훼 for the duration +distribution). Following the steps outlined by Clauset et al. (2009), we use the maximum-likelihood +estimator to determine the scaling exponent. This is the solution to the transcendental equation +휁′(̂훼, 푥min) +휁′(̂훼, 푥min) = −1 +푛 +푛 +∑ +푖=1 +ln 푥푖 +(7) +where 휁(훼, 푥min) is the Hurwitz zeta function. For values of 푥min < 6, a numerical look-up table based +on the built-in Hurwitz zeta function in the symbolic math toolbox was used (MATLAB2019b). For +Table 2. Simulation parameters for Fig. 2. +Parameter +Description +Value +휖 +bias towards silence +휖 = 8 (for 푁퐹 = 1) or +휖 = 12 (for 푁퐹 = 5) +휂 +variance multiplier +휂 = 4.0 +푁F +number of latent fields +푁F = 1 or 5 +휏퐹 +latent field time constant +휏 ∈  [log 103, log 105] +푁 +number of cells +푁 = 1024 +13 of 18 + +푥min > 6 we use an approximation (Clauset et al. (2009)), +̂훼 = 1 + 푛 +( +∑ +푖 +ln +푥푖 +푥min − 1 +2 +)−1 +. +(8) +To determine 푥min, we computed the maximum absolute difference between the empirical cu- +mulative density (푆(푥)) function and model’s cumulative density function 푃(푥) (the Kolmogorov- +Smirnov (KS) statistic; 퐷 = max푥≥푥min|푆(푥)−푃(푥)|). The KS statistic was computed between for power- +law models with scaling parameter ̂훼 and cutoffs 푥min. The value of 푥min that minimizes the KS statis- +tic was chosen. Occasionally the KS statistic had two local minima (as in Figure 2-Supplemental +Figure 1), indicating two different power-laws. In these cases, the minimum size and duration cut- +offs were the smallest values that were within 10% of the absolute minimum of the KS statistic. +Note that the statistic is computed for each model only on the power-law portion of the CDF (i.e. +푥푖 ≥ 푥min). We do not attempt to determine an upper cut-off value. +To assess the quality of the power-law fit, Clauset et al. (2009) compared the empirical observa- +tions to surrogate data generated from a semi-parametric power-law model. The semi-parametric +model sets the value of the CDF equal to the empirical CDF values up to 푥 = 푥min and then according +to the power-law model for 푥 > 푥min. If the KS statistic for the real data (relative to its fitted model) +is within the distribution of the KS statistics for surrogate datasets relative to their respective fitted +models, the power-law model was considered a reasonable fit. +Strict application of this methodology could give misleading results. Much of this is due to the +loss of statistical power when the minimum cutoff is so high that the number of observations drops. +For instance, in the simulations shown in Fig. 2, the one-variable duration distribution passed the +Clauset et al. (2009) criterion, with a minimum KS statistic of 0.03 when the duration cutoff was +18 time steps. However, for the five-variable simulation in Fig. 2, a power-law would be narrowly +rejected for both size and duration, despite having much smaller KS statistics: for 휏, the KS statistic +was 0.0087 (simulation range: 0.0008 to 0.0082; number of avalanches observed: 58, 787) and for 훼 it +was 0.0084 (simulation range: 0.0011 to 0.0075). Below we discuss this problem in more detail. +Determining range over which avalanche size scales with duration +For fitting 훾, our aim was to find the longest sampled range over which we have apparent power- +law scaling of size with duration. Because our sampled duration values have linear spacing, error +estimates are skewed if a naive goodness of fit criterion is used. We devised the following algorithm. +First, the simulation must have at least one avalanche of size 500. We fit 푆 = 푐퐷훾 over one decade +at a time. We chose as the lower duration cutoff the value of minimum duration for which the +largest number of subsequent (longer-duration) fits produced consistent fit parameters (Figure +2-Supp. Fig. 3 and 4, top row). Next, with the minimum duration set, we gradually increased the +maximum duration cut-off, and we determined whether there was a significant bias in the residual +over the first decade of the fit. We selected the highest duration cutoff for which there was no bias. +Finally, over this range, we re-fit the power law relationship and extracted confidence intervals. +Our analysis focused on finding the apparent power-law relationship that held over the largest +log-scale range. A common feature across simulation parameters (휏퐹 , 푁퐹 ) was the existence of +Table 3. Simulation parameters for Fig. 3 and 4. +Parameter +Description +Value +휖 +bias towards silence +휖 ∈ {2, 4, ...14} +휂 +variance multiplier +휂 ∈ {1, 2, ...10} +푁F +number of latent fields +푁F = 1 +휏퐹 +latent field time constant +quasistatic +푁 +number of cells +푁 = 128 +14 of 18 + +two distinct power-law regimes. This is apparent in Fig. 2I, which shows that when 푁퐹 = 1 at small +휏퐹 , the best-fit 훾 (that showing the largest range with power-law-consistent scaling) is much larger +(> 1.7), and then above 휏퐹 ∼ 3000, the best-fit 훾 drops to around 1.3. +Statistical power of power-law tests +In several cases, we found examples of power-law fits that passed the rejection criteria commonly +used to determine avalanche scaling relationships because of limited number of observations. A +key example is that of the single latent variable simulation shown in Fig. 2B, where we could not +reject a power law for the duration distribution. Conversely, strict application of the surrogate cri- +teria would reject a power law for distributions that were quantitatively much closer to a power-law +(i. e., lower KS statistic), but for which we had many more observations and thus a much stronger +surrogate test (Fig. 2). This points to the difficulty of applying a single criterion to determining a +power-law fit. In this work, we adhere to the criteria set forth in Clauset et al. (2009), with a mod- +ification to control for the unreasonably high statistical power of simulated data. Specifically, the +number of avalanches used for fitting and for surrogate analysis was capped at 500, 000, drawn +randomly from the entire pool of avalanches. +Additionally, we found examples in which a short simulation was rejected, but increasing the +simulation time by a factor of five yielded excellent power-law fits. We speculate that this arises +due to insufficient sampling of the latent space. These observations raise an important biological +point. Simulations provide the luxury of assuming the network is unchanging for as long as the +simulator cares to keep drawing samples. In a biological network, this is not the case. Over the +course of hours, the effective latent degrees of freedom could change drastically (e. g., due to +circadian effects (Aton et al., 2009), changes in behavioral state (Fu et al., 2014), plasticity (Hooks +and Chen, 2020), etc.), and the network itself (synaptic scaling, firing thresholds, etc.) could be +plastic (Hengen et al., 2016). All of these factors can be modeled in our framework by determining +appropriate cutoffs (in duration of recording, in time step sizes, for activity distributions) based on +specific experimental timescales. +Calculation of avalanche regimes +In the quasistatic model, we derive the dependence of the avalanche rate on 휂, 휖 and number of +neurons 푁, finding that there are two distinct regimes in which avalanches occur. Each time bin is +independent, conditioned on the value of ℎ. For an avalanche to occur, the probability of silence +in the population (i.e., all 푠푖 = 0) must not be too close to 0 (or there are no breaks in activity) or too +close to 1 (or there is no activity). At fixed ℎ, the probability of silence is +푃silence(휖, 휂; 퐽푖, 푁, ℎ) = +∏ +푖 +1 +1 + exp(−휂퐽푖ℎ + 휖). +(9) +An avalanche occurs when a silent time bin is followed by an active bin, which has probability +푃ava(휖, 휂; 퐽푖, 푁, ℎ) = 푃silence(1 − 푃silence). +Information calculation +Maximum-likelihood decoding +For large populations coupled to a single latent variable, we estimate the information between pop- +ulation spiking activity and the latent variable as the information between the maximum-likelihood +estimator ℎ∗ of the latent variable ℎ and the latent variable itself. This approximation fails at ex- +tremes of network activity levels (low or high). +Specifically, we approximate the log-likelihood of ℎ∗ given ℎtrue near ℎ∗ by log 퐿(ℎ−ℎ∗) ≈ log 퐿푚푎푥− +1 +2 +(ℎ−ℎ∗)2 +휎2 +ℎ∗ +, so we assume that ℎ∗ is normally distributed about ℎtrue with variance 휎2(ℎtrue). The variance +is then derived from the curvature of the log-likelihood at the maximum. The information between +15 of 18 + +two Gaussian variables, here 푃(ℎ∗|ℎ) = 푁(ℎ, 휎2 +ℎ∗) and 푝(ℎ) = 푁(0, 1), is +퐼(ℎ; ̄푠푖,푇 ) ≈ 1 +2 +⟨ +log +푇 +휎2 +ℎtrue +⟩ +ℎtrue +, +(10) +where the average is taken over ℎtrue ∼ 푁(0, 1). +Given a set of 푇 observations of the neurons {푠푖}, the likelihood is +푃({푠푖}푡|ℎ) = +푁,푇 +∏ +푖,푡 +푃(푠푖|ℎ) = +푁,푇 +∏ +푖,푡 +푒−휂푠푖퐽푖ℎ−휖푠푖 +1 + 푒−(휂퐽푖ℎ+휖) . +(11) +Maximizing the log likelihood gives the following condition: +0 = 휕(log 푃) +휕ℎ +||ℎ∗ = 휕 +휕ℎ +( +∑ +푖,푡 +((−휂푠푖퐽푖ℎ − 휖푠푖) − log(1 + 푒−(휂퐽푖ℎ+휖)) +) +||ℎ∗ +(12) += +∑ +푖 +−휂 ̄푠푖퐽푖푇 + +푇 퐽푖휂 +1 + 푒휂퐽푖ℎ∗+휖 , +(13) +where ̄푠푖 = 1 +푇 +∑ +푡 푠푖푡 is the average over observations 푡. The uncertainty in ℎ∗ is 휎ℎ, which was calcu- +lated from the second derivative of the log likelihood: +1 +휎2 +ℎ∗ += −휕2(log 푃) +휕ℎ2 +(14) += − 휕 +휕ℎ +( +∑ +푖 +−휂 ̄푠푖퐽푖푇 + +푇 퐽푖휂 +1 + 푒휂퐽푖ℎ+휖 +) +||ℎ∗ +(15) += +∑ +푖 +푇 (휂퐽푖)2푒휂퐽푖ℎ∗+휖 +(1 + 푒휂퐽푖ℎ∗+휖)2 +(16) += +∑ +푖 +푇 (휂퐽푖)2 +4 cosh2( 휂퐽푖ℎ∗+휖 +2 +) +. +(17) +This expression depends on the observations ̄푠푖 only through the maximum-likelihood estimate ℎ∗. +When ℎ∗ → ℎtrue, then the variance is +1 +휎2 +ℎ∗ += +∑ +푖 +푇 (휂퐽푖)2 +4 cosh2( 휂퐽푖ℎtrue+휖 +2 +) +≡ +푇 +휎2 +ℎtrue +. +(18) +To generate Figure 5, we evaluated Eqn. 10 using Eqn. 18. +Acknowledgments +IN was supported in part by the Simons Foundation Investigator program, the Simons-Emory Con- +sortium on Motor Control, NSF grant BCS/1822677 and NIH grant 2R01NS084844. AS was sup- +ported in part by NIH grant 1RF1MH130413-01 and by startup funds from the University of Min- +nesota Medical School. +References +Aitchison L, Corradi N, Latham PE. 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A-B: Probability density function for avalanche size (A) and duration (B) on +a log-log scale, with the best power law fit (red). C-D: In blue: Maximum likelihood exponent of a +power-law model as a function of the minimum (lower cutoff) size (C) and duration (D). In red: KS +statistics (see Methods) for each fit. “Best fit” is the power law with the minimum KS statistic. E-F: +Surrogate data procedure. To generate each surrogate, samples were drawn from a power law +with size / duration cutoff indicated (E, 푆푚푖푛 = 3; F, 퐷푚푖푛 = 18) and the KS statistic was computed. +Histograms illustrate KS statistic across surrogates (blue), while values derived from data are in +red. Because the red line does not fall within the blue histogram, the hypothesis that the data is +fitted well by a power law fit was rejected in E. At the same time, since the red line falls within the +blue histogram in F, the hypothesis was accepted. + +B +A +-10 +8 +15 +10 +0 +2 +og1 +size +0.07 +0.07 +0.06 +0.06 +KS +KS +0.05 +0.05 +statistic +statistic +0.04 +0.04 +王王 +0.03 +0.03 +0.02 +0.02 +0.5 +1.5 +2 +0.5 +1.5 +2 +log10 +log +E +min +250 +counts (2000 surrogates) +300 +200 +150 +00 +50 +50 +3 +2.5 +1.5 +1.8 +-1.6 +1.4 +og +KS statistic +log1 +KS statisticFigure 2–Figure supplement 2. Illustration of algorithm for determining 휏 and 훼, using example +in Fig. 2, five latent variables. Notation the same as in Fig. 2-Fig. Supplement 3. + +B +2 +6 +8 +10 +0.04 +0.05 +0.04 +0.03 +KS +0.03 +statistic +stat +0.02 +tist +0.02 +0.01 +0.01 +0.5 +1.5 +0.5 +1.5 +log10 +log +E +min +counts (2000 surrogates) +300 +counts (2000 surrogates) +250 +250 +200 +200 +150 +100 +100 +50 +50 +3 +-2.8-2.6-2.4-2.2 +2.8 +-2.6 +-2.4-2.2 +2 +log1 +KSstatistic +log10 +KS statisticFigure 2–Figure supplement 3. Illustration of algorithm for fitting the exponent 훾 and determining +the range, over which power law scaling of average size with duration is observed, using example in +Fig. 2(A-D). A-C: Determining the lower bound, the minimum duration 퐷푚푖푛. A: The relation log 푆 = +푏 + 훾 log 퐷 was fit using linear least-squares, restricted to (overlapping) 1-decade ranges (blue, red: +example decades). B: Confidence intervals for fit parameters (훾, 푏 for fits starting at each value +of 퐷푚푖푛. C: Best value of 퐷min was selected based on how many subsequent start points yielded +consistent slope/intercept values. D-F: Determining the upper bound, maximum duration 퐷푚푎푥. D: +Keeping 퐷푚푖푛 fixed based of value obtained in C, we test values of 퐷푚푎푥 up to the maximum duration +event, and fit over the range [퐷min, 퐷max]. E: Average residual over the fit range [퐷min, 퐷min + 1], +calculated for each fit and plotted against the value of 퐷max used for that fit. The largest value of +퐷max without evidence of bias in the residual was then selected. F: Final fit and range. + +alldurations +fit to dec. 1 +Estimate of value and power-law range +ex.decade 1 +fit to dec. 2 +ex. decade 2 +fit parameters for single-decade fits +fit range starting from each d. +min +8 +1.8 +0.5 +scale) +1.7 +t (b, 95% CI) +10 +6 +average size (log s +% 1.6 +8 +4 +0.5 +9 +intercept +2 +0 +1.3 +number +2 +-2 +1.2 +-1.5 +0 +0 +1 +2 +3 +4 +0 +1 +2 +3 +0 +2 +3 +duration (log scale) +d +(lower bound on fitted decade) +d. +min +min +all durations +fit to (i) +(i) dmax = 3.95 +fit to (i) +all end points +(i) dmax = 2.8 +all +final fit + within 95% CI +final fit range +Part 2: select upper cutoff (d, +)forS~D +SE) +residuals for fits ending at d. +max += 1.72, range: 2.15 +8 +0.01 +max ++/- 1 +8 + scale) + scale) +6 +0.005 +2.8] +average size (log +6o) azis +4 +4 +2 +80. -0.005 +average s +2 +0 +-0.01 +0 +error +-2 +-0.015 +-2 +0 +1 +2 +4 +3 +3.5 +4 +0 +1 +2 +3 +4 +duration (log scale) +d. +duration (log scale) +maxFigure 2–Figure supplement 4. Illustration of algorithm for fitting the exponent 훾 and determining +the range over which power-law scaling of average size with duration is observed, using example +in Fig. 2 E-H. See Fig. 2-Fig. Supplement 3 for caption. In this example, a lower value of 퐷min was +selected. Panel E, which was flat for Fig. 2-Fig. Supplement 3, now shows how extending the range +to high values of 퐷max can generate systematic errors at the low range of the fit, even while having +a high overall goodness of fit metric. +Figure 4–Figure supplement 1. Estimate of how long it takes to observe avalanche criticality at +each combination of 휂 and 휖. We took a parameter combination with a low rate of avalanches but +good apparent scaling (휂 = 4 and 휖 = −14) and assumed that this is a reasonable estimate of the +minimum number of observations (approximately 106 avalanches) required to observe scaling. To +translate to observation length (in hours), we divided the number of avalanches observed in each +full-length simulation by this minimum count and converted to a time using a time bin of 10 ms. +Simulations were for a recorded population of 128 neurons. For this size of population, 휖0 = 5.2. + +Reguired Sim Time (est.) +300h +6 +0 +30 h +-10 +14 +3 +4 +6 +8 +10 +n (gain)all durations +fit to dec. 1 +Estimate of value and power-law range +ex. decade 1 +fit to dec. 2 +ex. decade 2 +Part 1: select lower cutoff (d, +fit parameters for single-decade fits +fit range starting from each d. +min +6 +1.6 +0.5 +intercept (b, 95% ClI) +%S6 +10 +0.5 +5 +number of +1 +1.2 +0 +0 +1 +2 +3 +4 +0 +2 +3 +0 +1 +3 +2 +duration (log scale) +d +(lower bound on fitted decade) +d. +min +min +all durations +fit to (i) +fit to (i) +all end points +selectedd +dmin +max +all +final fit + within 95% CI +final fit range +Part 2: select upper cutoff (d, +)forS~D +SE) +residuals for fits ending at d. += 1.26, range: 2.15 +6 +6 +( +/- 1 +0.1 +50.08 +0.06 +uo +0.02 +ave error +1 +0 +0 +0 +2 +1 +4 +2 +3 +4 +0 +1 +2 +3 +4 +duration (log scale) +d +duration (log scale) +max \ No newline at end of file diff --git a/7tAyT4oBgHgl3EQf2_kh/content/tmp_files/load_file.txt b/7tAyT4oBgHgl3EQf2_kh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e5c7d123d68e2108fe93bfc02a1b2861e25d52e --- /dev/null +++ b/7tAyT4oBgHgl3EQf2_kh/content/tmp_files/load_file.txt @@ -0,0 +1,1126 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf,len=1125 +page_content='Avalanche scaling in large neural populations with distributed coupling to multiple dynamical latent variables Mia Morrell1, Ilya Nemenman2, Audrey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Sederberg3* For correspondence: sede0018@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='edu (AJS) 1Department of Physics, New York University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2Department of Physics, Department of Biology, Initiative in Theory and Modeling of Living Systems, Emory University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3Department of Neuroscience, University of Minnesota Medical School Abstract Observations of power laws in neural activity data have raised the intriguing notion that brains may operate in a critical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' One example of this critical state is “avalanche criticality,” which has been observed in a range of systems, including cultured neurons, zebrafish, and human EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' More recently, power laws have also been observed in neural populations in the mouse under a coarse-graining procedure, and they were explained as a consequence of the neural activity being coupled to multiple latent dynamical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' An intriguing possibility is that avalanche criticality emerges due to a similar mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Here, we determine the conditions under which dynamical latent variables give rise to avalanche criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We find that a single, quasi-static latent variable can generate critical avalanches, but that multiple latent variables lead to critical behavior in a broader parameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We identify two regimes of avalanches, both of which are critical, but differ in the amount of information carried about the latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Our results suggest that avalanche criticality arises in neural systems in which there is an emergent dynamical variable or shared inputs creating an effective latent dynamical variable, and when this variable can be inferred from the population activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Introduction The neural criticality hypothesis – the idea that neural systems operate close to a phase transition, perhaps for optimal information processing – is at the same time ambitious and banal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Measure- ments from biological systems are limited in the range of spatial and temporal scales that can be sampled, not only because of limits of recording techniques but also due to fundamentally non- stationary behavior of most, if not all, biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' These limitations make proving that an observation indicates critical behavior difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At the same time, the idea that brain networks are critical echoes the anthropic principle: tuned another way, a network becomes quiescent or epilep- tic, and in either case seems unlikely to support perception, thought, or flexible behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Further muddying the water, researchers have reported multiple kinds of criticality in neural networks, in- cluding through analysis of avalanches (Beggs and Plenz, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Plenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' O’Byrne and Jerbi, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Girardi-Schappo, 2021) and of coarse-grained activity (Meshulam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019), as well as of correlations (Dahmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' How these flavors of critical behavior relate to each other or to any functional network mechanism is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The phenomenon that we will refer to as “avalanche criticality” appears to be remarkably widespread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1 of 18 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='00759v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='NC] 2 Jan 2023 It was first observed in cultured neurons in a dish (Beggs and Plenz, 2003) and later studied in ze- brafish (Ponce-Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2018), turtles (Shew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2015), rodents (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019), monkeys (Petermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2009), and even humans (Poil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The standard analysis, described thoroughly later, requires extracting power-law exponents from a fit to a distribution of avalanche size and duration and assessing the relationship between exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' There is debate over whether these observations reflect true power laws, but within the resolution achievable from experiments, neural avalanches exhibit power laws with exponent relationships predicted from theory devel- oped in physical systems (Perković et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanche criticality is not the only form of criticality observed in neural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Zipf’s law (fre- quency of the network state being inversely proportional to its rank) appears in systems as diverse as fly motion estimation and salamander retina (Mora and Bialek, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Aitchi- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' More recently, Meshulam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2019) measured various statistics of population activity in a mouse hippocampus, including the eigenvalue spectrum of the covariance matrix and the variance of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' These were found to scale as populations were “coarse-grained” through a procedure in which neural activities were iteratively combined based on similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Neither the Zipf’s law nor the coarse-grained criticality can be explained by simple mechanistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Even though these three forms of criticality are observed through different analyses, it is pos- sible that they may originate from similar mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' While avalanche power laws may result from critical dynamics, they can also appear due to quasi-static latent variables, which can pro- duce power laws, but not the relationships expected between the critical exponents (Priesemann and Shriki, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We have previously shown that a dynamical latent variable (DLV) model, based on the coupling of neural populations to multiple dynamical latent variables, can reproduce scaling under coarse-graining analysis within experimental uncertainty (Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The Zipf’s law has been explained by a similar mechanism (Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Aitchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' However, it is not known under what conditions, if any, the DLV model generates avalanche criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In this paper, we systematically investigate avalanche statistics in the DLV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We show that a system coupled to multiple dynamical latent variables can generate avalanche criticality, and we examine the requirements for the number and timescale of variables for this criticality to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We find that avalanche criticality is observed over a wide range of parameters, some of which may be optimal for information representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Our results suggest that latent dynamical structure in large-scale neural recordings may be responsible for the observation of signatures of criticality across many systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Results Critical exponents values and crackling noise We begin by defining the metrics used to quantify avalanche statistics and briefly summarize ex- perimental observations, which have been reviewed in detail elsewhere (Plenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' O’Byrne and Jerbi, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Girardi-Schappo, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Activity is recorded across a set of neurons and binned in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanches are then defined as contiguous time bins in which at least one neuron in the population is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The duration of an avalanche is the number of contiguous time bins and the size is the summed activity during the avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The distributions of avalanche size and duration are fit to power laws (푃(푆) ∼ 푆−휏 for size 푆, and 푃(퐷) ∼ 퐷−훼 for duration 퐷) using standard methods (Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Power laws can be indicative of criticality, but they can also result from non-critical mechanisms (Touboul and Destexhe, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Priesemann and Shriki, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A more stringent test of criticality is the “crackling” relationship (Perković et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Touboul and Destexhe, 2017), which involves fitting a third power-law relationship, ̄푆(퐷) ∼ 퐷훾fit, and comparing 훾fit to the predicted exponent 훾pred, derived from the size and duration exponents, 휏 and 훼: 훾fit ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='= 훾pred ≡ 훼 − 1 휏 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (1) 2 of 18 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Dynamical Latent Variable model produces avalanche criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A: Model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Latent dynamical variables ℎ휇(푡) are broadly coupled to neurons 푠푖(푡) in the recorded population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B: Raster plot of a sample of activity binned at 3-ms resolution across 128 neurons with five latent variables, each with correlation timescale 휏퐹 = 15 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C: Projection of activity into a simulated field of view for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' D-F: Avalanche analysis in a network (parameters 푁퐹 = 5, 휏퐹 = 104, 휂 = 4 and 휖 = 12), showing size distribution (D), duration distribution (E), and size with duration scaling (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Lower cutoffs used in fitting are shown with vertical lines and their values are indicated in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' There are 푁obs = 42725 avalanches of size 푆 ≥ 푆min in this simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Estimated values of the critical exponents are shown in the titles of the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Previous work demonstrating approximate power laws in size and duration distributions through the mechanism of a slowly changing latent variable did not generate crackling (Touboul and Des- texhe, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Priesemann and Shriki, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Measuring power-laws in empirical data is challenging: it generally requires setting a lower cut- off in the size and duration, and the power-law behavior only has limited range due to the finite size and duration of the recording itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Nonetheless, there is some consensus (Shew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fontenele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019) that even if 휏 and 훼 vary over a wide range (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 to about 3) across recordings, the values of 훾fit and 훾pred stay in a relatively narrow range, from about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanche scaling in the Dynamical Latent Variable (DLV) model We studied a population of neurons that are coupled to dynamical latent variables but not coupled to each other (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We refer to this model as the Dynamical Latent Variable (DLV) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The latent variables determine the inputs to the simulated population of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We are agnostic as to the origin of these inputs: they may be externally driven from other brain areas, or they may arise from recurrent dynamics locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We have previously shown that the DLV model with at least about five latent variables can produce power laws under the coarse-graining analysis (Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In this paper, we examine avalanche criticality in the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Specifically, we model the neurons as binary units (푠푖) that are randomly (퐽푖휇 ∼ 푁(0, 1)) coupled to dynamical variables ℎ휇(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The probability of any pattern {푠푖}, given the current state of the latent 3 of 18 B h2(t) 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='. neurons ") 50 hm(t) S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='. time (s) 100 10° 103 Probability Density Probability Density Average Size 10 102 10 107 10 101 100 100 102 104 106 100 102 104 100 101 102 103 Avalanche Size S Avalanche Duration D Duration Dvariables, is 푃({푠푖}|ℎ휇(푡)) = 1 푍(ℎ휇(푡)) exp ( −휂 푁퐹 ∑ 휇=1 푠푖퐽푖휇ℎ휇(푡) − 휖푠푖 ) , (2) where the parameter 휂 controls the scaling of the variables and 휖 controls the overall activity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We modeled each latent variable as an Ornstein-Uhlenbeck process with the time scale 휏퐹 (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus our model has four parameters: 휂 (input scaling), 휖 (activity threshold), 휏퐹 (dynami- cal timescale), and 푁퐹 (number of neurons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Distributions of avalanche size and avalanche duration within this model followed approximate power laws (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In the example shown (푁퐹 = 5, 휏퐹 = 104, 휂 = 4 and 휖 = 12), we found exponents 휏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 (size) and 훼 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 (duration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Further, the average size of avalanches with fixed duration scaled as 푆 ∼ 퐷훾, with the fitted 훾fit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02, in agreement with the predicted value 훾pred = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus, our model could generate avalanche scaling, at least for some parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In the following sections, we examine how avalanche scaling depends on model parameters (푁퐹 , 휏퐹 , 휂 and 휖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We first focus on two sets of simulations: one set with 푁퐹 = 1 latent variable, which does not generate scaling under coarse-graining (Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021), and one set with 푁퐹 = 5 latent variables, which can generate such scaling for some values of parameters 휏퐹 , 휂, and 휖 (Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanche scaling depends on the number of latent variables We analyzed avalanches from one- and five-variable simulations, each with fixed latent dynamical timescale (휏퐹 = 5 × 103 time steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' see Table 2 for parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In the following sections, time is measured in simulation time steps, see Methods for converting time steps to seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We used es- tablished methods for measuring empirical power laws (Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The minimum cutoffs for size (푆min) and duration (퐷min) are indicated by vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For the population coupled to a single latent variable, the avalanche size distribution was not well fit by a power law (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' With a sufficiently high minimum cut-off (퐷min), the duration distribution was approximately power- law (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We next assessed whether the simulation produced crackling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' If so, the value 훾fit obtained by fitting ̄푆(퐷) ∼ 퐷훾fit would be similar to 훾pred = 훼−1 휏−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In many cases, such as the one-variable example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2C, the full range of avalanche durations were not fit by a single power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' There- fore, we determined the largest range over which a power law was a good fit to the simulated observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In this case, slightly over two decades of apparent scaling were observed starting from avalanches with minimum duration slightly less than 100 time steps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2C), with a best-fit value of 훾푓푖푡 ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='69, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' As we did not find a power-law in the size distribution, calculating 훾pred is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' If we do it anyway, we obtain 훾푝푟푒푑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03 (yellow line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2C), which clearly deviates from the fitted value of 훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus, for the single dynamical latent variable model (휏퐹 = 5000), power-law fits are poor, and there is no crackling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The five-variable model produces a different picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We now find avalanches for which size and duration distributions are much better fit by power-law models starting from very low minimum cutoffs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2D-E, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2-Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Average size scaled with duration, again over more than two decades, with 훾fit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03, which was in close agreement with 훾pred = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Holding other parameters constant, we thus found that scaling relationships and crackling arise in the multi-variable model but not the single-variable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanche scaling depends on the time scale of latent variables Based on simulations in the previous section, we surmised that the five-variable simulation gen- erated scaling more readily due to creating an “effective” latent variable that had slower dynam- ics than any individual latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We reasoned that at any moment in time, the latent vari- able state ℎ휇(푡) is a vector in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Because coupling to the latent variables is random throughout the population, only the length (∼ √ 푁퐹 ) and not the direction of this vector matters, 4 of 18 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Multiple latent variables generate avalanche scaling at shorter timescales than a single latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Parameters used for simulations for this figure are found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A-C: Scaling analysis for one variable models where the dynamic timescale is equal to 5 × 103 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A: Distribution of avalanche sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' MLE value of exponent for best-fit power law is 휏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='98 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 SE), with lower cutoff indicated by the vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B: Distribution of avalanche duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' MLE value of 훼 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='81 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C: Average size plotted against avalanche duration (blue points), with power-law fit (black line) and predicted relationship (yellow line) from MLE values for exponents in A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Gray bar on the horizontal axis indicates range over which a power law with 훾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='72 fits the data (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' D-F: Analysis of avalanches from a simulation of a population coupled to five independent latent variables where the dynamic timescale is equal to 5 × 103 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' G: Exponents 휏 for avalanche size distributions across timescales for one-variable (blue) and five-variable (red) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Each circle is a simulation with independently drawn coupling parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulations had to show scaling over at least two decades to be included in panels (G-J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' H: Exponents 훼 for avalanche duration distributions for simulations in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' I: Fitted values of 훾 for simulations in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' J: Difference between fitted and predicted 훾 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Five-variable simulations produce crackling over a wider range of timescales than single-variable simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Figure 2–Figure supplement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Methods, power law distribution fits, one variable example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Figure 2–Figure supplement 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Methods, power law distribution fits, five variable example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Figure 2–Figure supplement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Methods, gamma fit and range, one variable example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Figure 2–Figure supplement 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Methods, gamma fit and range, five variables example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 5 of 18 5 average size 4 2 2 601) ３ 4 pdf 4 ¥2 ¥6 6 S D min 8 8 avalanche size avalanche duration duration 5 (10g10) average size 4 2 ¥３ 4 pdf 4 2 ¥6 OZIs 6 D min 8 8 0 avalanche size avalanche duration duration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 CDAD D 團 ④ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4 D OD) 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 CD CRDD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 KED CTXD ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4 00 0 BXD ED 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 10000 00 1000 1000 10000 10000 10000 30000 30000 30000 100000 100000 100000 100000 dynamical timescale dynamical timescale dynamical timescale dynamical timescale TEand the timescale of changes in this length would be much slower than 휏퐹 , the timescale of each of the components ℎ휇(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We therefore speculated that increasing the timescale of dynamics of the latent variables should eventually lead to scaling and crackling in the single-variable model as well as the five-variable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To examine the dependence of avalanche scaling on this timescale, we simulated one-variable and five-variable networks at fixed 휂 and 휖 coupled to latent variables with the correlation time of their Ornstein-Uhlenbeck dynamics of 휏퐹 ∈ [103, 105] time steps, spanning from a factor of 10 faster to a factor of 10 slower than the original 휏퐹 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulations were replicated five times at each combination of parameters by drawing new latent variable coupling values (퐽푖휇), as well as new latent variable dynamics and instances of neural firing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For simulations that passed the criteria to be fitted by power laws, we plot the fitted values of 휏 , 훼, 훾fit and 훾fit − 훾pred (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2G-J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Missing points are those for which distributions did not pass the power law fit criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In the single-variable model, best-fit exponents changed abruptly for latent variable timescale around 휏퐹 = 104 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2G, H), while in the five-variable model, exponents tended to increase grad- ually (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2G, H, red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The discontinuity in the single-variable case reflected a change in the lower cutoff values in the power-law fits: size and duration distributions generated with faster latent dynamics could be fit reasonably well to a power law by using a high value of the lower cutoff (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2-Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For time scales greater than ∼ 104, the minimum cutoffs dropped, and the single-variable model generated power-law distributed avalanches and crackling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2J), similar to the five-variable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In summary, in the DLV model, introducing multiple variables gener- ated scaling at faster timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' However, by slowing the timescale of the latent dynamics, the DLV model generated signatures of critical avalanche scaling for both multi- and single-variable simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanche criticality, input scaling, and firing threshold In the previous section, we found that a very slow single DLV model generated scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus, from now on, we simplify the model in order to characterize avalanche statistics across values of input scaling 휂 and firing threshold 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Specifically, we modeled a population of 푁 = 128 neurons coupled to a single quasi-static latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We simulated 103 segments of 104 steps each and drew a new value of the latent variable (ℎ ∼ 푁(0, 1)) for each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Ten replicates of the simulation were generated at each of the combinations of 휂 and 휖 (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Almost independent of 휂 and 휖, we found quality power law fits and crackling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3 shows the average (across 푛 = 10 network realizations) of the exponents extracted from size (휏, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3A) and duration (훼, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3C) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At small firing threshold (휖 = 2), we do not observe scaling because the system is always active, and all avalanches merge into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At large firing threshold 휖 and low input scaling 휂, we do not observe scaling because activity is so sparse that all avalanches are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At intermediate values of the parameters, the simulations generated plausible scaling relationships in size and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The difference between 훾fit and 훾pred was typically less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4D-F) which was consistent with previously reported differences between fit and predicted exponents (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus, there appears to be no need for fine-tuning to generate apparent scaling in this model, at least in the limit of (near) infinite observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Wherever 휂 and 휖 generate avalanches, there are approximate power-law distributions and crackling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To determine where avalanches occur, we derive the avalanche rate across values of the latent variable ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In the quasi-static model, the probability of an avalanche initiation is the probability of a transition from the quiet to an active state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Because all neurons are conditionally independent, this is 푃ava = 푃silence(1 − 푃silence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Then the expected number of avalanches ̂푁ava is obtained by integrating 푃ava over ℎ at each value of 휂 and 휖: ̂푁ava = ∫ 푃ava(휖, 휂, ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 퐽푖, 푁)푝(ℎ)푑ℎ = ∫ ∏ 푖 ( 1 1 + 푒휂퐽푖ℎ+휖 ) ( 1 − ∏ 푖 ( 1 1 + 푒휂퐽푖ℎ+휖 )) 푝(ℎ)푑ℎ, (3) where 푝(ℎ) is the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This probability tracks the observed number of avalanches across simulations, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 6 of 18 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Exponents across network simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Each parameter combination 휂, 휖 was simulated for ten replicates, each time drawing randomly the couplings 퐽푖, the latent variable values, and the neural activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A: Average across replicates for the size exponent 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B: Scatter plot of 훼 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 휏 for each network replicate for parameter combinations indicated in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Linear relationships between 휏 and 훼, corresponding to the minimum and maximum values of 훾fit from panel E, are shown to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C: Same as A, for duration exponent 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' D: Predicted exponent, 훾pred, derived from A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' E: Value of 훾fit from fit to ̄푆퐷 ∼ 퐷훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' F: Difference between 훾pred and 훾fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 7 of 18 Average Exponents: Size .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 口 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 las 口 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 米 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 (duration exponent) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 2 4 6 10 米 米 n (gain) 米 柔 2 米 Average Exponents: Duration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 米 (sel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4 8 24680 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 3 n (gain) (sizeexponent) Crackling Exponents Predicted Difference 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' as Dias .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 10 810 4 aairTo gain an intuition for the conditions under which avalanches occur, we show two slices of the avalanche probability, at fixed 휂 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4B) and at fixed 휖 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Black regions indicate where avalanches are likely to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' If, for a given value of 휖 and 휂, there is no overlap between high avalanche probability regions and the distribution of ℎ, then there will be no avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For large 휖, avalanches occur because neurons with large coupling to the latent variable (휂|퐽푖| >> 1, recall 퐽푖 ∼ 푁(0, 1)) are occasionally activated by a value of the latent variable ℎ that is sufficient to exceed 휖 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus, the scaling parameter 휂 controls the value of ℎ for which avalanches occur most frequently (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' As 휖 decreases, avalanches occur for smaller and smaller ℎ until avalanches primarily occur when ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To calculate the probability of avalanches, we must integrate over all values of ℎ, but we can gain a qualitative understanding of which avalanche regime the system is in by examining the probability of avalanches at ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At ℎ = 0, the avalanche probability (see Methods) is 푃ava(휖, 휂, ℎ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 퐽푖, 푁) = ( 1 1 + 푒휖 )푁 ( 1 − ( 1 1 + 푒휖 )푁) , (4) which is maximized at 휖0 = − log(21∕푁 − 1), independent of 퐽푖 and 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The dependence on 푁 re- flects that a larger threshold is required for larger networks: large networks (푁 → ∞) are unlikely to achieve complete network silence, therefore preventing avalanches from occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Similarly, small networks (푁 ∼ 1) are unlikely to fire consecutively and thus are unlikely to avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We plot 푃ava(휖, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 퐽푖, 푁, ℎ = 0) as a function of 휖 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The peak at 휖0 divides the space into two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For 휖 < 휖0, a power-law is only observed in the large-size avalanches, which are rare (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4E, green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' By contrast, when 휖 > 휖0, minimum size cutoffs are low (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4F, orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Both regions, 휖 < 휖0 and 휖 > 휖0, exhibit crackling noise scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' If observation times are not sufficiently long (estimated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 4-Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1), then scaling will not be observed in the 휖 < 휖0 region, whose scaling relations consist of rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Insufficient observation times may explain experiments and simulations where avalanche scaling was not found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Inferring the latent variable Our analysis of 푃ava(휖, 휂, ℎ) at ℎ = 0 suggested that there are two types of avalanche regimes: one with high activity and high minimum cutoffs in the power law fit (Type 1), and the other with lower activity and size cutoffs (Type 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Further, when 푃ava drops to zero, avalanches disappear because the activity is too high or too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We now examine how information about the value of the latent variables represented in the network activity relates to the activity type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To delineate these types, we calculated numerically 휖∗(휂), the value of 휖 for which the probability of avalanches is maximized, and the contours of 푃ava (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Curves for 휖∗(휂) and 휖0 and 푃ava = 10−3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 5A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We expect that the more cells fire, the more information they would convey, until the firing rate saturates, and inferring the value of the latent variable becomes impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 5B supports the prediction: generally, information is higher in regions with more activity (lower 휖, higher 휂), but only up to a limit: as 휖 → 0, information decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This decrease begins approximately where the probability of avalanches drops to nearly zero (dashed black lines, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 5B-E) because all of the activity merges into a few very large avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In other words, the Type-1 avalanche region coincides with the highest information about the latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The critical brain hypothesis suggests that the brain operates in a critical state, and its func- tional role may be in optimizing information processing (Beggs, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Chialvo, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Under this hypothesis, we would expect the information conveyed by the network to be maximized in the regions we observe avalanche criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' However, we see that critical regions do not always have optimal information transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 5, the region that displays crackling noise is that where avalanches exist (푃ava > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='001), which corresponds to any 휂 value and 휖 ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This avalanche re- gion encompasses both networks with high information transmission and networks with low in- formation transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In summary, observing avalanche criticality in a system does not imply a high-information processing network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' However, the scaling can be seen at smaller cutoffs, 8 of 18 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanches in the DLV model with a single quasistatic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A: Number of avalanches in simulations as a function of the calculated probability of avalanches at fixed 휂 across values of 휖 and latent variable ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Line indicates equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B: Analytically calculated probability of avalanches with 휂 = 2 across values of 휖 and ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The latent variable ℎ is normally distributed with mean 0 and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Where the distribution of ℎ overlaps with regions of high probability (black), avalanches occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C: Analytically calculated probability of avalanches at 휖 = 8 across values of 휂 and ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Increasing 휂 narrows the range of ℎ that generates avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' D: Analytically calculated probability of avalanches at ℎ = 0 for a populations of 128 neurons (black line) and for a varying 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Size distributions corresponding to simulations marked by the green and orange crosses are in E, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' E: Example of size distribution with 휖 < 휖0 (orange marker in D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Size cutoff is close to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' F: Example of size distribution with 휖 > 휖0 (green marker in D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Size cutoff is < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Figure 4–Figure supplement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Estimated simulation time to observe avalanche criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' and hence with shorter recordings, in the high-information state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This parallels the discussion by Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2014), who noticed that the Zipf’s law always emerges in neural populations driven by quasi-stationary latent fields, but it emerges at smaller system sizes when the information about the latent variable is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Discussion Here we studied systems with distributed, random coupling to Dynamical Latent Variables (DLV) and we found that avalanche criticality is nearly always observed, with no fine-tuning required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Avalanche criticality was surprisingly robust to changes in input gain and firing rate threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Loss of avalanche criticality could occur if the latent process was not well-sampled, either because the simulation was not long enough or the dynamics of the latent variables were too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Finally, while information about the latent variables in the network activity was higher where avalanches were generated compared to when they were not, there was a range of information values across the critical avalanche regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Thus, avalanche criticality alone was not a predictor of optimal informa- tion transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Explaining experimental exponents A wide range of critical exponents have been found in ex vivo and in vivo recordings from various systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For instance, the seminal work on avalanche statistics in cultured neuronal networks by Beggs and Plenz (2003) found size and duration exponents of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0 respectively, along with 훾 = 2, when time was discretized with a time bin equal to the average inter-event interval in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' These values are predicted by a theoretical model of a critical branching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' By contrast, a survey of many in vivo and ex vivo recordings found power-law size distributions with exponents ranging from 1 to 3 depending on the system (Fontenele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Separately, Ma 9 of 18 A B c n= 2 E=8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='25 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='25 Avalanche count (Na) 2 2 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 (seIc a 6 1 8 m 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 2 0 14 0 2 5 0 5 5 0 5 7 P (1-P h h D silence silence E F T = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='01 SE) T = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='00 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='01 SE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='15 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 6 6 P°0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='05 0 8 8 10 15 0 E (bias) avalanche size avalanche sizeFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Information in the neural activity about the latent variable is higher in the low-휖 avalanche region, compared to high-휖 avalanche or high-rate avalanche-free activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A: Probability of avalanche per time step across values of 휂 and 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Solid magenta curve follows 휖∗(휂), the value of 휖 maximizing the probability of avalanches at fixed 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Dashed magenta line indicates 휖0, calculated analytically, which matches 휖∗ at 휂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B: Information about latent variable, calculated from maximum likelihood estimate of ℎ using population activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' MLE approximation is invalid in the dark-blue region bounded by gray curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Magenta line marks the maximum values of 푃ava, reproduced from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Dashed black curve indicates 푃ava = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The highest information region falls between 휖∗(휂) and the contour for 푃ava = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C - E: Slices of B, showing 퐼MLE(휖) for 휂 = {2, 5, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Magenta and dashed black lines again indicate 휖∗ and 푃ava = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='001, respectively, as in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Black dashed line marks the approximate boundary between the high-activity/no avalanche and the high-cutoff avalanche, and magenta line marks boundary between high-cutoff and low-cutoff avalanche regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 10 of 18 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='25 B 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 2 2 cells (MLE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='001 Probability of Avalanche 三 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 (bias) 6 (bias) 9 128 8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 for N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='08 1 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='04 10 D =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='001 ava 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='05 12 12 0 e(n) 14 0 14 0 0 2 4 6 8 10 2 4 6 8 10 n (input scaling) n (input scaling) c D n=2 n=5 E n=9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 128 128 128 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 = z z h*), h*), 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 0 0 0 0 5 10 0 5 10 0 5 10 E( (bias) E (bias) E (bias)et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2019) reported recordings in freely moving rats with size exponents ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In all of the these recordings, when the crackling relationship held, the reported value of 훾 was near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 (Fontenele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Our DLV model, across the parameters we tested that produced exponents consistent with the scaling relationship, generated 휏 values that ranged from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='9 to about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Across those simulations, we found values 훾 within a narrow band from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='3 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2I, J and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' While the exponent values our model produces are inconsistent with a critical branching process (훾 = 2), they match very closely the ranges of exponents reported by Fontenele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' One possible resolution to the discrepancy in exponents derives from how the system is sub- sampled in space or coarse-grained in time, both of which systematically change exponents 휏 and 훼 (Beggs and Plenz, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Shew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Were we to change the time bin, our modeling results would exhibit different exponent values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' However, neither manipulations of the latent variable timescale (휏퐹 or 푁퐹 ), nor of the overall activity level (휂, 휖) produced exponents close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0, despite maintaining the crackling relationship across many different choices of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A second possibility is that different experiments study similar, but distinct biological phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In other words, the underlying biology can differ between networks that were cultured in vitro and those that were not, whether they are in vivo or ex vivo (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', brain slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This could happen if cultured networks develop connections between neurons such that they truly do produce dy- namics that approximate a critical branching process, while brain networks that develop in a living brain have different structure and resulting dynamics and can be better understood as a system coupled to latent dynamical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This is especially true in sensory systems, where coupling to (latent) external stimuli in a way that the neural activity can be used to infer the stimuli is the reason for the networks’ existence (Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Relationship to past modeling work Our model is not the first to produce approximate power-law size and duration distributions for avalanches from a latent variable process (Touboul and Destexhe, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Priesemann and Shriki, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In particular, Priesemann and Shriki (2018) derived the conditions under which an inhomo- geneous Poisson process could produce such approximate scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The basic idea is to generate a weighted sum of exponentially distributed event sizes, each of which are generated from a homo- geneous Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' How each process is weighted in this sum determines the approximate power-law exponent, allowing one to tune the system to obtain the critical values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In- terestingly, this model did not generate non-trivial scaling of size with duration (푆 ∼ 퐷훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Instead, they found 훾 = 1, not the predicted 훾 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Our results differ significantly, in that 훾 was typically between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='3 and it was nearly always close to the prediction from 훼 and 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We speculate that this is due to nonlinearity in the mapping from latent variable to spiking activity, as doubling the latent field ℎ does not double the population activity, but doubling the rate of a homogeneous Poisson process does double the expected spike count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' As biological networks are likely to have such nonlinearities in their responses to common inputs, this scenario may be more applicable to certain kinds of recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Summary Latent variables – whether they are emergent from network dynamics (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Seder- berg and Nemenman, 2020) or derived from shared inputs – are ubiquitous in large-scale neural population recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This fact is reflected most directly in the relatively low-dimensional struc- ture in large-scale population recordings (Stringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Pandarinath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Nieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We previously used a model based on this observation to examine signatures of neural crit- icality under a coarse-graining analysis and found that coarse-grained criticality is generated by systems driven by many latent variables (Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Here we showed that the same model also generates avalanche criticality, and that when information about the latent variables can be inferred from the network, avalanche criticality is also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Crucially, finding signa- 11 of 18 tures of avalanche criticality required long observation times, such that the latent variable was well-sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Previous studies showed that Zipf’s law appears generically in systems coupled to a latent variable that changes slowly relative to the sampling time, and that the Zipf’s behavior is eas- ier to observe in the higher information regime (Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Aitchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' How- ever, this also suggests that observation of either scaling at modest data set sizes indeed points to some fine-tuning — namely to the increase of the information in the individual neurons (and, since neurons in these models are conditionally independent, also in the entire network) about the value of the latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In other words, one would expect a sensory part of the brain, if adapted to the statistics of the external stimuli, to exhibit all of these critical signatures at relatively modest data set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In monocular deprivation experiments, when the activity in the visual cor- tex is transiently not adapted to its inputs, scaling disappears, at least for recordings of a typical duration, and is restored as the system adapts to the new stimulus (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We speculate that the observed recovery of criticality by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2019) could be driven by neurons adapting to the reduced stimuli state, for instance, by adjusting 휂 (input scaling) and 휖 (firing rate threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Taken together, these results suggest that critical behavior in neural systems – whether based on the Zipf’s law, avalanches, or coarse-graining analysis – is expected whenever neural recordings ex- hibit some latent structure in population dynamics and this latent structure can be inferred from observations of the population activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Methods and Materials Simulation of Dynamic Latent Variable (DLV) model We study a model from Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2021), incorporating only latent variables (no place variables), and assuming that every cell is coupled to every latent variable with some randomly drawn coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The probability of observing a certain population state {푠푖} given latent variables {ℎ휇(푡)} at time 푡 is 푃({푠푖}|{ℎ휇}) = 1 푍({ℎ휇})푒퐻({푠푖},{ℎ휇}), (5) where 푍 is the normalization, and 퐻 is the “energy”: 퐻 = 푁,푁f ∑ 푖,푚=1 휂ℎ휇(푡)퐽푖휇푠푖 + 휖푠푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (6) The latent variables {ℎ휇(푡)} are Ornstein-Uhlenbeck processes with zero mean, unit variance, and time constant 휏푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Couplings 퐽푖휇 are drawn from the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The parameters 휂, 휖, and 휏푚 are constants, and we simulate 푁 = 1024 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For the infinite time constant simulation, we reset ℎ푛 ∼ \ue23a (0, 1) (for each of 푛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='.푁푛) and simulate for 10000 time steps, then repeat for 1000 draws of ℎ푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Time step units Most results were presented using arbitrary time units: all times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 휏퐹 and avalanche duration 퐷) are measured in units of an unspecified time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Specifying a time bin converts the probability Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulation parameters for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Parameter Description Value 휖 bias towards silence 휖 = 12 휂 variance multiplier 휂 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0 푁F number of latent fields 푁F = 5 휏퐹 latent field time constant 휏 = 104 푁 number of cells 푁 = 1024 12 of 18 of firing into actual firing rates, in spikes per second, and this choice determines which part of the 휂-휖 phase space is most relevant to a given experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The time step is the temporal resolution at which activity is discretized, which varies from sev- eral to hundreds of milliseconds across different experimental studies (Beggs and Plenz, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fontenele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In physical units and assuming a bin size of 3 ms to 10 ms, our choice of 휂 and 휖 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2 would yield physiologically realistic firing rate ranges (Hengen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2016), with high-firing neurons reaching averages rates of 20 − 50 spikes/second and median firing- rate neurons around 1 − 2 spikes/second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The timescales of latent variables examined range from about 3 seconds to 3000 seconds, assuming 3-ms bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulations were carried out for the same number of time steps (2 × 106), which would be approximately 1 to 2 “hours,” which is a reasonable duration for in vivo neural recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Note that at large values of 휏퐹 , the latent variable space is not well sampled during this time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Analysis of avalanche statistics Setting the threshold for observing avalanches In our model, we count avalanches as periods of continuous activity (in any subset of neurons) that is book-ended by time bins with no activity in the entire simulated neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For real neural populations of modest size, this method fails because there are no periods of quiescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The typical solution is to set a threshold, and to only count avalanches when the population activity exceeds that threshold, with the hope that results are relatively robust to that choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In our model, this operation is equivalent to changing 휖, which shifts the probability of firing up or down by a constant amount across all cells independent of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3 show that 훼 and 휏 decrease as the threshold for detection is increased (equivalent to large |휖|), but that the scaling relationship is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The model predicts that 훾pred − 훾fit would initially increase slightly with the detection threshold before decreasing back to near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Following the algorithm laid out in Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2009), we fit power laws to the size and dura- tion distributions from simulations generating avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We use least-squares fitting to estimate 훾fit, the scaling exponent for size with duration, assessing the consistency of the fit across decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Reading power laws from data We want, from each simulation, a quantification of the quality of scaling (how many decades, min- imally) and an estimate of the scaling exponents (휏 for the size distribution, 훼 for the duration distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Following the steps outlined by Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2009), we use the maximum-likelihood estimator to determine the scaling exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This is the solution to the transcendental equation 휁′(̂훼, 푥min) 휁′(̂훼, 푥min) = −1 푛 푛 ∑ 푖=1 ln 푥푖 (7) where 휁(훼, 푥min) is the Hurwitz zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For values of 푥min < 6, a numerical look-up table based on the built-in Hurwitz zeta function in the symbolic math toolbox was used (MATLAB2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulation parameters for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Parameter Description Value 휖 bias towards silence 휖 = 8 (for 푁퐹 = 1) or 휖 = 12 (for 푁퐹 = 5) 휂 variance multiplier 휂 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0 푁F number of latent fields 푁F = 1 or 5 휏퐹 latent field time constant 휏 ∈ \ue241 [log 103, log 105] 푁 number of cells 푁 = 1024 13 of 18 푥min > 6 we use an approximation (Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2009)), ̂훼 = 1 + 푛 ( ∑ 푖 ln 푥푖 푥min − 1 2 )−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (8) To determine 푥min, we computed the maximum absolute difference between the empirical cu- mulative density (푆(푥)) function and model’s cumulative density function 푃(푥) (the Kolmogorov- Smirnov (KS) statistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 퐷 = max푥≥푥min|푆(푥)−푃(푥)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The KS statistic was computed between for power- law models with scaling parameter ̂훼 and cutoffs 푥min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The value of 푥min that minimizes the KS statis- tic was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Occasionally the KS statistic had two local minima (as in Figure 2-Supplemental Figure 1), indicating two different power-laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In these cases, the minimum size and duration cut- offs were the smallest values that were within 10% of the absolute minimum of the KS statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Note that the statistic is computed for each model only on the power-law portion of the CDF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 푥푖 ≥ 푥min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We do not attempt to determine an upper cut-off value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To assess the quality of the power-law fit, Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2009) compared the empirical observa- tions to surrogate data generated from a semi-parametric power-law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The semi-parametric model sets the value of the CDF equal to the empirical CDF values up to 푥 = 푥min and then according to the power-law model for 푥 > 푥min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' If the KS statistic for the real data (relative to its fitted model) is within the distribution of the KS statistics for surrogate datasets relative to their respective fitted models, the power-law model was considered a reasonable fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Strict application of this methodology could give misleading results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Much of this is due to the loss of statistical power when the minimum cutoff is so high that the number of observations drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For instance, in the simulations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2, the one-variable duration distribution passed the Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2009) criterion, with a minimum KS statistic of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03 when the duration cutoff was 18 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' However, for the five-variable simulation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2, a power-law would be narrowly rejected for both size and duration, despite having much smaller KS statistics: for 휏, the KS statistic was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0087 (simulation range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0008 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0082;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' number of avalanches observed: 58, 787) and for 훼 it was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0084 (simulation range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0011 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='0075).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Below we discuss this problem in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Determining range over which avalanche size scales with duration For fitting 훾, our aim was to find the longest sampled range over which we have apparent power- law scaling of size with duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Because our sampled duration values have linear spacing, error estimates are skewed if a naive goodness of fit criterion is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We devised the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' First, the simulation must have at least one avalanche of size 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We fit 푆 = 푐퐷훾 over one decade at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We chose as the lower duration cutoff the value of minimum duration for which the largest number of subsequent (longer-duration) fits produced consistent fit parameters (Figure 2-Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3 and 4, top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Next, with the minimum duration set, we gradually increased the maximum duration cut-off, and we determined whether there was a significant bias in the residual over the first decade of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We selected the highest duration cutoff for which there was no bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Finally, over this range, we re-fit the power law relationship and extracted confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Our analysis focused on finding the apparent power-law relationship that held over the largest log-scale range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A common feature across simulation parameters (휏퐹 , 푁퐹 ) was the existence of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulation parameters for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Parameter Description Value 휖 bias towards silence 휖 ∈ {2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='14} 휂 variance multiplier 휂 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='10} 푁F number of latent fields 푁F = 1 휏퐹 latent field time constant quasistatic 푁 number of cells 푁 = 128 14 of 18 two distinct power-law regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This is apparent in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2I, which shows that when 푁퐹 = 1 at small 휏퐹 , the best-fit 훾 (that showing the largest range with power-law-consistent scaling) is much larger (> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='7), and then above 휏퐹 ∼ 3000, the best-fit 훾 drops to around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Statistical power of power-law tests In several cases, we found examples of power-law fits that passed the rejection criteria commonly used to determine avalanche scaling relationships because of limited number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A key example is that of the single latent variable simulation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2B, where we could not reject a power law for the duration distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Conversely, strict application of the surrogate cri- teria would reject a power law for distributions that were quantitatively much closer to a power-law (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', lower KS statistic), but for which we had many more observations and thus a much stronger surrogate test (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This points to the difficulty of applying a single criterion to determining a power-law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In this work, we adhere to the criteria set forth in Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (2009), with a mod- ification to control for the unreasonably high statistical power of simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Specifically, the number of avalanches used for fitting and for surrogate analysis was capped at 500, 000, drawn randomly from the entire pool of avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Additionally, we found examples in which a short simulation was rejected, but increasing the simulation time by a factor of five yielded excellent power-law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We speculate that this arises due to insufficient sampling of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' These observations raise an important biological point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulations provide the luxury of assuming the network is unchanging for as long as the simulator cares to keep drawing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In a biological network, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Over the course of hours, the effective latent degrees of freedom could change drastically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', due to circadian effects (Aton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2009), changes in behavioral state (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2014), plasticity (Hooks and Chen, 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' ), and the network itself (synaptic scaling, firing thresholds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=') could be plastic (Hengen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' All of these factors can be modeled in our framework by determining appropriate cutoffs (in duration of recording, in time step sizes, for activity distributions) based on specific experimental timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Calculation of avalanche regimes In the quasistatic model, we derive the dependence of the avalanche rate on 휂, 휖 and number of neurons 푁, finding that there are two distinct regimes in which avalanches occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Each time bin is independent, conditioned on the value of ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For an avalanche to occur, the probability of silence in the population (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=', all 푠푖 = 0) must not be too close to 0 (or there are no breaks in activity) or too close to 1 (or there is no activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At fixed ℎ, the probability of silence is 푃silence(휖, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 퐽푖, 푁, ℎ) = ∏ 푖 1 1 + exp(−휂퐽푖ℎ + 휖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (9) An avalanche occurs when a silent time bin is followed by an active bin, which has probability 푃ava(휖, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 퐽푖, 푁, ℎ) = 푃silence(1 − 푃silence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Information calculation Maximum-likelihood decoding For large populations coupled to a single latent variable, we estimate the information between pop- ulation spiking activity and the latent variable as the information between the maximum-likelihood estimator ℎ∗ of the latent variable ℎ and the latent variable itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' This approximation fails at ex- tremes of network activity levels (low or high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Specifically, we approximate the log-likelihood of ℎ∗ given ℎtrue near ℎ∗ by log 퐿(ℎ−ℎ∗) ≈ log 퐿푚푎푥− 1 2 (ℎ−ℎ∗)2 휎2 ℎ∗ , so we assume that ℎ∗ is normally distributed about ℎtrue with variance 휎2(ℎtrue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The variance is then derived from the curvature of the log-likelihood at the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The information between 15 of 18 two Gaussian variables, here 푃(ℎ∗|ℎ) = 푁(ℎ, 휎2 ℎ∗) and 푝(ℎ) = 푁(0, 1), is 퐼(ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' ̄푠푖,푇 ) ≈ 1 2 ⟨ log 푇 휎2 ℎtrue ⟩ ℎtrue , (10) where the average is taken over ℎtrue ∼ 푁(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Given a set of 푇 observations of the neurons {푠푖}, the likelihood is 푃({푠푖}푡|ℎ) = 푁,푇 ∏ 푖,푡 푃(푠푖|ℎ) = 푁,푇 ∏ 푖,푡 푒−휂푠푖퐽푖ℎ−휖푠푖 1 + 푒−(휂퐽푖ℎ+휖) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (11) Maximizing the log likelihood gives the following condition: 0 = 휕(log 푃) 휕ℎ ||ℎ∗ = 휕 휕ℎ ( ∑ 푖,푡 ((−휂푠푖퐽푖ℎ − 휖푠푖) − log(1 + 푒−(휂퐽푖ℎ+휖)) ) ||ℎ∗ (12) = ∑ 푖 −휂 ̄푠푖퐽푖푇 + 푇 퐽푖휂 1 + 푒휂퐽푖ℎ∗+휖 , (13) where ̄푠푖 = 1 푇 ∑ 푡 푠푖푡 is the average over observations 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The uncertainty in ℎ∗ is 휎ℎ, which was calcu- lated from the second derivative of the log likelihood: 1 휎2 ℎ∗ = −휕2(log 푃) 휕ℎ2 (14) = − 휕 휕ℎ ( ∑ 푖 −휂 ̄푠푖퐽푖푇 + 푇 퐽푖휂 1 + 푒휂퐽푖ℎ+휖 ) ||ℎ∗ (15) = ∑ 푖 푇 (휂퐽푖)2푒휂퐽푖ℎ∗+휖 (1 + 푒휂퐽푖ℎ∗+휖)2 (16) = ∑ 푖 푇 (휂퐽푖)2 4 cosh2( 휂퐽푖ℎ∗+휖 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (17) This expression depends on the observations ̄푠푖 only through the maximum-likelihood estimate ℎ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' When ℎ∗ → ℎtrue, then the variance is 1 휎2 ℎ∗ = ∑ 푖 푇 (휂퐽푖)2 4 cosh2( 휂퐽푖ℎtrue+휖 2 ) ≡ 푇 휎2 ℎtrue .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' (18) To generate Figure 5, we evaluated Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 10 using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Acknowledgments IN was supported in part by the Simons Foundation Investigator program, the Simons-Emory Con- sortium on Motor Control, NSF grant 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+page_content=' Shew WL, Clawson WP, Pobst J, Karimipanah Y, Wright NC, Wessel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Adaptation to sensory input tunes vi- sual cortex to criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Nature Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2015 Aug;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 11(8):659–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1038/nphys3370, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1038/nphys3370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Stringer C, Pachitariu M, Steinmetz N, Reddy CB, Carandini M, Harris KD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Spontaneous behaviors drive multidi- mensional, brainwide activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 364(6437):eaav7893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='org/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1126/ science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='aav7893, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='aav7893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Touboul J, Destexhe A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Power-law statistics and universal scaling in the absence of criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Phys Rev E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2017 Jan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 95:012413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='012413, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='012413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 18 of 18 Figure 2–Figure supplement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Illustration of algorithm for determining 휏 and 훼, using one vari- able example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A-B: Probability density function for avalanche size (A) and duration (B) on a log-log scale, with the best power law fit (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C-D: In blue: Maximum likelihood exponent of a power-law model as a function of the minimum (lower cutoff) size (C) and duration (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In red: KS statistics (see Methods) for each fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' “Best fit” is the power law with the minimum KS statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' E-F: Surrogate data procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To generate each surrogate, samples were drawn from a power law with size / duration cutoff indicated (E, 푆푚푖푛 = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' F, 퐷푚푖푛 = 18) and the KS statistic was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Histograms illustrate KS statistic across surrogates (blue), while values derived from data are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Because the red line does not fall within the blue histogram, the hypothesis that the data is fitted well by a power law fit was rejected in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' At the same time, since the red line falls within the blue histogram in F, the hypothesis was accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B A 10 8 15 10 0 2 og1 size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='06 KS KS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='05 statistic statistic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='04 王王 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 2 log10 log E min 250 counts (2000 surrogates) 300 200 150 00 50 50 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4 og KS statistic log1 KS statisticFigure 2–Figure supplement 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Illustration of algorithm for determining 휏 and 훼, using example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2, five latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Notation the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Supplement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B 2 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03 KS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='03 statistic stat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 tist 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 log10 log E min counts (2000 surrogates) 300 counts (2000 surrogates) 250 250 200 200 150 100 100 50 50 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 2 log1 KSstatistic log10 KS statisticFigure 2–Figure supplement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Illustration of algorithm for fitting the exponent 훾 and determining the range, over which power law scaling of average size with duration is observed, using example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2(A-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A-C: Determining the lower bound, the minimum duration 퐷푚푖푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' A: The relation log 푆 = 푏 + 훾 log 퐷 was fit using linear least-squares, restricted to (overlapping) 1-decade ranges (blue, red: example decades).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' B: Confidence intervals for fit parameters (훾, 푏 for fits starting at each value of 퐷푚푖푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' C: Best value of 퐷min was selected based on how many subsequent start points yielded consistent slope/intercept values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' D-F: Determining the upper bound, maximum duration 퐷푚푎푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' D: Keeping 퐷푚푖푛 fixed based of value obtained in C, we test values of 퐷푚푎푥 up to the maximum duration event, and fit over the range [퐷min, 퐷max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' E: Average residual over the fit range [퐷min, 퐷min + 1], calculated for each fit and plotted against the value of 퐷max used for that fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' The largest value of 퐷max without evidence of bias in the residual was then selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' F: Final fit and range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' alldurations fit to dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1 Estimate of value and power-law range ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='decade 1 fit to dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2 ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' decade 2 fit parameters for single-decade fits fit range starting from each d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' min 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 scale) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='7 t (b, 95% CI) 10 6 average size (log s % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 8 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 9 intercept 2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='3 number 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 0 0 1 2 3 4 0 1 2 3 0 2 3 duration (log scale) d (lower bound on fitted decade) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' min min all durations fit to (i) (i) dmax = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='95 fit to (i) all end points (i) dmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8 all final fit within 95% CI final fit range Part 2: select upper cutoff (d, )forS~D SE) residuals for fits ending at d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='72, range: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='15 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='01 max +/- 1 8 scale) scale) 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='005 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='8] average size (log 6o) azis 4 4 2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='005 average s 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='01 0 error 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='015 2 0 1 2 4 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 4 0 1 2 3 4 duration (log scale) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' duration (log scale) maxFigure 2–Figure supplement 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Illustration of algorithm for fitting the exponent 훾 and determining the range over which power-law scaling of average size with duration is observed, using example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2 E-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Supplement 3 for caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' In this example, a lower value of 퐷min was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Panel E, which was flat for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Supplement 3, now shows how extending the range to high values of 퐷max can generate systematic errors at the low range of the fit, even while having a high overall goodness of fit metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Figure 4–Figure supplement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Estimate of how long it takes to observe avalanche criticality at each combination of 휂 and 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' We took a parameter combination with a low rate of avalanches but good apparent scaling (휂 = 4 and 휖 = −14) and assumed that this is a reasonable estimate of the minimum number of observations (approximately 106 avalanches) required to observe scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' To translate to observation length (in hours), we divided the number of avalanches observed in each full-length simulation by this minimum count and converted to a time using a time bin of 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Simulations were for a recorded population of 128 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' For this size of population, 휖0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' Reguired Sim Time (est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=') 300h 6 0 30 h 10 14 3 4 6 8 10 n (gain)all durations fit to dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 1 Estimate of value and power-law range ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' decade 1 fit to dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' 2 ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' decade 2 Part 1: select lower cutoff (d, fit parameters for single-decade fits fit range starting from each d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' min 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 intercept (b, 95% ClI) %S6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='5 5 number of 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='2 0 0 1 2 3 4 0 2 3 0 1 3 2 duration (log scale) d (lower bound on fitted decade) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' min min all durations fit to (i) fit to (i) all end points selectedd dmin max all final fit within 95% CI final fit range Part 2: select upper cutoff (d, )forS~D SE) residuals for fits ending at d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='26, range: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='15 6 6 ( +/- 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='06 uo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} +page_content='02 ave error 1 0 0 0 2 1 4 2 3 4 0 1 2 3 4 duration (log scale) d duration (log scale) max' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf'} diff --git a/BtAzT4oBgHgl3EQfGPuX/content/tmp_files/2301.01025v1.pdf.txt b/BtAzT4oBgHgl3EQfGPuX/content/tmp_files/2301.01025v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fb6a19c4b7327b2ebe3ca5d455aabe32d801634 --- /dev/null +++ b/BtAzT4oBgHgl3EQfGPuX/content/tmp_files/2301.01025v1.pdf.txt @@ -0,0 +1,2287 @@ +Draft version January 4, 2023 +Typeset using LATEX default style in AASTeX63 +Gaussian Process Modeling Blazar Multiwavelength Variability: Indirectly Resolving Jet Structure +Haiyun Zhang (张海云),1 Dahai Yan (闫大海),1 and Li Zhang (张力)1 +1Department of Astronomy, Key Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University, Kunming 650091, China +ABSTRACT +Blazar jet structure can be indirectly resolved by analyzing the multiwavelength variability. In this +work, we analyze the long-term variability of blazars in radio, optical and X-ray energies with the +Gaussian process (GP) method. The multiwavelength variability can be successfully characterized by +the damped-random walk (DRW) model. The nonthermal optical characteristic timescales of 38 blazars +are statistically consistent with the γ-ray characteristic timescales of 22 blazars. For three individuals +(3C 273, PKS 1510-089, and BL Lac), the nonthermal optical, X-ray, and γ-ray characteristic timescales +are also consistent within the measured 95% errors, but the radio timescale of 3C 273 is too large to be +constrained by the decade-long light curve. The synchrotron and inverse-Compton emissions have the +same power spectral density, suggesting that the long-term jet variability is irrelevant to the emission +mechanism. In the plot of the rest-frame timescale versus black hole mass, the optical-γ-ray timescales +of the jet variability occupy almost the same space with the timescales of accretion disk emission from +normal quasars, which may imply that the long-term variabilities of the jet and accretion disk are +driven by the same physical process. It is suggested that the nonthermal optical-X-ray and γ-ray +emissions are produced in the same region, while the radio core which can be resolved by very-long- +baseline interferometry locates at a far more distant region from the black hole. Our study suggests +a new methodology for comparing thermal and nonthermal emissions, which is achieved by using the +standard GP method. +Keywords: Blazars (164), Jets (870), Light curves (918), Time series analysis (1916) +1. INTRODUCTION +Flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs) are classed into a special class of active +galactic nuclei (AGNs) called blazars, whose jets nearly point to the Earth. Blazars are highly variable over the entire +electromagnetic bands. One popular scenario is that the accretion onto a supermassive black hole is the central engine, +driving relativistic jet. But the detailed process is still unclear. Thanks to the high variability of blazars, one can +investigate the physical process close to the central engine (e.g., Rieger 2019), such as the location of the emitting +region and the jet-disk connection (e.g., Ackermann et al. 2016; Meyer et al. 2019; Zhang et al. 2022). +Using advanced interferometric instruments, blazar radio jet can be directly resovled on ∼parsec-scale (see Hovatta +& Lindfors 2019, for a recent review). This provides a calibrator for multi-band variability analysis. There have been +lots of works attempting to investigate the underlying physical process of blazar jet with multi-band variability (e.g., +Chatterjee et al. 2012; Nakagawa & Mori 2013; Xiong et al. 2017; Goyal et al. 2018, 2022). Max-Moerbeck et al. (2014) +investigated the time-domain relationship between radio and γ-ray emission of blazars, and found the correlations +only exist in a minority of the sources over a 4 yr period. They found radio variations lagging the γ-ray variations, +suggesting that the γ-ray emission originates upstream of the radio emission. This result is further verified by Liodakis +et al. (2018) who concluded that the radio variation is usually substantially delayed to the other wavelengths for +blazars. Bhatta (2021) analyzed the correlation between optical (V -band) and γ-ray variabilities for blazars and found +that the optical variability is highly correlated with the γ-ray variability except for 3C 273, however, no significant +Corresponding author: Dahai Yan +yandahai@ynu.edu.cn +Corresponding author: Li Zhang +lizhang@ynu.edu.cn +arXiv:2301.01025v1 [astro-ph.HE] 3 Jan 2023 + +2 +lagging is found. The multi-band variability analysis can be considered as an indirect approach for resolving blazar +jet. +The GP method becomes popular in modern time-domain astronomy (e.g., Ryan et al. 2019; Burke et al. 2021; Yang +et al. 2021; Griffiths et al. 2021; Covino et al. 2022; Rueda et al. 2022; Stone et al. 2022; Zhang et al. 2022). The GP +method enables us to effectively extract information from astronomical variability. For example, Zhang et al. (2022) +used a GP method to characterize the γ-ray variability of AGNs with stochastic process. It is found that the DRW +model can successfully fit the γ-ray variability, which is similar with the optical variability of AGN accretion disk +(Kelly et al. 2009; Li & Wang 2018; Burke et al. 2021). Moreover, Zhang et al. (2022) suggested a connection between +the jet and the accretion disk by comparing the rest-frame γ-ray timescales of blazars with the optical accretion disk +timescales of quasars. +In this work, we analyze the multi-band variability of blazars with the GP method, which is independent of the +temporal correlation analysis. We hope to extract additional information from the variability. Using the data from +Fermi-Large Area Telescope (Fermi-LAT), we carried out systematic research of γ-ray variability of AGNs recently +(Zhang et al. 2022). So far, the Small and Moderate Aperture Research Telescope System (SMARTS) monitoring +program1 (Bonning et al. 2012) and the Steward Observatory (SO) spectropolarimetric monitoring project2 (Smith +et al. 2009) can provide almost ten years’ (from 2008 to 2018) optical data of Fermi blazars. RXTE AGN Timing +& Spectral Database3 (Rivers et al. 2013) provides long-term X-ray data, and the Owens Valley Radio Observatory +(OVRO) 40 m program (Richards et al. 2011) provides radio light curves (LCs) from 2008 to 20204. Using these public +data, we analyze the radio, optical and X-ray variability of three individual blazars, as well as optical variability for a +sample including 38 Fermi blazars. The format of this paper is as follows. In Section 2, we describe the data as well +as the GP method. The modeling results of the three individual sources and 38 blazars are shown in Section 3. We +give discussions and physical interpretations of the results in Section 4. In Section 5, we conclude the paper with a +brief summary. +2. DATA AND GAUSSIAN PROCESS METHOD +2.1. Data and Sources +We use photometric data of blazars from the SMARTS and SO monitoring projects. The SMARTS program gives +photometric data at five wavelength bands (B, V, R, J, K), which were taken from the 1.3 m telescope at the Cerro +Tololo Inter-American Observatory. SO is a long-term optical program to support the Fermi Telescope, utilizing both +the 2.3 m Bok Telescope on Kitt Peak and the 1.54 m Kuiper Telescope on Mt.Bigelow in Arizona. The campaign +of the SO program spanned almost a decade from 2008 November to 2018 July. The X-ray data can be gained from +RXTE observation which provided us with 16 yr (1996-2012) data in 2-10 keV. OVRO 40 m program gives radio data +of blazars from 2008 to 2020, which is a large-scale, fast-cadence 15 GHz radio monitoring program. We select sources +having long-term continuous observations and a good sampling. For the source with a large gap in the LC, we only +use the data covering a longer period before or after the gap for analysis. Finally, We have 38 blazars in the optical +band, including 23 FSRQs and 15 BL Lacs. Three blazars (3C 273, BL Lac, and PKS 1510-089) have long-term RXTE +X-ray data. They are also in the sample of selected optical sources. Unfortunately, among the three sources, only 3C +273 has the OVRO LC. Table 1 gives the general information of these targets. +2.2. Gaussian Process Method +GPs are a class of statistical models, which are widely applied for modeling stochastic processes. For the one who is +interested in the stochastic behavior of astronomical variability, GP provides a flexible method to model the LC with +stochastic processes. The application of GPs for astronomical time-series is discussed in a recent review (Aigrain & +Foreman-Mackey 2022). Considering a data set of yn at coordinates xn, the GP model consists of a mean function +µθ(x) parameterized by θ and a kernel function (covariance function) kα(xn, xm) parameterized by parameters α +(Foreman-Mackey et al. 2017). For time-series data, the GP is one-dimensional, and the coordinates are time, xn=tn. +After obtaining the likelihood function with the above information, one can use Bayesian inference to estimate the +posterior distribution over the parameter space. +1 http://www.astro.yale.edu/smarts/glast/home.php +2 http://james.as.arizona.edu/∼psmith/Fermi/datause.html +3 https://cass.ucsd.edu/∼rxteagn/ +4 http://astro.caltech.edu/ovroblazars/ + +3 +Table 1. Information of 38 blazars. +Object +z +Type +logMBH/M⊙ +Ref. +(1) +(2) +(3) +(4) +(5) +1ES 1959+650 +0.048 +BLL +8.2 ± 0.17 +1 +1ES 2344+514 +0.044 +BLL +8.7 ± 0.18 +2 +3C 66A +0.37 +BLL +8.570.03 +0.6 +3,4 +3C 454.3 +0.859 +FSRQ +9.1 ± 0.5 +6 +PKS 0235+164 +0.94 +BLL +9.0 +7 +4C +38.41 +1.81396 +FSRQ +9.5 ± 0.5 +7 +CTA 102 +1.032 +FSRQ +8.7 +7 +Mkn 421 +0.03002 +BLL +8.3 ± 0.2 +6 +Mkn 501 +0.03298 +BLL +9.2 ± 0.2 +6 +OJ 287 +0.3056 +BLL +8.8 ± 0.5 +6 +4C +21.35 +0.43383 +FSRQ +8.9 ± 0.15 +8 +PKS 2155-304 +0.1167 +BLL +8.9 +9 +S5 0716+714 +0.31 +BLL +8.7 +10 +W Com +1.25813 +BLL +8.5 +14 +4C +01.02 +2.099 +FSRQ +9.5 +16 +PKS 0208-512 +1.003 +FSRQ +9.2 +7 +PKS 0235-618 +0.46657 +FSRQ +9.0 +14 +PKS 0402-362 +1.42284 +FSRQ +9.0 +14 +PKS 0426-380 +1.105 +BLL +8.6 +7 +PKS 0458-02 +2.286 +FSRQ +8.7 +11 +PKS 0502+049 +0.954 +FSRQ +8.9 ± 0.5 +12 +PKS 0528+134 +2.06 +FSRQ +9.0 +13 +PMN J0531-4827 +0.8116 +BLL +· · · +· · · +PMN J0850-1213 +0.566 +FSRQ +8.7 +14 +PKS 1144-379 +1.048 +BLL +8.5 +7 +PKS 1244-255 +0.633 +FSRQ +8.3 +14 +PKS B1406-076 +1.494 +FSRQ +9.4 +17 +PKS 1730-130 +0.902 +FSRQ +8.7 +14 +PKS 1954-388 +0.63 +FSRQ +8.0 ± 0.5 +5 +PKS 2142-75 +1.139 +FSRQ +9.7 +15 +PKS 2233-148 +0.33 +BLL +8.7 +14 +PKS 2326-502 +0.518 +FSRQ +9.3 +14 +PMN J2345-1555 +0.621 +FSRQ +8.2 ± 0.17 +8 +Ton 599 +0.72474 +FSRQ +8.5 ± 0.5 +5 +PKS 2052-47 +1.489 +FSRQ +8.9 +14 +3C 273 +0.15834 +FSRQ +8.9 ± 0.5 +5 +BL Lac +0.0686 +BLL +8.5 ± 0.2 +6 +PKS 1510-089 +0.36 +FSRQ +7.8+0.05 +−0.04 +18 +Note—(1) source name, (2) redshift, (3) source type, (4) black hole mass +(in solar mass) collected from the references in the last column. +X-ray +variability analysis is performed for the last three sources. References: (1) +Falomo et al. (2003), (2) Woo et al. (2005), (3) Kaur et al. (2017), (4) Gupta +et al. (2012), (5) Liu et al. (2006), (6) Wang et al. (2004), (7) Sbarrato et al. +(2012), (8) Shaw et al. (2012), (9) Ghisellini et al. (2010), (10) Kaur et al. +(2018), (11) Fan & Cao (2004), (12) Oshlack et al. (2002), (13) Palma et al. +(2011), (14) Paliya et al. (2017), (15) Dutka et al. (2013), (16) Schutte et al. +(2022), (17) Xue et al. (2016), (18) Rakshit (2020). + +4 +In practical application, the key point is choosing the kernel function. The DRW process (called Ornstein-Uhlenbeck +process in physics) is widely used to describe the variability of AGNs (e.g., Burke et al. 2021), and it is defined by an +exponential covariance function (e.g., Kelly et al. 2009; Zu et al. 2013), +k(tnm) = 2σ2 +DRW · exp(−tnm/τDRW) , +(1) +where tnm = |tn − tm| is the time lag between measurements m and n. The amplitude term (σDRW) represents the +amplitude of the random disturbance, and the damping (characteristic) timescale (τDRW) represents the timescale that +the system returns to the stability after experiencing a disturbance. Sometimes, an excess white noise term (σ2 +nδnm +where σn is the excess white noise amplitude and δnm is the Kronecker δ function) is needed in the situation that there +is a white noise in the LC in addition to the quoted measurement errors (Foreman-Mackey 2018; Burke et al. 2021). +A more complex kernel is the stochastically driven damped simple harmonic oscillator (SHO), which is described by +a second-order differential equation (Foreman-Mackey et al. 2017). The SHO kernel has been used to model the AGN +accretion disk (Yu et al. 2022) and jet (Zhang et al. 2022) variability. +Celerite software package is a GP tool for a stationary process (Foreman-Mackey et al. 2017). It uses the semi- +separated structure of a special covariance matrix to directly analyze and compute the GP likelihood for large data +sets. Yang et al. (2021) and Zhang et al. (2021, 2022) have tested the efficiency of this method for the study of AGN +jet variability, and suggested that celerite has a strong potentiality for studying AGN variability (also see Burke et al. +2021). Here, we use the DRW model implemented in celerite package to model the multi-band variability of blazars. +The Markov Chain Monte Carlo (MCMC) sampler emcee5 is adopted to perform posterior analysis. We assume +log-uniform priors on each of the parameters. The MCMC sampler is run for 50,000 iterations with 32 parallel walkers. +The first 20,000 steps are taken as burn-in. After modeling the LCs, we should estimate the fitting quality for assessing +whether the fitting results are reliable, e.g., whether the standardized residuals follow a Gaussian white-noise sequence. +The power spectral density (PSD) can be constructed by using the fitting results. The DRW PSD is in the form of +S(ω) = +� +8 +π σ2 +DRWτDRW +1 +1 + (ωτDRW)2 . +(2) +It is a broken power-law form with slope 0 below the broken frequency (fb) and slope -2 above the broken frequency. +The conversion between the timescale τDRW and fb is τDRW = 1/(2πfb). +The LC with large cadence or insufficient length leads to a large bias on the characteristic timescale derived from +modeling. +If the timescale is larger than the mean cadence of LC and less than 1/10 of the length of LC, the +measurement of the damping timescale from the LC is reliable (Burke et al. 2021). +3. RESULTS +3.1. Results of 3C 273, PKS 1510-089 and BL Lac +We first analyze the multi-band variability of the three individual sources, 3C 273, PKS 1510-089, and BL Lac. We +present the celerite fitting results of the LC for each source in the following. The measured timescales given in the +main text are with errors in 95% confidence intervals. +For 3C 273, the optical data in both B and V bands are available. We show the modeling results in Figure 1, in +which the left panel is for B-band LC and the right is for V -band LC. The DRW model can agree well with both +LCs. Looking at the distribution of standardized residuals and the auto-correlation function (ACF) of standardized +residuals (see details in Zhang et al. 2022), we believe the characteristic of each LC has been captured successfully. +Through MCMC sampling, we get the posterior probability density distributions of two parameters (σDRW and τDRW) +and show them in Figure 2. The values are listed in Table 2. The results are different between the two bands. The +parameters can be constrained by the B-band data but with large uncertainties, e.g., τDRW = 59+41 +−28 days. Comparing +the timescale with the cadence and the length of the LC, we believe that the B-band timescale is reliable. A broken +frequency corresponding to the characteristic timescale is shown in the B-band PSD (Figure 3). While the V -band +timescale is ≈ 3200 days, very close to the length of the LC. This means that the V -band timescale is unreliable, which +is also confirmed by the single power-law PSD (Figure 3). We show the modeling results of X-ray LC, the posterior +probability density distribution of parameters, and the PSD in the right panel in Figure 4, Figure 5 and Figure 6 +5 https://github.com/dfm/emcee + +5 +12.8 +12.9 +13.0 +13.1 +13.2 +13.3 +Magnitude +optical B-band +0 +500 +1000 +1500 +2000 +2500 +3000 +time-2454677.5 (JD) +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +Standardized Residuals +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +Standardized Residuals +0.0 +0.2 +0.4 +0.6 +0.8 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.5 +0.0 +0.5 +1.0 +ACF of Residuals +optical V-band +0 +500 +1000 +1500 +2000 +2500 +3000 +time-2454795.0 (JD) +Standardized Residuals +5 +0 +5 +Standardized Residuals +0.0 +0.2 +0.4 +0.6 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +3C 273 +Figure 1. DRW fitting results of 3C 273 in B-band (left panel) and V -band (right panel). For each column, the top panel +presents the observed LC (black points) and the modeled LC (orange/blue line). We show the standardized residuals (black +points) in the middle panel. In the bottom panel, there are two parts. The probability density of standardized residuals (black +ladder diagram) as well as the best-fit normal distribution (orange/blue solid line) are shown in the left part. The ACF of +residuals with the 95% confidence limits of the white noise (the gray region) are shown in the right part. +ln +DRW = +2.50+0.15 +0.12 +2.4 +1.6 +0.8 +0.0 +ln +DRW +4.5 +6.0 +7.5 +9.0 +ln +DRW(day) +4.5 +6.0 +7.5 +9.0 +ln +DRW(day) +ln +DRW(day) = 4.08+0.31 +0.26 +3C 273 +ln +DRW = +1.93+0.64 +0.46 +3.0 +2.4 +1.8 +1.2 +0.6 +ln +DRW +6 +7 +8 +9 +10 +ln +DRW(day) +6 +7 +8 +9 +10 +ln +DRW(day) +ln +DRW(day) = 8.06+1.29 +0.93 +3C 273 +Figure 2. Posterior probability densities of model parameters for 3C 273 in B-band (left) and V -band (right). The vertical +dotted lines mark the median value and 68% confidence intervals of the distribution of the parameter. +respectively. The values of the parameters can be found in Table 3. It is shown that the DRW model can describe the +X-ray variability of 3C 273. The parameters are well constrained. The X-ray PSD presents a broken frequency that +corresponds to a timescale of τDRW = 28+7 +−6 days. We give the radio results together with the X-ray results. The radio +PSD (the left panel of Figure 6) of 3C 273 is a single power law. The radio timescale is too large to be reliable. + +6 +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +Power(Magnitude2 day) +3C 273 +PSD obtained from B-band +PSD obtained from V-band +y = v +2 +Figure 3. B-band and V -band PSDs of 3C 273 constructed from the modeling results with DRW model. The orange line is +B-band PSD, and the blue line is V -band PSD. The corresponding color region denotes the 1σ confidence interval. The dashed +black line is a reference line with a slope of -2. +16 +18 +20 +22 +24 +26 +28 +30 +32 +Jy +radio +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +time-2454677.5 (JD) +6 +4 +2 +0 +2 +4 +Standardized Residuals +5 +0 +5 +Standardized Residuals +0.0 +0.1 +0.2 +0.3 +0.4 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +5 +10 +15 +20 +25 +30 +flux (×10 +11 erg cm +2 s +1) +X-ray +0 +1000 +2000 +3000 +4000 +5000 +time-2454795.0 (JD) +Standardized Residuals +2.5 +0.0 +2.5 +5.0 +7.5 +Standardized Residuals +0.0 +0.2 +0.4 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +3C 273 +Figure 4. DRW fitting results of 3C 273 for radio (left panel) and X-ray (right panel) data. The symbols and lines are the +same as those in Figure 1. +For PKS 1510-089, the V and B-band LCs can be described by the DRW model (Figure 7 and Figure 8). The +V -band τDRW of 39+18 +−14 days is larger than the B-band τDRW of 11 ± 3 days (Table 4). As expected, we get a smaller +value of fb in V -band PSD (Figure 9). The X-ray LC of PKS 1510-089 also can be fitted well by the DRW model +(Figure 10). The parameters are well constrained (Table 3), and the PSD is in the form of typical DRW PSD. A +trusted timescale τDRW = 26 ± 3 days is obtained. +For BL Lac, only the V -band and X-ray data are available. For the X-ray data, there are two large gaps in the first +2800 days of LC, we then take the following 2500 days of LC for analysis. When modeling the two LCs of BL Lac, +we get poor fitting (ACF of residuals deviating from the white noise) with the two-parameter DRW model. An excess +white noise term is then added to the DRW model, and we model the LCs with the three-parameter DRW model +again. The modeling of LC, the posterior distribution of parameters, and the broken power-law PSDs are shown in +Figure 11, Figure 12, and Figure 13, respectively. Optical results are shown in the left panels and the X-ray results + +7 +ln +DRW = 1.54+0.42 +0.41 +0.4 +0.8 +1.2 +1.6 +2.0 +ln +DRW +7 +8 +9 +10 +ln +DRW(day) +7 +8 +9 +10 +ln +DRW(day) +ln +DRW(day) = 8.95+0.84 +0.83 +3C 273 +ln +DRW = 0.91+0.06 +0.05 +0.75 +0.90 +1.05 +1.20 +ln +DRW +3.00 +3.25 +3.50 +3.75 +4.00 +ln +DRW(day) +3.00 +3.25 +3.50 +3.75 +4.00 +ln +DRW(day) +ln +DRW(day) = 3.34+0.12 +0.11 +3C 273 +Figure 5. Posterior probability densities of model parameters for 3C 273 in radio (left) and X-ray (right) energies. The vertical +dotted lines mark the median value and 68% confidence intervals of the distribution of the parameter. +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +Power(flux2 day) +3C 273 +PSD obtained from radio band +y = v +2 +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +Power(flux2 day) +3C 273 +PSD obtained from X-ray +y = v +2 +Figure 6. Radio and X-ray PSDs constructed from modeling LCs of 3C 273 with DRW model. The symbols and lines are the +same as those in Figure 3. +are shown in the right panels. One can see that the three-parameter DRW can fit the LCs well. Note that the highest +flux point in the LC is poorly fitted. After removing the highest flux point, the modeling results are unchanged. We +obtain the X-ray timescale of τDRW = 63+49 +−30 days and V -band τDRW of 47+26 +−19 days (Table 4). +We applied the SHO model to the optical and X-ray data of BL Lac. The fitting is not improved significantly, and +the parameters cannot be constrained. This suggests that the SHO model is not a good choice. The DRW including +an additional white noise can describe the variability behavior. The value of σ2 +n (0.01 for the V -band LC; 0.04 for +the X-ray LC) is larger than the squared of the mean size of the light curve uncertainties (σy2) where σy2=0.0001 for +V -band and 0.0036 for X-ray data. We have σ2 +DRW >σ2 +n+σy2 for both the V -band and X-ray data, which ensures that +the fitted DRW amplitude is reasonable (Burke et al. 2021). It is possible that the quoted measurement errors are +underestimated, and the excess white noise can account for excess measurement noise. +The γ-ray variability of the three sources has been analyzed in our previous work (Zhang et al. 2022) with the same +method. We give the multi-band timescales with the errors in 95% confidence intervals of the three sources in Table 4. +For 3C 273, the B-band, X-ray, and γ-ray timescales are consistent within the errors. The V -band and radio PSDs + +8 +14 +15 +16 +17 +18 +Magnitude +optical B-band +0 +500 +1000 +1500 +2000 +2500 +3000 +time-2454677.5 (JD) +5 +0 +5 +Standardized Residuals +5 +0 +5 +Standardized Residuals +0.0 +0.2 +0.4 +0.6 +0.8 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +optical V-band +0 +500 +1000 +1500 +2000 +2500 +3000 +time-2454795.0 (JD) +Standardized Residuals +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +Standardized Residuals +0.0 +0.2 +0.4 +0.6 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.0 +0.5 +1.0 +ACF of Residuals +PKS 1510-089 +Figure 7. DRW fitting results of B-band (left panel) and V -band (right panel) LCs for PKS 1510-089. The symbols and lines +are the same as those in Figure 1. +ln +DRW = +1.26+0.06 +0.05 +1.35 +1.20 +1.05 +0.90 +ln +DRW +2.1 +2.4 +2.7 +3.0 +ln +DRW(day) +2.1 +2.4 +2.7 +3.0 +ln +DRW(day) +ln +DRW(day) = 2.40+0.13 +0.12 +PKS 1510-089 +ln +DRW = +1.16+0.10 +0.09 +1.25 +1.00 +0.75 +0.50 +0.25 +ln +DRW +3.0 +3.6 +4.2 +4.8 +5.4 +ln +DRW(day) +3.0 +3.6 +4.2 +4.8 +5.4 +ln +DRW(day) +ln +DRW(day) = 3.67+0.22 +0.19 +PKS 1510-089 +Figure 8. Posterior probability densities of model parameters of B-band (left) and V -band (right) LCs for PKS 1510-089. The +symbols and lines are the same as those in Figure 2. +are single power-law, having no corresponding characteristic timescales. For PKS 1510-089, the V -band, X-ray and +γ-ray timescales are consistent within the errors but the B-band one has a smaller value. For BL Lac, the V -band, +X-ray and γ-ray timescales are also consistent within the errors. +3.2. Optical Results of 38 Blazars +The DRW model can successfully fit the long-term optical LCs of the 38 blazars. Based on the criteria of selecting +reliable measurements of the damping timescale, we get reliable optical timescales for the 38 blazars. +The basic +information of the 38 blazars and the modeling results are given in Table 1 and Table 2, respectively. Except for + +9 +Table 2. Modeling results of optical data for 38 blazars. +Object +Data sources +Waveband +Parameter of DRW +Damping timescale +Cadence +Length +ln σDRW +ln τDRW +(days) +(days) +(days) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +1ES 1959+650 +SO +V +−1.36+0.25 +−0.17 +5.13+0.53 +−0.39 +169+90 +−66 +23.5 +3548.2 +1ES 2344+514 +SO +V +−3.14+0.20 +−0.16 +4.92+0.58 +−0.46 +137+79 +−63 +17.35 +3539.1 +3C 66A +SO +V +−1.15+0.28 +−0.17 +5.35+0.57 +−0.36 +210+120 +−75 +9.04 +3217.9 +3C 454.3 +SO +V +−0.81+0.10 +−0.09 +3.82+0.21 +−0.18 +46+10 +−8 +5.97 +3563.3 +PKS 0235+164 +SO +V +−0.41+0.13 +−0.11 +3.93+0.28 +−0.24 +51+14 +−12 +14.0 +3417.9 +4C +38.41 +SO +V +−0.98+0.09 +−0.08 +3.51+0.19 +−0.17 +33+6 +−6 +9.7 +3561.2 +CTA 102 +SO +V +−0.10+0.15 +−0.12 +4.26+0.32 +−0.26 +71+23 +−18 +10.13 +3182.2 +Mkn 421 +SO +V +−1.19+0.19 +−0.14 +4.97+0.39 +−0.29 +144+56 +−42 +5.95 +3562.7 +Mkn 501 +SO +V +−3.13+0.10 +−0.09 +3.92+0.24 +−0.21 +50+12 +−11 +7.31 +3561.0 +OJ 287 +SO +B +−1.01+0.11 +−0.09 +3.57+0.23 +−0.19 +36+8 +−7 +5.32 +3079.7 +4C +21.35 +SO +V +−1.46+0.18 +−0.13 +4.79+0.37 +−0.28 +120+45 +−34 +7.70 +3357.9 +PKS 2155-304 +SO +V +−1.28+0.15 +−0.12 +4.42+0.32 +−0.26 +83+27 +−22 +11.03 +3561.2 +S5 0716+714 +SO +V +−0.96+0.11 +−0.09 +2.99+0.24 +−0.21 +20+5 +−4 +14.98 +3414.9 +W Com +SO +V +−1.13+0.18 +−0.13 +4.74+0.37 +−0.29 +114+42 +−33 +9.94 +3538.7 +4C +01.02 +S +B +−1.53+0.18 +−0.13 +3.27+0.40 +−0.32 +26+11 +−8 +10.35 +1408.2 +PKS 0208-512 +S +V +−0.56+0.15 +−0.12 +4.35+0.30 +−0.24 +77+23 +−19 +5.22 +3301.9 +PKS 0235-618 +S +R +−0.97+0.12 +−0.10 +2.73+0.29 +−0.25 +15+4 +−4 +6.76 +966.6 +PKS 0402-362 +S +V +−1.18+0.20 +−0.15 +3.91+0.42 +−0.32 +50+21 +−16 +8.39 +1459.0 +PKS 0426-380 +S +R +−0.34+0.31 +−0.19 +4.85+0.64 +−0.39 +128+82 +−50 +2.63 +1509.0 +PKS 0458-02 +S +R +−1.35+0.17 +−0.13 +3.20+0.38 +−0.31 +25+9 +−8 +10.53 +1148.1 +PKS 0502+049 +S +B +−0.56+0.16 +−0.13 +2.80+0.36 +−0.28 +16+6 +−5 +5.87 +769 +PKS 0528+134 +S +V +−1.38+0.20 +−0.15 +3.38+0.48 +−0.38 +29+14 +−11 +6.60 +956.6 +PMN J0531-4827 +S +V +0.02+0.23 +−0.17 +4.03+0.49 +−0.37 +56+28 +−21 +9.27 +1808.1 +PMN J0850-1213 +S +R +−0.79+0.14 +−0.11 +3.39+0.31 +−0.27 +30+9 +−8 +11.86 +1696.6 +PKS 1144-379 +S +B +−0.62+0.30 +−0.19 +4.72+0.62 +−0.41 +112+70 +−46 +8.99 +1590.7 +PKS 1244-255 +S +R +−0.99+0.12 +−0.10 +2.85+0.28 +−0.25 +17+5 +−4 +7.90 +1515.9 +PKS B1406-076 +S +R +−1.19+0.10 +−0.09 +3.24+0.22 +−0.20 +26+6 +−5 +130.9 +3365.9 +PKS 1730-130 +S +V +−1.59+0.14 +−0.11 +2.91+0.30 +−0.25 +18+6 +−5 +4.48 +1008.2 +PKS 1954-388 +S +R +−1.19+0.16 +−0.13 +3.57+0.39 +−0.34 +36+14 +−12 +20.12 +1851.0 +PKS 2142-75 +S +V +−2.03+0.11 +−0.10 +3.04+0.26 +−0.22 +21+5 +−5 +11.24 +1843.1 +PKS 2233-148 +S +R +−0.30+0.17 +−0.13 +3.37+0.36 +−0.28 +29+10 +−8 +6.34 +1110.2 +PKS 2326-502 +S +B +−0.38+0.22 +−0.16 +3.11+0.46 +−0.34 +22+10 +−8 +4.29 +720.1 +PMN J2345-1555 +S +R +−0.48+0.18 +−0.14 +3.44+0.39 +−0.30 +31+12 +−9 +6.59 +1424.1 +Ton 599 +SO +V +−0.39+0.20 +−0.15 +3.78+0.42 +−0.32 +44+18 +−14 +6.85 +1143.9 +PKS 2052-47 +S +V +−0.91+0.22 +−0.16 +4.41+0.54 +−0.40 +82+44 +−33 +10.25 +2121.2 +3C 273 +SO +B +−2.50+0.15 +−0.12 +4.08+0.31 +−0.26 +59+18 +−15 +8.9 +3179.12 +SO +V +−1.93+0.64 +−0.46 +8.06+1.29 +−0.93 +· · · +4.9 +3423.6 +PKS 1510-089 +SO +V +−1.16+0.10 +−0.09 +3.67+0.22 +−0.19 +39+9 +−7 +8.91 +3476.7 +SO +B +−1.26+0.06 +−0.05 +2.40+0.13 +−0.12 +11+1 +−1 +4.92 +3315.9 +BL Lac +SO +V +−0.91+0.10 +−0.08 +3.86+0.25 +−0.22 +47+12 +−10 +4.78 +3569.2 +Note— (1) source name, (2) data source, S is for SMARTS and SO is for Steward Observatory blazar data archive, +(3) wavebands of optical data, including B, V, R-band here, (4)(5) posterior parameters of modeling LC with DRW +model, (6) damping timescale in the observed frame. The uncertainties of model parameters and damping timescales +represent 1σ confidence intervals, (7) the mean cadence of the LC, and (8) the length of LC. + +10 +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +3 +10 +2 +10 +1 +100 +101 +102 +Power(Magnitude2 day) +PKS 1510-089 +PSD obtained from B-band +PSD obtained from V-band +y = v +2 +Figure 9. B and V -band PSDs of PKS 1510-089 constructed from the modeling results with the DRW model. The symbols +and lines are the same as those in Figure 3. +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Flux (×10 +11 ph cm +2 s +1) +X-ray +0 +1000 +2000 +3000 +4000 +5000 +time-50115.1 (MJD) +4 +2 +0 +2 +4 +Standardized Residuals +4 +2 +0 +2 +4 +Standardized Residuals +0.0 +0.1 +0.2 +0.3 +0.4 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +PKS 1510-089 +ln +DRW = +1.87+0.05 +0.05 +1.95 +1.80 +1.65 +ln +DRW +3.00 +3.25 +3.50 +3.75 +ln +DRW(day) +3.00 +3.25 +3.50 +3.75 +ln +DRW(day) +ln +DRW(day) = 3.25+0.12 +0.12 +PKS 1510-089 +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +Power(flux2 day) +PKS 1510-089 +PSD obtained from X-ray +y = v +2 +Figure 10. Modeling results of the X-ray LC (left), the posterior probability density distribution of parameters (middle), and +the X-ray PSD (right) for PKS 1510-089. +Table 3. Modeling results of X-ray data for the 3 blazars. +Object +Parameter of DRW +Damping timescale +Cadence +Length +ln σDRW +ln τDRW +ln σn +(days) +(days) +(days) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +3C 273 +0.91+0.06 +−0.05 +3.34+0.12 +−0.11 +· · · +28+3 +−3 +2.97 +5811.5 +PKS 1510-089 +−1.87+0.05 +−0.05 +3.25 ± 0.12 +· · · +26+3 +−3 +4.16 +5495.3 +BL Lac +−1.18+0.14 +−0.11 +4.15+0.33 +−0.26 +−1.64+0.03 +−0.04 +63+21 +−16 +2.19 +2493.1 +Note— (1) source name, (2)(3)(4) posterior parameters of modeling LC with DRW model, and (5) damping timescale in +the observed frame. The uncertainties of model parameters and damping timescales represent 1σ confidence intervals, +(6) the mean cadence of the LC, and (7) the length of LC. + +11 +13.5 +14.0 +14.5 +15.0 +15.5 +16.0 +Magnitude +optical V-band +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +time-2454677.5 (JD) +5 +0 +5 +10 +15 +Standardized Residuals +5 +0 +5 +Standardized Residuals +0.0 +0.2 +0.4 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +1 +2 +3 +4 +5 +flux (×10 +11 erg cm +2 s +1) +X-ray +0 +500 +1000 +1500 +2000 +time-2454795.0 (JD) +Standardized Residuals +5 +0 +5 +10 +Standardized Residuals +0.0 +0.2 +0.4 +Normalized Counts [a.u] +0 +10 +20 +30 +40 +50 +Time Lag +0.00 +0.25 +0.50 +0.75 +1.00 +ACF of Residuals +BL Lac +Figure 11. DRW fitting results of V -band (left panel) and X-ray data (right panel) for BL Lac. The symbols and lines are the +same as those in Figure 1. +ln +DRW = +0.91+0.10 +0.08 +3.2 +4.0 +4.8 +5.6 +ln +DRW(day) +ln +DRW(day) = 3.86+0.25 +0.22 +1.2 +0.9 +0.6 +0.3 +0.0 +ln +DRW +2.6 +2.4 +2.2 +2.0 +ln +n +3.2 +4.0 +4.8 +5.6 +ln +DRW(day) +2.6 +2.4 +2.2 +2.0 +ln +n +ln +n = +2.32+0.08 +0.09 +BL Lac +ln +DRW = +1.18+0.14 +0.11 +4.5 +6.0 +7.5 +9.0 +ln +DRW(day) +ln +DRW(day) = 4.15+0.33 +0.26 +0.8 +0.0 +0.8 +ln +DRW +1.76 +1.68 +1.60 +1.52 +ln +n +4.5 +6.0 +7.5 +9.0 +ln +DRW(day) +1.76 +1.68 +1.60 +1.52 +ln +n +ln +n = +1.64+0.03 +0.04 +BL Lac +Figure 12. V -band (left) and X-ray (right) posterior probability densities of model parameters for BL Lac. The symbols and +lines are the same as those in Figure 2. +3C 273 and PKS 1510-089 which are analyzed in Section 3.1, the timescales for different optical bands are consistent +for the other 36 sources. This indicates that the optical emission of the 36 blazars has the same origin, i.e., the jet +emission. In Table 2, we only list one optical band result for these sources. The timescale is between 10 days and 200 +days. +Some notes should be given on PKS 2052-47 and Ton 599. The fitting to the LC of PKS 2052-47 needs an additional +white noise, and the relation σ2 +DRW(0.16) > σ2 +n(0.026) + σy2(0.0016) still holds. Ton 599 has big gaps and few data in +the first half of its V -band LC, and we select the second half of the LC to analyze. + +12 +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +2 +10 +1 +100 +101 +102 +103 +Power(Magnitude2 day) +BL Lac +PSD obtained from V-band +y = v +2 +10 +3 +10 +2 +10 +1 +Frequency (day +1) +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Power(flux2 day) +BL Lac +PSD obtained from X-ray +y = v +2 +Figure 13. V -band (left) and X-ray (right) PSDs for BL Lac. The symbols and lines are the same as those in Figure 3. +Table 4. Damping timescale of 3C 273, PKS 1510-089 and BL Lac. +Object +B-band timescale +V -band timescale +X-ray timescale +γ-ray timescale +(days) +(days) +(days) +(days) +(1) +(2) +(3) +(4) +(5) +3C 273 +59+41 +−28 +unreliable +28+7 +−6 +31+12 +−10 +PKS 1510-089 +11+3 +−3 +39+18 +−14 +26+7 +−6 +40+14 +−12 +BL Lac +no data +47+26 +−19 +63+49 +−30 +69+36 +−25 +Note— (1) source name, (2)(3)(4)(5) multi-band damping timescales in the observed frame. The uncertainties +of the damping timescales represent 95% confidence intervals of the distribution of the parameter. +3.3. Origin of the optical emission from 3C 273 and PKS 1510-089 +The optical emission of 3C 273 and PKS 1510-089 is complicated. Blue bump can be seen in their multi-band spectral +energy distributions (SEDs; e.g., Abdo et al. 2010; Nalewajko et al. 2012; Castignani et al. 2017). SED modeling results +showed that the accretion disk has a significant contribution to the optical emissions of 3C 273 and PKS 1510-089 +(e.g., Nalewajko et al. 2012; Yan et al. 2012; Castignani et al. 2017). In addition, Zhang et al. (2019) found that a +long-term variation trend in the optical continuum LC of 3C 273 does not appear in the emission-line variation. This +suggests that the long-term variation trend is not contributed by the accretion disk, and it could originate from the +jet. Li et al. (2020) quantitatively decoupled the optical emissions from the jet and accretion disk in 3C 273 and found +that the jet emission accounts for 10%-40% of the total optical emission. Pandey et al. (2022) studied the correlation +between V -band flux and polarization degree (PD) variations using SO observation during 2008-2018. They found a +significant positive correlation only in two of the ten observing cycles. Note that the PD is quite small, and it changes +from 0.04% to 1.58% during 2008-2018. The V -band single power-law PSD we obtained here is different from the +typical PSD of the accretion disk (Suberlak et al. 2021; Burke et al. 2021) and jet variability (Zhang et al. 2022). The +complicated mixture of the jet and accretion disk emissions at the V -band may result in the single power-law PSD. +The mixed emission also results in the weak correlation between V -band and Fermi γ-ray variabilities reported by +Bhatta (2021). We find no significant correlation between B-band variability and γ-ray variability for 3C 273 and PKS +1510-089. Looking at the location of the blue bump in SED (Roy et al. 2021), we suggest that the B-band emission +of 3C 273 is dominated by the accretion disk photons. +For PKS 1510-089, the V and B-band timescales are clearly different, indicating different origins for the two bands’ +emissions. The V -band polarization of PSK 1510-089 is averagely greater than that of 3C 273, varying from 0.2% +to 25.82% (Pandey et al. 2022). Among the ten observing cycles during 2008-2018, a significant positive correlation + +13 +Table 5. Mean timescales (redshift-corrected) of blazars +in γ-ray and optical energies. +Waveband +logMBH/M⊙ +Mean timescale +(1) +(2) +(3) +γ-ray +8 − 9 +58+21 +−16 +9 − 10 +32+10 +−8 +8 − 10 +53+18 +−14 +optical +8 − 9 +51+23 +−11 +9 − 10 +19+6 +−5 +8 − 10 +42+18 +−13 +Note— (1) waveband, (2) the range of black hole +mass in solar mass, (3) the mean damping timescale +(redshift-corrected) with unit day. +The uncertainties +of timescales represent 1σ confidence intervals. +between V -band flux and PD variations is found in 5 cycles. Moreover, Castignani et al. (2017) found a good correlation +between the long-term SO V -band and γ-ray LCs. These results suggest that the V -band emission is dominated by +jet contribution. Also looking at the location of the blue bump in SED (Nalewajko et al. 2012), the B-band emission +with a smaller timescale of 11 days is suggested as the accretion disk contribution. +3.4. Comparing Optical and γ-ray results +Long-term Fermi γ-ray LCs of 22 blazars have been analyzed by Zhang et al. (2022) with the same GP method. The +optical timescale in this work is generally consistent with the γ-ray timescale (Figure 14). We examine the consistency +of the timescales in the two energy-bands by using a statistical significance test (T-test). We get t-statistic=1.1 and +p-value=0.28 (>0.05), which means that in statistic there is little difference between the two groups of timescales. The +optical amplitude term σDRW is less than one, and the γ-ray σDRW can be greater than 10. This means that γ-ray +variability can be more energetic than optical variability. +We separated the sources into two groups with MBH < 109M⊙ and MBH > 109M⊙. The mean timescales (redshift- +corrected) in different ranges of black hole mass are listed in Table 5. It is found that the mean timescale of the sources +in the mass range of 109-1010M⊙ is smaller in both γ-ray and optical energies. However, we have a few sources with +the mass of 109-1010M⊙, therefore this result may be tentative. +In Figure 15, we plot the relationship between the damping timescale in the rest frame (τ rest +damping) and the black hole +mass of blazars along with the results of normal quasars from Burke et al. (2021). The timescales should be modified +into the rest frame with the following formula: +τ rest +damping = τDRW δD +1 + z +. +(3) +An average Doppler factor of δD=10 is used here and the redshift z for each source is given in table 1. We show the +optical, X-ray, and γ-ray results in the plot. It is found that the nonthermal optical τ rest +damping of blazars and the thermal +optical timescale of normal quasars occupy the same space in the plot of τ rest +damping − MBH. +The X-ray results for the three individual blazars are also in the same area as the optical results. The B-band +timescale of 3C 273 is a typical value of accretion disk timescale. +The B-band timescale of PKS 1510-089 is an +outlier value among the accretion disk timescales. This value significantly deviates from the relation between damping +timescale and black hole mass reported by Burke et al. (2021). +4. DISCUSSION + +14 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +log( +DRW) +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +log( +damping/days) +Optical data +Gamma-ray data +0.0 +0.5 +1.0 +1.5 +2.0 +Normolized Counts [a,u] +0 +2 +4 +Normolized Counts [a,u] +Figure 14. Plot of the redshift-corrected timescale τDRW versus the amplitude σDRW. The red and blue points represent the +optical and γ-ray results, respectively. The side panels show the normalized histograms of the distributions of redshift-corrected +τDRW (right) and σDRW (top) for blazars. +104 +105 +106 +107 +108 +109 +1010 +MBH (M +) +100 +101 +102 +103 +rest +damping(days) +optical normal quasars +gamma-ray blazars +optical blazars +x-ray blazars +B-band PKS 1510-089 +Figure 15. +Plot of the rest-frame timescale versus black hole mass. +The gray data, lines, and area represent the optical +accretion disk results for normal quasars taken from Burke et al. (2021). Red data are γ-ray results for blazars taken from +Zhang et al. (2022), and the purple and blue data respectively represent the optical and X-ray results for blazars obtained in +this work. + +15 +It is difficult to directly resolve the inner jet structure of the blazar6. Especially, the location of the high-energy +emission region is still a hot open question (e.g., Madejski & Sikora 2016; B¨ottcher 2019). Multi-band variability +analysis provides an indirect approach to resolve the emission regions. The cross-correlation method is frequently used +in multi-band variability analysis (e.g., Liodakis et al. 2018; Bhatta 2021). +GP method has been wildly used to characterize the AGN accretion disk variability (Kelly et al. 2009; Zhang et al. +2018; Lu et al. 2019; Burke et al. 2021). In blazar science, it becomes popular in recent several years (e.g., Goyal et al. +2018; Ryan et al. 2019; Covino et al. 2020; Tarnopolski et al. 2020; Yang et al. 2021; Zhang et al. 2022). In this work, +we use the GP method to study the multi-band variability of the blazar. This provides results independent of the +cross-correlation method. +The γ-ray variability of the blazar has been studied by Zhang et al. (2022) with the GP method. Here we focus on +the X-ray and optical variability of the blazar. Multi-band emission from the blazar is dominated by the nonthermal +jet contribution. Two special blazars are 3C 273 and PKS 1510-089. An optical-ultraviolet bump appears in their +SED, which is associated with their thermal accretion disk emission (e.g., Nalewajko et al. 2012; Yan et al. 2012; +Castignani et al. 2017). +We fit the long-term optical LCs from the database of SO and SMARTS with the DRW model. Finally, 38 blazars +with a reliable characteristic timescale are selected. Except for 3C 273 and PKS 1510-089, the timescales in different +optical colors agree with each other for the remaining 36 blazars. This indicates that the emissions in different optical +colors of the 36 blazars have the same origin, i.e., the jet emission. +Ruan et al. (2012) modeled the optical LCs covering from 2002 December through 2008 March of 51 blazars using +the DRW model. They found that the observed damping timescale peaks at ∼80 days, and the intrinsic timescale +τ rest +damping peaks at ∼800 days7. The distribution of the optical timescale obtained in this work is flat (Figure 14), and +the average optical τ rest +damping is ∼400 days, which is smaller than the result of Ruan et al. (2012). All blazars in our +sample are Fermi-detected γ-ray sources. While the sample studied by Ruan et al. (2012) would be dominated by the +blazars of non-Fermi detection. Therefore, the results indicate that the optical timescale of the blazar of non-Fermi +detection may be longer than that of the blazar of Fermi detection. Xiong et al. (2015) found that the two population +blazars indeed have different physical properties, for example, the blazar of non-Fermi detection has a smaller Doppler +factor (Paliya et al. 2017). +In the reverberation mapping studies of 3C 273 and PSK 1510-089, a nonechoed long-term trend is found in the +optical continuum LC (Zhang et al. 2019; Li et al. 2020; Rakshit 2020). This reveals the mixed origin of their optical +emission. New clues on the origin of the optical emission can be found in our results. The V and B-band timescales of +PSK 1510-089 are different. Its long-term V -band variability is correlated with the γ-ray variability (Castignani et al. +2017), suggesting that the V -band emission is dominated by jet contribution. The long-term polarization variation +(Pandey et al. 2022) also supports that the nonthermal component is dominated at V -band. The V -band emission of +3C 273 seems to be more complicated. The jet contribution to V -band emission may be strongly time-dependent and +may vary in a large range. This complicated mixture of jet and accretion disk emission results in a single power-law +PSD. For the two sources, no significant correlation is found between B-band and γ-ray variabilities in our analysis. +The B-band emission is naturally considered as the accretion disk contribution. For 3C 273, the B-band timescale of +≈ 60 days is a typical value for the accretion disk emission of normal quasars. While the B-band timescale of ≈ 11 +days of PKS 1510-089 is significantly smaller, and it deviates from the τ rest +damping − MBH relation of Burke et al. (2021) +(Figure 15). This short timescale may imply special properties of its accretion disk. +The nonthermal optical, X-ray and γ-ray variabilities all have the typical DRW PSD. Namely, the PSD of synchrotron +emission is the same as that of inverse-Compton (IC) emission, consistent with the simulations with a time-dependent +one-zone leptonic blazar emission model (Thiersen et al. 2022). In other words, the long-term jet variability is irrelevant +to the underlying emission mechanism. +Burke et al. (2021) suggested that the DRW damping timescale measured from the accretion disk variability of +normal quasars could be associated with the thermal instability timescale expected in the AGN standard accretion +disk theory. Zhang et al. (2022) measured the γ-ray DRW damping timescale of AGNs from the Fermi-LAT data, and +found that the γ-ray timescales of 23 AGNs occupy almost the same space with the optical variability timescales of +normal quasars in the plot of τ rest +damping − MBH. In this work, we add the nonthermal optical timescale of blazars in this +6 The inner parsec jet of the blazar J19242914 has been resolved by the Event Horizon Telescope (Issaoun et al. 2022). +7 They also used δD = 10 for the Doppler effect correction. + +16 +plot. The nonthermal optical timescale of blazars also locates at the same region with the thermal optical timescale +of normal quasars in the plot (Figure 15). This implies that the jet variability is relevant to the accretion disk. The +thermal instability in accretion disk may not only cause the accretion disk variability but also the jet multi-band +variability. +Statistically, the nonthermal optical τ rest +damping of 38 blazars are consistent with the γ-ray τ rest +damping of 22 blazars. +Individually (3C 273, PKS 1510-089, and BL Lac), the damping timescales of the jet variability in optical, X-ray, +and γ-ray energies are consistent within the measured errors. Our results indicate that multi-band jet emissions are +produced in the same region. However, we still cannot know the distance from the emission region to the central black +hole. The radio observation is helpful to constrain this distance (Max-Moerbeck et al. 2014). We modeled the OVRO +radio LCs covering over ∼ten years, and we obtain a single power-law PSD. In this work, we only show the radio result +for 3C 273 as an example. We also modeled the 30-yr radio LCs of 3C 279 and 3C 454.3 obtained from Aalto University +Mets¨ahovi Radio Observatory, and we still get an unconstrained timescale. The results indicate the radio timescale is +very large and may be larger than 10 years. Through the very long baseline interferometry (VLBI) observation, one +can determine the distance from the radio core to the central black hole. Comparing the optical/X-ray/γ-ray timescale +and the radio timescale, we can infer that the optical/X-ray/γ-ray emission region is far upstream from the radio core. +5. SUMMARY +We analyze the blazar’s radio, optical, and X-ray variabilities using the GP tool celerite. The DRW model can +successfully fit the jet multi-band variabilities. The multi-band characteristic timescale is used to probe the structure +of the emission region in the blazar jet. Our main results are as follows. +(i) The synchrotron and IC emissions have the same PSD, i.e., the typical DRW PSD. This indicates that the jet’s +long-term variability is irrelevant to the underlying emission processes. In the plot of τ rest +damping−MBH, the jet timescales +locate at almost the same space as the accretion disk timescales of normal quasars, implying that the jet and accretion +disk variability is driven by the same physical process (Zhang et al. 2022). +(ii) The nonthermal optical, X-ray, and γ-ray variability has a consistent characteristic timescale. The radio char- +acteristic timescale is very long which cannot be constrained by decades-long LC. The results indicate that the non- +thermal optical-X-ray-γ-ray emission is produced in the same region, which is upstream and far from the radio core. +This supports the basic hypothesis of the standard Synchrotron-Self-Compton jet model. +The GP method provides a flexible approach to understand the variability pattern of AGN in the framework of +stochastic process. Adopting the standard GP tool (Foreman-Mackey et al. 2017), we build the link between accretion +disk (thermal emission) and the jet (nonthermal emission), i.e., Figure 15. This is a new methodology for comparing +thermal and nonthermal emissions, additional to the comparison between the thermal and nonthermal luminosities +(e.g, Ghisellini et al. 2011; Sbarrato et al. 2012; Ghisellini et al. 2014). +ACKNOWLEDGMENTS +We thank the referees’ valuable report. This work is partially supported by the National Key R & D Program of +China under grant No. 2018YFA0404204. H. Y. Zhang acknowledges the financial support from the Scientific Research +Fund project of Yunnan Education Department (2022Y053) and the Graduate Research innovation project of Yunnan +University (2021Y034). The work of D. H. Yan is also supported by the CAS Youth Innovation Promotion Association +and Basic research Program of Yunnan Province (202001AW070013). +Data from the Steward Observatory spectropolarimetric monitoring project were used. This program is supported +by Fermi Guest Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G. This re- +search has made use of up-to-date SMARTS optical/nearinfrared light curves. +This research has made use of +data from the OVRO 40-m monitoring program, which is supported by private funding from the California In- +situte of Technology and the Max Planck Institute for Radio Astronomy, and by NASA grants NNX08AW31G, +NNX11A043G, and NNX14AQ89G and NSF grants AST-0808050 and AST- 1109911. +This work also has made +use of {lightcurves} {spectral files} provided by the University of California, San Diego Center for Astrophysics and +Space Sciences, X-ray Group (R.E. Rothschild, A.G. Markowitz, E.S. Rivers, and B.A. McKim). +Facility: SMARTS. + +17 +Software: corner.py (Foreman-Mackey 2016), celerite (Foreman-Mackey et al. 2017), emcee (Foreman-Mackey et al. +2013), NumPy (Harris et al. 2020), Matplotlib (Hunter 2007), Astropy (Astropy Collaboration et al. 2013, 2018), SciPy +(Virtanen et al. 2020). +REFERENCES +Abdo, A. 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S., Koz�lowski, S., & Udalski, A. 2013, +ApJ, 765, 106, doi: 10.1088/0004-637X/765/2/106 + diff --git a/BtAzT4oBgHgl3EQfGPuX/content/tmp_files/load_file.txt b/BtAzT4oBgHgl3EQfGPuX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1a6f987439f3334082a1804367c7a5136ee75e7 --- /dev/null +++ b/BtAzT4oBgHgl3EQfGPuX/content/tmp_files/load_file.txt @@ -0,0 +1,1845 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf,len=1844 +page_content='Draft version January 4, 2023 Typeset using LATEX default style in AASTeX63 Gaussian Process Modeling Blazar Multiwavelength Variability: Indirectly Resolving Jet Structure Haiyun Zhang (张海云),1 Dahai Yan (闫大海),1 and Li Zhang (张力)1 1Department of Astronomy, Key Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University, Kunming 650091, China ABSTRACT Blazar jet structure can be indirectly resolved by analyzing the multiwavelength variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In this work, we analyze the long-term variability of blazars in radio, optical and X-ray energies with the Gaussian process (GP) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The multiwavelength variability can be successfully characterized by the damped-random walk (DRW) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The nonthermal optical characteristic timescales of 38 blazars are statistically consistent with the γ-ray characteristic timescales of 22 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For three individuals (3C 273, PKS 1510-089, and BL Lac), the nonthermal optical, X-ray, and γ-ray characteristic timescales are also consistent within the measured 95% errors, but the radio timescale of 3C 273 is too large to be constrained by the decade-long light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The synchrotron and inverse-Compton emissions have the same power spectral density, suggesting that the long-term jet variability is irrelevant to the emission mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In the plot of the rest-frame timescale versus black hole mass, the optical-γ-ray timescales of the jet variability occupy almost the same space with the timescales of accretion disk emission from normal quasars, which may imply that the long-term variabilities of the jet and accretion disk are driven by the same physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It is suggested that the nonthermal optical-X-ray and γ-ray emissions are produced in the same region, while the radio core which can be resolved by very-long- baseline interferometry locates at a far more distant region from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Our study suggests a new methodology for comparing thermal and nonthermal emissions, which is achieved by using the standard GP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Keywords: Blazars (164), Jets (870), Light curves (918), Time series analysis (1916) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' INTRODUCTION Flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs) are classed into a special class of active galactic nuclei (AGNs) called blazars, whose jets nearly point to the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Blazars are highly variable over the entire electromagnetic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' One popular scenario is that the accretion onto a supermassive black hole is the central engine, driving relativistic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' But the detailed process is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Thanks to the high variability of blazars, one can investigate the physical process close to the central engine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Rieger 2019), such as the location of the emitting region and the jet-disk connection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Using advanced interferometric instruments, blazar radio jet can be directly resovled on ∼parsec-scale (see Hovatta & Lindfors 2019, for a recent review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This provides a calibrator for multi-band variability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' There have been lots of works attempting to investigate the underlying physical process of blazar jet with multi-band variability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Chatterjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Nakagawa & Mori 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Max-Moerbeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2014) investigated the time-domain relationship between radio and γ-ray emission of blazars, and found the correlations only exist in a minority of the sources over a 4 yr period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' They found radio variations lagging the γ-ray variations, suggesting that the γ-ray emission originates upstream of the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This result is further verified by Liodakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2018) who concluded that the radio variation is usually substantially delayed to the other wavelengths for blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Bhatta (2021) analyzed the correlation between optical (V -band) and γ-ray variabilities for blazars and found that the optical variability is highly correlated with the γ-ray variability except for 3C 273, however, no significant Corresponding author: Dahai Yan yandahai@ynu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='cn Corresponding author: Li Zhang lizhang@ynu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='01025v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='HE] 3 Jan 2023 2 lagging is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The multi-band variability analysis can be considered as an indirect approach for resolving blazar jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The GP method becomes popular in modern time-domain astronomy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Griffiths et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Covino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Rueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The GP method enables us to effectively extract information from astronomical variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022) used a GP method to characterize the γ-ray variability of AGNs with stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It is found that the DRW model can successfully fit the γ-ray variability, which is similar with the optical variability of AGN accretion disk (Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Li & Wang 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Moreover, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022) suggested a connection between the jet and the accretion disk by comparing the rest-frame γ-ray timescales of blazars with the optical accretion disk timescales of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In this work, we analyze the multi-band variability of blazars with the GP method, which is independent of the temporal correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We hope to extract additional information from the variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Using the data from Fermi-Large Area Telescope (Fermi-LAT), we carried out systematic research of γ-ray variability of AGNs recently (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' So far, the Small and Moderate Aperture Research Telescope System (SMARTS) monitoring program1 (Bonning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012) and the Steward Observatory (SO) spectropolarimetric monitoring project2 (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2009) can provide almost ten years’ (from 2008 to 2018) optical data of Fermi blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' RXTE AGN Timing & Spectral Database3 (Rivers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2013) provides long-term X-ray data, and the Owens Valley Radio Observatory (OVRO) 40 m program (Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2011) provides radio light curves (LCs) from 2008 to 20204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Using these public data, we analyze the radio, optical and X-ray variability of three individual blazars, as well as optical variability for a sample including 38 Fermi blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The format of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In Section 2, we describe the data as well as the GP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The modeling results of the three individual sources and 38 blazars are shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We give discussions and physical interpretations of the results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In Section 5, we conclude the paper with a brief summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' DATA AND GAUSSIAN PROCESS METHOD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Data and Sources We use photometric data of blazars from the SMARTS and SO monitoring projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The SMARTS program gives photometric data at five wavelength bands (B, V, R, J, K), which were taken from the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 m telescope at the Cerro Tololo Inter-American Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' SO is a long-term optical program to support the Fermi Telescope, utilizing both the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 m Bok Telescope on Kitt Peak and the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='54 m Kuiper Telescope on Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='Bigelow in Arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The campaign of the SO program spanned almost a decade from 2008 November to 2018 July.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The X-ray data can be gained from RXTE observation which provided us with 16 yr (1996-2012) data in 2-10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' OVRO 40 m program gives radio data of blazars from 2008 to 2020, which is a large-scale, fast-cadence 15 GHz radio monitoring program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We select sources having long-term continuous observations and a good sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For the source with a large gap in the LC, we only use the data covering a longer period before or after the gap for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Finally, We have 38 blazars in the optical band, including 23 FSRQs and 15 BL Lacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Three blazars (3C 273, BL Lac, and PKS 1510-089) have long-term RXTE X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' They are also in the sample of selected optical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Unfortunately, among the three sources, only 3C 273 has the OVRO LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Table 1 gives the general information of these targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Gaussian Process Method GPs are a class of statistical models, which are widely applied for modeling stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For the one who is interested in the stochastic behavior of astronomical variability, GP provides a flexible method to model the LC with stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The application of GPs for astronomical time-series is discussed in a recent review (Aigrain & Foreman-Mackey 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Considering a data set of yn at coordinates xn, the GP model consists of a mean function µθ(x) parameterized by θ and a kernel function (covariance function) kα(xn, xm) parameterized by parameters α (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For time-series data, the GP is one-dimensional, and the coordinates are time, xn=tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' After obtaining the likelihood function with the above information, one can use Bayesian inference to estimate the posterior distribution over the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 1 http://www.' metadata={'source': 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+page_content='0686 BLL 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 6 PKS 1510-089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='36 FSRQ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='04 18 Note—(1) source name, (2) redshift, (3) source type, (4) black hole mass (in solar mass) collected from the references in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' X-ray variability analysis is performed for the last three sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' References: (1) Falomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2003), (2) Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2005), (3) Kaur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2017), (4) Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2012), (5) Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2006), (6) Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2004), (7) Sbarrato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2012), (8) Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2012), (9) Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2010), (10) Kaur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2018), (11) Fan & Cao (2004), (12) Oshlack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2002), (13) Palma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2011), (14) Paliya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2017), (15) Dutka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2013), (16) Schutte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022), (17) Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2016), (18) Rakshit (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 4 In practical application, the key point is choosing the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The DRW process (called Ornstein-Uhlenbeck process in physics) is widely used to describe the variability of AGNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021), and it is defined by an exponential covariance function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2013), k(tnm) = 2σ2 DRW · exp(−tnm/τDRW) , (1) where tnm = |tn − tm| is the time lag between measurements m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The amplitude term (σDRW) represents the amplitude of the random disturbance, and the damping (characteristic) timescale (τDRW) represents the timescale that the system returns to the stability after experiencing a disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Sometimes, an excess white noise term (σ2 nδnm where σn is the excess white noise amplitude and δnm is the Kronecker δ function) is needed in the situation that there is a white noise in the LC in addition to the quoted measurement errors (Foreman-Mackey 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' A more complex kernel is the stochastically driven damped simple harmonic oscillator (SHO), which is described by a second-order differential equation (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The SHO kernel has been used to model the AGN accretion disk (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022) and jet (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022) variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Celerite software package is a GP tool for a stationary process (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It uses the semi- separated structure of a special covariance matrix to directly analyze and compute the GP likelihood for large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021) and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021, 2022) have tested the efficiency of this method for the study of AGN jet variability, and suggested that celerite has a strong potentiality for studying AGN variability (also see Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Here, we use the DRW model implemented in celerite package to model the multi-band variability of blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The Markov Chain Monte Carlo (MCMC) sampler emcee5 is adopted to perform posterior analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We assume log-uniform priors on each of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The MCMC sampler is run for 50,000 iterations with 32 parallel walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The first 20,000 steps are taken as burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' After modeling the LCs, we should estimate the fitting quality for assessing whether the fitting results are reliable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', whether the standardized residuals follow a Gaussian white-noise sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The power spectral density (PSD) can be constructed by using the fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The DRW PSD is in the form of S(ω) = � 8 π σ2 DRWτDRW 1 1 + (ωτDRW)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2) It is a broken power-law form with slope 0 below the broken frequency (fb) and slope -2 above the broken frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The conversion between the timescale τDRW and fb is τDRW = 1/(2πfb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The LC with large cadence or insufficient length leads to a large bias on the characteristic timescale derived from modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' If the timescale is larger than the mean cadence of LC and less than 1/10 of the length of LC, the measurement of the damping timescale from the LC is reliable (Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Results of 3C 273, PKS 1510-089 and BL Lac We first analyze the multi-band variability of the three individual sources, 3C 273, PKS 1510-089, and BL Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We present the celerite fitting results of the LC for each source in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The measured timescales given in the main text are with errors in 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For 3C 273, the optical data in both B and V bands are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We show the modeling results in Figure 1, in which the left panel is for B-band LC and the right is for V -band LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The DRW model can agree well with both LCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Looking at the distribution of standardized residuals and the auto-correlation function (ACF) of standardized residuals (see details in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022), we believe the characteristic of each LC has been captured successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Through MCMC sampling, we get the posterior probability density distributions of two parameters (σDRW and τDRW) and show them in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The values are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The results are different between the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The parameters can be constrained by the B-band data but with large uncertainties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', τDRW = 59+41 −28 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Comparing the timescale with the cadence and the length of the LC, we believe that the B-band timescale is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' A broken frequency corresponding to the characteristic timescale is shown in the B-band PSD (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' While the V -band timescale is ≈ 3200 days, very close to the length of the LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This means that the V -band timescale is unreliable, which is also confirmed by the single power-law PSD (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We show the modeling results of X-ray LC, the posterior probability density distribution of parameters, and the PSD in the right panel in Figure 4, Figure 5 and Figure 6 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='com/dfm/emcee 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 Magnitude optical B-band 0 500 1000 1500 2000 2500 3000 time-2454677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 (JD) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 Standardized Residuals 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ACF of Residuals optical V-band 0 500 1000 1500 2000 2500 3000 time-2454795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 (JD) Standardized Residuals 5 0 5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals 3C 273 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' DRW fitting results of 3C 273 in B-band (left panel) and V -band (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For each column, the top panel presents the observed LC (black points) and the modeled LC (orange/blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We show the standardized residuals (black points) in the middle panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In the bottom panel, there are two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The probability density of standardized residuals (black ladder diagram) as well as the best-fit normal distribution (orange/blue solid line) are shown in the left part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The ACF of residuals with the 95% confidence limits of the white noise (the gray region) are shown in the right part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' ln DRW = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW(day) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW(day) ln DRW(day) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26 3C 273 ln DRW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 ln DRW 6 7 8 9 10 ln DRW(day) 6 7 8 9 10 ln DRW(day) ln DRW(day) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='06+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='93 3C 273 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Posterior probability densities of model parameters for 3C 273 in B-band (left) and V -band (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The vertical dotted lines mark the median value and 68% confidence intervals of the distribution of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The values of the parameters can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It is shown that the DRW model can describe the X-ray variability of 3C 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The parameters are well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The X-ray PSD presents a broken frequency that corresponds to a timescale of τDRW = 28+7 −6 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We give the radio results together with the X-ray results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The radio PSD (the left panel of Figure 6) of 3C 273 is a single power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The radio timescale is too large to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 6 10 3 10 2 10 1 Frequency (day 1) 10 5 10 4 10 3 10 2 10 1 100 101 Power(Magnitude2 day) 3C 273 PSD obtained from B-band PSD obtained from V-band y = v 2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' B-band and V -band PSDs of 3C 273 constructed from the modeling results with DRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The orange line is B-band PSD, and the blue line is V -band PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The corresponding color region denotes the 1σ confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The dashed black line is a reference line with a slope of -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 16 18 20 22 24 26 28 30 32 Jy radio 0 500 1000 1500 2000 2500 3000 3500 time-2454677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 (JD) 6 4 2 0 2 4 Standardized Residuals 5 0 5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals 5 10 15 20 25 30 flux (×10 11 erg cm 2 s 1) X-ray 0 1000 2000 3000 4000 5000 time-2454795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 (JD) Standardized Residuals 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals 3C 273 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' DRW fitting results of 3C 273 for radio (left panel) and X-ray (right panel) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For PKS 1510-089, the V and B-band LCs can be described by the DRW model (Figure 7 and Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The V -band τDRW of 39+18 −14 days is larger than the B-band τDRW of 11 ± 3 days (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' As expected, we get a smaller value of fb in V -band PSD (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The X-ray LC of PKS 1510-089 also can be fitted well by the DRW model (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The parameters are well constrained (Table 3), and the PSD is in the form of typical DRW PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' A trusted timescale τDRW = 26 ± 3 days is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For BL Lac, only the V -band and X-ray data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For the X-ray data, there are two large gaps in the first 2800 days of LC, we then take the following 2500 days of LC for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' When modeling the two LCs of BL Lac, we get poor fitting (ACF of residuals deviating from the white noise) with the two-parameter DRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' An excess white noise term is then added to the DRW model, and we model the LCs with the three-parameter DRW model again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The modeling of LC, the posterior distribution of parameters, and the broken power-law PSDs are shown in Figure 11, Figure 12, and Figure 13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Optical results are shown in the left panels and the X-ray results 7 ln DRW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='54+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW 7 8 9 10 ln DRW(day) 7 8 9 10 ln DRW(day) ln DRW(day) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='95+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='83 3C 273 ln DRW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='20 ln DRW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ln DRW(day) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ln DRW(day) ln DRW(day) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='11 3C 273 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Posterior probability densities of model parameters for 3C 273 in radio (left) and X-ray (right) energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The vertical dotted lines mark the median value and 68% confidence intervals of the distribution of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 10 3 10 2 10 1 Frequency (day 1) 10 3 10 2 10 1 100 101 102 103 Power(flux2 day) 3C 273 PSD obtained from radio band y = v 2 10 3 10 2 10 1 Frequency (day 1) 10 3 10 2 10 1 100 101 102 103 Power(flux2 day) 3C 273 PSD obtained from X-ray y = v 2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Radio and X-ray PSDs constructed from modeling LCs of 3C 273 with DRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' are shown in the right panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' One can see that the three-parameter DRW can fit the LCs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Note that the highest flux point in the LC is poorly fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' After removing the highest flux point, the modeling results are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We obtain the X-ray timescale of τDRW = 63+49 −30 days and V -band τDRW of 47+26 −19 days (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We applied the SHO model to the optical and X-ray data of BL Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The fitting is not improved significantly, and the parameters cannot be constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This suggests that the SHO model is not a good choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The DRW including an additional white noise can describe the variability behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The value of σ2 n (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='01 for the V -band LC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='04 for the X-ray LC) is larger than the squared of the mean size of the light curve uncertainties (σy2) where σy2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0001 for V -band and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0036 for X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We have σ2 DRW >σ2 n+σy2 for both the V -band and X-ray data, which ensures that the fitted DRW amplitude is reasonable (Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It is possible that the quoted measurement errors are underestimated, and the excess white noise can account for excess measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The γ-ray variability of the three sources has been analyzed in our previous work (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022) with the same method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We give the multi-band timescales with the errors in 95% confidence intervals of the three sources in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For 3C 273, the B-band, X-ray, and γ-ray timescales are consistent within the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The V -band and radio PSDs 8 14 15 16 17 18 Magnitude optical B-band 0 500 1000 1500 2000 2500 3000 time-2454677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 (JD) 5 0 5 Standardized Residuals 5 0 5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals optical V-band 0 500 1000 1500 2000 2500 3000 time-2454795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 (JD) Standardized Residuals 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ACF of Residuals PKS 1510-089 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' DRW fitting results of B-band (left panel) and V -band (right panel) LCs for PKS 1510-089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' ln DRW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='90 ln DRW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW(day) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW(day) ln DRW(day) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 PKS 1510-089 ln DRW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 ln DRW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 ln DRW(day) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 ln DRW(day) ln DRW(day) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='19 PKS 1510-089 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Posterior probability densities of model parameters of B-band (left) and V -band (right) LCs for PKS 1510-089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' are single power-law, having no corresponding characteristic timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For PKS 1510-089, the V -band, X-ray and γ-ray timescales are consistent within the errors but the B-band one has a smaller value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For BL Lac, the V -band, X-ray and γ-ray timescales are also consistent within the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Optical Results of 38 Blazars The DRW model can successfully fit the long-term optical LCs of the 38 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Based on the criteria of selecting reliable measurements of the damping timescale, we get reliable optical timescales for the 38 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The basic information of the 38 blazars and the modeling results are given in Table 1 and Table 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Except for 9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Modeling results of optical data for 38 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Object Data sources Waveband Parameter of DRW Damping timescale Cadence Length ln σDRW ln τDRW (days) (days) (days) (1) (2) (3) (4) (5) (6) (7) (8) 1ES 1959+650 SO V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='39 169+90 −66 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 3548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 1ES 2344+514 SO V −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='92+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='46 137+79 −63 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='35 3539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 3C 66A SO V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='36 210+120 −75 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='04 3217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 3C 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 SO V −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='18 46+10 −8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='97 3563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 PKS 0235+164 SO V −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='24 51+14 −12 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 3417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 4C +38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='41 SO V −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='51+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='17 33+6 −6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='7 3561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 CTA 102 SO V −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26 71+23 −18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='13 3182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 Mkn 421 SO V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='29 144+56 −42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='95 3562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='7 Mkn 501 SO V −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='13+0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26 59+18 −15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 3179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 SO V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='46 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='06+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='93 · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 3423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 PKS 1510-089 SO V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='19 39+9 −7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='91 3476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='7 SO B −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 11+1 −1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='92 3315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 BL Lac SO V −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='22 47+12 −10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='78 3569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 Note— (1) source name, (2) data source, S is for SMARTS and SO is for Steward Observatory blazar data archive, (3) wavebands of optical data, including B, V, R-band here, (4)(5) posterior parameters of modeling LC with DRW model, (6) damping timescale in the observed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The uncertainties of model parameters and damping timescales represent 1σ confidence intervals, (7) the mean cadence of the LC, and (8) the length of LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 10 10 3 10 2 10 1 Frequency (day 1) 10 3 10 2 10 1 100 101 102 Power(Magnitude2 day) PKS 1510-089 PSD obtained from B-band PSD obtained from V-band y = v 2 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' B and V -band PSDs of PKS 1510-089 constructed from the modeling results with the DRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 Flux (×10 11 ph cm 2 s 1) X-ray 0 1000 2000 3000 4000 5000 time-50115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 (MJD) 4 2 0 2 4 Standardized Residuals 4 2 0 2 4 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals PKS 1510-089 ln DRW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='65 ln DRW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 ln DRW(day) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 ln DRW(day) ln DRW(day) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 PKS 1510-089 10 3 10 2 10 1 Frequency (day 1) 10 5 10 4 10 3 10 2 10 1 100 101 Power(flux2 day) PKS 1510-089 PSD obtained from X-ray y = v 2 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Modeling results of the X-ray LC (left), the posterior probability density distribution of parameters (middle), and the X-ray PSD (right) for PKS 1510-089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Modeling results of X-ray data for the 3 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Object Parameter of DRW Damping timescale Cadence Length ln σDRW ln τDRW ln σn (days) (days) (days) (1) (2) (3) (4) (5) (6) (7) 3C 273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='11 · · 28+3 −3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='97 5811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 PKS 1510-089 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='12 · · 26+3 −3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='16 5495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 BL Lac −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='64+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='04 63+21 −16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='19 2493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 Note— (1) source name, (2)(3)(4) posterior parameters of modeling LC with DRW model, and (5) damping timescale in the observed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The uncertainties of model parameters and damping timescales represent 1σ confidence intervals, (6) the mean cadence of the LC, and (7) the length of LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 11 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 Magnitude optical V-band 0 500 1000 1500 2000 2500 3000 3500 time-2454677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 (JD) 5 0 5 10 15 Standardized Residuals 5 0 5 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals 1 2 3 4 5 flux (×10 11 erg cm 2 s 1) X-ray 0 500 1000 1500 2000 time-2454795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 (JD) Standardized Residuals 5 0 5 10 Standardized Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 Normalized Counts [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='u] 0 10 20 30 40 50 Time Lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 ACF of Residuals BL Lac Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' DRW fitting results of V -band (left panel) and X-ray data (right panel) for BL Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' ln DRW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 ln DRW(day) ln DRW(day) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln n 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 ln DRW(day) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln n ln n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='09 BL Lac ln DRW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW(day) ln DRW(day) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='8 ln DRW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='52 ln n 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 ln DRW(day) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='52 ln n ln n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='64+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='04 BL Lac Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' V -band (left) and X-ray (right) posterior probability densities of model parameters for BL Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 3C 273 and PKS 1510-089 which are analyzed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1, the timescales for different optical bands are consistent for the other 36 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This indicates that the optical emission of the 36 blazars has the same origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', the jet emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In Table 2, we only list one optical band result for these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The timescale is between 10 days and 200 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Some notes should be given on PKS 2052-47 and Ton 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The fitting to the LC of PKS 2052-47 needs an additional white noise, and the relation σ2 DRW(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='16) > σ2 n(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='026) + σy2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0016) still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Ton 599 has big gaps and few data in the first half of its V -band LC, and we select the second half of the LC to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 12 10 3 10 2 10 1 Frequency (day 1) 10 2 10 1 100 101 102 103 Power(Magnitude2 day) BL Lac PSD obtained from V-band y = v 2 10 3 10 2 10 1 Frequency (day 1) 10 4 10 3 10 2 10 1 100 101 102 Power(flux2 day) BL Lac PSD obtained from X-ray y = v 2 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' V -band (left) and X-ray (right) PSDs for BL Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The symbols and lines are the same as those in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Damping timescale of 3C 273, PKS 1510-089 and BL Lac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Object B-band timescale V -band timescale X-ray timescale γ-ray timescale (days) (days) (days) (days) (1) (2) (3) (4) (5) 3C 273 59+41 −28 unreliable 28+7 −6 31+12 −10 PKS 1510-089 11+3 −3 39+18 −14 26+7 −6 40+14 −12 BL Lac no data 47+26 −19 63+49 −30 69+36 −25 Note— (1) source name, (2)(3)(4)(5) multi-band damping timescales in the observed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The uncertainties of the damping timescales represent 95% confidence intervals of the distribution of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Origin of the optical emission from 3C 273 and PKS 1510-089 The optical emission of 3C 273 and PKS 1510-089 is complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Blue bump can be seen in their multi-band spectral energy distributions (SEDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Nalewajko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' SED modeling results showed that the accretion disk has a significant contribution to the optical emissions of 3C 273 and PKS 1510-089 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Nalewajko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In addition, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2019) found that a long-term variation trend in the optical continuum LC of 3C 273 does not appear in the emission-line variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This suggests that the long-term variation trend is not contributed by the accretion disk, and it could originate from the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2020) quantitatively decoupled the optical emissions from the jet and accretion disk in 3C 273 and found that the jet emission accounts for 10%-40% of the total optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022) studied the correlation between V -band flux and polarization degree (PD) variations using SO observation during 2008-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' They found a significant positive correlation only in two of the ten observing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Note that the PD is quite small, and it changes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='04% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='58% during 2008-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The V -band single power-law PSD we obtained here is different from the typical PSD of the accretion disk (Suberlak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021) and jet variability (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The complicated mixture of the jet and accretion disk emissions at the V -band may result in the single power-law PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The mixed emission also results in the weak correlation between V -band and Fermi γ-ray variabilities reported by Bhatta (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We find no significant correlation between B-band variability and γ-ray variability for 3C 273 and PKS 1510-089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Looking at the location of the blue bump in SED (Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021), we suggest that the B-band emission of 3C 273 is dominated by the accretion disk photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For PKS 1510-089, the V and B-band timescales are clearly different, indicating different origins for the two bands’ emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The V -band polarization of PSK 1510-089 is averagely greater than that of 3C 273, varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='2% to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='82% (Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Among the ten observing cycles during 2008-2018, a significant positive correlation 13 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Mean timescales (redshift-corrected) of blazars in γ-ray and optical energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Waveband logMBH/M⊙ Mean timescale (1) (2) (3) γ-ray 8 − 9 58+21 −16 9 − 10 32+10 −8 8 − 10 53+18 −14 optical 8 − 9 51+23 −11 9 − 10 19+6 −5 8 − 10 42+18 −13 Note— (1) waveband, (2) the range of black hole mass in solar mass, (3) the mean damping timescale (redshift-corrected) with unit day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The uncertainties of timescales represent 1σ confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' between V -band flux and PD variations is found in 5 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Moreover, Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2017) found a good correlation between the long-term SO V -band and γ-ray LCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' These results suggest that the V -band emission is dominated by jet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Also looking at the location of the blue bump in SED (Nalewajko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012), the B-band emission with a smaller timescale of 11 days is suggested as the accretion disk contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Comparing Optical and γ-ray results Long-term Fermi γ-ray LCs of 22 blazars have been analyzed by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022) with the same GP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The optical timescale in this work is generally consistent with the γ-ray timescale (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We examine the consistency of the timescales in the two energy-bands by using a statistical significance test (T-test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We get t-statistic=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='1 and p-value=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='28 (>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='05), which means that in statistic there is little difference between the two groups of timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The optical amplitude term σDRW is less than one, and the γ-ray σDRW can be greater than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This means that γ-ray variability can be more energetic than optical variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We separated the sources into two groups with MBH < 109M⊙ and MBH > 109M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The mean timescales (redshift- corrected) in different ranges of black hole mass are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It is found that the mean timescale of the sources in the mass range of 109-1010M⊙ is smaller in both γ-ray and optical energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' However, we have a few sources with the mass of 109-1010M⊙, therefore this result may be tentative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In Figure 15, we plot the relationship between the damping timescale in the rest frame (τ rest damping) and the black hole mass of blazars along with the results of normal quasars from Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The timescales should be modified into the rest frame with the following formula: τ rest damping = τDRW δD 1 + z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (3) An average Doppler factor of δD=10 is used here and the redshift z for each source is given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We show the optical, X-ray, and γ-ray results in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' It is found that the nonthermal optical τ rest damping of blazars and the thermal optical timescale of normal quasars occupy the same space in the plot of τ rest damping − MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The X-ray results for the three individual blazars are also in the same area as the optical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The B-band timescale of 3C 273 is a typical value of accretion disk timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The B-band timescale of PKS 1510-089 is an outlier value among the accretion disk timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This value significantly deviates from the relation between damping timescale and black hole mass reported by Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' DISCUSSION 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 log( DRW) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='50 log( damping/days) Optical data Gamma-ray data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='0 Normolized Counts [a,u] 0 2 4 Normolized Counts [a,u] Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Plot of the redshift-corrected timescale τDRW versus the amplitude σDRW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The red and blue points represent the optical and γ-ray results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The side panels show the normalized histograms of the distributions of redshift-corrected τDRW (right) and σDRW (top) for blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 104 105 106 107 108 109 1010 MBH (M ) 100 101 102 103 rest damping(days) optical normal quasars gamma-ray blazars optical blazars x-ray blazars B-band PKS 1510-089 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Plot of the rest-frame timescale versus black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The gray data, lines, and area represent the optical accretion disk results for normal quasars taken from Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Red data are γ-ray results for blazars taken from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022), and the purple and blue data respectively represent the optical and X-ray results for blazars obtained in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 15 It is difficult to directly resolve the inner jet structure of the blazar6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Especially, the location of the high-energy emission region is still a hot open question (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Madejski & Sikora 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' B¨ottcher 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Multi-band variability analysis provides an indirect approach to resolve the emission regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The cross-correlation method is frequently used in multi-band variability analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Liodakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Bhatta 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' GP method has been wildly used to characterize the AGN accretion disk variability (Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In blazar science, it becomes popular in recent several years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Covino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Tarnopolski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In this work, we use the GP method to study the multi-band variability of the blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This provides results independent of the cross-correlation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The γ-ray variability of the blazar has been studied by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022) with the GP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Here we focus on the X-ray and optical variability of the blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Multi-band emission from the blazar is dominated by the nonthermal jet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Two special blazars are 3C 273 and PKS 1510-089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' An optical-ultraviolet bump appears in their SED, which is associated with their thermal accretion disk emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Nalewajko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We fit the long-term optical LCs from the database of SO and SMARTS with the DRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Finally, 38 blazars with a reliable characteristic timescale are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Except for 3C 273 and PKS 1510-089, the timescales in different optical colors agree with each other for the remaining 36 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This indicates that the emissions in different optical colors of the 36 blazars have the same origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', the jet emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2012) modeled the optical LCs covering from 2002 December through 2008 March of 51 blazars using the DRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' They found that the observed damping timescale peaks at ∼80 days, and the intrinsic timescale τ rest damping peaks at ∼800 days7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The distribution of the optical timescale obtained in this work is flat (Figure 14), and the average optical τ rest damping is ∼400 days, which is smaller than the result of Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' All blazars in our sample are Fermi-detected γ-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' While the sample studied by Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2012) would be dominated by the blazars of non-Fermi detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Therefore, the results indicate that the optical timescale of the blazar of non-Fermi detection may be longer than that of the blazar of Fermi detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2015) found that the two population blazars indeed have different physical properties, for example, the blazar of non-Fermi detection has a smaller Doppler factor (Paliya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In the reverberation mapping studies of 3C 273 and PSK 1510-089, a nonechoed long-term trend is found in the optical continuum LC (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Rakshit 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This reveals the mixed origin of their optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' New clues on the origin of the optical emission can be found in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The V and B-band timescales of PSK 1510-089 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Its long-term V -band variability is correlated with the γ-ray variability (Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017), suggesting that the V -band emission is dominated by jet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The long-term polarization variation (Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022) also supports that the nonthermal component is dominated at V -band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The V -band emission of 3C 273 seems to be more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The jet contribution to V -band emission may be strongly time-dependent and may vary in a large range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This complicated mixture of jet and accretion disk emission results in a single power-law PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For the two sources, no significant correlation is found between B-band and γ-ray variabilities in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The B-band emission is naturally considered as the accretion disk contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' For 3C 273, the B-band timescale of ≈ 60 days is a typical value for the accretion disk emission of normal quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' While the B-band timescale of ≈ 11 days of PKS 1510-089 is significantly smaller, and it deviates from the τ rest damping − MBH relation of Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021) (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This short timescale may imply special properties of its accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The nonthermal optical, X-ray and γ-ray variabilities all have the typical DRW PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Namely, the PSD of synchrotron emission is the same as that of inverse-Compton (IC) emission, consistent with the simulations with a time-dependent one-zone leptonic blazar emission model (Thiersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In other words, the long-term jet variability is irrelevant to the underlying emission mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2021) suggested that the DRW damping timescale measured from the accretion disk variability of normal quasars could be associated with the thermal instability timescale expected in the AGN standard accretion disk theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (2022) measured the γ-ray DRW damping timescale of AGNs from the Fermi-LAT data, and found that the γ-ray timescales of 23 AGNs occupy almost the same space with the optical variability timescales of normal quasars in the plot of τ rest damping − MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In this work, we add the nonthermal optical timescale of blazars in this 6 The inner parsec jet of the blazar J19242914 has been resolved by the Event Horizon Telescope (Issaoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 7 They also used δD = 10 for the Doppler effect correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 16 plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The nonthermal optical timescale of blazars also locates at the same region with the thermal optical timescale of normal quasars in the plot (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This implies that the jet variability is relevant to the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The thermal instability in accretion disk may not only cause the accretion disk variability but also the jet multi-band variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Statistically, the nonthermal optical τ rest damping of 38 blazars are consistent with the γ-ray τ rest damping of 22 blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Individually (3C 273, PKS 1510-089, and BL Lac), the damping timescales of the jet variability in optical, X-ray, and γ-ray energies are consistent within the measured errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Our results indicate that multi-band jet emissions are produced in the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' However, we still cannot know the distance from the emission region to the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The radio observation is helpful to constrain this distance (Max-Moerbeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We modeled the OVRO radio LCs covering over ∼ten years, and we obtain a single power-law PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In this work, we only show the radio result for 3C 273 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' We also modeled the 30-yr radio LCs of 3C 279 and 3C 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='3 obtained from Aalto University Mets¨ahovi Radio Observatory, and we still get an unconstrained timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The results indicate the radio timescale is very large and may be larger than 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Through the very long baseline interferometry (VLBI) observation, one can determine the distance from the radio core to the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Comparing the optical/X-ray/γ-ray timescale and the radio timescale, we can infer that the optical/X-ray/γ-ray emission region is far upstream from the radio core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' SUMMARY We analyze the blazar’s radio, optical, and X-ray variabilities using the GP tool celerite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The DRW model can successfully fit the jet multi-band variabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The multi-band characteristic timescale is used to probe the structure of the emission region in the blazar jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Our main results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (i) The synchrotron and IC emissions have the same PSD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', the typical DRW PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This indicates that the jet’s long-term variability is irrelevant to the underlying emission processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' In the plot of τ rest damping−MBH, the jet timescales locate at almost the same space as the accretion disk timescales of normal quasars, implying that the jet and accretion disk variability is driven by the same physical process (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' (ii) The nonthermal optical, X-ray, and γ-ray variability has a consistent characteristic timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The radio char- acteristic timescale is very long which cannot be constrained by decades-long LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The results indicate that the non- thermal optical-X-ray-γ-ray emission is produced in the same region, which is upstream and far from the radio core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This supports the basic hypothesis of the standard Synchrotron-Self-Compton jet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The GP method provides a flexible approach to understand the variability pattern of AGN in the framework of stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Adopting the standard GP tool (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2017), we build the link between accretion disk (thermal emission) and the jet (nonthermal emission), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=', Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This is a new methodology for comparing thermal and nonthermal emissions, additional to the comparison between the thermal and nonthermal luminosities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='g, Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Sbarrato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the referees’ valuable report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This work is partially supported by the National Key R & D Program of China under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 2018YFA0404204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Zhang acknowledges the financial support from the Scientific Research Fund project of Yunnan Education Department (2022Y053) and the Graduate Research innovation project of Yunnan University (2021Y034).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' The work of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Yan is also supported by the CAS Youth Innovation Promotion Association and Basic research Program of Yunnan Province (202001AW070013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Data from the Steward Observatory spectropolarimetric monitoring project were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This program is supported by Fermi Guest Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This re- search has made use of up-to-date SMARTS optical/nearinfrared light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This research has made use of data from the OVRO 40-m monitoring program, which is supported by private funding from the California In- situte of Technology and the Max Planck Institute for Radio Astronomy, and by NASA grants NNX08AW31G, NNX11A043G, and NNX14AQ89G and NSF grants AST-0808050 and AST- 1109911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' This work also has made use of {lightcurves} {spectral files} provided by the University of California, San Diego Center for Astrophysics and Space Sciences, X-ray Group (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Rothschild, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Markowitz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' Rivers, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' McKim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfGPuX/content/2301.01025v1.pdf'} +page_content=' 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Chang-Dong Wang3 +1*College of Mathematics and Informatics, South China +Agricultural University, Guangzhou, China. +2Key Laboratory of Smart Agricultural Technology in Tropical +South China, Ministry of Agriculture and Rural Affairs, China. +3School of Computer Science and Engineering, Sun Yat-sen +University, Guangzhou, China. +*Corresponding author(s). E-mail(s): huangdonghere@gmail.com; +Contributing authors: dengxiaozhi45@gmail.com; +changdongwang@hotmail.com; +Abstract +Contrastive deep clustering has recently gained significant attention +with its ability of joint contrastive learning and clustering via deep +neural networks. Despite the rapid progress, previous works mostly +require both positive and negative sample pairs for contrastive cluster- +ing, which rely on a relative large batch-size. Moreover, they typically +adopt a two-stream architecture with two augmented views, which over- +look the possibility and potential benefits of multi-stream architectures +(especially with heterogeneous or hybrid networks). In light of this, +this paper presents a new end-to-end deep clustering approach termed +Heterogeneous Tri-stream Clustering Network (HTCN). The tri-stream +architecture in HTCN consists of three main components, including two +weight-sharing online networks and a target network, where the param- +eters of the target network are the exponential moving average of that +of the online networks. Notably, the two online networks are trained by +simultaneously (i) predicting the instance representations of the target +network and (ii) enforcing the consistency between the cluster repre- +sentations of the target network and that of the two online networks. +Experimental results on four challenging image datasets demonstrate the +superiority of HTCN over the state-of-the-art deep clustering approaches. +The code is available at https://github.com/dengxiaozhi/HTCN. +Keywords: Data clustering, Image clustering, Deep clustering, Deep neural +network, Contrastive learning +1 +arXiv:2301.04451v1 [cs.LG] 11 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Heterogeneous Tri-stream Clustering Network +1 Introduction +Data clustering is the process of grouping data samples into multiple clus- +ters in an unsupervised manner, which is a fundamental task in a variety +of applications [1–3]. The traditional clustering algorithms typically focus +on some low-level information and lack the representation learning ability, +which may lead to sub-optimal performance when dealing with some complex +high-dimensional data like images. +In recent years, the deep learning has gained tremendous progress [4–6], +which has also been exploited for tackling the clustering task, giving rise to the +rapid development of the deep clustering algorithms [7–11]. For example, Xie +et al. [7] presented a deep clustering method called Deep Embedded Clustering +(DEC), which simultaneously learns representations and cluster assignments +with an objective loss based on Kullback-Leibler (KL) divergence. Guo et al. +[8] extended DEC by incorporating the reconstruction loss (via autoencoder) +to preserve local structures. Ji et al. [10] sought to learn invariant information +of data by maximizing the mutual information between paired samples. More +recently, the contrastive learning has emerged as a promising technique for +exploiting sample-wise (or augmentation-wise) contrastiveness to improve the +deep clustering performance. Van Gansbeke et al. [12] presented the Semantic +Clustering by Adopting Nearest neighbors (SCAN) method, which first adopts +contrastive learning to learn discriminant features and then performs semantic +clustering with the K-nearest neighbors exploited. Dang et al. [13] matched +local-level and global-level nearest neighbors to further improve clustering per- +formance. Li et al. [14] presented the Contrastive Clustering (CC) method to +perform feature learning and clustering with simultaneous instance-level and +cluster-level contrastive learning. +Despite significant success, these contrastive deep clustering methods [12– +14] are mostly faced with two limitations. On the one hand, they typically +requires both positive sample pairs and negative sample pairs during their +contrastive learning process, which rely on a relatively large batch-size (for +sufficient negative pairs) and may bring in a heavier computational burden. +On the other hand, these prior works generally adopt a two-stream architec- +ture (with two weight-sharing augmented views), which neglect the possibility +of going beyond the two-stream architecture to utilize three or even more +streams of networks (with heterogeneous or hybrid structures). Recently Grill +et al. [15] presented the Bootstrap Your Own Latent (BYOL) method, which +adopts an asymmetric two-stream architecture (with an online network and a +target network) and conducts the contrastive learning without negative pairs, +where the online network is trained by predicting the feature representations +of the target network. Though the requirement for negative sample pairs is +remedied, BYOL still complies with the two-stream architecture and also lacks +the ability of directly learning the clustering structure. It remains a challeng- +ing problem how to incorporate contrastive learning into multiple streams of +heterogeneous networks while alleviating the dependence on negative sample +pairs for strengthened deep clustering performance. + +Springer Nature 2021 LATEX template +Heterogeneous Tri-stream Clustering Network +3 +In light of this, this paper presents a novel deep clustering approach +termed Heterogeneous Tri-stream Clustering Network (HTCN), which lever- +ages three streams of heterogeneous networks for simultaneous cluster-level +and instance-level contrastive learning without requiring negative sample pairs +(as illustrated in Fig. 1). Inspired by BYOL [15], we design a novel tri-stream +architecture with three augmented views, corresponding to two online net- +works and a target network, respectively. Note that the online network and +the target network are heterogeneous, which differ from each other in the net- +work structure and the updating mechanism. The two online networks share +the same parameters, while the parameters of the target network are the +exponential moving average of that of the online networks. Here, the exponen- +tial moving average is a type of moving average that places a greater weight +and significance on the most recent data samples [15]. Each online network +is associated with an instance predictor and a cluster predictor, which pro- +duce the instance-level representations and the cluster-level representations, +respectively. Different from the online networks, the target network utilizes a +cluster predictor to generate the cluster-level representations while producing +the instance-level representations by the projector directly. The incorporation +of an instance predictor in the online networks is meant to prevent the potential +collapse where the networks produce the same feature representations for most +samples. Then we train the two online networks by (i) predicting the target +network’s representation of the same image via the mean squared error (MSE) +loss (for the instance-level contrastive learning) and (ii) enforcing the consis- +tency between the predicted cluster distributions of the two online networks +and that of the target network via the information noise contrastive estimation +(InfoNCE) [16] loss (for the cluster-level contrastive learning). Experiments +conducted on four image datasets demonstrate the superiority of our approach +over the state-of-the-art deep clustering approaches. +For clarity, the contributions of this work are summarized below. +• A heterogeneous tri-stream architecture is designed, where two online net- +works and a target network are jointly leveraged for instance-level and +cluster-level contrastive learning. +• A novel deep clustering approach termed HTCN is proposed, which utilizes +three augmented views for contrastive learning without requiring negative +sample pairs. +• Experimental results on four image datasets confirm the advantegeous clus- +tering performance of our HTCN approach over the state-of-the-art deep +clustering approaches. +The rest of the paper is organized as follows. The related works on deep +clustering and self-supervised learning are reviewed in Section 2. The proposed +HTCN framework is described in Section 3. The experiments are reported in +Section 4. Finally, Section 5 concludes the paper. + +Springer Nature 2021 LATEX template +4 +Heterogeneous Tri-stream Clustering Network +2 Related Work +In this section, we will introduce the related works on deep clustering and +self-supervised learning. +2.1 Deep Clustering +Traditional clustering methods such as K-means [17] and spectral cluster- +ing (SC) [18] have achieved promising results in handling low-dimensional +data, but they may result in sub-optimal performance when faced with high- +dimensional data (e.g., images and videos) due to the lack of the representation +learning ability. To address this, the deep learning based clustering methods, +referred to as the deep clustering methods, have recently achieved significant +success, which leverage the power of feature learning of deep neural networks +for the clustering task [7–10, 12–14, 19–27]. +Previous deep clustering methods can be divided into two main categories, +namely, the one-stage methods and the two-stage methods. The goal of the +one-stage approach is to perform feature representation learning and clustering +assignment simultaneously. Xie et al. [7] proposed a Deep Embedding Cluster- +ing (DEC) method, which jointly optimizes feature learning and clustering with +a KL-divergence loss. Caron et al. [21] iteratively clustered the learned features +with K-means and regarded the cluster assignments as supervisory signals to +optimize the network. Li et al.[14] presented a Contrastive Clustering (CC) +method that performs contrastive learning at instance-level and cluster-level +for deep clustering. Besides the one-stage methods, some researchers have also +made considerable efforts to the two-stage clustering methods. Van Gansbeke +et al.[12] proposed a two-stage clustering method called Semantic Clustering +by Adopting Nearest neighbors (SCAN), which first learns the semantic fea- +tures via contrastive learning and then utilizes the features for clustering in +the next stage. To extend SCAN, Dang et al. [13] designed a Nearest Neighbor +Matching (NNM) method, which selects both local and global nearest neigh- +bors to optimize the network, where the neighbors are forced to be close to +each other. +2.2 Self-supervised Learning +Self-supervised learning has recently emerged as a powerful technique with the +ability to learn representation from raw data without human supervision, in +which the contrastive learning methods [28–31] have been a representative and +promising category. +The goal of contrastive learning is to minimize the distance between posi- +tive sample pairs while maximizing the distance between negative sample pairs +in a self-supervised manner, where positive pairs and negative pairs are defined +through data augmentations. In particular, some researchers maintained a +memory bank [28, 29] that contains large amounts of representations of nega- +tive samples to achieve high performance. However, these methods that utilize +memory banks to store and update representations may be computationally + +Springer Nature 2021 LATEX template +Heterogeneous Tri-stream Clustering Network +5 +expensive. To address the problems with memory banks, He et al. [30] pro- +posed a Momentum Contrast (MoCo) method that trains an encoder by the +momentum update mechanism maintaining a long queue of negative examples. +Following the MoCo method, Chen et al. [31] proposed a Simple framework +for Contrastive LeaRning (SimCLR) method which carefully designs the strat- +egy of data augmentation and a non-linear transformation head. In addition, +the clustering based methods [32, 33] adopt a clustering approach to group +similar features together, which address the issue that every sample is consid- +ered as a discrete class in previous works. More recently, some self-supervised +learning methods that only rely on positive pairs and directly predict the out- +put of one augmented view from another augmented view [15, 34, 35] have +been developed, among which a representative method is the BYOL method +[15]. The BYOL method [15] adopts an asymmetric two-stream architecture, +which, however, lacks the ability to learn the clustering structure directly and +also overlooks the opportunities and potential benefits of going beyond the +two-stream architecture to three or more streams of networks (even with het- +erogeneous or hybrid structures) to further enhance the contrastive learning +and clustering performance. +3 Proposed Framework +3.1 Framework Overview +This paper presents a heterogeneous tri-stream network architecture termed +HTCN for contrastive deep clustering (as illustrated in Fig. 1), which goes +beyond the traditional two-stream architecture to explore the constrastive net- +work in a multi-stream manner. Also, HTCN doesn’t require negative sample +pairs, which makes it more resilient to different batch-size. Specifically, HTCN +consists of three main components, including two online networks and a target +network. The online networks and the target network are respectively parame- +terized by different sets of weights, where the parameters of the target network +are an exponential moving average of that of the online networks. +Given a batch of N images, we perform three types of augmentations on +each image, denoted as xi with i ∈ [1, N], to generate 3 · N augmented (or +distorted) images, denoted as {xa +1, . . . , xa +N, xb +1, . . . , xb +N, xc +1, . . . , xc +N}. The back- +bones (i.e., fθ and fξ) and projectors (i.e., gθ and gξ) are adopted to extract +features from the distorted images via za +i = gθ(fθ(xa +i )), zb +i = gξ(fξ(xb +i)) and +zc +i = gθ(fθ(xc +i)). Then the instance predictors transform za +i and zc +i to ya +i +and yc +i , respectively, while the cluster predictors transform za +i , zb +i and zc +i to +˜qa +i , ˜qb +i and ˜qc +i , respectively. Note that, similar to the asymmetric architecture +of BYOL, the target network is not associated with an instance predictor, +and the representations generated by its projector are used to guide the +instance-level learning of the two online networks. The row space of the feature +matrix learned by the projector or the instance predictor is expressed as the +instance-level representations, while the column space of the feature matrix +learned by the cluster predictor is expressed as the cluster-level representations. + +Springer Nature 2021 LATEX template +6 +Heterogeneous Tri-stream Clustering Network +𝑓! +𝑓" +𝑔! +𝑔" +ℎ" +ℎ! +𝑝! +𝑥# +𝑥$ +𝑧# +𝑧$ +𝑦# +𝑞# +𝑞$ +𝑥 +𝑞$ +𝑞# +Minimize InfoNCE Loss +Distorted images +Backbone +Cluster predictor +Projector +Online network +Target network +Instance predictor +𝑦# +𝑧$ +Minimize MSE Loss +𝑦% +𝑧$ +Input images +𝑞$ +𝑞% +Minimize InfoNCE Loss +Cluster predictor +Distorted images +Backbone +Projector +𝑇# +𝑇$ +𝑇% +𝑓! +𝑔! +ℎ! +𝑝! +𝑥% +𝑧% +𝑦% +𝑞% +Cluster predictor +Distorted images +Backbone +Projector +Instance predictor +sg +Online network +Target network +Fig. 1 Illustration of the proposed HTCN framework. The tri-stream network consists of +two weight-sharing online networks and a target network, where the parameters of the target +network is an exponential moving average of that of the online networks. Instance predictors +and cluster predictors are incorporated in the three networks, after which the MSE loss +and the InfoNCE loss are ultilized for instance-level contrastive learning and cluster-level +contrastive learning, respectively. The network architecture can be trained in an end-to-end +manner, where the final clustering is obtained via the cluster predictor of the target network. +The instance-level representations are utilized to enforce the instance-level +contrastive learning with an MSE loss optimized, while the cluster-level repre- +sentations are utilized to enforce the cluster-level contrastive learning with an +InfoNCE loss optimized. Finally, the instance-level and cluster-level contrastive +losses are simultaneously utilized to optimize the tri-stream network. +3.2 Instance-level Contrastiveness +Our HTCN approach simultaneously performs feature learning and clustering +without requiring negative sample pairs. In instance-level contrastive learn- +ing, we aim to train the two online networks by predicting the instance +representations of target network. Specifically, let ya +i and zb +i be the instance +representations of xi in the first online network and the target network, +respectively. The instance-level contrastive loss between them is defined as +La,b,i = ∥ya +i − zb +i ∥2 +2 = 2 − 2 · +⟨ya +i , zb +i ⟩ +∥ya +i ∥2 · ∥zb +i ∥2 +, +(1) + +Springer Nature 2021 LATEX template +Heterogeneous Tri-stream Clustering Network +7 +where ya +i and zb +i are the normalized representations. Thus the loss between the +first and second views can be expressed as +La,b = 1 +N +N +� +i=1 +La,b,i +(2) +Similar to BYOL [15], the exchange of the online and target views is performed +during each training step. Also, we utilize another online network to predict +the representations produced by the target network, whose loss is defined as +Lb,c,i = ∥yc +i − zb +i ∥2 +2 =2 − 2 · +⟨yc +i , zb +i ⟩ +∥yc +i ∥2 · ∥zb +i ∥2 +, i ∈ [1, N], +(3) +Lb,c = 1 +N +N +� +i=1 +Lb,c,i, +(4) +Therefore, the instance-level contrastive loss among the three streams of +networks is defined as +Linstance = La,b + Lb,c. +(5) +3.3 Cluster-level Contrastiveness +The cluster predictor maps the representations produced by the projector to +M-dimensional probability vectors, where M is the number of clusters. These +probability vectors, whose i-th element denotes how likely the image belongs to +the i-th cluster, can be interpreted as the soft label. Let qa, qb, qc ∈ RN×M be +the feature matrices produced by the cluster predictors of the three networks, +respectively. Each column of the feature matrix denotes an N-dimensional clus- +ter representation, denoted as qk +i , while the each row denotes a M-dimensional +probability vector, denoted as ˜qk +i (for k ∈ {a, b, c}). +For a cluster representation qa +i , we regard qa +i and qb +i as a positive cluster +pair, and the other 2·M −2 pairs (in the first and second views) as the negative +cluster pairs. The pair-wise similarity is defined as +s(qa +i , qb +j) = ⟨qa +i , qb +j⟩ +∥qa +i ∥∥qb +j∥, +i, j ∈ [1, M] +(6) +Then the InfoNCE loss for qa +i is computed by +ℓa +i = − log +exp(s(qa +i , qb +j)/τ) +�M +j=1[exp(s(qa +i , qa +j )/τ) + exp(s(qa +i , qb +j)/τ)] +, +(7) +where τ is the temperature parameter. After traversing all cluster representa- +tions, the cluster-level contrastive loss between the first and second augmented + +Springer Nature 2021 LATEX template +8 +Heterogeneous Tri-stream Clustering Network +views can be obtained as +ˆLa,b = +1 +2M +M +� +i=1 +(ℓa +i + ℓb +i) − H(Q), +(8) +where H(Q) is the entropy of the cluster-assignment probability, which helps +to avoid a degenerate solution that most images fall into the same cluster and +is computed as +H(Q) = − +M +� +i=1 +[P(qa +i ) log P(qa +i ) + P(qb +i ) log P(qb +i )], +(9) +P(qk +i ) = +N +� +j=1 +qk +ji +∥q∥1 +, +k ∈ {a, b} +(10) +For each batch of images, a view pair is formed between each online network +and the target network, leading to a total of two view pairs for the cluster- +level contrastive learning. Therefore, the cluster-level contrastive loss can be +defined as +Lcluster = ˆLa,b + ˆLb,c. +(11) +3.4 Overall Loss Function +The tri-stream network of HTCN is trained by simultaneously considering the +instance-level contrastiveness and the cluster-level contrastiveness. The overall +loss function is defined as +L = Linstance + Lcluster. +(12) +At each training step, we optimize the overall loss function w.r.t. the online +networks’ parameters θ only, but not the target network’s parameters ξ. The +parameters of the target is updated as an exponential moving average of that +of the online networks. That is +θ ← optimizer(θ, ∇θL, η), +(13) +ξ ← αξ + (1 − α)θ. +(14) +where η is the learning rate and α is the momentum coefficient. After the +training, we only keep the target network to perform clustering, which can be +obtained in the cluster predictor. + +Springer Nature 2021 LATEX template +Heterogeneous Tri-stream Clustering Network +9 +Table 1 Description of the benchmark image datasets. +Dataset +#Images +#Classes +CIFAR-100 +60,000 +20 +ImageNet-10 +13,000 +10 +ImageNet-Dogs +19,500 +15 +Tiny-ImageNet +100,000 +200 +(a) CIFAR-100 +(b) ImageNet-10 +(c) ImageNet-Dogs +(d) Tiny-ImageNet +Fig. 2 Some examples of the four image datasets. +3.5 Implementation Details +In HTCN, we use the ResNet34 [36] as the backbone. The projectors and the +instance predictors have the same network structure, each of which is a multi- +layer perceptron (MLP) with 256-dimensional output units. Each of the cluster +predictors is a two-layer MLP, whose output dimension is equal to the desired +number of clusters. +Three augmented (or distorted) views are generated by applying a family of +transformations to each input image. Five types of augmentations are utilized, +including ResizedCrop, HorizontalFlip, ColorJitter, Grayscale and Gaussian- +Blur [14]. As each transformation has a probability of being adopted, the +distortions of the three streams can thus be randomly decided. During opti- +mization, we use the Adam optimizer and train the model for 1000 epochs. +The learning rate is set to 0.0003.The batch size is set to 128. +4 Experiments +4.1 Datasets and Evaluation Metrics +The experiments are conducted on four widely-used image datasets, namely, +CIFAR-100 [37], ImageNet-10 [38], ImageNet-Dogs [38], and Tiny-ImageNet +[39]. The statistics of these benchmark datasets are given in Table 1, and some +sample images of these datasets are illustrated in Fig. 2. +To compare the clustering results of different clustering methods, three +evaluation metrics are adopted, including normalized mutual information + +Springer Nature 2021 LATEX template +10 +Heterogeneous Tri-stream Clustering Network +Table 2 The NMI(%) scores by different clustering methods (The best score in each +column is in bold). +Dataset +CIFAR-100 +ImageNet-10 +ImageNet-Dogs +Tiny-ImageNet +K-means [17] +8.4 +11.9 +5.5 +6.5 +SC [18] +9.0 +15.1 +3.8 +6.3 +AC [43] +9.8 +13.8 +3.7 +6.9 +NMF [44] +7.9 +13.2 +4.4 +7.2 +AE [45] +10.0 +21.0 +10.4 +13.1 +DAE [46] +11.1 +20.6 +10.4 +12.7 +DCGAN [47] +12.0 +22.5 +12.1 +13.5 +DeCNN [48] +9.2 +18.6 +9.8 +11.1 +VAE [49] +10.8 +19.3 +10.7 +11.3 +JULE [22] +10.3 +17.5 +5.4 +10.2 +DEC [7] +13.6 +28.2 +12.2 +11.5 +DAC [38] +18.5 +39.4 +21.9 +19.0 +DCCM [50] +28.5 +60.8 +32.1 +22.4 +GATC [51] +28.5 +59.4 +28.1 +- +PICA [19] +31.0 +80.2 +35.2 +27.7 +DRC [26] +35.6 +83.0 +38.4 +32.1 +CC [14] +43.1 +85.9 +44.5 +34.0 +HTCN +46.5 +87.5 +49.4 +35.6 +(NMI) [40], clustering accuracy (ACC) [41], and adjusted rand index (ARI) +[42]. +4.2 Comparison with State-of-the-Art +In this section, we compare the proposed method against four non-deep +clustering methods, namely, K-means [17], Spectral Clustering (SC) [18], +Agglomerative Clustering (AC) [43], and Nonnegative Matrix Factorization +(NMF) [44], and thirteen deep clustering methods, namely, Auto-Encoder +(AE) [45], Denoising Auto-Encoder (DAE) [46], Deep Convolutional Genera- +tive Adversarial Networks (DCGAN) [47], DeConvolutional Neural Networks +(DeCNN) [48], Aariational Auto-Encoder (VAE) [49], Joint Unsupervised +LEarning (JULE) [22], Deep Embedded Clustering (DEC) [7], Deep Adap- +tive Clustering (DAC) [38], Deep Comprehensive Correlation Mining (DCCM) +[50], Gaussian ATtention Network for image Clustering (GATC) [51], PartI- +tion Confidence mAximization (PICA) [19], Deep Robust Clustering (DRC) +[26] and Contrastive Clustering (CC) [14]. +As shown in Table 2, 3 and 4, our HTCN method achieves the best scores +on all the four benchmark datasets w.r.t. NMI, ACC, and ARI. Notably, on +the ImageNet-Dogs dataset, our HTCN method obtains NMI(%),ACC(%) and +ARI(%) scores of 49.4, 49.3, and 35.2, respectively, which significantly outper- +forms the second best method (i.e., CC) that obtains NMI(%),ACC(%) and +ARI(%) scores of 44.5, 42.9, and 27.4. The experimental results in Table 2, + +Springer Nature 2021 LATEX template +Heterogeneous Tri-stream Clustering Network +11 +Table 3 The ACC(%) scores by different clustering methods (The best score in each +column is in bold). +Dataset +CIFAR-100 +ImageNet-10 +ImageNet-Dogs +Tiny-ImageNet +K-means [17] +13.0 +24.1 +10.5 +2.5 +SC [18] +13.6 +27.4 +11.1 +2.2 +AC [43] +13.8 +24.2 +13.9 +2.7 +NMF [44] +11.8 +23.0 +11.8 +2.9 +AE [45] +16.5 +31.7 +18.5 +4.1 +DAE [46] +15.1 +30.4 +19.0 +3.9 +DCGAN [47] +15.3 +34.6 +17.4 +4.1 +DeCNN [48] +13.3 +31.3 +17.5 +3.5 +VAE [49] +15.2 +33.4 +17.9 +3.6 +JULE [22] +13.7 +30.0 +13.8 +3.3 +DEC [7] +18.5 +38.1 +19.5 +3.7 +DAC [38] +23.8 +52.7 +27.5 +6.6 +DCCM [50] +32.7 +71.0 +38.3 +10.8 +GATC [51] +32.7 +73.9 +32.2 +- +PICA [19] +33.7 +87.0 +35.2 +9.8 +DRC [26] +36.7 +88.4 +38.9 +13.9 +CC [14] +42.9 +89.3 +42.9 +14.0 +HTCN +47.2 +90.5 +49.3 +16.0 +Table 4 The ARI(%) scores by different clustering methods (The best score in each +column is in bold). +Dataset +CIFAR-100 +ImageNet-10 +ImageNet-Dogs +Tiny-ImageNet +K-means [17] +2.8 +5.7 +2.0 +0.5 +SC [18] +2.2 +7.6 +1.3 +0.4 +AC [43] +3.4 +6.7 +2.1 +0.5 +NMF [44] +2.6 +6.5 +1.6 +0.5 +AE [45] +4.8 +15.2 +7.3 +0.7 +DAE [46] +4.6 +13.8 +7.8 +0.7 +DCGAN [47] +4.5 +15.7 +7.8 +0.7 +DeCNN [48] +3.8 +14.2 +7.3 +0.6 +VAE [49] +4.0 +16.8 +7.9 +0.6 +JULE [22] +3.3 +13.8 +2.8 +0.6 +DEC [7] +5.0 +20.3 +7.9 +0.7 +DAC [38] +8.8 +30.2 +11.1 +1.7 +DCCM [50] +17.3 +55.5 +18.2 +3.8 +GATC [51] +17.3 +55.2 +16.3 +- +PICA [19] +17.1 +76.1 +20.1 +4.0 +DRC [26] +20.8 +79.8 +23.3 +5.6 +CC [14] +26.6 +82.2 +27.4 +7.1 +HTCN +30.5 +83.9 +35.2 +7.6 + +Springer Nature 2021 LATEX template +12 +Heterogeneous Tri-stream Clustering Network +Table 5 The clustering performance of HTCN using different combinations of network +architectures. +Architecture +NMI +ACC +ARI +Tri-stream architecture +46.5 +47.2 +30.5 +Dual-stream (Online+Target) +42.2 +42.5 +26.8 +Dual-stream (Online+Online) +39.9 +40.2 +24.4 +Table 6 The clustering performance of HTCN using different loss functions. +Loss function +NMI +ACC +ARI +With instance and cluster losses +46.5 +47.2 +30.5 +With only instance loss +43.3 +35.6 +14.6 +With only cluster loss +38.4 +36.4 +22.7 +3 and 4 confirm the advantageous clustering performance of HTCN over the +baseline methods. +4.3 Influence of the Tri-stream Architecture +In the proposed framework, we present a tri-stream architecture which consists +of two online networks and a target network. In this section, we test the influ- +ence of the three streams of networks. As shown in Table 5, using an online +network and a target network leads to better clustering results than using +two online networks, while using three streams of networks outperforms both +variants of using two streams, which shows the benefits of the heterogeneous +tri-stream architecture. +4.4 Influence of Two Types of Contrastive losses +In the section, we test the influence of the two types of contrastive losses, +i.e., the instance-level contrastive loss and the cluster-level contrastive loss. As +shown in Table 6, training with both types of losses can lead to better clustering +performance than training with only one of them, which confirm the joint +contribution of the instance-level and cluster-level losses in the self-supervised +training. +4.5 Influence of the Asymmetric Settings +Two symmetry-breaking mechanisms are enforced between the online and tar- +get networks [15]. First, an instance predictor is incorporated in each online +network, which does not exist in the target network. Second, the so-called +stop-gradient is incorporated in the target network, which indicates that this +network is not updated using backpropagation. We test the influence of the + +Springer Nature 2021 LATEX template +Heterogeneous Tri-stream Clustering Network +13 +Table 7 The NMI(%), ACC(%), and ARI(%) by HTCN removing different asymmetric +settings. +Asymmetric settings +NMI +ACC +ARI +HTCN +46.5 +47.2 +30.5 +No predictor +40.9 +41.2 +25.0 +No stop-gradient +39.2 +39.3 +23.7 +0 +200 +400 +600 +800 +1000 +Epochs +0 +10 +20 +30 +40 +50 +55 +NMI +(a) CIFAR-100 +0 +200 +400 +600 +800 +1000 +Epochs +0 +20 +40 +60 +80 +100 +NMI +(b) ImageNet-10 +0 +200 +400 +600 +800 +1000 +Epochs +0 +10 +20 +30 +40 +50 +55 +NMI +(c) ImageNet-dogs +0 +200 +400 +600 +800 +1000 +Epochs +0 +5 +10 +15 +20 +25 +30 +35 +40 +NMI +(d) Tiny-ImageNet +Fig. 3 Illustration of the convergence of HTCN (w.r.t. its NMI performance) on the four +benchmark datasets. +asymmetric settings by removing one of the instance predictor and the stop- +gradient. As shown in Table 7, training with both asymmetric settings leads +to better performance than training with only one of them. +4.6 Convergence Analysis +In this section, we test the convergence of the proposed HTCN method as +the number of epochs increases. As shown in Fig. 3, the clustering scores +(w.r.t. NMI) of the proposed HTCN method rapidly increase during the first +200 epochs on the benchmark datasets. When going beyond 200 epochs, the +increase of epochs still benefits the clustering performance consistently. In this +paper, the number of epochs is set to 1000 on all benchmark datasets. +5 Conclusion and Future Work +The paper develops a new deep clustering approach termed HTCN, which +breaks through the conventional two-stream contrastive architecture to explore +the rich possibilities in heterogeneous multi-stream contrastive learning and +clustering. In HTCN, the two weight-sharing online networks are trained by +predicting the instance representations of the target network and enforcing +the consistency between the cluster representations of the target and online +networks. Thus the tri-stream network architecture can be optimized in an +end-to-end manner via simultaneous instance-level and cluster-level contrastive +learning. Experimental results on four challenging image datasets have shown +the superior performance of our HTCN approach over the state-of-the-art deep + +Springer Nature 2021 LATEX template +14 +Heterogeneous Tri-stream Clustering Network +clustering approaches. In this paper, we mainly focus on the deep clustering +task for images. In the future work, a possible direction is to extend the pro- +posed framework to the deep clustering tasks for more complex data types, +such as time series data and document data. +Declarations +• Funding. +This +work +was +supported +by +the +NSFC +(61976097 +& +61876193) and the Natural Science Foundation of Guangdong Province +(2021A1515012203). +• Conflict of interest. The authors declare that they have no conflict of +interest. +• Ethical approval. This article does not contain any studies with human +participants or animals performed by any of the authors. +• Consent to participate. Informed consent to participate was obtained +from all individual participants included in the study. +• Consent for publication. Informed consent for publication was obtained +from all individual participants included in the study. +• Availability of data and materials. All datasets used in this paper are +publicly-available datasets. +• Code +availability. +The +code +is +available +at +https://github.com/ +dengxiaozhi/HTCN. +• Authors’ contributions. XD: Conceptualization, Methodology, Writing– +Original Draft. DH: Conceptualization, Writing–Review & Editing. CDW: +Optimization, Writing–Review & Editing. +References +[1] Frey, B.J., Dueck, D.: Clustering by passing messages between data points. +Science 315, 972–976 (2007) +[2] Huang, D., Wang, C.-D., Lai, J.-H.: Locally weighted ensemble clustering. +IEEE Transactions on Cybernetics 48(5), 1460–1473 (2018) +[3] Huang, D., Wang, C.-D., Wu, J.-S., Lai, J.-H., Kwoh, C.-K.: Ultra- +scalable spectral clustering and ensemble clustering. IEEE Transactions +on Knowledge and Data Engineering 32(6), 1212–1226 (2020) +[4] Yu, J., Tan, M., Zhang, H., Rui, Y., Tao, D.: Hierarchical deep click +feature prediction for fine-grained image recognition. 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In: Proc. of European +Conference on Computer Vision (ECCV), pp. 735–751 (2020) + diff --git a/CtE3T4oBgHgl3EQfUgpf/content/tmp_files/load_file.txt b/CtE3T4oBgHgl3EQfUgpf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..516ae50f617528960a0e0e63681a0edfef44a477 --- /dev/null +++ b/CtE3T4oBgHgl3EQfUgpf/content/tmp_files/load_file.txt @@ -0,0 +1,873 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf,len=872 +page_content='Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network Xiaozhi Deng1, Dong Huang1,2* and Chang-Dong Wang3 1*College of Mathematics and Informatics, South China Agricultural University, Guangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 2Key Laboratory of Smart Agricultural Technology in Tropical South China, Ministry of Agriculture and Rural Affairs, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 3School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' E-mail(s): huangdonghere@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Contributing authors: dengxiaozhi45@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' changdongwang@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Abstract Contrastive deep clustering has recently gained significant attention with its ability of joint contrastive learning and clustering via deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Despite the rapid progress, previous works mostly require both positive and negative sample pairs for contrastive cluster- ing, which rely on a relative large batch-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Moreover, they typically adopt a two-stream architecture with two augmented views, which over- look the possibility and potential benefits of multi-stream architectures (especially with heterogeneous or hybrid networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In light of this, this paper presents a new end-to-end deep clustering approach termed Heterogeneous Tri-stream Clustering Network (HTCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The tri-stream architecture in HTCN consists of three main components, including two weight-sharing online networks and a target network, where the param- eters of the target network are the exponential moving average of that of the online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Notably, the two online networks are trained by simultaneously (i) predicting the instance representations of the target network and (ii) enforcing the consistency between the cluster repre- sentations of the target network and that of the two online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Experimental results on four challenging image datasets demonstrate the superiority of HTCN over the state-of-the-art deep clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='com/dengxiaozhi/HTCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Keywords: Data clustering, Image clustering, Deep clustering, Deep neural network, Contrastive learning 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='04451v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='LG] 11 Jan 2023 Springer Nature 2021 LATEX template 2 Heterogeneous Tri-stream Clustering Network 1 Introduction Data clustering is the process of grouping data samples into multiple clus- ters in an unsupervised manner, which is a fundamental task in a variety of applications [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The traditional clustering algorithms typically focus on some low-level information and lack the representation learning ability, which may lead to sub-optimal performance when dealing with some complex high-dimensional data like images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In recent years, the deep learning has gained tremendous progress [4–6], which has also been exploited for tackling the clustering task, giving rise to the rapid development of the deep clustering algorithms [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' For example, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [7] presented a deep clustering method called Deep Embedded Clustering (DEC), which simultaneously learns representations and cluster assignments with an objective loss based on Kullback-Leibler (KL) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [8] extended DEC by incorporating the reconstruction loss (via autoencoder) to preserve local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [10] sought to learn invariant information of data by maximizing the mutual information between paired samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' More recently, the contrastive learning has emerged as a promising technique for exploiting sample-wise (or augmentation-wise) contrastiveness to improve the deep clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Van Gansbeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [12] presented the Semantic Clustering by Adopting Nearest neighbors (SCAN) method, which first adopts contrastive learning to learn discriminant features and then performs semantic clustering with the K-nearest neighbors exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [13] matched local-level and global-level nearest neighbors to further improve clustering per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [14] presented the Contrastive Clustering (CC) method to perform feature learning and clustering with simultaneous instance-level and cluster-level contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Despite significant success, these contrastive deep clustering methods [12– 14] are mostly faced with two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' On the one hand, they typically requires both positive sample pairs and negative sample pairs during their contrastive learning process, which rely on a relatively large batch-size (for sufficient negative pairs) and may bring in a heavier computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' On the other hand, these prior works generally adopt a two-stream architec- ture (with two weight-sharing augmented views), which neglect the possibility of going beyond the two-stream architecture to utilize three or even more streams of networks (with heterogeneous or hybrid structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Recently Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [15] presented the Bootstrap Your Own Latent (BYOL) method, which adopts an asymmetric two-stream architecture (with an online network and a target network) and conducts the contrastive learning without negative pairs, where the online network is trained by predicting the feature representations of the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Though the requirement for negative sample pairs is remedied, BYOL still complies with the two-stream architecture and also lacks the ability of directly learning the clustering structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' It remains a challeng- ing problem how to incorporate contrastive learning into multiple streams of heterogeneous networks while alleviating the dependence on negative sample pairs for strengthened deep clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network 3 In light of this, this paper presents a novel deep clustering approach termed Heterogeneous Tri-stream Clustering Network (HTCN), which lever- ages three streams of heterogeneous networks for simultaneous cluster-level and instance-level contrastive learning without requiring negative sample pairs (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Inspired by BYOL [15], we design a novel tri-stream architecture with three augmented views, corresponding to two online net- works and a target network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Note that the online network and the target network are heterogeneous, which differ from each other in the net- work structure and the updating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The two online networks share the same parameters, while the parameters of the target network are the exponential moving average of that of the online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Here, the exponen- tial moving average is a type of moving average that places a greater weight and significance on the most recent data samples [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Each online network is associated with an instance predictor and a cluster predictor, which pro- duce the instance-level representations and the cluster-level representations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Different from the online networks, the target network utilizes a cluster predictor to generate the cluster-level representations while producing the instance-level representations by the projector directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The incorporation of an instance predictor in the online networks is meant to prevent the potential collapse where the networks produce the same feature representations for most samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Then we train the two online networks by (i) predicting the target network’s representation of the same image via the mean squared error (MSE) loss (for the instance-level contrastive learning) and (ii) enforcing the consis- tency between the predicted cluster distributions of the two online networks and that of the target network via the information noise contrastive estimation (InfoNCE) [16] loss (for the cluster-level contrastive learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Experiments conducted on four image datasets demonstrate the superiority of our approach over the state-of-the-art deep clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' For clarity, the contributions of this work are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' A heterogeneous tri-stream architecture is designed, where two online net- works and a target network are jointly leveraged for instance-level and cluster-level contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' A novel deep clustering approach termed HTCN is proposed, which utilizes three augmented views for contrastive learning without requiring negative sample pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Experimental results on four image datasets confirm the advantegeous clus- tering performance of our HTCN approach over the state-of-the-art deep clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The related works on deep clustering and self-supervised learning are reviewed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The proposed HTCN framework is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The experiments are reported in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Finally, Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 Heterogeneous Tri-stream Clustering Network 2 Related Work In this section, we will introduce the related works on deep clustering and self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 Deep Clustering Traditional clustering methods such as K-means [17] and spectral cluster- ing (SC) [18] have achieved promising results in handling low-dimensional data, but they may result in sub-optimal performance when faced with high- dimensional data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=', images and videos) due to the lack of the representation learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' To address this, the deep learning based clustering methods, referred to as the deep clustering methods, have recently achieved significant success, which leverage the power of feature learning of deep neural networks for the clustering task [7–10, 12–14, 19–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Previous deep clustering methods can be divided into two main categories, namely, the one-stage methods and the two-stage methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The goal of the one-stage approach is to perform feature representation learning and clustering assignment simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [7] proposed a Deep Embedding Cluster- ing (DEC) method, which jointly optimizes feature learning and clustering with a KL-divergence loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [21] iteratively clustered the learned features with K-means and regarded the cluster assignments as supervisory signals to optimize the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [14] presented a Contrastive Clustering (CC) method that performs contrastive learning at instance-level and cluster-level for deep clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Besides the one-stage methods, some researchers have also made considerable efforts to the two-stage clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Van Gansbeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [12] proposed a two-stage clustering method called Semantic Clustering by Adopting Nearest neighbors (SCAN), which first learns the semantic fea- tures via contrastive learning and then utilizes the features for clustering in the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' To extend SCAN, Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [13] designed a Nearest Neighbor Matching (NNM) method, which selects both local and global nearest neigh- bors to optimize the network, where the neighbors are forced to be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 Self-supervised Learning Self-supervised learning has recently emerged as a powerful technique with the ability to learn representation from raw data without human supervision, in which the contrastive learning methods [28–31] have been a representative and promising category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The goal of contrastive learning is to minimize the distance between posi- tive sample pairs while maximizing the distance between negative sample pairs in a self-supervised manner, where positive pairs and negative pairs are defined through data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In particular, some researchers maintained a memory bank [28, 29] that contains large amounts of representations of nega- tive samples to achieve high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' However, these methods that utilize memory banks to store and update representations may be computationally Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network 5 expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' To address the problems with memory banks, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [30] pro- posed a Momentum Contrast (MoCo) method that trains an encoder by the momentum update mechanism maintaining a long queue of negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Following the MoCo method, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' [31] proposed a Simple framework for Contrastive LeaRning (SimCLR) method which carefully designs the strat- egy of data augmentation and a non-linear transformation head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In addition, the clustering based methods [32, 33] adopt a clustering approach to group similar features together, which address the issue that every sample is consid- ered as a discrete class in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' More recently, some self-supervised learning methods that only rely on positive pairs and directly predict the out- put of one augmented view from another augmented view [15, 34, 35] have been developed, among which a representative method is the BYOL method [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The BYOL method [15] adopts an asymmetric two-stream architecture, which, however, lacks the ability to learn the clustering structure directly and also overlooks the opportunities and potential benefits of going beyond the two-stream architecture to three or more streams of networks (even with het- erogeneous or hybrid structures) to further enhance the contrastive learning and clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 3 Proposed Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 Framework Overview This paper presents a heterogeneous tri-stream network architecture termed HTCN for contrastive deep clustering (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 1), which goes beyond the traditional two-stream architecture to explore the constrastive net- work in a multi-stream manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Also, HTCN doesn’t require negative sample pairs, which makes it more resilient to different batch-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Specifically, HTCN consists of three main components, including two online networks and a target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The online networks and the target network are respectively parame- terized by different sets of weights, where the parameters of the target network are an exponential moving average of that of the online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Given a batch of N images, we perform three types of augmentations on each image, denoted as xi with i ∈ [1, N], to generate 3 · N augmented (or distorted) images, denoted as {xa 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' , xa N, xb 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' , xb N, xc 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' , xc N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The back- bones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=', fθ and fξ) and projectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=', gθ and gξ) are adopted to extract features from the distorted images via za i = gθ(fθ(xa i )), zb i = gξ(fξ(xb i)) and zc i = gθ(fθ(xc i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Then the instance predictors transform za i and zc i to ya i and yc i , respectively, while the cluster predictors transform za i , zb i and zc i to ˜qa i , ˜qb i and ˜qc i , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Note that, similar to the asymmetric architecture of BYOL, the target network is not associated with an instance predictor, and the representations generated by its projector are used to guide the instance-level learning of the two online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The row space of the feature matrix learned by the projector or the instance predictor is expressed as the instance-level representations, while the column space of the feature matrix learned by the cluster predictor is expressed as the cluster-level representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 6 Heterogeneous Tri-stream Clustering Network 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑓" 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑔" ℎ" ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑝!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑥# 𝑥$ 𝑧# 𝑧$ 𝑦# 𝑞# 𝑞$ 𝑥 𝑞$ 𝑞# Minimize InfoNCE Loss Distorted images Backbone Cluster predictor Projector Online network Target network Instance predictor 𝑦# 𝑧$ Minimize MSE Loss 𝑦% 𝑧$ Input images 𝑞$ 𝑞% Minimize InfoNCE Loss Cluster predictor Distorted images Backbone Projector 𝑇# 𝑇$ 𝑇% 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑝!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 𝑥% 𝑧% 𝑦% 𝑞% Cluster predictor Distorted images Backbone Projector Instance predictor sg Online network Target network Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 1 Illustration of the proposed HTCN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The tri-stream network consists of two weight-sharing online networks and a target network, where the parameters of the target network is an exponential moving average of that of the online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Instance predictors and cluster predictors are incorporated in the three networks, after which the MSE loss and the InfoNCE loss are ultilized for instance-level contrastive learning and cluster-level contrastive learning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The network architecture can be trained in an end-to-end manner, where the final clustering is obtained via the cluster predictor of the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The instance-level representations are utilized to enforce the instance-level contrastive learning with an MSE loss optimized, while the cluster-level repre- sentations are utilized to enforce the cluster-level contrastive learning with an InfoNCE loss optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Finally, the instance-level and cluster-level contrastive losses are simultaneously utilized to optimize the tri-stream network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 Instance-level Contrastiveness Our HTCN approach simultaneously performs feature learning and clustering without requiring negative sample pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In instance-level contrastive learn- ing, we aim to train the two online networks by predicting the instance representations of target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Specifically, let ya i and zb i be the instance representations of xi in the first online network and the target network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The instance-level contrastive loss between them is defined as La,b,i = ∥ya i − zb i ∥2 2 = 2 − 2 · ⟨ya i , zb i ⟩ ∥ya i ∥2 · ∥zb i ∥2 , (1) Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network 7 where ya i and zb i are the normalized representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Thus the loss between the first and second views can be expressed as La,b = 1 N N � i=1 La,b,i (2) Similar to BYOL [15], the exchange of the online and target views is performed during each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Also, we utilize another online network to predict the representations produced by the target network, whose loss is defined as Lb,c,i = ∥yc i − zb i ∥2 2 =2 − 2 · ⟨yc i , zb i ⟩ ∥yc i ∥2 · ∥zb i ∥2 , i ∈ [1, N], (3) Lb,c = 1 N N � i=1 Lb,c,i, (4) Therefore, the instance-level contrastive loss among the three streams of networks is defined as Linstance = La,b + Lb,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 Cluster-level Contrastiveness The cluster predictor maps the representations produced by the projector to M-dimensional probability vectors, where M is the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' These probability vectors, whose i-th element denotes how likely the image belongs to the i-th cluster, can be interpreted as the soft label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Let qa, qb, qc ∈ RN×M be the feature matrices produced by the cluster predictors of the three networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Each column of the feature matrix denotes an N-dimensional clus- ter representation, denoted as qk i , while the each row denotes a M-dimensional probability vector, denoted as ˜qk i (for k ∈ {a, b, c}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' For a cluster representation qa i , we regard qa i and qb i as a positive cluster pair, and the other 2·M −2 pairs (in the first and second views) as the negative cluster pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The pair-wise similarity is defined as s(qa i , qb j) = ⟨qa i , qb j⟩ ∥qa i ∥∥qb j∥, i, j ∈ [1, M] (6) Then the InfoNCE loss for qa i is computed by ℓa i = − log exp(s(qa i , qb j)/τ) �M j=1[exp(s(qa i , qa j )/τ) + exp(s(qa i , qb j)/τ)] , (7) where τ is the temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' After traversing all cluster representa- tions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' the cluster-level contrastive loss between the first and second augmented Springer Nature 2021 LATEX template 8 Heterogeneous Tri-stream Clustering Network views can be obtained as ˆLa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='b = 1 2M M � i=1 (ℓa i + ℓb i) − H(Q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' (8) where H(Q) is the entropy of the cluster-assignment probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' which helps to avoid a degenerate solution that most images fall into the same cluster and is computed as H(Q) = − M � i=1 [P(qa i ) log P(qa i ) + P(qb i ) log P(qb i )],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' (9) P(qk i ) = N � j=1 qk ji ∥q∥1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' k ∈ {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' b} (10) For each batch of images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' a view pair is formed between each online network and the target network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' leading to a total of two view pairs for the cluster- level contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Therefore, the cluster-level contrastive loss can be defined as Lcluster = ˆLa,b + ˆLb,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' (11) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 Overall Loss Function The tri-stream network of HTCN is trained by simultaneously considering the instance-level contrastiveness and the cluster-level contrastiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The overall loss function is defined as L = Linstance + Lcluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' (12) At each training step, we optimize the overall loss function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' the online networks’ parameters θ only, but not the target network’s parameters ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The parameters of the target is updated as an exponential moving average of that of the online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' That is θ ← optimizer(θ, ∇θL, η), (13) ξ ← αξ + (1 − α)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' (14) where η is the learning rate and α is the momentum coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' After the training, we only keep the target network to perform clustering, which can be obtained in the cluster predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network 9 Table 1 Description of the benchmark image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Dataset #Images #Classes CIFAR-100 60,000 20 ImageNet-10 13,000 10 ImageNet-Dogs 19,500 15 Tiny-ImageNet 100,000 200 (a) CIFAR-100 (b) ImageNet-10 (c) ImageNet-Dogs (d) Tiny-ImageNet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 2 Some examples of the four image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 Implementation Details In HTCN, we use the ResNet34 [36] as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The projectors and the instance predictors have the same network structure, each of which is a multi- layer perceptron (MLP) with 256-dimensional output units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Each of the cluster predictors is a two-layer MLP, whose output dimension is equal to the desired number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Three augmented (or distorted) views are generated by applying a family of transformations to each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Five types of augmentations are utilized, including ResizedCrop, HorizontalFlip, ColorJitter, Grayscale and Gaussian- Blur [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' As each transformation has a probability of being adopted, the distortions of the three streams can thus be randomly decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' During opti- mization, we use the Adam optimizer and train the model for 1000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='The batch size is set to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 Datasets and Evaluation Metrics The experiments are conducted on four widely-used image datasets, namely, CIFAR-100 [37], ImageNet-10 [38], ImageNet-Dogs [38], and Tiny-ImageNet [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The statistics of these benchmark datasets are given in Table 1, and some sample images of these datasets are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' To compare the clustering results of different clustering methods, three evaluation metrics are adopted, including normalized mutual information Springer Nature 2021 LATEX template 10 Heterogeneous Tri-stream Clustering Network Table 2 The NMI(%) scores by different clustering methods (The best score in each column is in bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Dataset CIFAR-100 ImageNet-10 ImageNet-Dogs Tiny-ImageNet K-means [17] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 SC [18] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 AC [43] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 NMF [44] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 AE [45] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 DAE [46] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 DCGAN [47] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 DeCNN [48] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 VAE [49] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 JULE [22] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 DEC [7] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 DAC [38] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 DCCM [50] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 GATC [51] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 PICA [19] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 DRC [26] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 CC [14] 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 HTCN 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 (NMI) [40], clustering accuracy (ACC) [41], and adjusted rand index (ARI) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 Comparison with State-of-the-Art In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' we compare the proposed method against four non-deep clustering methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' K-means [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Spectral Clustering (SC) [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Agglomerative Clustering (AC) [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' and Nonnegative Matrix Factorization (NMF) [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' and thirteen deep clustering methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Auto-Encoder (AE) [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Denoising Auto-Encoder (DAE) [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Deep Convolutional Genera- tive Adversarial Networks (DCGAN) [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' DeConvolutional Neural Networks (DeCNN) [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Aariational Auto-Encoder (VAE) [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Joint Unsupervised LEarning (JULE) [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Deep Embedded Clustering (DEC) [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Deep Adap- tive Clustering (DAC) [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Deep Comprehensive Correlation Mining (DCCM) [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Gaussian ATtention Network for image Clustering (GATC) [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' PartI- tion Confidence mAximization (PICA) [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Deep Robust Clustering (DRC) [26] and Contrastive Clustering (CC) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' As shown in Table 2, 3 and 4, our HTCN method achieves the best scores on all the four benchmark datasets w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' NMI, ACC, and ARI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Notably, on the ImageNet-Dogs dataset, our HTCN method obtains NMI(%),ACC(%) and ARI(%) scores of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4, 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3, and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2, respectively, which significantly outper- forms the second best method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=', CC) that obtains NMI(%),ACC(%) and ARI(%) scores of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9, and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The experimental results in Table 2, Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network 11 Table 3 The ACC(%) scores by different clustering methods (The best score in each column is in bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Dataset CIFAR-100 ImageNet-10 ImageNet-Dogs Tiny-ImageNet K-means [17] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 SC [18] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 AC [43] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 NMF [44] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 AE [45] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 DAE [46] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 DCGAN [47] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 DeCNN [48] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 VAE [49] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 33.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 CC [14] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 HTCN 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 Table 4 The ARI(%) scores by different clustering methods (The best score in each column is in bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Dataset CIFAR-100 ImageNet-10 ImageNet-Dogs Tiny-ImageNet K-means [17] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 NMF [44] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 AE [45] 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 DCGAN [47] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 DeCNN [48] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 DEC [7] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 DAC [38] 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 GATC [51] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 PICA [19] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 DRC [26] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 CC [14] 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='1 HTCN 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 Springer Nature 2021 LATEX template 12 Heterogeneous Tri-stream Clustering Network Table 5 The clustering performance of HTCN using different combinations of network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Architecture NMI ACC ARI Tri-stream architecture 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 Dual-stream (Online+Target) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='8 Dual-stream (Online+Online) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 Table 6 The clustering performance of HTCN using different loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Loss function NMI ACC ARI With instance and cluster losses 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 With only instance loss 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 With only cluster loss 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 3 and 4 confirm the advantageous clustering performance of HTCN over the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 Influence of the Tri-stream Architecture In the proposed framework, we present a tri-stream architecture which consists of two online networks and a target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In this section, we test the influ- ence of the three streams of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' As shown in Table 5, using an online network and a target network leads to better clustering results than using two online networks, while using three streams of networks outperforms both variants of using two streams, which shows the benefits of the heterogeneous tri-stream architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='4 Influence of Two Types of Contrastive losses In the section, we test the influence of the two types of contrastive losses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=', the instance-level contrastive loss and the cluster-level contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' As shown in Table 6, training with both types of losses can lead to better clustering performance than training with only one of them, which confirm the joint contribution of the instance-level and cluster-level losses in the self-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 Influence of the Asymmetric Settings Two symmetry-breaking mechanisms are enforced between the online and tar- get networks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' First, an instance predictor is incorporated in each online network, which does not exist in the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Second, the so-called stop-gradient is incorporated in the target network, which indicates that this network is not updated using backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' We test the influence of the Springer Nature 2021 LATEX template Heterogeneous Tri-stream Clustering Network 13 Table 7 The NMI(%), ACC(%), and ARI(%) by HTCN removing different asymmetric settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Asymmetric settings NMI ACC ARI HTCN 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='5 No predictor 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='0 No stop-gradient 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='7 0 200 400 600 800 1000 Epochs 0 10 20 30 40 50 55 NMI (a) CIFAR-100 0 200 400 600 800 1000 Epochs 0 20 40 60 80 100 NMI (b) ImageNet-10 0 200 400 600 800 1000 Epochs 0 10 20 30 40 50 55 NMI (c) ImageNet-dogs 0 200 400 600 800 1000 Epochs 0 5 10 15 20 25 30 35 40 NMI (d) Tiny-ImageNet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 3 Illustration of the convergence of HTCN (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' its NMI performance) on the four benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' asymmetric settings by removing one of the instance predictor and the stop- gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' As shown in Table 7, training with both asymmetric settings leads to better performance than training with only one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='6 Convergence Analysis In this section, we test the convergence of the proposed HTCN method as the number of epochs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 3, the clustering scores (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' NMI) of the proposed HTCN method rapidly increase during the first 200 epochs on the benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' When going beyond 200 epochs, the increase of epochs still benefits the clustering performance consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In this paper, the number of epochs is set to 1000 on all benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' 5 Conclusion and Future Work The paper develops a new deep clustering approach termed HTCN, which breaks through the conventional two-stream contrastive architecture to explore the rich possibilities in heterogeneous multi-stream contrastive learning and clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In HTCN, the two weight-sharing online networks are trained by predicting the instance representations of the target network and enforcing the consistency between the cluster representations of the target and online networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Thus the tri-stream network architecture can be optimized in an end-to-end manner via simultaneous instance-level and cluster-level contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Experimental results on four challenging image datasets have shown the superior performance of our HTCN approach over the state-of-the-art deep Springer Nature 2021 LATEX template 14 Heterogeneous Tri-stream Clustering Network clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In this paper, we mainly focus on the deep clustering task for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' In the future work, a possible direction is to extend the pro- posed framework to the deep clustering tasks for more complex data types, such as time series data and document data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Declarations Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' This work was supported by the NSFC (61976097 & 61876193) and the Natural Science Foundation of Guangdong Province (2021A1515012203).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Ethical approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' This article does not contain any studies with human participants or animals performed by any of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Consent to participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Informed consent to participate was obtained from all individual participants included in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Consent for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Informed consent for publication was obtained from all individual participants included in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Availability of data and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' All datasets used in this paper are publicly-available datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Code availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content='com/ dengxiaozhi/HTCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' Authors’ contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' XD: Conceptualization, Methodology, Writing– Original Draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' DH: Conceptualization, Writing–Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} +page_content=' CDW: Optimization, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE3T4oBgHgl3EQfUgpf/content/2301.04451v1.pdf'} diff --git a/E9AyT4oBgHgl3EQf4_pw/content/tmp_files/2301.00796v1.pdf.txt b/E9AyT4oBgHgl3EQf4_pw/content/tmp_files/2301.00796v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..084b7b67fa84e4edc7bb2a4bce7f9aee74330b4b --- /dev/null +++ b/E9AyT4oBgHgl3EQf4_pw/content/tmp_files/2301.00796v1.pdf.txt @@ -0,0 +1,779 @@ +Direct lattice calculation of inclusive hadronic decay rates +of the 𝝉 lepton +A. Evangelista,𝑎,∗ R. Frezzotti,𝑎 G. Gagliardi,𝑏 V. Lubicz,𝑐 F. Sanfilippo,𝑏 S. Simula𝑏 +and N. Tantalo𝑎 +𝑎Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata", Via della Ricerca Scientifica 1, +I-00133 Rome, Italy +𝑏Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, +Italy +𝑐Dipartimento di Matematica e Fisica, Università di Roma Tre and INFN, Sezione di Roma Tre, Via della +Vasca Navale 84, I-00146 Rome, Italy +E-mail: antonio.evangelista@roma2.infn.it +The inclusive hadronic decay–rates of the 𝜏 lepton are particularly interesting from the phe- +nomenological point of view since they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠. In +this talk, we discuss how a recent method for the extraction of smeared spectral densities from Eu- +clidean lattice correlators can be used to obtain a direct lattice determination of inclusive hadronic +𝜏 decay rates. We also present preliminary numerical results obtained by applying this method +to correlators measured on two gauge ensembles produced by the ETMC with 𝑁 𝑓 = 2 + 1 + 1 +dynamical flavours at physical pion masses, lattice spacing 𝑎 ≃ 0.08 fm and volumes 𝐿 ≃ 5.1 fm +and 𝐿 ≃ 7.6 fm. +The 39th International Symposium on Lattice Field Theory (Lattice2022), +8-13 August, 2022 +Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.00796v1 [hep-lat] 2 Jan 2023 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +𝜏 +𝜈𝜏 +𝑓 +¯𝑔 +𝑝𝜏 +𝑝𝜈 +𝑋 𝑓 𝑔 +Figure 1: The 𝜏 → 𝜈𝜏𝑋 𝑓 𝑔 Feynman diagram (with no gluons). The hadronic final state 𝑋 𝑓 𝑔, with flavour +quantum numbers 𝑓 and 𝑔, has fixed 4-momentum 𝑝𝑋 = 𝑝𝜏 − 𝑝𝜈. +1. +Introduction +Inclusive hadronic 𝜏 decays are particularly interesting from the phenomenological viewpoint since +they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠. The determinations of 𝑉𝑢𝑠 from leptonic +and semileptonic kaon decays [1] are in fairly good agreement with the one of Ref. [2] but, for many +years, a puzzling tension with other determinations obtained from inclusive hadronic 𝜏 decays has +been observed and debated [1, 3]. On the lattice, hadronic 𝜏 decays have been studied by using +dispersion relations and by combining non-perturbative lattice inputs with perturbative and/or OPE +calculations (see for example [4]). +Here we present a method to perform a fully non-perturbative direct lattice calculation of the +𝜏 ↦→ 𝑋𝜈𝜏 decay rate. In our method, the decay rate is extracted from the two-point Euclidean +correlators of the hadronic weak currents that mediate the decay. This is done by using the algorithm +of Ref. [5] that allows to extract smeared spectral densities from Euclidean lattice correlators and, +building on Refs. [6, 7], by using as smearing kernels smoothed versions of the step-functions that +define the physical phase-space integration domain. +We also present preliminary numerical results obtained by applying this method to the relevant cor- +relators measured on two gauge ensembles produced by the Extended Twisted–Mass Collaboration +(ETMC) with 𝑁 𝑓 = 2 + 1 + 1 dynamical flavours with physical pion mass. The two ensembles, +corresponding to the cB211.072.64 (B64 in short) and cB211.072.96 (B96 in short) entries in +TABLE V of Ref. [8], have the same lattice spacing, 𝑎 = 0.07957(13) fm, and differ only for the +physical volumes that are 𝐿 = 5.09 fm and 𝐿 = 7.64 fm respectively. +2. +Reconstruction of the inclusive rate using the HLT method +By relying on the Fermi effective theory for the weak interactions and by neglecting long–distance +QED radiative corrections, the ratio 𝑅 𝑓 𝑔 of the inclusive hadronic decay rate Γ[𝜏 ↦→ 𝑋 𝑓 𝑔𝜈𝜏] with +the leptonic decay rate Γ[𝜏 ↦→ 𝑒 ¯𝜈𝑒𝜈𝜏] can be expressed as +𝑅 𝑓 𝑔 = 12𝜋 𝑆𝐸𝑊 +��𝑉𝑓 𝑔 +��2 ∫ 1 +𝑟 𝑓 𝑔 +d𝜔 𝜔 +� +1 − 𝜔2�2 � +𝜌𝐿 +𝑓 𝑔(𝜔) + 𝜌𝑇 +𝑓 𝑔(𝜔) +� +1 + 2𝜔2�� +. +(1) +In the previous formula, 𝑓 and 𝑔 label the flavour quantum numbers of the final hadronic states +𝑋 𝑓 𝑔 having four–momentum 𝑝𝑋, 𝑟 𝑓 𝑔 = 𝑚 𝑓 𝑔/𝑚𝜏 is the ratio of the mass of the lightest hadronic +state and the 𝜏-mass, 𝑆𝐸𝑊 = 1.0201(3) is the short–distance electroweak correction [9]. The +2 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +longitudinal and transverse form factors 𝜌𝐿 +𝑓 𝑔(𝜔) and 𝜌𝑇 +𝑓 𝑔(𝜔) parametrize the hadronic spectral +density +H 𝜇𝜈 +𝑓 𝑔(𝑝𝑋) = (2𝜋)4 ⟨0| 𝐻𝜇 +𝑓 𝑔(0) 𝛿(4) (P − 𝑝𝑋) 𝐻𝜈† +𝑓 𝑔(0) |0⟩ += 𝑝𝜇 +𝑋 𝑝𝜈 +𝑋 𝜌𝐿 +𝑓 𝑔(𝜔) + +� +𝑝𝜇 +𝑋 𝑝𝜈 +𝑋 − 𝑔𝜇𝜈𝑝2 +𝑋 +� +𝜌𝑇 +𝑓 𝑔(𝜔) , +𝜔2 = 𝑝2 +𝑋 +𝑚2𝜏 +, +(2) +where P = (H, �P) is the QCD four–momentum operator and 𝐻𝜇 +𝑓 𝑔 = 𝑉 𝜇 +𝑓 𝑔 − 𝐴𝜇 +𝑓 𝑔 is the hadronic weak +current that mediates the decay. +In the following, we concentrate on the 𝑢𝑑-flavour channel and omit the 𝑓 𝑔 flavour indexes in +intermediate expressions. Moreover, we study separately the longitudinal (𝐿) and transverse (𝑇) +contributions to 𝑅𝑢𝑑 and also the contributions coming from the vector (𝑉 𝜇) and axial-vector (𝐴𝜇) +currents. To this end we introduce the indexes +𝐼 = {𝐿,𝑇} , +𝐽 = {𝑉, 𝐴} , +(3) +and the different components of the spectral density H 𝜇𝜈(𝑝𝑋) according to +H 𝐿 +𝐽 (𝜔) ≡ H00 +𝐽 (𝜔) , +H𝑇 +𝐽 (𝜔) ≡ 1 +3 +3 +∑︁ +𝑖=1 +H𝑖𝑖 +𝐽 (𝜔) , +H 𝐼 (𝜔) ≡ H 𝐼 +𝑉 (𝜔) + H 𝐼 +𝐴(𝜔) , +(4) +with +H 𝜇𝜈 +𝐽 (𝑝𝑋) ≡ (2𝜋)4 ⟨0| 𝐽𝜇(0) 𝛿(4) (P − 𝑝𝑋) 𝐽𝜈†(0) |0⟩ . +(5) +By working in the reference frame where the final hadronic state is at rest, +𝑝𝑋 = (𝑚𝜏𝜔, �0) , +(6) +we have +𝑅𝐼 +𝐽 (𝜎) = 12𝜋 𝑆𝐸𝑊 |𝑉𝑢𝑑|2 +∫ ∞ +𝑟𝑢𝑑 +d𝜔 H 𝐼 +𝐽 (𝜔) 𝐾𝐼 +𝜎(𝜔) , +𝑅𝑢𝑑 = lim +𝜎→0 𝑅𝑢𝑑(𝜎) = lim +𝜎→0 +� +𝑅𝐿 +𝑉 (𝜎) + 𝑅𝐿 +𝐴(𝜎) + 𝑅𝑇 +𝑉 (𝜎) + 𝑅𝑇 +𝐴(𝜎) +� +, +(7) +where, in analogy to Refs. [6, 7], we have introduced the longitudinal (𝐾 𝐿 +𝜎) and transverse (𝐾𝑇 +𝜎) +smearing kernels +𝐾 𝐿 +𝜎(𝜔) = (1 − 𝜔2)2 +𝜔 +Θ𝜎(1 − 𝜔) , +𝐾𝑇 +𝜎(𝜔) = (1 − 𝜔2)2(1 + 2𝜔2) +𝜔 +Θ𝜎(1 − 𝜔) . +(8) +The function Θ𝜎(𝑥) appearing in the previous formula can be any 𝐶∞ smoothed version of the +step-function 𝜃(𝑥) such that lim𝜎↦→0 Θ𝜎(𝑥) = 𝜃(𝑥). In the following, we will consider the three +different choices given by +Θ(1) +𝜎 (𝑥) = +1 +1 + 𝑒−𝑥/𝜎 , +Θ(2) +𝜎 (𝑥) = +1 +1 + 𝑒− sinh(𝑥/𝜎) , +Θ(3) +𝜎 (𝑥) = 1 + Erf (𝑥/𝜎) +2 +. +(9) +3 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +Under the assumption that the spectral densities H 𝐼 (𝜔) are regular at the end-point of the phase- +space, i.e. 𝜔 = 1, an analytical calculation shows that the corrections to the 𝜎 ↦→ 0 limit are even +functions of 𝜎, starting at O�𝜎4�, i.e. +∫ ∞ +𝑟𝑢𝑑 +d𝜔 H 𝐼 (𝜔) +� +𝐾𝐼 +𝜎(𝜔) − 𝐾𝐼 +0 (𝜔) +� += O(𝜎4) , +(10) +This assumption is of course not valid on a finite volume where the spectral densities are not regular. +Indeed, because of the quantization of the spectrum, the finite–volume spectral densities H 𝐼 (𝜔) +are sums of Dirac 𝛿-functions localized in correspondence of the eigenvalues of the finite–volume +Hamiltonian. However, precisely for this reason and as emphasized in Ref. [5], the 𝜎 → 0 limit in +Eq. (7) has to be taken after performing the necessary 𝐿 → ∞ extrapolation of the lattice data. A +detailed numerical investigation of the dependence upon the volume of our results is postponed to a +future publication. Here, see below, we simply check that the results obtained on the two ensembles +with volumes 𝐿 ≃ 5.1 fm and 𝐿 ≃ 7.6 fm are compatible within the statistical uncertainties and +then attempt a 𝜎 → 0 extrapolation by relying on Eq. (10). +The representation of 𝑅𝑢𝑑 given in Eq. (7) allows for a straightforward application of the method +developed in Ref. [5] along the lines of Ref. [7]. The starting point is the relation between the +hadronic spectral density H 𝜇𝜈 +𝐽 (𝑝𝑋) and the Euclidean two-point correlator 𝐶𝜇𝜈 +𝐽 +at vanishing three- +momentum (our lattice input), i.e. +𝐶𝜇𝜈 +𝐽 (𝑡) ≡ +∫ +d3𝑥 T ⟨0| 𝐽𝜇(𝑎𝑡, �𝑥)𝐽𝜈†(0) |0⟩ = 𝑚𝜏 +2𝜋 +∫ ∞ +𝑟𝑢𝑑 +𝑑𝜔 H 𝜇𝜈 +𝐽 (𝑝𝑋) 𝑒−𝑎𝑚𝜏 𝜔𝑡, +𝑝𝑋 = (𝑚𝜏𝜔, �0), +(11) +where 𝑡 is the Euclidean time in units of the lattice spacing 𝑎.1 The main idea is then to express the +smeared-kernels 𝐾 𝐿 +𝜎(𝜔) and 𝐾𝑇 +𝜎(𝜔) in terms of the basis function {𝑒−𝑎𝑚𝜏 𝜔𝑡}𝑡=1,...,∞, i.e. +𝐾𝐼 +𝜎(𝜔) = +∞ +∑︁ +𝑡=1 +𝑔𝐼 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 . +(12) +In this way, once the coefficients 𝑔𝐼 (𝑡, 𝜎) are known, the longitudinal (𝑅𝐿 +𝐽 ) and transverse (𝑅𝑇 +𝐽 ) +contributions to 𝑅𝑢𝑑 can be computed from the knowledge of +𝐶𝐿 +𝐽 (𝑡) = − 2𝜋 +𝑚𝜏 +𝐶00 +𝐽 (𝑡) , +𝐶𝑇 +𝐽 (𝑡) = 2𝜋 +3𝑚𝜏 +3 +∑︁ +𝑖=1 +𝐶𝑖𝑖 +𝐽 (𝑡) , +(13) +by using +∞ +∑︁ +𝑡=1 +𝑔𝐼 (𝑡, 𝜎)𝐶𝐼 +𝐽 (𝑡) = +∫ ∞ +𝑟𝑢𝑑 +d𝜔 H 𝐼 +𝐽 (𝜔) 𝐾𝐼 +𝜎(𝜔) , +(14) +and inserting the result in Eqs. (7). However, as discussed thoroughly in Refs. [5], the problem +of finding the coefficients 𝑔𝐼 (𝑡, 𝜎) presents a certain number of technical difficulties. First of all, +1On a lattice having a finite temporal extent 𝑇, Eq. (11) must be modified replacing in the r.h.s. 𝑒−𝑎𝑚𝜏 𝜔𝑡 with +𝑒−𝑎𝑚𝜏 𝜔𝑡 + 𝑒−𝑎𝑚𝜏 𝜔(𝑇 −𝑡). +4 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +the sums appearing on the r.h.s. of Eqs. (12) need necessarily to be truncated at a finite value +𝑡 = 𝑡𝑚𝑎𝑥, hence the goal is to find a finite set of coefficients 𝑔𝐼 (𝑡, 𝜎), with 𝑡 ∈ {1, . . . , 𝑡𝑚𝑎𝑥}, +such that both the statistical (due to the fluctuation of 𝐶𝐼 +𝐽 (𝑡)) and the systematic errors (due to the +inexact reconstruction of the kernels) in the resulting determination of 𝑅𝐼 +𝐽 are under control. If +we were only concerned with systematic errors, the best coefficients 𝑔𝐼 (𝑡, 𝜎) could be obtained by +minimizing the quadratic form +𝐴𝐼 +𝛼[𝒈] = +∫ ∞ +𝐸0 +d𝜔 𝑒𝑎𝑚𝜏 𝜔𝛼�� 𝑓 (𝜔; 𝒈) − 𝐾𝐼 +𝜎(𝜔) +��2 , +(15) +with +𝑓 (𝜔; 𝒈) ≡ +𝑡max +∑︁ +𝑡=1 +𝑔(𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 . +(16) +Indeed, for any 𝛼 < 2 and 0 < 𝐸0 < 𝑟𝑢𝑑, the functional in Eq. (15) corresponds to a weighted +𝐿2-norm in the functional space defined in the interval [𝐸0, ∞]. However, for small values of +𝜎, the coefficients 𝑔𝐼 (𝑡, 𝜎) resulting from the minimization of 𝐴𝐼 +𝛼[𝒈] turn out to be very large +in magnitude and oscillating in sign, strongly amplifying the statistical errors of 𝐶𝐼 +𝐽 (𝑡) when the +𝑡max-truncated version of the sum in Eq. (14) is evaluated (see Ref. [5] for more details on this +point). +The method of Ref. [5], provides a regularization mechanism to this problem, enabling to find an +optimal balance between statistical and systematic errors. This is achieved by minimizing a linear +combination +𝑊 𝐼 𝐽 +𝛼 [𝒈] ≡ 𝐴𝐼 +𝛼[𝒈] +𝐴𝐼𝛼[0] + 𝜆𝐵𝐼 𝐽 [𝒈] , +(17) +of the norm-functional 𝐴𝐼 +𝛼[𝒈] and of the error-functional +𝐵𝐼 𝐽 [𝒈] = +1 +(𝐶𝐼 +𝐽 (0))2 +𝑡𝑚𝑎𝑥 +∑︁ +𝑡1,𝑡2=1 +𝑔(𝑡1, 𝜎) 𝑔(𝑡2, 𝜎) CovI +J(𝑡1, 𝑡2) , +(18) +where CovI +J(𝑡1, 𝑡2) is the covariance matrix of the lattice correlator 𝐶𝐽 +𝐼 (𝑡), and 𝜆 is the so-called +trade-off parameter [5]. For any fixed value of the algorithmic parameters 𝒑 ≡ {𝛼, 𝐸0, 𝜆, 𝑡𝑚𝑎𝑥}, the +minimization +𝜕𝑊 𝐼 𝐽 +𝛼 [𝒈] +𝜕𝑔(𝑡, 𝜎) +���� +𝒈=𝒈𝐼 𝐽 +𝒑 += 0 , +(19) +defines the coefficients 𝒈𝐼 𝐽 +𝒑 . The systematic error associated to the inexact reconstruction of the +smeared kernel, +𝐾𝐼 𝐽 +𝒑 (𝜔) ≡ 𝑓 (𝜔; 𝒈𝐼 𝐽 +𝒑 ) = +𝑡max +∑︁ +𝑡=1 +𝑔𝐼 𝐽 +𝒑 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 , +(20) +5 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +0.005 +0.006 +0.007 +0.008 +0.009 +0.010 +dT(gTV +p ) +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +RT +V( ) |Vud|2 +ensemble=B64 += 0 += 1 += 2 +0.005 +0.006 +0.007 +0.008 +0.009 +0.010 +dT(gTV +p ) +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +RT +V( ) |Vud|2 +ensemble=B96 += 0 += 1 += 2 +0 +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +KT( +) +KT( +) += 0 += 1 += 2 +0 +2 +4 +6 +8 +10 +0.04 +0.02 +0.00 +0.02 +0.04 +(KT( +) +KT, V +* +( +)) += 0 += 1 += 2 +reg=TM += 0.0500 +Figure 2: Top: the contribution 𝑅𝑇 +𝑉 /|𝑉𝑢𝑑|2 obtained using 𝛼 = 0 (green), 𝛼 = 1 (yellow) and 𝛼 = 2− +(blue), is plotted against 𝑑𝑇 (𝒈𝑇 𝑉 +𝒑 +) for 𝜎 = 0.05. For 𝛼 = 2−, the rightmost (leftmost) vertical dashed line +indicates the point satisfying Eq. (23) with 𝑟 = 104 (103), while the horizontal blue band corresponds to our +final determination obtained combining in quadrature the statistical and the systematic errors. The results +are shown in the TM lattice regularization for both the B64 (top-left figure) and the B96 (top-right figure) +ensembles at 𝜎 = 0.05. Bottom: the reconstructed smearing kernels 𝐾𝑇 𝑉 +∗ +(𝜔), obtained using the coefficients +𝒈𝑇 𝑉 +∗ +of Eq. (23) are compared, for 𝛼 = 0, 1, 2−, with the target one 𝐾𝑇 +𝜎(𝜔) for 𝜎 = 0.05 (bottom-left figure). +In the bottom-right figure we show 𝜔 · (𝐾𝑇 +𝜎(𝜔) − 𝐾𝑇 𝑉 +∗ +(𝜔)). +can be quantified through the quantity +𝑑𝐼 (𝒈) = +� +� +𝐴𝐼 +0 [𝒈] +𝐴𝐼 +0 [0] . +(21) +In the following, we will quote our best estimate for the four contributions 𝑅𝐿,𝑇 +𝑉 ,𝐴(𝜎), see Eq. (7), +performing the so-called stability analysis (see Ref. [10] and also the Supplementary Material of +Ref. [11]), which amounts to select the algorithmic parameters 𝒑 in such a way that the corresponding +𝑑𝐼 (𝒈𝐼 𝐽 +𝒑 ) is sufficiently small and the results stable, within statistical errors, under variations of 𝒑 +(the so-called statistically dominated regime).2 +3. +Numerical results +In this section, we present our preliminary results for 𝑅𝑢𝑑. These have been obtained by using +the Euclidean lattice correlators 𝐶𝐽 +𝐼 (𝑡) produced by the ETMC on the two ensembles B64 and +B96. We have considered two different discretized versions of the local weak current, peculiar to +our twisted-mass LQCD setup, that in the following will be indicated as twisted-mass (TM) and +Osterwalder-Seiler (OS) [12]. The results obtained using the two discretizations only differ by O(𝑎2) +cut-off effects, enabling us to approach the continuum limit in two different ways. Furthermore, we +2More numerical details on this point will be given in a forthcoming publication. +6 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0 +1 +2 +3 +4 +Rud( ) |Vud|2 +RL +A( ) |Vud|2 +RT +A( ) |Vud|2 +RL +V( ) |Vud|2 +RT +V( ) |Vud|2 +ens=B64 reg=TM +(1) +(2) +(3) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0 +1 +2 +3 +4 +Rud( ) |Vud|2 +RL +A( ) |Vud|2 +RT +A( ) |Vud|2 +RL +V( ) |Vud|2 +RT +V( ) |Vud|2 +ens=B96 reg=TM +(1) +(2) +(3) +Figure 3: The decay rate 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 as a function of 𝜎 in the range [0.0044, 0.1]. The results have +been obtained in the TM regularization and are shown for both the volumes (B64 top, B96 bottom) and for +the three choices of Θ(𝜔) in Eqs. (9). In the case Θ(1) +𝜎 (𝜔) we also show, separately, the four contributions +𝑅𝐿,𝑇 +𝑉 ,𝐴(𝜎)/|𝑉𝑢𝑑|2. +considered three different values, +𝛼 = {0, 1, 2−} , +(22) +for the parameter 𝛼 appearing in Eq. (15), where 𝛼 = 2− in practice means 𝛼 = 1.99. We set the +parameter 𝐸0 in Eq. (15) to 𝐸0 = 0.05 ≃ 0.6 𝑚 𝜋/𝑚𝜏 and use 𝑡𝑚𝑎𝑥 = 64, 96 respectively for the +ensembles B64 and B96. +In Figure 2 we show our determination of 𝑅𝑇 +𝑉 (𝜎)/|𝑉𝑢𝑑|2 in the TM regularization and at 𝜎 = 0.05, +obtained employing the three values of 𝛼 and the smeared kernel Θ(1) +𝜎 , see Eqs. (9). The results are +shown as a function of the parameter 𝑑𝑇 (𝒈𝑇 𝑉 +𝒑 +) defined in Eq. (21) and provide an illustrative example +of our stability analysis. For large values of 𝑑𝑇 (𝒈𝑇 𝑉 +𝒑 +) the results corresponding to different values +of 𝛼 are substantially different because in this regime the reconstruction of the smearing kernel +is very bad. At very small values of 𝑑𝑇 (𝒈𝑇 𝑉 +𝒑 +), where the quality of the reconstruction becomes +excellent, the results corresponding to the different values of 𝛼 become compatible because the +statistical errors are quite large. We observe that the results corresponding to 𝛼 = 1, 2− stabilize at +much larger values of 𝑑𝑇 (𝒈𝑇 𝑉 +𝒑 +) than the 𝛼 = 0 ones. This behaviour, already observed in Ref. [11] +where the same 𝐿2-norms have been used, can be explained by noticing that for 𝛼 > 0 the presence +7 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +5 +10 +15 +20 +25 +Rud( ) |Vud|2 +2 dof = 0.065 +Rud( += 0) |Vud|2 = 3.675 ± 0.072 +ens=B64 reg=TM +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +3.50 +3.75 +4.00 +4.25 +4.50 +(1) +(2) +(3) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +5 +10 +15 +20 +25 +Rud( ) |Vud|2 +2 dof = 0.058 +Rud( += 0) |Vud|2 = 3.562 ± 0.057 +ens=B96 reg=TM +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +3.50 +3.75 +4.00 +4.25 +4.50 +(1) +(2) +(3) +Figure 4: Combined 𝜎 → 0 extrapolations of our results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regulariza- +tion for both volumes. The datasets corresponding to the three choices of Θ(𝜔) appearing in Eqs. (9) have +different colours. Assuming negligible finite-volume effects, these are expected to have the same 𝜎 → 0 +limit and to differ at finite 𝜎 with leading corrections of 𝑂(𝜎4). The data have been fitted using the ansatz of +Eq. (25). The green point is the result of the extrapolation while the solid curves are the fitted curves 𝑅𝑘 (𝜎) +for 𝑘 = 1 (red), 𝑘 = 2 (blue) and 𝑘 = 3 (yellow). +of the exponential 𝑒𝑎𝑚𝜏 𝜔𝛼 in Eq. (15) improves the quality of the reconstruction in the large-𝜔 +region. Indeed, the errors in the reconstruction of the smearing kernels (e.g. 𝐾𝑇 +𝜎(𝜔)) for large +values of 𝜔 get amplified in the corresponding smeared quantities (e.g. 𝑅𝑇 +𝐽 (𝜎)) because, in general, +spectral densities grow asymptotically with the energy (e.g. H𝑇 +𝐽 (𝜔) ∝ 𝜔2). +For 𝛼 = 1, 2−, we found that the results obtained at the point 𝒈𝐼 𝐽 +∗ +such that the condition +𝐴𝐼 +𝛼[𝒈𝐼 𝐽 +∗ ] +𝐴𝐼𝛼[0] += 𝑟𝐵𝐼 𝐽 [𝒈𝐼 𝐽 +∗ ] , +𝑟 = 104 , +(23) +holds true, are in the statistically dominated regime. In what follows, the central values of the +four contributions to 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 in Eq. (7) are estimated by using the 𝛼 = 2− results (that are +remarkably stable) and the coefficients 𝒈𝐼 𝐽 +∗ . Residual systematic errors are instead evaluated by +re-performing the analysis using 𝑟 = 103 (see the vertical lines in Figure 2). Any variation of the +result corresponding to the choice 𝑟 = 103 w.r.t. the result corresponding to 𝑟 = 104 that goes +beyond a mere statistical fluctuation is added in quadrature to the statistical error. +In Figure 3 we show our preliminary results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regularization +8 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +𝐿 +TM (𝑎 = 0.08 fm) +OS (𝑎 = 0.08 fm) +HFLAV+HT (𝑎 = 0) +5.1 fm +3.675(72) +3.550(60) +7.6 fm +3.562(57) +3.676(236) +∞ +3.6615(78) +Table 1: Preliminary results for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 obtained in this work at fixed lattice spacing 𝑎 ≃ 0.08 fm in +both the TM and OS lattice regularizations on the volumes 𝐿 ≃ 5.1 fm (ensemble B64) and 𝐿 ≃ 7.6 fm +(ensemble B96). For comparison, we also show the result obtained by taking 𝑅𝑢𝑑 form Ref. [13] (HFLAV) +and 𝑉𝑢𝑑 from Ref. [14] (HT). +by using the three different smearing kernels of Eq. (9) and 23 values of 𝜎 in the range +𝜎 ∈ [0.0044, 0.2] . +(24) +We observe a remarkably flat behaviour for 𝜎 < 0.05 3. Moreover, the results corresponding to the +two volumes 𝐿 = 5.1 fm and 𝐿 = 7.6 fm are compatible at all values of 𝜎 within less than 1.5 standard +deviations. This implies that finite-volume effects are negligible within the quoted errors, even at +the smallest value of 𝜎 that we have considered. In the light of these observations, we attempted a +combined 𝜎 → 0 extrapolation of our results by relying on the infinite-volume asymptotic formula +of Eq. (10). On each ensemble and for each regularization, the results corresponding to the three +smearing kernels Θ(𝑘) +𝜎 (𝑘 = 1, 2, 3) have been fitted by using the following ansatz +𝑅𝑘(𝜎) = 𝑅 + 𝑐1,𝑘 · 𝜎4 + 𝑐2,𝑘 · 𝜎6 , +(25) +where 𝑐1,𝑘 and 𝑐2,𝑘 are free fit parameters which depend on the smearing kernel while 𝑅 ≡ +𝑅𝑢𝑑/|𝑉𝑢𝑑|2 is the common 𝜎 = 0 extrapolation. The quality of the fits is excellent on both volumes +and for both regularizations. In the case of the TM regularization, the results of these extrapolations +are shown in Figure. 4, again for the two volumes. +In Table. 1, we report our final determination, for the two considered volumes and for both the +TM and OS regularization. For comparison, we also reported in the table the result for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 +obtained by taking 𝑅𝑢𝑑 form Ref. [13] (HFLAV) and 𝑉𝑢𝑑 from Ref. [14] (HT). Although our OS +results on the B96 ensemble are still affected by a quite large uncertainty, the spread between the TM +and OS results on the B64 ensemble, where the accuracy is less than 2%, gives a first encouraging +indication about the size of the cut-off effects. We are currently performing a more detailed analysis +of all systematic effects and plan to extend the analysis to all the physical point ETMC ensembles +in order to carry out a reliable continuum limit extrapolation. +4. +Conclusion and Outlooks +We illustrated a method that allows to compute on the lattice the inclusive hadronic decay rates of +the 𝜏 lepton without the need of perturbative and/or OPE inputs. We also presented preliminary +3According to our experience, see e.g. Refs. [7, 10, 11], the numerical reconstruction of smearing kernels corre- +sponding to 𝜃-functions in the 𝜎 → 0 limit is much easier than in the case of kernels corresponding to Dirac 𝛿-functions. +9 + +Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton +A. Evangelista +results for the inclusive decay rate in the 𝑢𝑑-flavour channel. These have been obtained by applying +this method on two 𝑁 𝑓 = 2 + 1 + 1 QCD gauge ensembles produced by the ETMC with physical +pion masses, at fixed cutoff 𝑎 = 0.07957(13) fm and with volumes 𝐿 = 5.09 fm and 𝐿 = 7.64 fm. +In our method, as originally proposed in Refs. [6, 7], the step-functions that define the physical +phase-space integration domain are smoothed and used as smearing kernels in the algorithm of +Ref. [5]. Controlling the limit in which the smearing radius goes to zero is a crucial step of the +method, to be performed after the necessary infinite-volume extrapolations. Our numerical results +provide a rather solid evidence that this limit can be taken with controlled theoretical and numerical +errors. +We postpone to future work a more detailed illustration of the theoretical analysis of the vanishing +smearing width limit and a thorough investigation of all systematic uncertainties, including the +required continuum extrapolations. +We also plan to extend our computation to the more phe- +nomenologically relevant 𝑢𝑠-flavour channel. +References +[1] Flavour Lattice Averaging Group (FLAG) collaboration, FLAG Review 2021, Eur. Phys. J. C 82 +(2022) 869 [2111.09849]. +[2] R.J. Hudspith, R. Lewis, K. Maltman and J. Zanotti, A resolution of the inclusive flavor-breaking 𝜏 +|𝑉𝑢𝑠| puzzle, Phys. Lett. B 781 (2018) 206 [1702.01767]. +[3] K. Maltman et al., Current Status of inclusive hadronic 𝜏 determinations of |V𝑢𝑠|, SciPost Phys. 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C 102 (2020) 045501. +10 + diff --git a/E9AyT4oBgHgl3EQf4_pw/content/tmp_files/load_file.txt b/E9AyT4oBgHgl3EQf4_pw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef2bad2a9dda75be27516aaf768ec4e8bb165060 --- /dev/null +++ b/E9AyT4oBgHgl3EQf4_pw/content/tmp_files/load_file.txt @@ -0,0 +1,426 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf,len=425 +page_content='Direct lattice calculation of inclusive hadronic decay rates of the 𝝉 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista,𝑎,∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Frezzotti,𝑎 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Gagliardi,𝑏 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Lubicz,𝑐 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Sanfilippo,𝑏 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Simula𝑏 and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Tantalo𝑎 𝑎Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata", Via della Ricerca Scientifica 1, I-00133 Rome, Italy 𝑏Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy 𝑐Dipartimento di Matematica e Fisica, Università di Roma Tre and INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy E-mail: antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='evangelista@roma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='it The inclusive hadronic decay–rates of the 𝜏 lepton are particularly interesting from the phe- nomenological point of view since they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In this talk, we discuss how a recent method for the extraction of smeared spectral densities from Eu- clidean lattice correlators can be used to obtain a direct lattice determination of inclusive hadronic 𝜏 decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We also present preliminary numerical results obtained by applying this method to correlators measured on two gauge ensembles produced by the ETMC with 𝑁 𝑓 = 2 + 1 + 1 dynamical flavours at physical pion masses, lattice spacing 𝑎 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 fm and volumes 𝐿 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1 fm and 𝐿 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory (Lattice2022), 8-13 August, 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00796v1 [hep-lat] 2 Jan 2023 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista 𝜏 𝜈𝜏 𝑓 ¯𝑔 𝑝𝜏 𝑝𝜈 𝑋 𝑓 𝑔 Figure 1: The 𝜏 → 𝜈𝜏𝑋 𝑓 𝑔 Feynman diagram (with no gluons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The hadronic final state 𝑋 𝑓 𝑔, with flavour quantum numbers 𝑓 and 𝑔, has fixed 4-momentum 𝑝𝑋 = 𝑝𝜏 − 𝑝𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Introduction Inclusive hadronic 𝜏 decays are particularly interesting from the phenomenological viewpoint since they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The determinations of 𝑉𝑢𝑠 from leptonic and semileptonic kaon decays [1] are in fairly good agreement with the one of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [2] but, for many years, a puzzling tension with other determinations obtained from inclusive hadronic 𝜏 decays has been observed and debated [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' On the lattice, hadronic 𝜏 decays have been studied by using dispersion relations and by combining non-perturbative lattice inputs with perturbative and/or OPE calculations (see for example [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Here we present a method to perform a fully non-perturbative direct lattice calculation of the 𝜏 ↦→ 𝑋𝜈𝜏 decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In our method, the decay rate is extracted from the two-point Euclidean correlators of the hadronic weak currents that mediate the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' This is done by using the algorithm of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5] that allows to extract smeared spectral densities from Euclidean lattice correlators and, building on Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [6, 7], by using as smearing kernels smoothed versions of the step-functions that define the physical phase-space integration domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We also present preliminary numerical results obtained by applying this method to the relevant cor- relators measured on two gauge ensembles produced by the Extended Twisted–Mass Collaboration (ETMC) with 𝑁 𝑓 = 2 + 1 + 1 dynamical flavours with physical pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The two ensembles, corresponding to the cB211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='64 (B64 in short) and cB211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='96 (B96 in short) entries in TABLE V of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [8], have the same lattice spacing, 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='07957(13) fm, and differ only for the physical volumes that are 𝐿 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='09 fm and 𝐿 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='64 fm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Reconstruction of the inclusive rate using the HLT method By relying on the Fermi effective theory for the weak interactions and by neglecting long–distance QED radiative corrections, the ratio 𝑅 𝑓 𝑔 of the inclusive hadronic decay rate Γ[𝜏 ↦→ 𝑋 𝑓 𝑔𝜈𝜏] with the leptonic decay rate Γ[𝜏 ↦→ 𝑒 ¯𝜈𝑒𝜈𝜏] can be expressed as 𝑅 𝑓 𝑔 = 12𝜋 𝑆𝐸𝑊 ��𝑉𝑓 𝑔 ��2 ∫ 1 𝑟 𝑓 𝑔 d𝜔 𝜔 � 1 − 𝜔2�2 � 𝜌𝐿 𝑓 𝑔(𝜔) + 𝜌𝑇 𝑓 𝑔(𝜔) � 1 + 2𝜔2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (1) In the previous formula, 𝑓 and 𝑔 label the flavour quantum numbers of the final hadronic states 𝑋 𝑓 𝑔 having four–momentum 𝑝𝑋, 𝑟 𝑓 𝑔 = 𝑚 𝑓 𝑔/𝑚𝜏 is the ratio of the mass of the lightest hadronic state and the 𝜏-mass, 𝑆𝐸𝑊 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0201(3) is the short–distance electroweak correction [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The 2 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista longitudinal and transverse form factors 𝜌𝐿 𝑓 𝑔(𝜔) and 𝜌𝑇 𝑓 𝑔(𝜔) parametrize the hadronic spectral density H 𝜇𝜈 𝑓 𝑔(𝑝𝑋) = (2𝜋)4 ⟨0| 𝐻𝜇 𝑓 𝑔(0) 𝛿(4) (P − 𝑝𝑋) 𝐻𝜈† 𝑓 𝑔(0) |0⟩ = 𝑝𝜇 𝑋 𝑝𝜈 𝑋 𝜌𝐿 𝑓 𝑔(𝜔) + � 𝑝𝜇 𝑋 𝑝𝜈 𝑋 − 𝑔𝜇𝜈𝑝2 𝑋 � 𝜌𝑇 𝑓 𝑔(𝜔) , 𝜔2 = 𝑝2 𝑋 𝑚2𝜏 , (2) where P = (H, �P) is the QCD four–momentum operator and 𝐻𝜇 𝑓 𝑔 = 𝑉 𝜇 𝑓 𝑔 − 𝐴𝜇 𝑓 𝑔 is the hadronic weak current that mediates the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In the following, we concentrate on the 𝑢𝑑-flavour channel and omit the 𝑓 𝑔 flavour indexes in intermediate expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Moreover, we study separately the longitudinal (𝐿) and transverse (𝑇) contributions to 𝑅𝑢𝑑 and also the contributions coming from the vector (𝑉 𝜇) and axial-vector (𝐴𝜇) currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' To this end we introduce the indexes 𝐼 = {𝐿,𝑇} , 𝐽 = {𝑉, 𝐴} , (3) and the different components of the spectral density H 𝜇𝜈(𝑝𝑋) according to H 𝐿 𝐽 (𝜔) ≡ H00 𝐽 (𝜔) , H𝑇 𝐽 (𝜔) ≡ 1 3 3 ∑︁ 𝑖=1 H𝑖𝑖 𝐽 (𝜔) , H 𝐼 (𝜔) ≡ H 𝐼 𝑉 (𝜔) + H 𝐼 𝐴(𝜔) , (4) with H 𝜇𝜈 𝐽 (𝑝𝑋) ≡ (2𝜋)4 ⟨0| 𝐽𝜇(0) 𝛿(4) (P − 𝑝𝑋) 𝐽𝜈†(0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (5) By working in the reference frame where the final hadronic state is at rest, 𝑝𝑋 = (𝑚𝜏𝜔, �0) , (6) we have 𝑅𝐼 𝐽 (𝜎) = 12𝜋 𝑆𝐸𝑊 |𝑉𝑢𝑑|2 ∫ ∞ 𝑟𝑢𝑑 d𝜔 H 𝐼 𝐽 (𝜔) 𝐾𝐼 𝜎(𝜔) , 𝑅𝑢𝑑 = lim 𝜎→0 𝑅𝑢𝑑(𝜎) = lim 𝜎→0 � 𝑅𝐿 𝑉 (𝜎) + 𝑅𝐿 𝐴(𝜎) + 𝑅𝑇 𝑉 (𝜎) + 𝑅𝑇 𝐴(𝜎) � , (7) where, in analogy to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [6, 7], we have introduced the longitudinal (𝐾 𝐿 𝜎) and transverse (𝐾𝑇 𝜎) smearing kernels 𝐾 𝐿 𝜎(𝜔) = (1 − 𝜔2)2 𝜔 Θ𝜎(1 − 𝜔) , 𝐾𝑇 𝜎(𝜔) = (1 − 𝜔2)2(1 + 2𝜔2) 𝜔 Θ𝜎(1 − 𝜔) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (8) The function Θ𝜎(𝑥) appearing in the previous formula can be any 𝐶∞ smoothed version of the step-function 𝜃(𝑥) such that lim𝜎↦→0 Θ𝜎(𝑥) = 𝜃(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In the following, we will consider the three different choices given by Θ(1) 𝜎 (𝑥) = 1 1 + 𝑒−𝑥/𝜎 , Θ(2) 𝜎 (𝑥) = 1 1 + 𝑒− sinh(𝑥/𝜎) , Θ(3) 𝜎 (𝑥) = 1 + Erf (𝑥/𝜎) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (9) 3 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista Under the assumption that the spectral densities H 𝐼 (𝜔) are regular at the end-point of the phase- space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝜔 = 1, an analytical calculation shows that the corrections to the 𝜎 ↦→ 0 limit are even functions of 𝜎, starting at O�𝜎4�, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' ∫ ∞ 𝑟𝑢𝑑 d𝜔 H 𝐼 (𝜔) � 𝐾𝐼 𝜎(𝜔) − 𝐾𝐼 0 (𝜔) � = O(𝜎4) , (10) This assumption is of course not valid on a finite volume where the spectral densities are not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Indeed, because of the quantization of the spectrum, the finite–volume spectral densities H 𝐼 (𝜔) are sums of Dirac 𝛿-functions localized in correspondence of the eigenvalues of the finite–volume Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' However, precisely for this reason and as emphasized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5], the 𝜎 → 0 limit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (7) has to be taken after performing the necessary 𝐿 → ∞ extrapolation of the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' A detailed numerical investigation of the dependence upon the volume of our results is postponed to a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Here, see below, we simply check that the results obtained on the two ensembles with volumes 𝐿 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1 fm and 𝐿 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 fm are compatible within the statistical uncertainties and then attempt a 𝜎 → 0 extrapolation by relying on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The representation of 𝑅𝑢𝑑 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (7) allows for a straightforward application of the method developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5] along the lines of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The starting point is the relation between the hadronic spectral density H 𝜇𝜈 𝐽 (𝑝𝑋) and the Euclidean two-point correlator 𝐶𝜇𝜈 𝐽 at vanishing three- momentum (our lattice input), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝐶𝜇𝜈 𝐽 (𝑡) ≡ ∫ d3𝑥 T ⟨0| 𝐽𝜇(𝑎𝑡, �𝑥)𝐽𝜈†(0) |0⟩ = 𝑚𝜏 2𝜋 ∫ ∞ 𝑟𝑢𝑑 𝑑𝜔 H 𝜇𝜈 𝐽 (𝑝𝑋) 𝑒−𝑎𝑚𝜏 𝜔𝑡, 𝑝𝑋 = (𝑚𝜏𝜔, �0), (11) where 𝑡 is the Euclidean time in units of the lattice spacing 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1 The main idea is then to express the smeared-kernels 𝐾 𝐿 𝜎(𝜔) and 𝐾𝑇 𝜎(𝜔) in terms of the basis function {𝑒−𝑎𝑚𝜏 𝜔𝑡}𝑡=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=',∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝐾𝐼 𝜎(𝜔) = ∞ ∑︁ 𝑡=1 𝑔𝐼 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (12) In this way, once the coefficients 𝑔𝐼 (𝑡, 𝜎) are known, the longitudinal (𝑅𝐿 𝐽 ) and transverse (𝑅𝑇 𝐽 ) contributions to 𝑅𝑢𝑑 can be computed from the knowledge of 𝐶𝐿 𝐽 (𝑡) = − 2𝜋 𝑚𝜏 𝐶00 𝐽 (𝑡) , 𝐶𝑇 𝐽 (𝑡) = 2𝜋 3𝑚𝜏 3 ∑︁ 𝑖=1 𝐶𝑖𝑖 𝐽 (𝑡) , (13) by using ∞ ∑︁ 𝑡=1 𝑔𝐼 (𝑡, 𝜎)𝐶𝐼 𝐽 (𝑡) = ∫ ∞ 𝑟𝑢𝑑 d𝜔 H 𝐼 𝐽 (𝜔) 𝐾𝐼 𝜎(𝜔) , (14) and inserting the result in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' However, as discussed thoroughly in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5], the problem of finding the coefficients 𝑔𝐼 (𝑡, 𝜎) presents a certain number of technical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' First of all, 1On a lattice having a finite temporal extent 𝑇, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (11) must be modified replacing in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝑒−𝑎𝑚𝜏 𝜔𝑡 with 𝑒−𝑎𝑚𝜏 𝜔𝑡 + 𝑒−𝑎𝑚𝜏 𝜔(𝑇 −𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 4 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista the sums appearing on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (12) need necessarily to be truncated at a finite value 𝑡 = 𝑡𝑚𝑎𝑥, hence the goal is to find a finite set of coefficients 𝑔𝐼 (𝑡, 𝜎), with 𝑡 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' , 𝑡𝑚𝑎𝑥}, such that both the statistical (due to the fluctuation of 𝐶𝐼 𝐽 (𝑡)) and the systematic errors (due to the inexact reconstruction of the kernels) in the resulting determination of 𝑅𝐼 𝐽 are under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' If we were only concerned with systematic errors, the best coefficients 𝑔𝐼 (𝑡, 𝜎) could be obtained by minimizing the quadratic form 𝐴𝐼 𝛼[𝒈] = ∫ ∞ 𝐸0 d𝜔 𝑒𝑎𝑚𝜏 𝜔𝛼�� 𝑓 (𝜔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝒈) − 𝐾𝐼 𝜎(𝜔) ��2 , (15) with 𝑓 (𝜔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝒈) ≡ 𝑡max ∑︁ 𝑡=1 𝑔(𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (16) Indeed, for any 𝛼 < 2 and 0 < 𝐸0 < 𝑟𝑢𝑑, the functional in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (15) corresponds to a weighted 𝐿2-norm in the functional space defined in the interval [𝐸0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' However, for small values of 𝜎, the coefficients 𝑔𝐼 (𝑡, 𝜎) resulting from the minimization of 𝐴𝐼 𝛼[𝒈] turn out to be very large in magnitude and oscillating in sign, strongly amplifying the statistical errors of 𝐶𝐼 𝐽 (𝑡) when the 𝑡max-truncated version of the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (14) is evaluated (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5] for more details on this point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The method of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5], provides a regularization mechanism to this problem, enabling to find an optimal balance between statistical and systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' This is achieved by minimizing a linear combination 𝑊 𝐼 𝐽 𝛼 [𝒈] ≡ 𝐴𝐼 𝛼[𝒈] 𝐴𝐼𝛼[0] + 𝜆𝐵𝐼 𝐽 [𝒈] , (17) of the norm-functional 𝐴𝐼 𝛼[𝒈] and of the error-functional 𝐵𝐼 𝐽 [𝒈] = 1 (𝐶𝐼 𝐽 (0))2 𝑡𝑚𝑎𝑥 ∑︁ 𝑡1,𝑡2=1 𝑔(𝑡1, 𝜎) 𝑔(𝑡2, 𝜎) CovI J(𝑡1, 𝑡2) , (18) where CovI J(𝑡1, 𝑡2) is the covariance matrix of the lattice correlator 𝐶𝐽 𝐼 (𝑡), and 𝜆 is the so-called trade-off parameter [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' For any fixed value of the algorithmic parameters 𝒑 ≡ {𝛼, 𝐸0, 𝜆, 𝑡𝑚𝑎𝑥}, the minimization 𝜕𝑊 𝐼 𝐽 𝛼 [𝒈] 𝜕𝑔(𝑡, 𝜎) ���� 𝒈=𝒈𝐼 𝐽 𝒑 = 0 , (19) defines the coefficients 𝒈𝐼 𝐽 𝒑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The systematic error associated to the inexact reconstruction of the smeared kernel, 𝐾𝐼 𝐽 𝒑 (𝜔) ≡ 𝑓 (𝜔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝒈𝐼 𝐽 𝒑 ) = 𝑡max ∑︁ 𝑡=1 𝑔𝐼 𝐽 𝒑 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 , (20) 5 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='010 dT(gTV p ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2 RT V( ) |Vud|2 ensemble=B64 = 0 = 1 = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='010 dT(gTV p ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2 RT V( ) |Vud|2 ensemble=B96 = 0 = 1 = 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0 KT( ) KT( ) = 0 = 1 = 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='04 (KT( ) KT, V ( )) = 0 = 1 = 2 reg=TM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0500 Figure 2: Top: the contribution 𝑅𝑇 𝑉 /|𝑉𝑢𝑑|2 obtained using 𝛼 = 0 (green), 𝛼 = 1 (yellow) and 𝛼 = 2− (blue), is plotted against 𝑑𝑇 (𝒈𝑇 𝑉 𝒑 ) for 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' For 𝛼 = 2−, the rightmost (leftmost) vertical dashed line indicates the point satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (23) with 𝑟 = 104 (103), while the horizontal blue band corresponds to our final determination obtained combining in quadrature the statistical and the systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The results are shown in the TM lattice regularization for both the B64 (top-left figure) and the B96 (top-right figure) ensembles at 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Bottom: the reconstructed smearing kernels 𝐾𝑇 𝑉 ∗ (𝜔), obtained using the coefficients 𝒈𝑇 𝑉 ∗ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (23) are compared, for 𝛼 = 0, 1, 2−, with the target one 𝐾𝑇 𝜎(𝜔) for 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='05 (bottom-left figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In the bottom-right figure we show 𝜔 · (𝐾𝑇 𝜎(𝜔) − 𝐾𝑇 𝑉 ∗ (𝜔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' can be quantified through the quantity 𝑑𝐼 (𝒈) = � � 𝐴𝐼 0 [𝒈] 𝐴𝐼 0 [0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (21) In the following, we will quote our best estimate for the four contributions 𝑅𝐿,𝑇 𝑉 ,𝐴(𝜎), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (7), performing the so-called stability analysis (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [10] and also the Supplementary Material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [11]), which amounts to select the algorithmic parameters 𝒑 in such a way that the corresponding 𝑑𝐼 (𝒈𝐼 𝐽 𝒑 ) is sufficiently small and the results stable, within statistical errors, under variations of 𝒑 (the so-called statistically dominated regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Numerical results In this section, we present our preliminary results for 𝑅𝑢𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' These have been obtained by using the Euclidean lattice correlators 𝐶𝐽 𝐼 (𝑡) produced by the ETMC on the two ensembles B64 and B96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We have considered two different discretized versions of the local weak current, peculiar to our twisted-mass LQCD setup, that in the following will be indicated as twisted-mass (TM) and Osterwalder-Seiler (OS) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The results obtained using the two discretizations only differ by O(𝑎2) cut-off effects, enabling us to approach the continuum limit in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Furthermore, we 2More numerical details on this point will be given in a forthcoming publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 6 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='10 0 1 2 3 4 Rud( ) |Vud|2 RL A( ) |Vud|2 RT A( ) |Vud|2 RL V( ) |Vud|2 RT V( ) |Vud|2 ens=B64 reg=TM (1) (2) (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='10 0 1 2 3 4 Rud( ) |Vud|2 RL A( ) |Vud|2 RT A( ) |Vud|2 RL V( ) |Vud|2 RT V( ) |Vud|2 ens=B96 reg=TM (1) (2) (3) Figure 3: The decay rate 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 as a function of 𝜎 in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0044, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The results have been obtained in the TM regularization and are shown for both the volumes (B64 top, B96 bottom) and for the three choices of Θ(𝜔) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In the case Θ(1) 𝜎 (𝜔) we also show, separately, the four contributions 𝑅𝐿,𝑇 𝑉 ,𝐴(𝜎)/|𝑉𝑢𝑑|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' considered three different values, 𝛼 = {0, 1, 2−} , (22) for the parameter 𝛼 appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (15), where 𝛼 = 2− in practice means 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We set the parameter 𝐸0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (15) to 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='05 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 𝑚 𝜋/𝑚𝜏 and use 𝑡𝑚𝑎𝑥 = 64, 96 respectively for the ensembles B64 and B96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In Figure 2 we show our determination of 𝑅𝑇 𝑉 (𝜎)/|𝑉𝑢𝑑|2 in the TM regularization and at 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='05, obtained employing the three values of 𝛼 and the smeared kernel Θ(1) 𝜎 , see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The results are shown as a function of the parameter 𝑑𝑇 (𝒈𝑇 𝑉 𝒑 ) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (21) and provide an illustrative example of our stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' For large values of 𝑑𝑇 (𝒈𝑇 𝑉 𝒑 ) the results corresponding to different values of 𝛼 are substantially different because in this regime the reconstruction of the smearing kernel is very bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' At very small values of 𝑑𝑇 (𝒈𝑇 𝑉 𝒑 ), where the quality of the reconstruction becomes excellent, the results corresponding to the different values of 𝛼 become compatible because the statistical errors are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We observe that the results corresponding to 𝛼 = 1, 2− stabilize at much larger values of 𝑑𝑇 (𝒈𝑇 𝑉 𝒑 ) than the 𝛼 = 0 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' This behaviour, already observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [11] where the same 𝐿2-norms have been used, can be explained by noticing that for 𝛼 > 0 the presence 7 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='200 5 10 15 20 25 Rud( ) |Vud|2 2 dof = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='065 Rud( = 0) |Vud|2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='675 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='072 ens=B64 reg=TM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='50 (1) (2) (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='200 5 10 15 20 25 Rud( ) |Vud|2 2 dof = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='058 Rud( = 0) |Vud|2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='562 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='057 ens=B96 reg=TM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='50 (1) (2) (3) Figure 4: Combined 𝜎 → 0 extrapolations of our results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regulariza- tion for both volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The datasets corresponding to the three choices of Θ(𝜔) appearing in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (9) have different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Assuming negligible finite-volume effects, these are expected to have the same 𝜎 → 0 limit and to differ at finite 𝜎 with leading corrections of 𝑂(𝜎4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The data have been fitted using the ansatz of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The green point is the result of the extrapolation while the solid curves are the fitted curves 𝑅𝑘 (𝜎) for 𝑘 = 1 (red), 𝑘 = 2 (blue) and 𝑘 = 3 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' of the exponential 𝑒𝑎𝑚𝜏 𝜔𝛼 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (15) improves the quality of the reconstruction in the large-𝜔 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Indeed, the errors in the reconstruction of the smearing kernels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝐾𝑇 𝜎(𝜔)) for large values of 𝜔 get amplified in the corresponding smeared quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 𝑅𝑇 𝐽 (𝜎)) because, in general, spectral densities grow asymptotically with the energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' H𝑇 𝐽 (𝜔) ∝ 𝜔2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' For 𝛼 = 1, 2−, we found that the results obtained at the point 𝒈𝐼 𝐽 ∗ such that the condition 𝐴𝐼 𝛼[𝒈𝐼 𝐽 ∗ ] 𝐴𝐼𝛼[0] = 𝑟𝐵𝐼 𝐽 [𝒈𝐼 𝐽 ∗ ] , 𝑟 = 104 , (23) holds true, are in the statistically dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In what follows, the central values of the four contributions to 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (7) are estimated by using the 𝛼 = 2− results (that are remarkably stable) and the coefficients 𝒈𝐼 𝐽 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Residual systematic errors are instead evaluated by re-performing the analysis using 𝑟 = 103 (see the vertical lines in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Any variation of the result corresponding to the choice 𝑟 = 103 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' the result corresponding to 𝑟 = 104 that goes beyond a mere statistical fluctuation is added in quadrature to the statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In Figure 3 we show our preliminary results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regularization 8 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista 𝐿 TM (𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 fm) OS (𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 fm) HFLAV+HT (𝑎 = 0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1 fm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='675(72) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='550(60) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 fm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='562(57) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='676(236) ∞ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6615(78) Table 1: Preliminary results for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 obtained in this work at fixed lattice spacing 𝑎 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='08 fm in both the TM and OS lattice regularizations on the volumes 𝐿 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1 fm (ensemble B64) and 𝐿 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 fm (ensemble B96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' For comparison, we also show the result obtained by taking 𝑅𝑢𝑑 form Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [13] (HFLAV) and 𝑉𝑢𝑑 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [14] (HT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' by using the three different smearing kernels of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (9) and 23 values of 𝜎 in the range 𝜎 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='0044, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (24) We observe a remarkably flat behaviour for 𝜎 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Moreover, the results corresponding to the two volumes 𝐿 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='1 fm and 𝐿 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='6 fm are compatible at all values of 𝜎 within less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' This implies that finite-volume effects are negligible within the quoted errors, even at the smallest value of 𝜎 that we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In the light of these observations, we attempted a combined 𝜎 → 0 extrapolation of our results by relying on the infinite-volume asymptotic formula of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' On each ensemble and for each regularization, the results corresponding to the three smearing kernels Θ(𝑘) 𝜎 (𝑘 = 1, 2, 3) have been fitted by using the following ansatz 𝑅𝑘(𝜎) = 𝑅 + 𝑐1,𝑘 · 𝜎4 + 𝑐2,𝑘 · 𝜎6 , (25) where 𝑐1,𝑘 and 𝑐2,𝑘 are free fit parameters which depend on the smearing kernel while 𝑅 ≡ 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 is the common 𝜎 = 0 extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' The quality of the fits is excellent on both volumes and for both regularizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In the case of the TM regularization, the results of these extrapolations are shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 4, again for the two volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 1, we report our final determination, for the two considered volumes and for both the TM and OS regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' For comparison, we also reported in the table the result for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 obtained by taking 𝑅𝑢𝑑 form Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [13] (HFLAV) and 𝑉𝑢𝑑 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [14] (HT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Although our OS results on the B96 ensemble are still affected by a quite large uncertainty, the spread between the TM and OS results on the B64 ensemble, where the accuracy is less than 2%, gives a first encouraging indication about the size of the cut-off effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We are currently performing a more detailed analysis of all systematic effects and plan to extend the analysis to all the physical point ETMC ensembles in order to carry out a reliable continuum limit extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Conclusion and Outlooks We illustrated a method that allows to compute on the lattice the inclusive hadronic decay rates of the 𝜏 lepton without the need of perturbative and/or OPE inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We also presented preliminary 3According to our experience, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [7, 10, 11], the numerical reconstruction of smearing kernels corre- sponding to 𝜃-functions in the 𝜎 → 0 limit is much easier than in the case of kernels corresponding to Dirac 𝛿-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' 9 Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Evangelista results for the inclusive decay rate in the 𝑢𝑑-flavour channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' These have been obtained by applying this method on two 𝑁 𝑓 = 2 + 1 + 1 QCD gauge ensembles produced by the ETMC with physical pion masses, at fixed cutoff 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='07957(13) fm and with volumes 𝐿 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='09 fm and 𝐿 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content='64 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' In our method, as originally proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [6, 7], the step-functions that define the physical phase-space integration domain are smoothed and used as smearing kernels in the algorithm of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Controlling the limit in which the smearing radius goes to zero is a crucial step of the method, to be performed after the necessary infinite-volume extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' Our numerical results provide a rather solid evidence that this limit can be taken with controlled theoretical and numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We postpone to future work a more detailed illustration of the theoretical analysis of the vanishing smearing width limit and a thorough investigation of all systematic uncertainties, including the required continuum extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} +page_content=' We also plan to extend our computation to the more 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'} diff --git a/ENA0T4oBgHgl3EQfA_81/content/tmp_files/2301.01969v1.pdf.txt b/ENA0T4oBgHgl3EQfA_81/content/tmp_files/2301.01969v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6205502c7052207190d237cca28796dcb88129f --- /dev/null +++ b/ENA0T4oBgHgl3EQfA_81/content/tmp_files/2301.01969v1.pdf.txt @@ -0,0 +1,1530 @@ +Prepared for submission to JINST +Imaging of muon track in CsI(Tl) crystal with +single photon sensitive camera +Zhimin Wang,a,b,c,1 Min Li,a,b Diru Wu,a,b Jinchang Liu,a,c Yongpeng Zhang,a,c +Xiangcheng Meng,a Caimei Liu,a,b Changgen Yanga,b +aInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China +bUniversity of Chinese Academy of Sciences, Beijing 100049, China +cState Key Laboratory of Particle Detection and Electronics, Beijing 100049, China +E-mail: wangzhm@ihep.ac.cn +Abstract: As a novel approach on visual photon imaging by a single photon sensitive +camera and PMTs, this work is trying to measure and identify muon tracks from the 2-D +images of CsI(Tl) crystal (scintillator detectors). It is possible that muon tracks can be +seen directly with a good signal-to-noise ratio neither with further amplification nor external +light, which provides an evolution method for particle measurement in the photon-starved +regime of scintillation detectors. The setup of the crystal and camera testing system and +the identification algorithm of muon track will be discussed in detail including the system +calibration, identification model, signal-to-noise ratio, muon track confirmation, and an +expectation on further improvements and applications. +Keywords: photon detectors, scintillator detector, imaging, single photon, camera, muon +track +ArXiv ePrint: 1234.56789 +1Corresponding author. +arXiv:2301.01969v1 [physics.ins-det] 5 Jan 2023 + +Contents +1 +Introduction +1 +2 +CsI(Tl) crystal with camera +2 +2.1 +Setup +2 +2.2 +Calibration +3 +3 +Muon track +5 +3.1 +Signal vs. Noise +5 +3.2 +Image track survey +8 +3.3 +Muon identification +10 +4 +Discussion +12 +4.1 +Muon tracks or not? +12 +4.2 +Possible system optimization +13 +4.3 +Further applications +13 +5 +Summary +14 +1 +Introduction +Vertex and track reconstruction are critical for most particle physics experiments, such as +studies on neutrino, dark matter, and others. There is a long list of related technologies +including but not limited to emulsion film[1–3], cloud chamber[4], bubble chamber[5], spark +chamber[6], multi-wire proportional chamber[7], TPC[8], Si strip[9] and Si pixel[10] etc. In +the case of photon-based detection, in particular, PMT or SiPM is the commonly used sensor +for timing, intensity, and crude spatial reconstruction, such as JUNO[11, 12], Darkside[13], +JUNO-TAO[14], SNO+[15, 16], and DUNE[17] etc., where computer algorithms are further +used to have a better reconstruction on the vertex or track[18–20]. +Recently, many efforts are focusing on photon imaging-related projects following the +new development of sensors, where the critical challenges are the need for high spatial +resolution over large volumes[21] and better effective signal-to-noise ratio under the photon- +starved regime. +For many years classical emulsion film radiography is being replaced by digital detec- +tor imaging, especially in medical applications due to faster and more reliable diagnostics +and computed tomography and tomosynthesis capabilities[22]. The single photon count- +ing X-ray CCD camera spectrometer is used in laser-plasma interaction experiments as a +simple tool to study the K-shell X-ray generation. A CCD detector enables the spectrum +of the impinging X-ray radiation to be obtained without further dispersive devices[23]. +– 1 – + +Among the imaging systems used for thermal neutron imaging worldwide, the most preva- +lent configuration is CCD camera based[24]. +Single-photon light detection and ranging +(lidar) offers single-photon sensitivity and picosecond timing resolution, which is desirable +for high-precision three-dimensional (3D) imaging over long distances[25]. Single image 3D +photography enables viewers to view a still image from novel viewpoints[26]. Some good +sensors are developed too, such as SPC3[27], a single photon counting camera based on a +2-D imaging array. A small, high resolution, high signal-to-noise GEM-based TPC with a +2-D CCD readout designed to provide a benchmark for background discrimination and di- +rectional sensitivity that could be used for future optimization studies for directional dark +matter experiments [28, 29]. A skipper CCD was also developed for very low noise and +directly measured a muon track through ionization inside the sensor[30]. +But, generally, it is not suitable to directly image of vertex or track in case of a starved- +photon regime and uniform angular distribution of the photons[21, 31]. Photography by +CCD or other technologies, in particular single photon imaging, provides another new pos- +sibility, such as our previous study for particle imaging by event[32]. +In this article, we will try to have a further detailed check on the imaging of muon track +in CsI(Tl) crystal with a single photon sensitive camera and PMTs. Sec.2 will introduce +the system setup and calibration. Sec.3 will discuss the expected features of muon tracks, +measurements, and track surveys. +Sec.4 will provide further discussions on the results, +possible improvements of the system, and further expectations. And a short summary is in +Sec.5. +2 +CsI(Tl) crystal with camera +2.1 +Setup +An imaging system, as in [32], is set up with a single photon sensitive camera of ORCA- +Quest qCMOS C15550-20UP, which is a new product of Hamamatsu Photonics [33]. The +detailed layout of the system is shown in Fig. 1. The output of the camera will save in tif +format with 16 bits of each of the 4096(H)×2304(V) pixels and the volume of each photo is +around 16 MB. The CsI(Tl) crystal (7.5×7.5×15 cm3) is located in front of the camera and +the two 3-inch PMTs, where the distance between them can be adjusted. An alpha source +of 241Am is used and put on the top surface of the crystal. The two 3-inch PMTs are used +to calibrate and monitor the signal intensity of the crystal, the coincidence of which is used +as a trigger of the CAEN DT5751 (1 GS/s with 1 V p-p dynamic range, [34]) for waveform +data taking. The threshold of each 3-inch PMT is set to around 1 p.e. (photon-electron). +The maximum rate of the data-taking system is limited by the DT5751, which is generally +lower than 100 Hz with data saving of four channels and 10000 samples of each channel. +Here the window length of the waveform recording is set to 6 µs (6000 samples/waveform), +and the maximum data-taking rate is around 70-80 Hz. +In order to increase the acceptance of the emitted photons from the crystal, a lens with +a much short focal length and small number aperture (1/2”, C type, 6-∞ mm, f/1.4) is used. +The images of the crystal with different distances are shown in Fig. 2, which are taken with +illumination before the dark box is closed. The field of view with the used lens is in a circle +– 2 – + +Figure 1: Layout of the imaging measurement system. +and much smaller than the full size of the camera sensor. The circle shape and its outside +of the field of view will be considered in the following measurement and analysis. Please +note that there is a clear distortion around the edge of the field of view (crystal region) +when the object distance is too small as in Fig. 2b. +(a) 15 cm +(b) 4 cm +Figure 2: Crystal photos when the dark box is open with natural illumination and different +object distances. +2.2 +Calibration +Fig. 3a shows the measured charge spectra with the alpha source located when the object +distance is around 15 cm: the two 3-inch PMTs and the sum of them. A long tail can be +found at the right of the spectrum, which is known as cosmic muons. A 400 p.e. cut to the +sum spectrum is used to select the events of muons. A factor of particle identification (PID) +is calculated by each waveform of each PMT, and it is defined by the ratio of the charge in +the first 300 ns to the whole window, as shown in Fig. 3b. A 2-D cut on the PID is used to +identify the events from the alpha source, the red dash line (PID_pmt_0+PID_pmt_1) +as shown in Fig. 3c. +The selected spectra are shown in Fig. 3d, where the blue curve +is selected as the alpha-like events by (PID_pmt_0+PID_pmt_1>0.5), the Magenta +– 3 – + +couple +Source +Camera +Lens +Power +USB cable +Crystal +Dark Boxcurve is selected for the muon events candidates by (PID_pmt_0+PID_pmt_1<0.5 and +charge_pmt_0+charge_pmt_1>400 p.e.), and the green curve is assumed as the gamma- +like events by (PID_pmt_0+PID_pmt_1<0.5). +(a) Charge spectra of eac PMT and sum +(b) PID of each event of each PMT +(c) 2-D distribution of PID of the two PMTs +(d) Selected events by PID and charge +Figure 3: Measured charge spectra by the 3-inch PMTs, and the PID distrbution of the +waveform. +Taking into account the dead time of the data-taking system, the actual event rate of +each measurement is re-normalized according to the selected muon rate, where the reference +muon rate of the crystal is from the measurement and selection of without source with object +distance of 15 cm. +It is around 3 Hz for selected muons, 2.6 Hz for gamma-like events, +and 2.2 Hz for alpha-like events of the measurement without source and object distance of +15 cm. The data-taking rate and the re-normalized rate are listed in Table 1 for different +configurations. The rate of alpha-like events is around 100 Hz and increases from 93 Hz to +188 Hz following the shortening of the object distance from 15 cm to 4 cm. The contribution +from the alpha source is much higher than that from the background of around 2.2 Hz. +Following the classification of the events, the mean charge of each kind of event is +calculated too. The mean charge of the alpha-like events is 130 p.e. of 15 cm, 150 p.e. of +10 cm, and 159 p.e. of 4 cm, respectively. The signal intensity is not following the solid angle +simply, for the 4 cm, in particular, which is because the distance to the 3-inch PMTs is +rather small than the object distance of the camera to the crystal according to the layout +of the system. The mean intensity of the muon events is around 2100 p.e., which suffers +from statistic uncertainty and solid angle issues too. +The images of the crystal with the alpha source are taken by the camera with different +exposure times and different object distances. The region of the source is selected and shown +in Fig. 4, where the selected image dimension of the sensor is around 0.23 mm ×0.28 mm +– 4 – + +Count +Sum single_S +sPMT single 0 +102 +sPMT single1 +10 +10 +102 +103 +Charge/peCount +700 +h1D_pidpmto +600 +h1D pid pmt1 +500 +400 +300 +200 +100 +0.2 +0.4 +0.6 +0.8 +PIDPMT_ +16 +14 +0.8 +12 +0.6 +10 +8 +0.4 +6 +4 +0.2 +0.2 +0.4 +0.6 +0.8 +C +0 +PMT 0Count +10 +Sum singleS +Sum_single_S_high +Sum_single_S_low +Sumsingle_S_low_gamma +102 +SumsingleSlowmuon +10 +10 +102 +103 +Charge/peTable 1: Event rate and charge intensity of crystal with two 3-inch PMTs. The distance +is between the camera and the crystal front surface. +The events are measured by the +coincidence by the two 3-inch PMTs, and the charge is from the sum of the two PMTs +of each event. The alpha-like events are selected by the sum of PID, and the separation +between muon and gamma-like is by a charge cut after the PID cut. +Type +DAQ Rate +Normalized +Rate (Hz) +Mean Charge (p.e.) +(Distance) +(Hz) +Rate (Hz) +Muon +Gamma-like +Alpha-like +Muon +Gamma-like +Alpha-like +w/o source 15 cm +∼8 +7.8 +3.0 +2.6 +2.2 +2099.0 +181.3 +124.5 +w/ source 15 cm +∼80 +93.1 +3.0 +6.2 +83.9 +2108.9 +184.0 +130.2 +w/ source 10 cm +∼70 +160.1 +3.0 +18.2 +138.9 +2057.4 +204.0 +150.4 +w/ source 4 cm +∼70 +188.9 +3.0 +21.3 +164.6 +2062.8 +207.1 +159.5 +with 50 pixel (V) ×60 pixels (H) and 4.6 µm × 4.6 µm per pixel. The regime of the alpha +source can be identified as around 3 mm scale of an object distance of 15 cm, 2 mm scale +of 10 cm, and 1 mm scale of 4 cm, and the light intensity gradually dims when shortening +the exposure time. It is almost identified to event level with 0.05 s exposure time but on a +higher noise background, where only a few alphas occur during the time. The dimension of +the image of the source is enlarging when the object distance reduces as expected. +The intensity of the source region is integrated and converted into p.e. as shown in +Fig. 5, where the noise (baseline is around 200 ADC) of the camera is subtracted according +to a parallel region of the source with equal area [32]. +The conversion factor is around +7.8 ADC/p.e. The diameter of the source region is 21 pixels for an object distance of 15 cm, +22 pixels for 10 cm, and 33 pixels for 4 cm. The fitted intensity per second by a linear curve +is around 314 p.e. of an object distance of 15 cm, 862 p.e. of 10 cm, and 2255 p.e. of 4 cm, +respectively. +Considering the rate of the alpha source measured under different object +distances as in Tab. 1, the ratio of measured charge intensity of the camera and the PMTs +is around 3% of an object distance of 15 cm, 4% of 10 cm, and 8% of 4 cm, respectively. +The expected typical charge intensity of alpha-like event viewed by the camera is around +3.9 p.e. of an object distance of 15 cm, 6.0 p.e. of 10 cm, and 12.8 p.e. of 4 cm, respectively. +The expected typical charge intensity of each muon viewed by the camera is around 60 p.e. of +an object distance of 15 cm, 85 p.e. of 10 cm, and 177 p.e. of 4 cm, respectively. +3 +Muon track +3.1 +Signal vs. Noise +As stated in [32], the noise of the camera is still much higher than the traditional used +PMT or SiPM, which is much worse when we are trying to use many pixels for imaging +measurement. It can be expected that it will help to identify the target by a smaller area +and stronger intensity of the same object, as seen in the left plateau of the curves in Fig. 5, +where the difference of the plateau level (noise) is mainly from the dimension of the imaging +area. The minimum of the plateau is from the object distance of 15 cm configuration, which +is proportional to the ratio squared of the focal length to object distance, even the final +– 5 – + +Figure 4: 2-D images of alpha source with object distances of 15 cm (50 pixel (V) ×60 pixels +(H), left), 10 cm (50 pixel (V) ×60 pixels (H), middle) and 4 cm (50 pixel (V) ×60 pixels (H), +right) versus exposure time of 60 s, 1 s, 100 ms, 50 ms and 20 ms. The pixel size is 4.6 µm +× 4.6 µm. The z-axis is in the unit of ADC which is the camera output per pixel directly +relative to the light intensity. The baseline of each pixel is around 200 ADC, and the gain +factor is around 7.8 ADC/p.e. +signal-to-noise (SN) ratio is also proportional to the square of the reciprocal of focal length +except for the effective aperture. +A calculation is further executed to compare the effect of the imaging shape under the +same noise level. Here a pure statistic model is used to check the mean of each pixel and +total sum with an assumption of 0.3 p.e. noise level per pixel (sigma of Gaussian) in a circle +(diameter) or line (in length and in width of one pixel) shape. The uncertainty of the mean +of each pixel is inversely proportional to the square root of total pixel numbers as shown in +Fig. 6a, where more pixels will have smaller fluctuation comparing circles to lines. While +the total sum of all pixels is proportional to the square root of pixel numbers shown in +Fig. 6b, where the required sum in lines is much smaller than that of in circles to identify +a signal. If we aim to identify a signal by a three-times signal-to-noise ratio with similar +– 6 – + +15cm +10cm +183 +1695 +- +1515 +1825 +4cm +1400 +169 +1510 +60s +1820 +60s +1200 +1685 +1815 +168 +1810 +1000 +1675 +1805 +800 +1670 +0081 +1665 +1795 +600 +1660 +1790 +400 +1655 +1785 +1968 +1780 +1830 +4cm +212 +1695 +15cm +1515 +10cm +1825 +211 +1690 +1s +1510 +1820 +1s +1685 +210 +1505 +1815 +1680 +1500 +1810 +209 +1675 +1495 +208 +1670 +1490E +1800 +207 +1665 +1485E +1795 +206 +1660 +1480 +1790 +1655 +1475 +1785 +205 +160 +1790 +15cm +1520 +10cm +- +1695 +: +4cm +1515 +211 +1690 +100ms +100ms +100ms +1685 +1505 +1815 +1810月 +209 +1680 +1500 +1675 +208 +1670 +1490 +1800E +1665E +1485 +1795 +: +206 +1660 +1480 +1790 +1785 +205 +1655 +1475 +190 +204 +1520 +- +212 +10cm +-4cm +1695 +15cm +1515 +211 +1510 +50ms +1820 +50ms +50ms +210 +1685 +: +1505 +- +1815 +1500 +1810 +209 +1680 +1675 +- +1495 +: +1805 +208 +1670 +1490 +1800 +207 +1665E +1485 +1795 +206 +1660 +1480 +1790 +1655 +1475 +1785 +205 +1R50E +1470元 +- +AEA +AA7A +178 +70 +04 +1700 +1520 +1830 +212 +15cm +- +1695 +1515 +10cm +： +1825 +211 +1690 +20ms +1510 +20ms +1820 +20ms +210 +1685 +1505 +1815 +1680 +1500 +1810 +209 +1675 +1805 +208 +1670E +1490 +1800E +207 +1665 +1485 +- +1795 +206 +1660 +■ +1480 +1790 +1655E +1475 +1785 +- +: +1479020 +165010 +2030 +2040 +2020 +2030 +2040 +2050 +2060 +2060 +2070 +179640 +2050 +2060 +2070 +2080 +Hpixel +Hpie +2090 +2100Figure 5: Measured intensity of alpha source by camera versus exposure time under dif- +ferent object distances. +total intensity, the signal in a line is much more effective than that in a circle. +(a) Uncertainty of the mean of each pixel: in +p.e. +(b) Sum of noise vs. dimension: in p.e. +Figure 6: Pure noise versus image dimension in circles or lines. +Following this strategy, an alpha-like event in a circle is difficult to be identified directly +according to their aimed imaging area and captured light intensity if we only can locate +a range of an image by one camera as in [32]. While, in another hand, a muon track is +a good candidate to check if we assume its image follows a straight line with tiny width. +With the configurations of the crystal system (20000 photons/MeV, 2 MeV/cm, focal length +6 mm, sensor quantum efficiency 30%, a 4 cm track, around 8 p.e./cm and 330 pixels/cm at +an object distance of 4 cm), we can anticipate the averaged intensity of each pixel along the +muon track imaging versus object distance and effective numerical aperture (ap), as shown +in Fig. 7a. A smaller object distance and effective numerical aperture mean more light is +collected with the same focal length and lens diameter. The expected signal-to-noise ratio +with the assumption of 0.3 p.e. noise, and 4 cm track length can be further checked as shown +in Fig. 7b, and it is around three times SN when the object distance is 4 cm. +– 7 – + +Average inpeperpixel +average_circle_pixel +10 +average_line_pixel +10-2 +10-3 +10 +0 +100 +200 +300 +400 +500 +DimensioninpixelTotalsuminpe +102 +10 + sum_pe circle +sum pe line +0 +100 +200 +300 +400 +500 +DimensioninpixelIntensity +105 +1 2inch 6mm 4cm +104 + 1 2inch 6mm 10cm +1 2inch 6mm 15cm +103 +102 +10 +10-3 +10-2 +10-1 +10 +Exposuretime/s(a) Averaged intensity of muon track in pixel +(b) S/N expectation following the model +Figure 7: +Intensity and signal-to-noise ratio of a muon track: +20000 photons/MeV, +2 MeV/cm, focal length 6 mm, sensor quantum efficiency 30%, a 4 cm track, around +8 p.e./cm and 330 pixels/cm at object distance of 4 cm. +The measured total intensity of a track will increase following the track length as shown +in Fig. 8a, where a length of 10 mm track means around 330 pixels with 6 mm focal length +and 4 cm object distance. The signal-to-noise ratio also will improve following the track +length increasing as in Fig. 8b. The effect of the noise level in each pixel is further evaluated +in Fig. 8c. It will reach around three times the signal-to-noise level with 0.3 p.e. noise level, +4 cm track length, 6 mm focal length, 1.4 aperture, and 4 cm object distance. A smaller +noise per pixel and smaller aperture will help improve the signal-to-noise ratio as well as +larger optical lens dimensions. +3.2 +Image track survey +The images of the crystal system with the alpha source and 1 s exposure time are taken. +Fig. 9 shows an example of the image, where the location of the alpha source can be identified +clearly, and it will be used as an online anchor for light intensity and location. +While +according to the calculation in Sec. 3.1, the object distance of 4 cm configuration is run +continuously for 30 s to check out possible muon racks with better signal-to-noise ratio. +With the images (object distance of 4 cm as an example), a survey was done to each +assumed line in aimed pixel range (from vertical line 1700 to vertical line 1600, the minimum +track length is around 100 pixels) following the strategy, where the averaged intensity of each +pixel is calculated among each assumed lines as shown in Fig. 10a and Fig. 10c. In Fig. 10a, a +green dotted line is also plotted, which is a used cut of five times of noise uncertainty which +is relative to the pixel number in a model (0.3*7.8 ADC/sqrt(pixel number)). Some tracks +can be identified as tagged in red and shown in Fig. 10c. The sum of each assumed track +is also plotted in Fig. 10b and Fig. 10d including the selected possible track candidates. An +offset of the pixel average is related to the baseline of each pixel which is further corrected +during the sum calculation. The five-times cut selects the possible tracks efficiently from +noise, which is better for long track and higher than the calculated three-times to avoid +more noise as seen. +The identified candidate of tracks by the survey can be found in Fig. 11, while it seems +too many than the expectation which is around three muon per second hitting the crystal. +– 8 – + +Average inpeperpixel +0. +0.09 +— ap_1.0 +0.08 +0.07 +ap_1.4 +0.06 +0.05 +0.04 +0.03 +0.02 +0.01 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Objectdistanceinmm3 +12 +10 +sn_ap_1.0 +8 +sn_ap _1.4 +6 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Objectdistanceinmm(a) Total intensity versus track length +(b) S/N versus track length +(c) S/N versus noise level per pixel +Figure 8: Total intensity and signal-to-noise ratio versus track length, and signal-to- +noise ratio versus noise level per pixel and effective numerical aperture with a 4 cm track: +20000 photons/MeV, 2 MeV/cm, focal length 6 mm, sensor quantum efficiency 30%, around +8 p.e./cm and 330 pixels/cm at an object distance of 4 cm. +(a) Object distance of 15 cm +(b) Object distance of 10 cm +(c) Object distance of 4 cm +Figure 9: Full image of the crystal with alpha source under 1 s exposure time. +– 9 – + +Total light intensity in pe +180 +160 +sum line ap1.0 +140 +sum line ap1.4 +120 +100 +80 +60 +40 +20 +0 +20 +40 +60 +80 +100 +Tracklengthinmm3 +10 +-SNap1.0 +SN ap1.4 +20 +40 +60 +80 +100 +Track length inmm3 +20 +18 +16 +- SN ap1.0 +14 +SN_ap1.4 +12 +10 +8 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Noiselevelperpixelinpepixel +212 +2200 +> +2000 +211 +1800 +210 +1600 +1400 +209 +1200 +208 +1000 +207 +800 +600 +206 +400 +205 +200 +3500 +204 +500 +1000 +1500 +2000 +2500 +3000 +4000 +H pixelpixel +212 +2200 +> +2000 +211 +1800 +210 +1600 +1400 +209 +1200 +208 +1000 +207 +800 +600 +206 +400 +205 +200 +204 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +H pixelpixel +212 +2200 +> +2000 +211 +1800 +210 +1600 +1400 +209 +1200 +208 +1000 +207 +800 +600 +206 +400 +205 +200 +204 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +H pixel(a) Averaged intensity of each pixel 2-D +(b) Intensity sum 2-D +(c) Averaged intensity of each pixel 1-D +(d) Intensity sum 1-D +Figure 10: Averaged intensity of each pixel and the sum of the assumed tracks. +The direction of the candidate tracks and locations also exceed the range of the view field +of the lens in Fig. 2. It still needs further checking on the quality of the identification even +if a five-times cut is used. +Figure 11: Selected tracks under 1 s exposure time. +3.3 +Muon identification +The identified track candidates need to be further checked as muon track candidates. Check- +ing the uniform trend along the tracking candidate is a good way to avoid the noise effect. +It is extended to another 2000 pixels along the track in maximum (Fig. 12a) to check the +summed intensity (Fig. 12b) and its averaged intensity per pixel versus the track’s length +– 10 – + +h2Dtracklengthintensity +2.8 +1200 +2.6 +1000 +2.4 +2.2 +800 +2 +600 +1.8 +1.6 +400 +1.4 +200 +1.2 +500 +1500 +3000 +0 +1000 +2000 +2500 +tracklengthinpixelh2Dtrack_sumlength +track intensityinADC +800 +4000 +700 +3500 +600 +3000 +500 +2500 +400 +2000 +300 +1500 +200 +1000 +100 +500 +500 +1000 +1500 +2000 +2500 +3000 +tracklengthinpixelh1Dtrackpixel_average +Count +h1D_track_pixel_average +105 +h1D_track_pixel_average_select +104 +103 +102 +10 +0.5 +1.5 +2.5 +3 +3.5 +Pixel average ADCh1Dtracksum +Count +104 +h1Dtrack_sum +h1Dtracksumselect +103 +102 +10 +400 +200 +0 +200 +400 +600 +800 +1000 +tracksuminADCh2Dtrack +/pixel +230 +2200 +> +2000 +225 +1800 +1600 +220 +1400 +1200 +215 +1000 +800 +210 +600 +400 +205 +200 +00 +200 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +H pixel(Fig. 12c). As shown in Fig. 12b, most of the candidates are excluded after the extension, +and the sum of a few candidates keeps increasing versus the length of a true track candidate +as expected in Fig. 8a. As shown in Fig. 12c, most of the candidates are excluded too, and +the average of a few candidates keeps increasing or stable versus the length which is over +the cut. +(a) Extended tracks +(b) Extended sum +(c) Extended average +Figure 12: Track candidate checking by extension. +The tracks after further checking are drawn in the 3-D plots as shown in Fig. 13 and +Fig. 14. Fig. 13 shows a candidate with a short length, the direction (Fig. 13a), and intensity +(Fig. 13b) distribution is reasonable. +Fig. 14 shows a candidate with a long length, the +direction (Fig. 14a) and intensity (Fig. 14b) distribution is reasonable too. They are good +candidates for muon tracks. +(a) Extended track 1-D +(b) Extended track in 3-D +Figure 13: Track candidates one in 3-D +(a) Extended track: 1-D +(b) extended track: 3-D +Figure 14: Track candidates two in 3-D +– 11 – + +h2D trackext only +pixel +230 +2200 +> +2000 +225 +1800 +1600 +220 +1400 +1200 +215 +1000 +800 +210 +600 +400 +205 +200 +200 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Hpixelh2D_track_sumlength +track intensity in ADC +1000 +4000 +900 +3500 +800 +700 +3000 +600 +2500 +500 +2000 +400 +1500 +300 +1000 +200 +100 +500 +500 +1000 +1500 +2000 +2500 +3000 +- +track length in pixelh2D_tracklength_intensity +track intensity in ADC +: +2.8 +1200 +2.6 +1000 +2.4 +2.2 +800 +600 +1.8 +1.6 +400 +1.4 +200 +1.2 +500 +1000 +1500 +2000 +2500 +3000 +track length in pixelpixelV:pixelH(trackNum==50) +2000 +1900 +1800 +1700 +1600 +1500 +1400 +1300 +1200 +1100 +100600 +1500 +2000 +2500 +3000 +3500 +4000pixe/Value:pixe/V:pixelH{trackNum==50 +214 +212 +210 +208 +206 +204 +202 +200 +198 +196 +4000 +3500 +1650 +1640 +3000 +1630 +1620 +2500 +1610 +1600 +2000 +1590 +15801500pixelV:pixelH (trackNum==2) +2000 +1800 +1600 +1400 +1200 +1000 +800 +E +600 +400E +- +2500 +2600 +2700 +2800 +3000 +3100 +3200 +3300 +3400 +3500pixelValue:pixelV:pixelH(trackNum==2) +214 +212 +210 +208 +206 +204 +202 +200 +198 +196 +2890 +2900 +2870 +2880 +2850 +2860 +02830 +28404 +Discussion +4.1 +Muon tracks or not? +The angle distribution of the selected muon track candidates is further plotted as in Fig. 15 +after the track extension check as shown in Fig. 15b, where the theta angle is defined in +the range [-90,+90]◦ to distinguish left and right relative to the camera. There are still too +many abnormal tracks around 0 or |90|◦ as seen in Fig. 15a and Fig. 15c, which are related to +some kind of systematic readout noise of the camera. After excluding the abnormal tracks +around 0 or |90|◦, the theta distribution of the track candidates is basically consistent with +expectation and a peak around 40◦ (cos(θ)∼0.7) as a hint (Fig. 15c and Fig. 15d). But the +statistics are still not enough to have a good check by the muon angle distribution, even the +selected muon candidates are much more than the expectation during the 30 s data-taking +period (around 3 Hz × 30 s). +(a) All selected tracks +(b) Total intensity in ADC vs. track length +(c) Theta in degree +(d) Cos(theta) +Figure 15: Distribution of selected tracks +According to [35, 36], the quenching factor of the alpha of 241Am in CsI(Tl) is taken +as 0.5 with an energy of 5.4 MeV, then the light intensity viewed by the camera is around +1.4 p.e./MeV at an object distance of 15 cm, 2.2 p.e./MeV of 10 cm, and 4.7 p.e./MeV of 4 cm +following the measurement in Sec. 2.2, respectively. Assuming the energy deposit of muon +is around 2 MeV per cm, it means 2.8 p.e./400µm at an object distance of 15 cm (around +80 pixels, 0.035 p.e./pixel) on camera, 4.4 p.e./625µm at an object distance of 10 cm (around +105 pixels, 0.042 p.e./pixel) on camera, and 9.4 p.e./1300µm at an object distance of 4 cm +(around 260 pixels, 0.036 p.e./pixel) on camera. With the data in Fig.15b, the intensity per +pixel of the selected muon track candidates is from 0.1 to 1 ADC or 0.01 to 0.1 p.e., which +– 12 – + +pixel +230 +2200 +2000 +225 +1800 +1600 +220 +1400 +1200 +215 +1000 +800 +210 +600 +400 +205 +200 +200 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +H pixelSum +1200 +1000 +800 +600 +400 +200 +500 +1000 +1500 +2000 +2500 +3000 +Track length in pixelCount +10° +102 +10 +80 +60 +40 +-20 +20 +40 +60 +80 +Theta/degreeCount +103 +102 +10 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Cos(theta)is wide than the expectation from the measurement in Sec. 2.2 and could be related the +variation of muon location. +4.2 +Possible system optimization +Taking more data to accumulate the muon tracks is an effective solution for more precise +angle checking to find out more features of the noise, but the data volume will increase +too. Following the issues of muon track identification, shorter exposure time is one of the +effective methods to reduce the camera noise as known, while the data volume will increase +too for similar statistics. +To realize a good muon tagging and reconstruction method by visual photons directly, +except to find a camera with a further lower noise level as the skipper CCD[30], it is +possible to further improve the system by updating crystals with higher light yield, and +better apertures, and to reduce the distortion of the image by the lens. +Improving the track identification algorithms including distortion identification is also +a valuable solution for track identification and better data compression. The coincidence +with more than one camera is also another effective way to reduce the noise and improve +the measurement as suggested in [32]. +4.3 +Further applications +With the novel method to measure the muon track or realize a similar topology of a particle +measurement in a scintillation detector, it can be used in a huge LS detector to measure the +track of muons precisely as shown in Fig. 16, including to identify possible showers along +the tracks as in JUNO, which is valuable to further study relevant background and suppress +their contribution to neutrino measurement. +Figure 16: Simulated 1 GeV Muon in liquid scintilator +– 13 – + +4000 +2000 +0 +1000 +y/m500 +0 +1000 +500 +x/mm +-500 +0 +-500 +-1000-1000With further improved sensitivity of the system, it is possible to be used to tag the +topology of different particles to do particle identification as simulated in Fig. 17, where +electron, positron, gamma, alpha, proton, pion of 1 GeV in liquid scintillator can be further +identified through their topology for further direction or physics study. +(a) 1 GeV Gamma +(b) 1 GeV proton +(c) 1 GeV π0 +(d) 1 GeV π+ +Figure 17: Simulated 1 GeV particle in liquid scintillator. +5 +Summary +With the crystal system viewed by the single photon sensitive camera and PMTs, system +calibration was further discussed. The muon track imaging was tested further, and the +data were analyzed according to the understanding of the characteristic expectation. Some +possible tracks are identified with the averaged signal intensity cuts. But there still are a few +critical items that need to be finalized. Some improvements are proposed and suggested. +The realization of muon track direct measurement is valuable for future experiments and +applications. +Acknowledgments +This work was supported by the National Natural Science Foundation of China (NSFC) +No. 11875282 and 11475205, the State Key Laboratory of Particle Detection and Electronics, +SKLPDE-ZZ-202208. +– 14 – + +N +4000 +2000 +0 +1000 +y/m200 +0 +1000 +500 +-500 +0 +x/mm +-1000-1000 +500/mm +1000 +N +500 +0 +1000 +y/mm +0 +1000 +-1000 +500 +x/mm +0 +-2000-1000 +-5006000 +mm +N +4000 +2000 +0 +1500 +yx900 +him +500 +1000 +0 +500 +x/mm +-500 +0 +-1000-1000 +5004000 +N3000 +2000 +1000 +0 +1000 +2000 +-2000 +1000 +x/mm +-3000 +0 +-4000-2000 +-1000References +[1] J. Knuesel. The photographic emulsion technology of the OPERA experiment on its way to +find the oscillation. Nuclear Physics B-proceedings Supplements - NUCL PHYS B-PROC +SUPPL, 215:66–68, 06 2011. doi: 10.1016/j.nuclphysbps.2011.03.136. +[2] Kunihiro Morishima. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Min Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='b Diru Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='b Jinchang Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='c Yongpeng Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='c Xiangcheng Meng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='a Caimei Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='b Changgen Yanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='b aInstitute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' China bUniversity of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' China cState Key Laboratory of Particle Detection and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' China E-mail: wangzhm@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='cn Abstract: As a novel approach on visual photon imaging by a single photon sensitive camera and PMTs, this work is trying to measure and identify muon tracks from the 2-D images of CsI(Tl) crystal (scintillator detectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It is possible that muon tracks can be seen directly with a good signal-to-noise ratio neither with further amplification nor external light, which provides an evolution method for particle measurement in the photon-starved regime of scintillation detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The setup of the crystal and camera testing system and the identification algorithm of muon track will be discussed in detail including the system calibration, identification model, signal-to-noise ratio, muon track confirmation, and an expectation on further improvements and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Keywords: photon detectors, scintillator detector, imaging, single photon, camera, muon track ArXiv ePrint: 1234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='56789 1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='01969v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='ins-det] 5 Jan 2023 Contents 1 Introduction 1 2 CsI(Tl) crystal with camera 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Setup 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Calibration 3 3 Muon track 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Signal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Noise 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Image track survey 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 Muon identification 10 4 Discussion 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Muon tracks or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Possible system optimization 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 Further applications 13 5 Summary 14 1 Introduction Vertex and track reconstruction are critical for most particle physics experiments, such as studies on neutrino, dark matter, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' There is a long list of related technologies including but not limited to emulsion film[1–3], cloud chamber[4], bubble chamber[5], spark chamber[6], multi-wire proportional chamber[7], TPC[8], Si strip[9] and Si pixel[10] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' In the case of photon-based detection, in particular, PMT or SiPM is the commonly used sensor for timing, intensity, and crude spatial reconstruction, such as JUNO[11, 12], Darkside[13], JUNO-TAO[14], SNO+[15, 16], and DUNE[17] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=', where computer algorithms are further used to have a better reconstruction on the vertex or track[18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Recently, many efforts are focusing on photon imaging-related projects following the new development of sensors, where the critical challenges are the need for high spatial resolution over large volumes[21] and better effective signal-to-noise ratio under the photon- starved regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' For many years classical emulsion film radiography is being replaced by digital detec- tor imaging, especially in medical applications due to faster and more reliable diagnostics and computed tomography and tomosynthesis capabilities[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The single photon count- ing X-ray CCD camera spectrometer is used in laser-plasma interaction experiments as a simple tool to study the K-shell X-ray generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A CCD detector enables the spectrum of the impinging X-ray radiation to be obtained without further dispersive devices[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' – 1 – Among the imaging systems used for thermal neutron imaging worldwide, the most preva- lent configuration is CCD camera based[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Single-photon light detection and ranging (lidar) offers single-photon sensitivity and picosecond timing resolution, which is desirable for high-precision three-dimensional (3D) imaging over long distances[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Single image 3D photography enables viewers to view a still image from novel viewpoints[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Some good sensors are developed too, such as SPC3[27], a single photon counting camera based on a 2-D imaging array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A small, high resolution, high signal-to-noise GEM-based TPC with a 2-D CCD readout designed to provide a benchmark for background discrimination and di- rectional sensitivity that could be used for future optimization studies for directional dark matter experiments [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A skipper CCD was also developed for very low noise and directly measured a muon track through ionization inside the sensor[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' But, generally, it is not suitable to directly image of vertex or track in case of a starved- photon regime and uniform angular distribution of the photons[21, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Photography by CCD or other technologies, in particular single photon imaging, provides another new pos- sibility, such as our previous study for particle imaging by event[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' In this article, we will try to have a further detailed check on the imaging of muon track in CsI(Tl) crystal with a single photon sensitive camera and PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 will introduce the system setup and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 will discuss the expected features of muon tracks, measurements, and track surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 will provide further discussions on the results, possible improvements of the system, and further expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' And a short summary is in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2 CsI(Tl) crystal with camera 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Setup An imaging system, as in [32], is set up with a single photon sensitive camera of ORCA- Quest qCMOS C15550-20UP, which is a new product of Hamamatsu Photonics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The detailed layout of the system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The output of the camera will save in tif format with 16 bits of each of the 4096(H)×2304(V) pixels and the volume of each photo is around 16 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The CsI(Tl) crystal (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5×15 cm3) is located in front of the camera and the two 3-inch PMTs, where the distance between them can be adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' An alpha source of 241Am is used and put on the top surface of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The two 3-inch PMTs are used to calibrate and monitor the signal intensity of the crystal, the coincidence of which is used as a trigger of the CAEN DT5751 (1 GS/s with 1 V p-p dynamic range, [34]) for waveform data taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The threshold of each 3-inch PMT is set to around 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (photon-electron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The maximum rate of the data-taking system is limited by the DT5751, which is generally lower than 100 Hz with data saving of four channels and 10000 samples of each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Here the window length of the waveform recording is set to 6 µs (6000 samples/waveform), and the maximum data-taking rate is around 70-80 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' In order to increase the acceptance of the emitted photons from the crystal, a lens with a much short focal length and small number aperture (1/2”, C type, 6-∞ mm, f/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The images of the crystal with different distances are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2, which are taken with illumination before the dark box is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The field of view with the used lens is in a circle – 2 – Figure 1: Layout of the imaging measurement system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' and much smaller than the full size of the camera sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The circle shape and its outside of the field of view will be considered in the following measurement and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Please note that there is a clear distortion around the edge of the field of view (crystal region) when the object distance is too small as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) 15 cm (b) 4 cm Figure 2: Crystal photos when the dark box is open with natural illumination and different object distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Calibration Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3a shows the measured charge spectra with the alpha source located when the object distance is around 15 cm: the two 3-inch PMTs and the sum of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A long tail can be found at the right of the spectrum, which is known as cosmic muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A 400 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' cut to the sum spectrum is used to select the events of muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A factor of particle identification (PID) is calculated by each waveform of each PMT, and it is defined by the ratio of the charge in the first 300 ns to the whole window, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A 2-D cut on the PID is used to identify the events from the alpha source, the red dash line (PID_pmt_0+PID_pmt_1) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The selected spectra are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3d, where the blue curve is selected as the alpha-like events by (PID_pmt_0+PID_pmt_1>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5), the Magenta – 3 – couple Source Camera Lens Power USB cable Crystal Dark Boxcurve is selected for the muon events candidates by (PID_pmt_0+PID_pmt_1<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 and charge_pmt_0+charge_pmt_1>400 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' ), and the green curve is assumed as the gamma- like events by (PID_pmt_0+PID_pmt_1<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) Charge spectra of eac PMT and sum (b) PID of each event of each PMT (c) 2-D distribution of PID of the two PMTs (d) Selected events by PID and charge Figure 3: Measured charge spectra by the 3-inch PMTs, and the PID distrbution of the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Taking into account the dead time of the data-taking system, the actual event rate of each measurement is re-normalized according to the selected muon rate, where the reference muon rate of the crystal is from the measurement and selection of without source with object distance of 15 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It is around 3 Hz for selected muons, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 Hz for gamma-like events, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Hz for alpha-like events of the measurement without source and object distance of 15 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The data-taking rate and the re-normalized rate are listed in Table 1 for different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The rate of alpha-like events is around 100 Hz and increases from 93 Hz to 188 Hz following the shortening of the object distance from 15 cm to 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The contribution from the alpha source is much higher than that from the background of around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Following the classification of the events, the mean charge of each kind of event is calculated too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The mean charge of the alpha-like events is 130 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 15 cm, 150 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 10 cm, and 159 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 4 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The signal intensity is not following the solid angle simply, for the 4 cm, in particular, which is because the distance to the 3-inch PMTs is rather small than the object distance of the camera to the crystal according to the layout of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The mean intensity of the muon events is around 2100 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=', which suffers from statistic uncertainty and solid angle issues too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The images of the crystal with the alpha source are taken by the camera with different exposure times and different object distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The region of the source is selected and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 4, where the selected image dimension of the sensor is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='23 mm ×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='28 mm – 4 – Count Sum single_S sPMT single 0 102 sPMT single1 10 10 102 103 Charge/peCount 700 h1D_pidpmto 600 h1D pid pmt1 500 400 300 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 PIDPMT_ 16 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 10 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 6 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 C 0 PMT 0Count 10 Sum singleS Sum_single_S_high Sum_single_S_low Sumsingle_S_low_gamma 102 SumsingleSlowmuon 10 10 102 103 Charge/peTable 1: Event rate and charge intensity of crystal with two 3-inch PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The distance is between the camera and the crystal front surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The events are measured by the coincidence by the two 3-inch PMTs, and the charge is from the sum of the two PMTs of each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The alpha-like events are selected by the sum of PID, and the separation between muon and gamma-like is by a charge cut after the PID cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Type DAQ Rate Normalized Rate (Hz) Mean Charge (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=') (Distance) (Hz) Rate (Hz) Muon Gamma-like Alpha-like Muon Gamma-like Alpha-like w/o source 15 cm ∼8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 2099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 w/ source 15 cm ∼80 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='9 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='9 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 w/ source 10 cm ∼70 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='9 2057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 w/ source 4 cm ∼70 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 2062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 with 50 pixel (V) ×60 pixels (H) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 µm × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 µm per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The regime of the alpha source can be identified as around 3 mm scale of an object distance of 15 cm, 2 mm scale of 10 cm, and 1 mm scale of 4 cm, and the light intensity gradually dims when shortening the exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It is almost identified to event level with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='05 s exposure time but on a higher noise background, where only a few alphas occur during the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The dimension of the image of the source is enlarging when the object distance reduces as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The intensity of the source region is integrated and converted into p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 5, where the noise (baseline is around 200 ADC) of the camera is subtracted according to a parallel region of the source with equal area [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The conversion factor is around 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 ADC/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The diameter of the source region is 21 pixels for an object distance of 15 cm, 22 pixels for 10 cm, and 33 pixels for 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The fitted intensity per second by a linear curve is around 314 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of an object distance of 15 cm, 862 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 10 cm, and 2255 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 4 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Considering the rate of the alpha source measured under different object distances as in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 1, the ratio of measured charge intensity of the camera and the PMTs is around 3% of an object distance of 15 cm, 4% of 10 cm, and 8% of 4 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The expected typical charge intensity of alpha-like event viewed by the camera is around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='9 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of an object distance of 15 cm, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 10 cm, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 4 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The expected typical charge intensity of each muon viewed by the camera is around 60 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of an object distance of 15 cm, 85 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 10 cm, and 177 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' of 4 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3 Muon track 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Signal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Noise As stated in [32], the noise of the camera is still much higher than the traditional used PMT or SiPM, which is much worse when we are trying to use many pixels for imaging measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It can be expected that it will help to identify the target by a smaller area and stronger intensity of the same object, as seen in the left plateau of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 5, where the difference of the plateau level (noise) is mainly from the dimension of the imaging area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The minimum of the plateau is from the object distance of 15 cm configuration, which is proportional to the ratio squared of the focal length to object distance, even the final – 5 – Figure 4: 2-D images of alpha source with object distances of 15 cm (50 pixel (V) ×60 pixels (H), left), 10 cm (50 pixel (V) ×60 pixels (H), middle) and 4 cm (50 pixel (V) ×60 pixels (H), right) versus exposure time of 60 s, 1 s, 100 ms, 50 ms and 20 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The pixel size is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 µm × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The z-axis is in the unit of ADC which is the camera output per pixel directly relative to the light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The baseline of each pixel is around 200 ADC, and the gain factor is around 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 ADC/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' signal-to-noise (SN) ratio is also proportional to the square of the reciprocal of focal length except for the effective aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A calculation is further executed to compare the effect of the imaging shape under the same noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Here a pure statistic model is used to check the mean of each pixel and total sum with an assumption of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' noise level per pixel (sigma of Gaussian) in a circle (diameter) or line (in length and in width of one pixel) shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The uncertainty of the mean of each pixel is inversely proportional to the square root of total pixel numbers as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 6a, where more pixels will have smaller fluctuation comparing circles to lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' While the total sum of all pixels is proportional to the square root of pixel numbers shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 6b, where the required sum in lines is much smaller than that of in circles to identify a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' If we aim to identify a signal by a three-times signal-to-noise ratio with similar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='– 6 – ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2050 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2060 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2070 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2080 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Hpixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Hpie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2090 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2100Figure 5: Measured intensity of alpha source by camera versus exposure time under dif- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='ferent object distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' total intensity, the signal in a line is much more effective than that in a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) Uncertainty of the mean of each pixel: in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (b) Sum of noise vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' dimension: in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Figure 6: Pure noise versus image dimension in circles or lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Following this strategy, an alpha-like event in a circle is difficult to be identified directly according to their aimed imaging area and captured light intensity if we only can locate a range of an image by one camera as in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' While, in another hand, a muon track is a good candidate to check if we assume its image follows a straight line with tiny width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' With the configurations of the crystal system (20000 photons/MeV, 2 MeV/cm, focal length 6 mm, sensor quantum efficiency 30%, a 4 cm track, around 8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/cm and 330 pixels/cm at an object distance of 4 cm), we can anticipate the averaged intensity of each pixel along the muon track imaging versus object distance and effective numerical aperture (ap), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A smaller object distance and effective numerical aperture mean more light is collected with the same focal length and lens diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The expected signal-to-noise ratio with the assumption of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' noise, and 4 cm track length can be further checked as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 7b, and it is around three times SN when the object distance is 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='– 7 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Average inpeperpixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='average_circle_pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='average_line_pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='DimensioninpixelTotalsuminpe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='sum_pe circle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='sum pe line ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='DimensioninpixelIntensity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 2inch 6mm 4cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 2inch 6mm 10cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 2inch 6mm 15cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Exposuretime/s(a) Averaged intensity of muon track in pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(b) S/N expectation following the model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Figure 7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Intensity and signal-to-noise ratio of a muon track: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='20000 photons/MeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2 MeV/cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' focal length 6 mm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' sensor quantum efficiency 30%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' a 4 cm track,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' around 8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/cm and 330 pixels/cm at object distance of 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The measured total intensity of a track will increase following the track length as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 8a, where a length of 10 mm track means around 330 pixels with 6 mm focal length and 4 cm object distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The signal-to-noise ratio also will improve following the track length increasing as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The effect of the noise level in each pixel is further evaluated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It will reach around three times the signal-to-noise level with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' noise level, 4 cm track length, 6 mm focal length, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 aperture, and 4 cm object distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' A smaller noise per pixel and smaller aperture will help improve the signal-to-noise ratio as well as larger optical lens dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Image track survey The images of the crystal system with the alpha source and 1 s exposure time are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 9 shows an example of the image, where the location of the alpha source can be identified clearly, and it will be used as an online anchor for light intensity and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' While according to the calculation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1, the object distance of 4 cm configuration is run continuously for 30 s to check out possible muon racks with better signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' With the images (object distance of 4 cm as an example), a survey was done to each assumed line in aimed pixel range (from vertical line 1700 to vertical line 1600, the minimum track length is around 100 pixels) following the strategy, where the averaged intensity of each pixel is calculated among each assumed lines as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 10a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 10c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 10a, a green dotted line is also plotted, which is a used cut of five times of noise uncertainty which is relative to the pixel number in a model (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3*7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 ADC/sqrt(pixel number)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Some tracks can be identified as tagged in red and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 10c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The sum of each assumed track is also plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 10b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 10d including the selected possible track candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' An offset of the pixel average is related to the baseline of each pixel which is further corrected during the sum calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The five-times cut selects the possible tracks efficiently from noise, which is better for long track and higher than the calculated three-times to avoid more noise as seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The identified candidate of tracks by the survey can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 11, while it seems too many than the expectation which is around three muon per second hitting the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' – 8 – Average inpeperpixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='09 — ap_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='07 ap_1.' metadata={'source': 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Objectdistanceinmm3 12 10 sn_ap_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 8 sn_ap _1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 6 0 20 40 60 80 100 120 140 160 180 Objectdistanceinmm(a) Total intensity versus track length (b) S/N versus track length (c) S/N versus noise level per pixel Figure 8: Total intensity and signal-to-noise ratio versus track length, and signal-to- noise ratio versus noise level per pixel and effective numerical aperture with a 4 cm track: 20000 photons/MeV, 2 MeV/cm, focal length 6 mm, sensor quantum efficiency 30%, around 8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/cm and 330 pixels/cm at an object distance of 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) Object distance of 15 cm (b) Object distance of 10 cm (c) Object distance of 4 cm Figure 9: Full image of the crystal with alpha source under 1 s exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' – 9 – Total light intensity in pe 180 160 sum line ap1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='0 140 sum line ap1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 120 100 80 60 40 20 0 20 40 60 80 100 Tracklengthinmm3 10 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='H pixel(a) Averaged intensity of each pixel 2-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(b) Intensity sum 2-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(c) Averaged intensity of each pixel 1-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(d) Intensity sum 1-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Figure 10: Averaged intensity of each pixel and the sum of the assumed tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The direction of the candidate tracks and locations also exceed the range of the view field of the lens in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It still needs further checking on the quality of the identification even if a five-times cut is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Figure 11: Selected tracks under 1 s exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 Muon identification The identified track candidates need to be further checked as muon track candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Check- ing the uniform trend along the tracking candidate is a good way to avoid the noise effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' It is extended to another 2000 pixels along the track in maximum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 12a) to check the summed intensity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 12b) and its averaged intensity per pixel versus the track’s length – 10 – h2Dtracklengthintensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 1200 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 800 2 600 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 500 1500 3000 0 1000 2000 2500 tracklengthinpixelh2Dtrack_sumlength track intensityinADC 800 4000 700 3500 600 3000 500 2500 400 2000 300 1500 200 1000 100 500 500 1000 1500 2000 2500 3000 tracklengthinpixelh1Dtrackpixel_average Count h1D_track_pixel_average 105 h1D_track_pixel_average_select 104 103 102 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 Pixel average ADCh1Dtracksum Count 104 h1Dtrack_sum h1Dtracksumselect 103 102 10 400 200 0 200 400 600 800 1000 tracksuminADCh2Dtrack /pixel 230 2200 > 2000 225 1800 1600 220 1400 1200 215 1000 800 210 600 400 205 200 00 200 500 1000 1500 2000 2500 3000 3500 4000 H pixel(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 12c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 12b, most of the candidates are excluded after the extension, and the sum of a few candidates keeps increasing versus the length of a true track candidate as expected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 12c, most of the candidates are excluded too, and the average of a few candidates keeps increasing or stable versus the length which is over the cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) Extended tracks (b) Extended sum (c) Extended average Figure 12: Track candidate checking by extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The tracks after further checking are drawn in the 3-D plots as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 13 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 13 shows a candidate with a short length, the direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 13a), and intensity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 13b) distribution is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 14 shows a candidate with a long length, the direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 14a) and intensity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 14b) distribution is reasonable too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' They are good candidates for muon tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(a) Extended track 1-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(b) Extended track in 3-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Figure 13: Track candidates one in 3-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(a) Extended track: 1-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='(b) extended track: 3-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Figure 14: Track candidates two in 3-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='– 11 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='h2D trackext only ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2860 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='02830 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='28404 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='Discussion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 Muon tracks or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The angle distribution of the selected muon track candidates is further plotted as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 15 after the track extension check as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 15b, where the theta angle is defined in the range [-90,+90]◦ to distinguish left and right relative to the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' There are still too many abnormal tracks around 0 or |90|◦ as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 15a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 15c, which are related to some kind of systematic readout noise of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' After excluding the abnormal tracks around 0 or |90|◦, the theta distribution of the track candidates is basically consistent with expectation and a peak around 40◦ (cos(θ)∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='7) as a hint (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 15c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 15d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' But the statistics are still not enough to have a good check by the muon angle distribution, even the selected muon candidates are much more than the expectation during the 30 s data-taking period (around 3 Hz × 30 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) All selected tracks (b) Total intensity in ADC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' track length (c) Theta in degree (d) Cos(theta) Figure 15: Distribution of selected tracks According to [35, 36], the quenching factor of the alpha of 241Am in CsI(Tl) is taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 with an energy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 MeV, then the light intensity viewed by the camera is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/MeV at an object distance of 15 cm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/MeV of 10 cm, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='7 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/MeV of 4 cm following the measurement in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Assuming the energy deposit of muon is around 2 MeV per cm, it means 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/400µm at an object distance of 15 cm (around 80 pixels, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='035 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/pixel) on camera, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/625µm at an object distance of 10 cm (around 105 pixels, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='042 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/pixel) on camera, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/1300µm at an object distance of 4 cm (around 260 pixels, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='036 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='/pixel) on camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' With the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='15b, the intensity per pixel of the selected muon track candidates is from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 to 1 ADC or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=', which – 12 – pixel 230 2200 2000 225 1800 1600 220 1400 1200 215 1000 800 210 600 400 205 200 200 0 500 1000 1500 2000 2500 3000 3500 4000 H pixelSum 1200 1000 800 600 400 200 500 1000 1500 2000 2500 3000 Track length in pixelCount 10° 102 10 80 60 40 20 20 40 60 80 Theta/degreeCount 103 102 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='9 Cos(theta)is wide than the expectation from the measurement in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 and could be related the variation of muon location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='2 Possible system optimization Taking more data to accumulate the muon tracks is an effective solution for more precise angle checking to find out more features of the noise, but the data volume will increase too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Following the issues of muon track identification, shorter exposure time is one of the effective methods to reduce the camera noise as known, while the data volume will increase too for similar statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' To realize a good muon tagging and reconstruction method by visual photons directly, except to find a camera with a further lower noise level as the skipper CCD[30], it is possible to further improve the system by updating crystals with higher light yield, and better apertures, and to reduce the distortion of the image by the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Improving the track identification algorithms including distortion identification is also a valuable solution for track identification and better data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The coincidence with more than one camera is also another effective way to reduce the noise and improve the measurement as suggested in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='3 Further applications With the novel method to measure the muon track or realize a similar topology of a particle measurement in a scintillation detector, it can be used in a huge LS detector to measure the track of muons precisely as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 16, including to identify possible showers along the tracks as in JUNO, which is valuable to further study relevant background and suppress their contribution to neutrino measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Figure 16: Simulated 1 GeV Muon in liquid scintilator – 13 – 4000 2000 0 1000 y/m500 0 1000 500 x/mm 500 0 500 1000-1000With further improved sensitivity of the system, it is possible to be used to tag the topology of different particles to do particle identification as simulated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 17, where electron, positron, gamma, alpha, proton, pion of 1 GeV in liquid scintillator can be further identified through their topology for further direction or physics study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' (a) 1 GeV Gamma (b) 1 GeV proton (c) 1 GeV π0 (d) 1 GeV π+ Figure 17: Simulated 1 GeV particle in liquid scintillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 5 Summary With the crystal system viewed by the single photon sensitive camera and PMTs, system calibration was further discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The muon track imaging was tested further, and the data were analyzed according to the understanding of the characteristic expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Some possible tracks are identified with the averaged signal intensity cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' But there still are a few critical items that need to be finalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Some improvements are proposed and suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The realization of muon track direct measurement is valuable for future experiments and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' 11875282 and 11475205, the State Key Laboratory of Particle Detection and Electronics, SKLPDE-ZZ-202208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' – 14 – N 4000 2000 0 1000 y/m200 0 1000 500 500 0 x/mm 1000-1000 500/mm 1000 N 500 0 1000 y/mm 0 1000 1000 500 x/mm 0 2000-1000 5006000 mm N 4000 2000 0 1500 yx900 him 500 1000 0 500 x/mm 500 0 1000-1000 5004000 N3000 2000 1000 0 1000 2000 2000 1000 x/mm 3000 0 4000-2000 1000References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Knuesel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' The photographic emulsion technology of the OPERA experiment on its way to find the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' Nuclear Physics B-proceedings Supplements - NUCL PHYS B-PROC SUPPL, 215:66–68, 06 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='nuclphysbps.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content='com/science/article/pii/S0927650509001650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} +page_content=' – 17 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENA0T4oBgHgl3EQfA_81/content/2301.01969v1.pdf'} diff --git a/IdE0T4oBgHgl3EQfiAEY/content/tmp_files/2301.02438v1.pdf.txt b/IdE0T4oBgHgl3EQfiAEY/content/tmp_files/2301.02438v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..670160e536f92ab4e9a84b73f4f873b7093d9ae0 --- /dev/null +++ b/IdE0T4oBgHgl3EQfiAEY/content/tmp_files/2301.02438v1.pdf.txt @@ -0,0 +1,971 @@ +Experimental demonstration of position-controllable topological interface states +in high-frequency topological integrated circuits +Tetsuya Iizuka,∗ Haochen Yuan,† Yoshio Mita,† Akio Higo,‡ Shun Yasunaga,† and Motohiko Ezawa§ +(Dated: January 9, 2023) +Topological integrated circuits are integrated circuit realizations of topological systems. We perform an ex- +perimental demonstration by taking instances of the Su-Schrieffer-Heeger model and the Kitaev topological +superconductor model. They are found to realize high frequency resonances around 17GHz. We explicitly +observe the spatial profile of a topological edge state. In particular, the topological interface state between a +topological segment and a trivial segment is the Majorana-like state in the Kitaev model. We construct a switch- +able structure in the integrated circuit, which enables us to control the position of a Majorana-like interface state +arbitrarily along a chain. Our results will open topological electronics with high frequency integrated circuits. +Topological insulators and superconductors are fascinat- +ing new states of matter[1, 2]. +The Su-Schrieffer-Heeger +(SSH) model[3] and the Kitaev topological superconduc- +tor model[4] are simplest one-dimensional (1D) systems re- +alizing topological insulators and superconductors, respec- +tively. +Especially, topological superconductors host Ma- +jorana edge states[5–8], which are the key elements of a +topological quantum computer[9, 10]. +The area of topo- +logical physics is expanded nowadays to photonic[11–16], +acoustic[17–21], mechanical[22–29] and electronic-circuit +systems[30–37]. +They are called artificial topological sys- +tems. +There are several merits which are difficult to be +achieved in inorganic crystals: 1) It is possible to make a fine +tuning of the system, which is crucial for observing topologi- +cal edge states. 2) It is possible to construct a few site systems. +3) It is possible to directly measure the site dependent infor- +mation. +It is relatively easy to materialize the SSH model because it +involves only real hoppings. On the other hand, this is not the +case with respect to the Kitaev model because it is a p-wave +topological superconductor, although the Majorana edge state +itself can be generated in a s-wave superconductor with the +aid of a topological insulator nanowire[38, 39]. +Electronic circuits present an ideal platform to realize var- +ious topological phases[30–37, 40–46]. +The emergence of +topological edge states are observed by means of impedance +resonance. However, experimental demonstrations have so far +been restricted to printed circuit boards with discrete com- +ponents. Indeed, to the best of our knowledge, there is no +integrated-circuit realization working at high resonant fre- +quency, although a simulation of the SSH model was done +recently[47]. The integrated circuit realization is an important +step toward industrial applications of topological electronics. +In order to generate Majorana-like states, it is necessary to +simulate electron and hole bands in electronic circuits. Al- +though there is a proposal with the use of chains of capacitors +∗ Systems Design Lab., School of Engineering, The University of Tokyo. +iizuka@vdec.u-tokyo.ac.jp (corresponding author) +† Department of Electrical Engineering and Information Systems, The Uni- +versity of Tokyo. +‡ Systems Design Lab., School of Engineering, The University of Tokyo. +§ Department of Applied Physics, The University of Tokyo. ezawa@ap.t.u- +tokyo.ac.jp (corresponding author) +and inductors[44, 45], there is so far no experimental demon- +stration of this theoretical proposal. +Most of previous experiments were carried out based on +patterned structures, where it is impossible to control the topo- +logical and trivial phases once the sample is manufactured. +Actually, it is very hard to introduce switch structures in in- +organic materials, photonic crystals and acoustic systems. On +the other hand, transistors act as switches in electronic cir- +cuits and hence, there is a possibility to construct a switchable +topological system based on electronic circuits. +In this paper, we perform for the first time an experimen- +tal demonstration of topological integrated circuits, which are +integrated circuit realizations of topological systems, by tak- +ing instances of the SSH model and the Kitaev model. An +integrated circuit implementation enables us to realize very +high resonant frequency as large as 17GHz. We explicitly ob- +serve the spatial profile of a topological edge state and deter- +mine its penetration length. The system may contain several +topological and trivial segments simultaneously along a chain. +In particular, we observe the signal of a Majorana-like state +emerging at the interface of a topological segment and a triv- +ial segment. It is topologically protected since it necessarily +emerges between the topological and trivial segments. These +two topologically different segments are interchangeable sim- +ply by exchanging inductors and capacitors. +SSH chain +The SSH chain is the basic model of a topological insula- +tor. It was implemented in a printed circuit board with dis- +crete components[31] a few years ago. The electronic circuit +is illustrated in Fig.1a. Capacitances are alternating along the +chain, and each node is grounded via an inductor. +We have implemented the SSH model in two 32-unit cell +chains on chips in an integrated circuit as shown in Fig.1b. +Both capacitors and inductors are implemented by metallic +wires. An inductor is materialized by a swirling structure of +wire as shown in Fig.1c, while a capacitor is materialized by +a comb-teeth structure as shown in Fig.1d. The comb-teeth +structure is prepared in order to increase the capacitance with +small occupation area. The explicit values of the capacitance +and the inductance are shown in Table.1, which is very tiny +compared with printed circuit boards with discrete compo- +nents. The small capacitance and inductance lead to a high +resonant frequency ωresonant = 1/ +√ +LC. +The impedance as a function of the input frequency is +arXiv:2301.02438v1 [cond-mat.mes-hall] 6 Jan 2023 + +2 +FIG. 1. SSH chain. a, An illustration of an electronic-circuit representing the SSH chain. b, A picture of a 32-unit cell integrated circuit for +the SSH chain. c, A zoom-in view of its unit cell layout. d, A picture of the comb teeth capacitor. Frequency dependence of the impedance +measured from the left edge of the 17.2 GHz chain for e, all-topological and f, all-trivial setups. g, The spatial profile of the impedance values +for all-topological mode measured from the left edge at the resonant frequency of 13.1 GHz. In e and f, solid and dashed lines show the +measured and simulated results of the SSH chain, respectively. +shown in Fig.1e and f. +The impedance is evaluated with +the two-point impedance between two nodes. The solid and +dashed lines show measurement and simulation results, re- +spectively. A peak emerges at the characteristic frequency for +a topological phase as shown in Fig.1e. The measured res- +onant frequency is 10.7GHz. On the other hand, there is no +peak for a trivial phase as shown in Fig.1f. +The node-dependent impedance is shown in Fig.1g. The +impedance decays exponentially. The penetration length of +the topological edge state is estimated as 0.414, which is in +good agreement with the theoretical value 1/ log(C2/C1) = +0.449. See Supplementary Information IV for details. +We have also carried out measurements on the SSH chain +with 8.8 GHz. See Supplementary Information I for details. +Kitaev chain +The Kitaev chain model is the basic model of a topologi- +cal superconductor. Our main result is its implementation in +an integrated electronic circuit. To realize a Cooper pair it +is necessary to prepare an electron band and a hole band to- +gether with cross terms between these two bands, as shown in +TABLE I. Parameters used for the SSH chain (left table) and the +Kitaev chain (right table). +8.8 GHz 17.2 GHz +C1 +42 fF +22 fF +C2 +414 fF +204 fF +L +721 pH +378 pH +8.8 GHz 17.2 GHz +C +440 fF +220 fF +L +747 pH +384 pH +Cx +396 fF +204 fF +Lx 830 pH +427 pH +C0 +880 fF +440 fF +L0 374 pH +192 pH +Methods. +We first illustrate an electronic circuit for the Kitaev +chain [44, 45] in Fig. 2a, b and c. The capacitor channel (in- +dicated in red) corresponds to the electron band, while the +inductor channel (in blue) corresponds to the hole band. The +two main channels are crosslinked through Cx and Lx. Each +node is connected to the ground via an inductor L0 or a ca- +pacitor C0 to realize a topological state or a trivial state, re- +spectively, as shown in Figs. 2a and b. The topological phase + +400μm +c +Unit cell +Unit cell +C2 +C1 +forProbing Test +H&} +ContactPads +区HH区 +HH区 +240μm +000 +GNDLine +b +Unit cell +d +480μm +3300μm +e +f +Topological phase +Trivial phase +g +1000 +1000 +Two-Point Impedance [Q] +Two-Point Impedance [2] +a +100 +10.7GHz +Two-Point Impedance +100 +100 +Node +10 +Node0 +10 +10 +Node +Node.2 +1 +0.1 +0.1 +0.1 +Node +Penetrationlength=0.414 +Node6 +0.01 +Node7 +0.01 +1 +2 +3456 +10 +20 +1 +2 +3456 +10 +20 +0 +1 +2 +3 +4 +5 +6 +Frequency [GHz] +Frequency [GHz] +Node Number3 +FIG. 2. Kitaev chain. a, b and c, The electronic-circuit representation of the Kitaev chain. a, All-topological configuration where the +topological edge state emerges at both the left and right edges of the chain. b, All-trivial configuration that does not have a topological edge +state. c, The implemented state-configurable Kitaev chain circuit. d, By using two SPDT switches with inverters in the unit cell, the connection +of L0 and C0 can be swapped to change its topological/trivial state. The SPDT switch is realized by two CMOS transmission gate switches. e, +A picture of an 16-unit cell integrated circuit for the SSH chain. f, A picture of a unit cell. g, A zoom of SPDT switches in Fig.2f. Each SPDT +switch is composed of an inverter and two transmission gates with n-type and p-type MOS FETs as in Fig.2d. h and i, Frequency dependence +of the impedance measured from the right edge of the electronic circuit with the characteristic frequency ωresonant =17.2 GHz Kitaev chain +for all-topological and all-trivial setups, respectively. Solid and dashed lines show the measured and simulated results of the Kitaev chain, +respectively. j, The spatial profile of the impedance values for all-topological mode measured from both left and right edges at the resonant +frequency of 13.1 GHz. +is realized by the configuration shown in Fig. 2a, while the +trivial phase is realized by the configuration shown in Fig. 2b. +A single Kitaev chain may accommodate several segments +which are either topological or trivial. A Majorana-like state +emerges at an interface between the two phases. We introduce +two single-pole double-throw (SPDT) switches in each unit +cell as illustrated in Fig. 2c. The electric circuit for the SPDT +switch is shown in Fig. 2d. The switching is done by swapping +the connection of L0 and C0, by way of which the position of +a Kitaev interface state is controlled. In the integrated circuits, +the SPDT switch is simply implemented with an inverter and +two CMOS transmission gates, composed of n-type and p- +type metal oxide semiconductor (MOS) field-effect transistors +(FETs) as shown in Fig. 2f and g. +The Kitaev chain circuit shown in Fig. 2c is implemented +onto the chip using 180 nm CMOS technology as shown in + +a +Topological phase +b +Trivial phase +Unit cell +g +Switches +220μm +GNDLine +Transmission +SPDTSwitch +Gate +forProbingTest +C +ContactPads +Transmission +Gate +000 +000 +000 +400μm +Inverter +000 +000 +Switches +C +d +SPDT switch +Probingpoints +SPDTSwitch +IN +000 +区 +OUT1 +OUT2 +Co +SPDT +switch +GNDLine +000 +SW +OUT1 +OUT2 +e +Unit cell +Unit cell +400μm +3800μm +h +Topological phase +Trivial phase +J +100 +100 +Two-Point Impedance [Q] +Two-Point Impedance [Ω] +Two-Point Impedance [Ω] +100 +13.1GHZ +D.. +From-leftedge +Node 16 +Penetration length=0.660 +10 +10 +10 +Node16 +Node 15 +Fromirightedge +Penetration length=0.666 +Node.15 +Node 14 +Node. +1 +lode +0.1 +0.1 +0.1 +Nodel +Node 12 +0.01 +0.01 +0.01 +2 +3 +456 +10 +20 +2 +3 +456 +10 +20 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Frequency [GHz] +Freguency[GHz] +Node Number4 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 +Impedance [Ω] +Node Number +3 trivial +3 trivial +2 trivial +4 topological +4 topological +8.8GHz chain +17.2GHz chain +From node 4 +From node 7 +From node 10 From node 13 +From node 4 +From node 7 +From node 10 From node 13 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 +Impedance [Ω] +Node Number +4 trivial +3 trivial +1 trivial +4 topological +4 topological +8.8GHz chain +17.2GHz chain +From node 5 +From node 8 +From node 10 +From node 13 +From node 5 +From node 8 +From node 10 +From node 13 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 +Impedance [Ω] +Node Number +4 trivial +4 trivial +8 topological +8.8GHz chain +17.2GHz chain +From node 5 +From node 12 +From node 5 +From node 12 +a +b +c +FIG. 3. Topological interface states. Measurement results of the +topological edge state locations depending on different Kitaev chain +configurations. +"n trivial (topological)" indicates that the trivial +(topological) segment contains n unit cells. Blue (red) data points +are for 8.8 GHz (17.2 GHz) chain. a, The topological edge states +emerge at 4th, 7th, 10th and 11th nodes. b, The locations of the edge +states move to the 5th, 8th, 10th and 11th nodes. c, When two topo- +logical segments combine to one segment, the edge states emerge +only at 5th and 12th nodes. +Fig. 2e. On a 5 mm×5 mm chip, two 16-unit cell Kitaev chain +circuits were integrated for two different target resonant fre- +quencies, 8.8 GHz and 17.2 GHz. We show a zoom-in view of +the unit cell layout in Fig. 2f, which shows that it includes 3 +inductors L, Lx and L0, 3 capacitors C, Cx and C0, 2 SPDT +switches, and a contact pad at each node for direct probing +measurement with GSG (Ground, Signal, Ground) probes. A +photo of the SPDT switches is shown in Fig. 2g. Two trans- +mission gates and an inverter are integrated for each SPDT +switch. The values for the capacitors and inductors are sum- +marized in TABLE I. +Fig. 2h, i and j summarizes the impedance measurement re- +sults of the Kitaev chain designed for 17.2 GHz resonant fre- +quency. Figs. 2h and i shows the frequency dependence of the +impedance measured from the right edge of the chain for topo- +logical and trivial setups, respectively. The solid and dashed +lines show measurement and simulation results, respectively. +We have also carried out a measurement for the Kitaev +chain with 8.8 GHz. See Supplementary Information II for +details. +As we can see from the impedance peak of the rightmost +edge in Fig. 2h, the measured resonant frequency is shifted +down from the calculated value of 17.2 GHz to 13.1 GHz. +This is mainly caused by the parasitic inductance of the metal +wires in the unit cell to connect the circuit elements. With- +out considering the wires, the simulated resonant frequency is +16.4 GHz, which is much closer to the theoretical value. For +both topological and trivial setups, the measurement results +agree well with the simulation especially around the resonant +frequency. +Fig. 2j summarizes the two-point impedance values at the +measured resonant frequency 13.1 GHz. +The blue and red +lines show the impedance measured from the left and the right +edges, respectively. The leftmost (0-th) and rightmost (16- +th) node impedance correspond to Z11 value of the 2 × 2 +impedance matrix. +In the topological setup, the impedance peaks are observed +at both the edges. The penetration length of the topological +edge state is 0.660 unit cell for the left edge and 0.666 unit +cell for the right edge, which show a good agreement with the +theoretical value 0.610 unit cell. See Supplementary Informa- +tion V for details. +We have so far observed the topological edge states. There +is also a topological interface state between topological and +trivial phases. +It is possible to switch the topological and +trivial phases for each segment. Fig. 3 summarizes the 2- +point impedance at the resonant frequency with 3 different +switch configurations for the Kitaev chains with 8.8 GHz and +17.2 GHz designs, where one point is fixed at the topologi- +cal/trivial interface and the other point is moved from 1 to +16. In Fig. 3a we divided the chain into 4 segments as shown +Fig. 3a. The impedance peak that corresponds to the topo- +logical interface state emerges at the edges of the topological +segments. When we move the left topological segment to the +right by one unit, the location of the edge states moves ac- +cordingly as shown in Fig. 3b. Then if the two separated topo- +logical segments are combined into one segment as shown in +Fig. 3c, we observe only two impedance peaks at the left and +right edges of the single topological segments. This clearly +demonstrates the movement of the topological interface state +that emerges on the electronic-circuit realization of the Kitaev +chain implemented onto the integrated circuit. We also ob- +serve the same behavior for two chains with different resonant +frequencies, which proves that the topological interface state +emerges independent of the designed resonant frequency. +Conclusion +We have materialized the SSH model and the Kitaev model +in integrated circuits. These models have topological and triv- +ial phases. It is possible to create several segments which +are either topological or trivial in a single chain. Topological +edge states emerge at both the edges of a topological segment, +which are observable by mean of the impedance resonance. +We have demonstrated that the segment size can be as small +as one unit cell because the penetration length can be made +smaller than one unit cell: See Fig.3b. Furthermore, we have +equipped our integrated circuit with a switchable structure, +which enables us to control the position of a topological in- +terface state arbitrarily along a chain. Such a possibility is a +great merit of topological electric circuits over other artificial + +5 +FIG. 4. Setup. a, A microphotograph of the chip that integrates the SSH model and the Kitaev chain, where A (B) shows the circuits for the +32-stage SSH model with 17.2GHz (8.8GHz), while C (D) shows the circuits for the 16-stage Kitaev model with 8.8GHz(17.2GHz). b, A +photo of the measurement setup. c, A block diagram of the measurement setup. +topological systems, where an integrated topological pattern +is printed once and for all. +We have observed that the resonant frequency is lower than +the theoretical value estimated from ωresonant = 1/ +√ +LC. This +is due to the parasitic inductance present in the wires. Details +are shown in Supplementary Information III. +The integrated circuit has small inductance and capaci- +tance, which leads to high frequency operation. The size of +the unit cell is 200µm and hence, largely integrated circuits +are possible. Furthermore, mass production is possible in in- +tegrated circuits, which will benefit for future industrial appli- +cations of topological electronics. +Methods +Measurements. A block diagram and a photo of the mea- +surement setup are shown in Fig.4. We observed the topolog- +ical edge state based on two-point impedance measurement. +We observe two-point impedance with a vector network +analyzer (VNA), Keysight N5222B. The chip measurement +is done on the probe station, Formfactor Summit11000. A +2×2 Z-matrix is derived from the 2×2 S-parameter measured +by the VNA. The chain configuration (the state of the SPDT +switches) is controlled by the serial-parallel interface (SPI) in- +tegrated on the same chip, whose configuration data are writ- +ten from an external PC. +Simulation is done with a circuit simulator, Cadence Spec- +tre. The S-parameters of the passive components such as ca- +pacitors and inductors are extracted for circuit simulation with +Cadence EMX, which is a planar 3D electromagnetic simula- +tor based on the Fast Multipole Method (FMM) designed for +high-frequency integrated circuits. +SSH model. +The SSH is defined by the following 1D +Hamiltonian, +H = +N +� +x=1 +tA +� +c† +2x−1c2x + c† +2xc2x−1 +� ++tB +� +c† +2xc2x+1 + c† +2x+1c2x +� +. +(1) +It is realized by an LC circuit as shown in Fig.1a. When we ap- +ply an AC source with frequency ω, with Kirchhoff’s current +law, the sum of currents from all adjacent nodes m flowing +into node n leads to the following formula, +In(ω) = +� +m +Jnm(ω)Vm(ω), +(2) +where Jnm(ω) is the circuit Laplacian. By Fourier transform- +ing from the node x to the momentum k, it is summarized +as +� +IA (k) +IB (k) +� += JAB(ω) +� +VA (k) +VB (k) +� +, +(3) +where +JAB(ω) = iω +� +1 +ω2L − (C1 + C2) +C1 + C2e−ik +C1 + C2eik +1 +ω2L − (C1 + C2) +� +(4) +is the circuit Laplacian. The condition for the impedance reso- +nance is determined by the condition where the diagonal term +is zero at the resonant frequency and the resonant frequency +is determined as +ωresonant = 1/ +� +L (C1 + C2) +(5) +for the topological phase. +On the other hand, there is no +impedance resonance for the trivial phase. +1D p-wave Kitaev topological superconductor model. +The original Kitaev p-wave superconductor model is defined +on the 1D lattice as +H = −µ +� +x +c† +xcx − t +2 +� +x +� +c† +xcx+1 + c† +x+1cx +� +−1 +2 +� +x +� +∆eiφcxcx+1 + ∆e−iφc† +x+1c† +x +� +, +(6) +where µ is the chemical potential, t > 0 is the nearest- +neighbor hopping strength and ∆ > 0 is the p-wave pairing +amplitude of the superconductor. +By introducing the Nambu representation Ψ† +k = +� +c† +k, c−k +� +and Ψk = +� +ck, c† +−k +�T +one can write the Hamiltonian in the + +a +b +c +5mm +NetworkAnalyzer +Vecto Network Analyzer +Screen +（KeysightN5222B） +Port1 +Port2 +ABCD +Network Analyzer +Chip +(Behindthe Microscope) +UnderTest +5mm +A +GPLLLLLL +Microscopic View +oftheChip +Powersupplyand +Serial-Parallel Interface +Chipon the Probe Station +(Tocontrolswitches) +SPI control signals6 +Bogoliubov-de Gennes form +H = 1 +2 +� +k +Ψ† +kH(k)Ψk, +(7) +with a 2×2 form Hamiltonian +H(k) = 1 +2 +� +−t cos k − µ +i∆0 sin k +−i∆0 sin k +t cos k + µ +� +. +(8) +The zero-energy state of the Bogoliubov-de Gennes Hamilto- +nian is a Majorana state, and hence, there appear Majorana +edge states in the topological phase of the Kitaev model. +Here, t, µ, σi and ∆i represent the hopping amplitude, the +chemical potential, the spin degree of freedom, and the su- +perconducting gap parameter, respectively. It is well known +that the system is topological for |µ| < |2t| and trivial for +|µ| > |2t| irrespective of ∆i provided ∆i ̸= 0. +We then realize this p-wave Kitaev model by way of an +electronic circuit. As shown in Fig.2a, this circuit chain con- +tains two main lines, one connected by a series of capacitors C +implementing the electrons band, while another connected by +a series of inductors L implementing the holes band, respec- +tively. Pairing interaction between the two bands is simulated +by bridging capacitors Cx and inductors Lx. Each electron +node and each hole node is connected to the ground via a ca- +pacitor C0 and inductors L0, respectively. The hopping am- +plitudes t realized in the electrons band and holes band are op- +posite since the capacitors C contained in the electrons band +contribute the terms iωC while the inductors L contained in +the holes band contribute the terms 1/(iωL). +The circuit Laplacian is given by +Jab(ω) = +� +f1 g1 +g2 f2 +� +, +(9) +where +f1 = −2C cos k + 2C − +� +ω2L0 +�−1 +f2 = 2 +� +ω2L +�−1 cos k − 2 +� +ω2L +�−1 + C0 +g1 = −Cxeik + +� +ω2Lx +�−1 e−ik +g2 = +� +ω2Lx +�−1 eik − Cxe−ik, +(10) +for topological phase and +f1 = −2C cos k + 2C + C0 +f2 = 2 +� +ω2L +�−1 cos k − 2 +� +ω2L +�−1 − +� +ω2L0 +�−1 +g1 = −Cxeik + +� +ω2Lx +�−1 e−ik +g2 = +� +ω2Lx +�−1 eik − Cxe−ik, +(11) +for trivial phase. +The essence to realize the 1D model in circuit form is +to make the circuit Laplacian equal to the system Hamilto- +nian. +Clearly, to make it possible, particle-hole symmetry +(PHS) must be respected, which requires these three pairs +of LC resonators shares the same resonant frequency, that is, +ωresonant ≡ 1/ +√ +LC = 1/√L0C0 = 1/√LxCx. 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M.E. and T.I. wrote the +manuscript with input from H.Y., Y.M., A.H. and S.Y. All the au- +thors discussed the project and the results. +Additional information +Supplementary information is available. +Competing financial interests +The authors declare no competing financial interests. + diff --git a/IdE0T4oBgHgl3EQfiAEY/content/tmp_files/load_file.txt b/IdE0T4oBgHgl3EQfiAEY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dddd29303560c3baaccec76b369be57b9c70f126 --- /dev/null +++ b/IdE0T4oBgHgl3EQfiAEY/content/tmp_files/load_file.txt @@ -0,0 +1,760 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf,len=759 +page_content='Experimental demonstration of position-controllable topological interface states in high-frequency topological integrated circuits Tetsuya Iizuka,∗ Haochen Yuan,† Yoshio Mita,† Akio Higo,‡ Shun Yasunaga,† and Motohiko Ezawa§ (Dated: January 9, 2023) Topological integrated circuits are integrated circuit realizations of topological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We perform an ex- perimental demonstration by taking instances of the Su-Schrieffer-Heeger model and the Kitaev topological superconductor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' They are found to realize high frequency resonances around 17GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We explicitly observe the spatial profile of a topological edge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In particular, the topological interface state between a topological segment and a trivial segment is the Majorana-like state in the Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We construct a switch- able structure in the integrated circuit, which enables us to control the position of a Majorana-like interface state arbitrarily along a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Our results will open topological electronics with high frequency integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Topological insulators and superconductors are fascinat- ing new states of matter[1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The Su-Schrieffer-Heeger (SSH) model[3] and the Kitaev topological superconduc- tor model[4] are simplest one-dimensional (1D) systems re- alizing topological insulators and superconductors, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Especially, topological superconductors host Ma- jorana edge states[5–8], which are the key elements of a topological quantum computer[9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The area of topo- logical physics is expanded nowadays to photonic[11–16], acoustic[17–21], mechanical[22–29] and electronic-circuit systems[30–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' They are called artificial topological sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' There are several merits which are difficult to be achieved in inorganic crystals: 1) It is possible to make a fine tuning of the system, which is crucial for observing topologi- cal edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2) It is possible to construct a few site systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3) It is possible to directly measure the site dependent infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It is relatively easy to materialize the SSH model because it involves only real hoppings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' On the other hand, this is not the case with respect to the Kitaev model because it is a p-wave topological superconductor, although the Majorana edge state itself can be generated in a s-wave superconductor with the aid of a topological insulator nanowire[38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Electronic circuits present an ideal platform to realize var- ious topological phases[30–37, 40–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The emergence of topological edge states are observed by means of impedance resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' However, experimental demonstrations have so far been restricted to printed circuit boards with discrete com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Indeed, to the best of our knowledge, there is no integrated-circuit realization working at high resonant fre- quency, although a simulation of the SSH model was done recently[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The integrated circuit realization is an important step toward industrial applications of topological electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In order to generate Majorana-like states, it is necessary to simulate electron and hole bands in electronic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Al- though there is a proposal with the use of chains of capacitors ∗ Systems Design Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=', School of Engineering, The University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' iizuka@vdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='jp (corresponding author) † Department of Electrical Engineering and Information Systems, The Uni- versity of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' ‡ Systems Design Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=', School of Engineering, The University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' § Department of Applied Physics, The University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' ezawa@ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='u- tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='jp (corresponding author) and inductors[44, 45], there is so far no experimental demon- stration of this theoretical proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Most of previous experiments were carried out based on patterned structures, where it is impossible to control the topo- logical and trivial phases once the sample is manufactured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Actually, it is very hard to introduce switch structures in in- organic materials, photonic crystals and acoustic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' On the other hand, transistors act as switches in electronic cir- cuits and hence, there is a possibility to construct a switchable topological system based on electronic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In this paper, we perform for the first time an experimen- tal demonstration of topological integrated circuits, which are integrated circuit realizations of topological systems, by tak- ing instances of the SSH model and the Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' An integrated circuit implementation enables us to realize very high resonant frequency as large as 17GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We explicitly ob- serve the spatial profile of a topological edge state and deter- mine its penetration length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The system may contain several topological and trivial segments simultaneously along a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In particular, we observe the signal of a Majorana-like state emerging at the interface of a topological segment and a triv- ial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It is topologically protected since it necessarily emerges between the topological and trivial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' These two topologically different segments are interchangeable sim- ply by exchanging inductors and capacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' SSH chain The SSH chain is the basic model of a topological insula- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It was implemented in a printed circuit board with dis- crete components[31] a few years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The electronic circuit is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Capacitances are alternating along the chain, and each node is grounded via an inductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We have implemented the SSH model in two 32-unit cell chains on chips in an integrated circuit as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Both capacitors and inductors are implemented by metallic wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' An inductor is materialized by a swirling structure of wire as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1c, while a capacitor is materialized by a comb-teeth structure as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The comb-teeth structure is prepared in order to increase the capacitance with small occupation area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The explicit values of the capacitance and the inductance are shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1, which is very tiny compared with printed circuit boards with discrete compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The small capacitance and inductance lead to a high resonant frequency ωresonant = 1/ √ LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The impedance as a function of the input frequency is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='02438v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='mes-hall] 6 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' SSH chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' a, An illustration of an electronic-circuit representing the SSH chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' b, A picture of a 32-unit cell integrated circuit for the SSH chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' c, A zoom-in view of its unit cell layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' d, A picture of the comb teeth capacitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Frequency dependence of the impedance measured from the left edge of the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz chain for e, all-topological and f, all-trivial setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' g, The spatial profile of the impedance values for all-topological mode measured from the left edge at the resonant frequency of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In e and f, solid and dashed lines show the measured and simulated results of the SSH chain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1e and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The impedance is evaluated with the two-point impedance between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The solid and dashed lines show measurement and simulation results, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' A peak emerges at the characteristic frequency for a topological phase as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The measured res- onant frequency is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='7GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' On the other hand, there is no peak for a trivial phase as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The node-dependent impedance is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The impedance decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The penetration length of the topological edge state is estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='414, which is in good agreement with the theoretical value 1/ log(C2/C1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' See Supplementary Information IV for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We have also carried out measurements on the SSH chain with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' See Supplementary Information I for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Kitaev chain The Kitaev chain model is the basic model of a topologi- cal superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Our main result is its implementation in an integrated electronic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' To realize a Cooper pair it is necessary to prepare an electron band and a hole band to- gether with cross terms between these two bands, as shown in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Parameters used for the SSH chain (left table) and the Kitaev chain (right table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz C1 42 fF 22 fF C2 414 fF 204 fF L 721 pH 378 pH 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz C 440 fF 220 fF L 747 pH 384 pH Cx 396 fF 204 fF Lx 830 pH 427 pH C0 880 fF 440 fF L0 374 pH 192 pH Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We first illustrate an electronic circuit for the Kitaev chain [44, 45] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2a, b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The capacitor channel (in- dicated in red) corresponds to the electron band, while the inductor channel (in blue) corresponds to the hole band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The two main channels are crosslinked through Cx and Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Each node is connected to the ground via an inductor L0 or a ca- pacitor C0 to realize a topological state or a trivial state, re- spectively, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The topological phase 400μm c Unit cell Unit cell C2 C1 forProbing Test H&} ContactPads 区HH区 HH区 240μm 000 GNDLine b Unit cell d 480μm 3300μm e f Topological phase Trivial phase g 1000 1000 Two-Point Impedance [Q] Two-Point Impedance [2] a 100 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='7GHz Two-Point Impedance 100 100 Node 10 Node0 10 10 Node Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 Node Penetrationlength=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='414 Node6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='01 Node7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='01 1 2 3456 10 20 1 2 3456 10 20 0 1 2 3 4 5 6 Frequency [GHz] Frequency [GHz] Node Number3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Kitaev chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' a, b and c, The electronic-circuit representation of the Kitaev chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' a, All-topological configuration where the topological edge state emerges at both the left and right edges of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' b, All-trivial configuration that does not have a topological edge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' c, The implemented state-configurable Kitaev chain circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' d, By using two SPDT switches with inverters in the unit cell, the connection of L0 and C0 can be swapped to change its topological/trivial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The SPDT switch is realized by two CMOS transmission gate switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' e, A picture of an 16-unit cell integrated circuit for the SSH chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' f, A picture of a unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' g, A zoom of SPDT switches in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Each SPDT switch is composed of an inverter and two transmission gates with n-type and p-type MOS FETs as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' h and i, Frequency dependence of the impedance measured from the right edge of the electronic circuit with the characteristic frequency ωresonant =17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz Kitaev chain for all-topological and all-trivial setups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Solid and dashed lines show the measured and simulated results of the Kitaev chain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' j, The spatial profile of the impedance values for all-topological mode measured from both left and right edges at the resonant frequency of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' is realized by the configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2a, while the trivial phase is realized by the configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' A single Kitaev chain may accommodate several segments which are either topological or trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' A Majorana-like state emerges at an interface between the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We introduce two single-pole double-throw (SPDT) switches in each unit cell as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The electric circuit for the SPDT switch is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The switching is done by swapping the connection of L0 and C0, by way of which the position of a Kitaev interface state is controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In the integrated circuits, the SPDT switch is simply implemented with an inverter and two CMOS transmission gates, composed of n-type and p- type metal oxide semiconductor (MOS) field-effect transistors (FETs) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The Kitaev chain circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2c is implemented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='onto the chip using 180 nm CMOS technology as shown in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Topological phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Trivial phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Unit cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Switches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='220μm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='GNDLine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='SPDTSwitch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Gate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='forProbingTest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='ContactPads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Gate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='400μm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Inverter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Switches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='SPDT switch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Probingpoints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='SPDTSwitch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='IN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='区 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='OUT1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='OUT2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Co ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='SPDT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='switch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='GNDLine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='SW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='OUT1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='OUT2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Unit cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Unit cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='400μm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='3800μm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Topological phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Trivial phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Two-Point Impedance [Q] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Two-Point Impedance [Ω] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Two-Point Impedance [Ω] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1GHZ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='. From-leftedge Node 16 Penetration length=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='660 10 10 10 Node16 Node 15 Fromirightedge Penetration length=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='666 Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='15 Node 14 Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 1 lode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 Nodel Node 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='01 2 3 456 10 20 2 3 456 10 20 0 2 4 6 8 10 12 14 16 Frequency [GHz] Freguency[GHz] Node Number4 0 5 10 15 20 25 30 35 40 45 0 2 4 6 8 10 12 14 16 Impedance [Ω] Node Number 3 trivial 3 trivial 2 trivial 4 topological 4 topological 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8GHz chain 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2GHz chain From node 4 From node 7 From node 10 From node 13 From node 4 From node 7 From node 10 From node 13 0 5 10 15 20 25 30 35 40 45 0 2 4 6 8 10 12 14 16 Impedance [Ω] Node Number 4 trivial 3 trivial 1 trivial 4 topological 4 topological 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8GHz chain 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2GHz chain From node 5 From node 8 From node 10 From node 13 From node 5 From node 8 From node 10 From node 13 0 5 10 15 20 25 30 35 40 45 0 2 4 6 8 10 12 14 16 Impedance [Ω] Node Number 4 trivial 4 trivial 8 topological 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8GHz chain 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2GHz chain From node 5 From node 12 From node 5 From node 12 a b c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Topological interface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Measurement results of the topological edge state locations depending on different Kitaev chain configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' "n trivial (topological)" indicates that the trivial (topological) segment contains n unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Blue (red) data points are for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz) chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' a, The topological edge states emerge at 4th, 7th, 10th and 11th nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' b, The locations of the edge states move to the 5th, 8th, 10th and 11th nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' c, When two topo- logical segments combine to one segment, the edge states emerge only at 5th and 12th nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' On a 5 mm×5 mm chip, two 16-unit cell Kitaev chain circuits were integrated for two different target resonant fre- quencies, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We show a zoom-in view of the unit cell layout in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2f, which shows that it includes 3 inductors L, Lx and L0, 3 capacitors C, Cx and C0, 2 SPDT switches, and a contact pad at each node for direct probing measurement with GSG (Ground, Signal, Ground) probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' A photo of the SPDT switches is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Two trans- mission gates and an inverter are integrated for each SPDT switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The values for the capacitors and inductors are sum- marized in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2h, i and j summarizes the impedance measurement re- sults of the Kitaev chain designed for 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz resonant fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2h and i shows the frequency dependence of the impedance measured from the right edge of the chain for topo- logical and trivial setups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The solid and dashed lines show measurement and simulation results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We have also carried out a measurement for the Kitaev chain with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' See Supplementary Information II for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' As we can see from the impedance peak of the rightmost edge in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2h, the measured resonant frequency is shifted down from the calculated value of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' This is mainly caused by the parasitic inductance of the metal wires in the unit cell to connect the circuit elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' With- out considering the wires, the simulated resonant frequency is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='4 GHz, which is much closer to the theoretical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' For both topological and trivial setups, the measurement results agree well with the simulation especially around the resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 2j summarizes the two-point impedance values at the measured resonant frequency 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The blue and red lines show the impedance measured from the left and the right edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The leftmost (0-th) and rightmost (16- th) node impedance correspond to Z11 value of the 2 × 2 impedance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In the topological setup, the impedance peaks are observed at both the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The penetration length of the topological edge state is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='660 unit cell for the left edge and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='666 unit cell for the right edge, which show a good agreement with the theoretical value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='610 unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' See Supplementary Informa- tion V for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We have so far observed the topological edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' There is also a topological interface state between topological and trivial phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It is possible to switch the topological and trivial phases for each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3 summarizes the 2- point impedance at the resonant frequency with 3 different switch configurations for the Kitaev chains with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8 GHz and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2 GHz designs, where one point is fixed at the topologi- cal/trivial interface and the other point is moved from 1 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3a we divided the chain into 4 segments as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The impedance peak that corresponds to the topo- logical interface state emerges at the edges of the topological segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' When we move the left topological segment to the right by one unit, the location of the edge states moves ac- cordingly as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Then if the two separated topo- logical segments are combined into one segment as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 3c, we observe only two impedance peaks at the left and right edges of the single topological segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' This clearly demonstrates the movement of the topological interface state that emerges on the electronic-circuit realization of the Kitaev chain implemented onto the integrated circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We also ob- serve the same behavior for two chains with different resonant frequencies, which proves that the topological interface state emerges independent of the designed resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Conclusion We have materialized the SSH model and the Kitaev model in integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' These models have topological and triv- ial phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It is possible to create several segments which are either topological or trivial in a single chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Topological edge states emerge at both the edges of a topological segment, which are observable by mean of the impedance resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We have demonstrated that the segment size can be as small as one unit cell because the penetration length can be made smaller than one unit cell: See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Furthermore, we have equipped our integrated circuit with a switchable structure, which enables us to control the position of a topological in- terface state arbitrarily along a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Such a possibility is a great merit of topological electric circuits over other artificial 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' a, A microphotograph of the chip that integrates the SSH model and the Kitaev chain, where A (B) shows the circuits for the 32-stage SSH model with 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2GHz (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8GHz), while C (D) shows the circuits for the 16-stage Kitaev model with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='8GHz(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' b, A photo of the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' c, A block diagram of the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' topological systems, where an integrated topological pattern is printed once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We have observed that the resonant frequency is lower than the theoretical value estimated from ωresonant = 1/ √ LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' This is due to the parasitic inductance present in the wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Details are shown in Supplementary Information III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The integrated circuit has small inductance and capaci- tance, which leads to high frequency operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The size of the unit cell is 200µm and hence, largely integrated circuits are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Furthermore, mass production is possible in in- tegrated circuits, which will benefit for future industrial appli- cations of topological electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Methods Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' A block diagram and a photo of the mea- surement setup are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We observed the topolog- ical edge state based on two-point impedance measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We observe two-point impedance with a vector network analyzer (VNA), Keysight N5222B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The chip measurement is done on the probe station, Formfactor Summit11000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' A 2×2 Z-matrix is derived from the 2×2 S-parameter measured by the VNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The chain configuration (the state of the SPDT switches) is controlled by the serial-parallel interface (SPI) in- tegrated on the same chip, whose configuration data are writ- ten from an external PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Simulation is done with a circuit simulator, Cadence Spec- tre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The S-parameters of the passive components such as ca- pacitors and inductors are extracted for circuit simulation with Cadence EMX, which is a planar 3D electromagnetic simula- tor based on the Fast Multipole Method (FMM) designed for high-frequency integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' SSH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The SSH is defined by the following 1D Hamiltonian, H = N � x=1 tA � c† 2x−1c2x + c† 2xc2x−1 � +tB � c† 2xc2x+1 + c† 2x+1c2x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' (1) It is realized by an LC circuit as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' When we ap- ply an AC source with frequency ω, with Kirchhoff’s current law, the sum of currents from all adjacent nodes m flowing into node n leads to the following formula, In(ω) = � m Jnm(ω)Vm(ω), (2) where Jnm(ω) is the circuit Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' By Fourier transform- ing from the node x to the momentum k, it is summarized as � IA (k) IB (k) � = JAB(ω) � VA (k) VB (k) � , (3) where JAB(ω) = iω � 1 ω2L − (C1 + C2) C1 + C2e−ik C1 + C2eik 1 ω2L − (C1 + C2) � (4) is the circuit Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The condition for the impedance reso- nance is determined by the condition where the diagonal term is zero at the resonant frequency and the resonant frequency is determined as ωresonant = 1/ � L (C1 + C2) (5) for the topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' On the other hand, there is no impedance resonance for the trivial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 1D p-wave Kitaev topological superconductor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The original Kitaev p-wave superconductor model is defined on the 1D lattice as H = −µ � x c† xcx − t 2 � x � c† xcx+1 + c† x+1cx � −1 2 � x � ∆eiφcxcx+1 + ∆e−iφc† x+1c† x � , (6) where µ is the chemical potential, t > 0 is the nearest- neighbor hopping strength and ∆ > 0 is the p-wave pairing amplitude of the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' By introducing the Nambu representation Ψ† k = � c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' c−k � and Ψk = � ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' c† −k �T one can write the Hamiltonian in the a b c 5mm NetworkAnalyzer Vecto Network Analyzer Screen （KeysightN5222B） Port1 Port2 ABCD Network Analyzer Chip (Behindthe Microscope) UnderTest 5mm A GPLLLLLL Microscopic View oftheChip Powersupplyand Serial-Parallel Interface Chipon the Probe Station (Tocontrolswitches) SPI control signals6 Bogoliubov-de Gennes form H = 1 2 � k Ψ† kH(k)Ψk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' (7) with a 2×2 form Hamiltonian H(k) = 1 2 � −t cos k − µ i∆0 sin k −i∆0 sin k t cos k + µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' (8) The zero-energy state of the Bogoliubov-de Gennes Hamilto- nian is a Majorana state, and hence, there appear Majorana edge states in the topological phase of the Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Here, t, µ, σi and ∆i represent the hopping amplitude, the chemical potential, the spin degree of freedom, and the su- perconducting gap parameter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It is well known that the system is topological for |µ| < |2t| and trivial for |µ| > |2t| irrespective of ∆i provided ∆i ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' We then realize this p-wave Kitaev model by way of an electronic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='2a, this circuit chain con- tains two main lines, one connected by a series of capacitors C implementing the electrons band, while another connected by a series of inductors L implementing the holes band, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Pairing interaction between the two bands is simulated by bridging capacitors Cx and inductors Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Each electron node and each hole node is connected to the ground via a ca- pacitor C0 and inductors L0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The hopping am- plitudes t realized in the electrons band and holes band are op- posite since the capacitors C contained in the electrons band contribute the terms iωC while the inductors L contained in the holes band contribute the terms 1/(iωL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The circuit Laplacian is given by Jab(ω) = � f1 g1 g2 f2 � , (9) where f1 = −2C cos k + 2C − � ω2L0 �−1 f2 = 2 � ω2L �−1 cos k − 2 � ω2L �−1 + C0 g1 = −Cxeik + � ω2Lx �−1 e−ik g2 = � ω2Lx �−1 eik − Cxe−ik, (10) for topological phase and f1 = −2C cos k + 2C + C0 f2 = 2 � ω2L �−1 cos k − 2 � ω2L �−1 − � ω2L0 �−1 g1 = −Cxeik + � ω2Lx �−1 e−ik g2 = � ω2Lx �−1 eik − Cxe−ik, (11) for trivial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The essence to realize the 1D model in circuit form is to make the circuit Laplacian equal to the system Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Clearly, to make it possible, particle-hole symmetry (PHS) must be respected, which requires these three pairs of LC resonators shares the same resonant frequency, that is, ωresonant ≡ 1/ √ LC = 1/√L0C0 = 1/√LxCx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Once PHS is respected, the relationship between circuit components and Hamiltonian parameters could be induced and expressed as follows: � � � t = −C, µ = −2C + C0, ∆0 = −Cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' (12) To make the 1D circuit chain topological, we set µ to 0 to meet the topological mode requirements of |µ| < |2t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' This topological property is satisfied by the emergence of grounded capacitors C0 and inductors L0, since the system will be pre- cisely located at the critical point between the topological and trivial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Therefore, by exchanging the connections of C0 and L0, we could perform transitions between these two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Impedance resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The emergence of a topological edge states is observed via impedance resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The topo- logical edge state is a zero-energy eigenstate of the Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' It corresponds to the zero admittance, and hence, the emergence is observable by the divergence in the impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The two-point impedance between the a and b nodes is given by[32] Zab ≡ Va/Ib = Gab, (13) where G is the Green function defined by the inverse of the Laplacian J, G ≡ J−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' 7 [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Hasan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Kane, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Mod.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Molenkamp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Kiessling, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Schindler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Greiter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} 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+page_content=' Wang, L Yang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Zhang, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='12, 13410 (2022) Acknowledgments This work is supported by CREST, JST (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' JPMJCR20T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The authors would like to thank Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Li and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Chen for their measurement support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The on-chip passive components are designed based on RF cell library developed by Umeda laboratory, Tokyo Uni- versity of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' The LSI chip in this study was designed and fab- ricated through the activities of VDEC, The University of Tokyo, in collaboration with Cadence Design Systems, Rohm Corporation and Toppan Printing Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Author contributions M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' planned the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' designed the topological circuits and performed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' collected and analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' wrote the manuscript with input from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' All the au- thors discussed the project and the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Additional information Supplementary information is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} +page_content=' Competing financial interests The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfiAEY/content/2301.02438v1.pdf'} diff --git a/ONAyT4oBgHgl3EQfgvjr/content/tmp_files/2301.00366v1.pdf.txt b/ONAyT4oBgHgl3EQfgvjr/content/tmp_files/2301.00366v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c3120f2667a283013ad0d5ddbc1b05d7065a9c7 --- /dev/null +++ b/ONAyT4oBgHgl3EQfgvjr/content/tmp_files/2301.00366v1.pdf.txt @@ -0,0 +1,799 @@ +Self-Supervised Object Segmentation with a Cut-and-Pasting GAN +Kunal Chaturvedi, Ali Braytee, Jun Li, Mukesh Prasad +aSchool of Computer Science, University of Technology Sydney, Ultimo, 2007, NSW, Australia +Abstract +This paper proposes a novel self-supervised based Cut-and-Paste GAN to perform foreground object seg- +mentation and generate realistic composite images without manual annotations. We accomplish this goal by +a simple yet effective self-supervised approach coupled with the U-Net based discriminator. The proposed +method extends the ability of the standard discriminators to learn not only the global data representations +via classification (real/fake) but also learn semantic and structural information through pseudo labels created +using the self-supervised task. The proposed method empowers the generator to create meaningful masks +by forcing it to learn informative per-pixel as well as global image feedback from the discriminator. Our +experiments demonstrate that our proposed method significantly outperforms the state-of-the-art methods on +the standard benchmark datasets. +Keywords: Generative adversarial networks, Self-supervised learning, Cut-and-Paste, Segmentation +1. Introduction +Generative adversarial networks (GANs) [1] have +become a popular class of image synthesis meth- +ods due to their demonstrated ability to create high- +dimensional samples with desired data distribution. +The primary objective of GANs is to generate di- +verse, high-quality images while also ensuring the +stability of GAN training [2] [3]. GAN consists of +generator and discriminator networks trained in an +adversarial manner. The generator attempts to syn- +thesize the real data distribution to fool the discrim- +inator, whereas the discriminator’s goal is to distin- +guish between the generator’s real and fake data. In +image segmentation, several compositional genera- +tive models have been proposed [4, 5, 6, 7] , where +the generator creates a synthesized composite image +by copying the object from one image and pasting +it in another to fool the discriminator about think- +ing the synthesized composite image is real. But, +the generator may not perform any segmentation, +and the background may look realistic. Therefore, +Email address: Kunal.Chaturvedi, Ali.Braytee, +Jun.Li, Mukesh.Prasad@uts.edu.au (Mukesh Prasad) +for effective training, the discriminator needs to pro- +vide the generator with informative learning signals +by learning relevant semantics and structures of the +data that may result in more effective generators. +However, the current state-of-the-art GANs employ +discriminators based on the classification network, +which learn only a single discriminative signal such +as the difference between real and fake images. In +such a non-stationary environment, the generator be- +comes prone to catastrophic forgetting and may lead +to training instability or mode collapse [8]. +To address the aforementioned issues, additional +discriminatory signals are required to guide the train- +ing mechanism and assist the generator in producing +high-quality images. This can be accomplished by +increasing the capacity of the discriminator with aux- +iliary tasks and signals. These auxiliary tasks on the +labeled datasets resist the forgetting issues and im- +prove the training stability of GANs, but it suffers +with unlabeled datasets. +Recently, self-supervised +learning has been explored on numerous GANs +methods [8],[9],[10],[11]. The self-supervised tasks +provide the learning environment with additional +guidance to the standard training mechanism. Most +of the recent self-supervision methods based GANs +Preprint submitted to Nuclear Physics B +January 3, 2023 +arXiv:2301.00366v1 [cs.CV] 1 Jan 2023 + +use auxiliary tasks based on transformation. For ex- +ample, SS-GAN developed by Chen et al. [8] uses +rotation prediction as an auxiliary task. In FX-GAN, +Huang et al. +[10] use the pretext task of predic- +tion on corrupted real images, and in LT-GAN [9], +the authors use distinguishing GAN-induced trans- +formation as a pretext task. However, the goals of +these transformation-based self-supervised tasks are +inconsistent with the GAN’s goal of mimicking the +real data distribution. Moreover, this problem ampli- +fies when the generator’s task is to construct segmen- +tation masks from the foreground images. +In order to maintain an enriched real data repre- +sentation and improve the quality of generated seg- +mentation masks, we propose a Self-Supervised Cut- +and-Paste GAN (SS-CPGAN) based on U-net archi- +tecture [12], which unifies cut-and-paste adversar- +ial training with a self-supervised task. +It allows +the discriminator to learn both local and global dif- +ferences between real and fake data. +In contrast +to the existing transformation based self-supervision +methods, our self-supervision learning method cre- +ates pseudo labels using unsupervised segmentation +methods. Then, it simultaneously forces the discrim- +inator to provide the generator with global feedback +(real or fake) and the per-pixel feedback of the syn- +thesized images with the help of pseudo labels. +To sum up, our main contributions are: +• We propose a novel Self-Supervised Cut-and- +Paste GAN (SS-CPGAN), that unifies cut-and- +paste adversarial training with a segmentation- +based self-supervised task. SS-CPGAN lever- +age unlabeled data to maximize segmentation +performance and generate highly realistic com- +posite images. +• The proposed self-supervised task in SS- +CPGAN improves the discriminator’s represen- +tation ability by enhancing structure learning +with global and local feedback. This enables the +generator with additional discriminatory sig- +nals to achieve superior results and stabilize the +training process. +• We perform a comprehensive analysis on the +benchmark datasets and compare our proposed +method with the baseline method. +2. Related Works +2.1. Unsupervised Object Segmentation via GANs +Unsupervised segmentation using GANs is an im- +portant topic in research. Several works [4, 5, 6, 7] +investigate the use of compositional generative mod- +els to obtain high quality segmentation masks. Copy- +pasting GAN [7] performs unsupervised object dis- +covery by extracting foreground objects and then +copying and pasting them onto the different back- +grounds. +Similarly, PerturbGAN [5] generates a +foreground mask along with a background and fore- +ground image in an adversarial manner. Recently, +Abdal et al. +(2021) [6] propose a method to use +an alpha network that includes two pretrained gen- +erators and a discriminator based on the StyleGAN +to generate high quality masks. +These methods +learn object segmentation without needing to use +annotations. However, they are prone to degener- +ate solutions or other trivial cases. +For example, +the generator may not perform any segmentation, +and the background looks realistic or the generator +may segment foreground masks consisting of all- +ones. To avoid such problems, special care needs +to be taken while training the compositional genera- +tive models. Copy-pasting GAN uses anti-shortcut, +border-zeroing, blur, and grounded fakes to prevent +trivial solutions [7]. +PerturbGAN avoids such so- +lutions by randomly shifting object segments rel- +ative to the background [5]. +However, Abdal et +al. (2021) [6] make several changes to the original +StyleGAN and use truncation trick along with regu- +larization to avoid degenerate solutions. +2.2. Self-supervised learning +Self-supervised learning learns useful feature rep- +resentations from data with the help of pretext tasks. +Recently, many pretext tasks coupled with adversar- +ial training have been introduced [8]. The motiva- +tion for using self-supervised learning is to: (1) pre- +vent discriminator forgetting [13]; (2) improve train- +ing stability [14]; (3) and ensure high quality of im- +ages generated [15]. The self-supervision techniques +rely on pretext tasks on geometric transformations +(e.g., prediction on rotated images[8], corrupted im- +ages [10], GAN-induced transformations [9], or a +deshuffling task that predicts the shuffled orders [16]) +2 + +to increase the discriminator’s representation power. +Unlike the aforementioned methods, we incorporate +segmentation-based self-supervised learning coupled +with the Cut-and-Paste GAN to obtain high-quality +segmentation masks. +Most importantly, with our +self-supervised approach, no extra care is needed to +deal with the trivial solutions prevalent in composi- +tional generative models. +3. Method +In this section, we first present the standard ter- +minology of adversarial training and the encoder- +decoder based discriminator. +We then introduce +our Self-Supervised Cut-and-Paste GAN built upon +the cut-and-paste adversarial training. +The uni- +fied framework with the segmentation-based self- +supervised task encourages the generator to empha- +size local and global structures while synthesizing +masks. +3.1. Adversarial Training +As shown in Fig. 2, we build a generative model +in which the generator takes the foreground image +as the input and generates a composite image using +a combination of the predicted mask, source fore- +ground image and the background image to fool the +discriminator. Formally, we define the input fore- +ground source image as If ∈ Pdata and background +image as Ib ∈ Pdata where Pdata denotes the set of +input images. Now, we define a generator (G) that +is trained in an adversarial manner against the dis- +criminator (D). +During the training process, the +generator predicts a segmentation mask defined by +mg(If) = G(I f) where mg(I f) ∈ [0, 1]. Then, using +the predicted mask mg(If), foreground source image +If, and background image Ib, we define composite +image as follows +IC = mg(I f)I f + (1 − mg(I f))Ib +(1) +The discriminator’s objective is to classify the +composite image as real or fake. As a result, the stan- +dard objective of the discriminator and the generator +of the CPGAN is defined as follows +LD = max +D +E +� +log D(If) + log(1 − D(IC)) +� +(2) +LG = min +G E �log D(IC)� +(3) +The discriminator works as a classification network +that is restricted to learn only through the discrim- +inative differences between the real and fake sam- +ples. +Thus, the discriminator fails to provide any +useful information to the generator. Therefore, we +use an encoder-decoder-based discriminator network +with self-supervised learning to mitigate this prob- +lem. +3.2. Encoder-Decoder based Discriminator +In this work, we replace standard classification- +based discriminator with the U-net based discrimi- +nator [12]. The U-net is an encoder-decoder-based +architecture that consists of a network of convolu- +tional layers, skip connections for semantic segmen- +tation. It was initially proposed for biomedical im- +age segmentation, which achieved precise segmen- +tation results with few training images. Further, it +demonstrates good results in other applications, in- +cluding ArcGIS [17], remote sensing [18], and oth- +ers. Its architecture (see Figure 1) is symmetric and +consists of two paths, an Encoder that extracts spatial +features from the input image (downscaling process), +and a Decoder that constructs the segmentation map +from the extracted feature maps (upscaling process). +We use the encoder part of the U-net as the standard +classification-based discriminator that performs the +binary decision on real/fake composite images. And +the decoder part of the U-net architecture is utilized +by the self-supervised task to give per-pixel feedback +of the synthesized images with the help of pseudo la- +bels. This allows the discriminator to learn both rel- +evant local and global differences between real/fake +images. +3.3. Self-Supervised +Cut-and-Paste +GAN +(SS- +CPGAN) +To improve the representation learning ability of +the CPGAN, the discriminator must be able to learn +semantic as well as structural information from the +synthesized images. +Therefore, we focus our ap- +proach on using self-supervised learning to build +comprehensive representations for the CPGAN. In +this work, we employ a segmentation based self- +supervised task, with the primary goal of enabling +3 + +Figure 1: An overview of U-net architecture. The different ar- +rows denote the different operations used in the encode-decoder +based architecture. +the discriminator with enhanced learned features that +ultimately empower the generator to create consis- +tent and structurally coherent masks. +The pseudo +segmentation masks mUS(I f) ∈ [0, 1] are created +using graph-based unsupervised segmentation algo- +rithm [16]. These masks obtained by the GrabCut +technique acts as a good prior for the U-net based +discriminator (see, Figure 2). Here, the discriminator +performs two important tasks, i.e., 1) classification +of real/fake compositing images and 2) performing +per-pixel based classification on I f ∈ Pdata to gener- +ate segmentation masks. Given the self-supervised +pseudo labels, we train the discriminator for accu- +rate pixel-level prediction. The introduction of self- +supervisory signals empowers the discriminator by +enhancing its localization ability and forces the dis- +criminator it to learn useful semantic representations. +This mechanism enables the generator to achieve op- +timized results and makes the training process more +stabilized. +Formally, we define I f +∈ Pdata as the source +image containing foreground object, and Pdata de- +notes the set of input images. Given a source fore- +ground image I f, we create pseudo label denoted by +mUS(I f) ∈ [0, 1], using an unsupervised segmenta- +tion algorithm. Then, we define mw(IC) ∈ [0, 1] as +the pixel-wise segmentation mask produced by the +decoder of the discriminator. Hereafter, we optimize +the overall discriminator loss function (Eq. 5) by +augmenting a new self-supervision based loss (Eq. 4) +Lsel f−supervised = L +� +mw(IC), 1 − mUS(If) +� +(4) +L′ +D = LD + λLsel f−supervised +(5) +where L is the cross-entropy loss, and λ denotes the +loss weight for the self-supervision based loss. This +hyperparameter is updated based on the compari- +son between mUS(I f) and mw(IC), using intersection- +over-union (IoU). The details of the hyperparameter +chosen are explained in the implementation details +section. The framework of the self-supervised learn- +ing is shown in Figure 2. +4. Experimentation +This section discusses the implementation details +of the proposed method and an extensive set of ex- +periments on various datasets. +4.1. Datasets +We utilize five different datasets for the foreground +and background set to train our SS-CPGAN as de- +scribed below: +• Caltech-UCSD Birds (CUB) 200-2011 is a fre- +quently used benchmark for unsupervised im- +age segmentation. It consists of 11,788 images +from 200 birds species. +• Oxford 102 Flowers consists of 8,189 images +from 102 flower classes. +• FGVC Aircraft (Airplanes) contains 102 dif- +ferent aircraft model variants with 100 images +of each. This dataset was originally used for the +purpose of fine-grained visual categorization. +• MIT Places2 is a scene-centric dataset with +more than 10 million images consisting of over +400 unique scene classes. However, in the ex- +periments, we use the classes: rainforest, for- +est, sky, and swamp as a background set for +the Caltech-UCSD Birds dataset, and the Ox- +ford 102 Flowers, we use the class: herb garden +as a background set. +4 + +64 +64 +128 +t9 +64 +Input +indno +Image +abe +128 +128 +256 +128 +256 +256 +512 +256 +Conv 3 X 3 +Max pool2x 2 +Up-conv 2 × 2 +512 +512 +1024512 +Concatenation +1024 +Finalconv 1x 1Figure 2: The proposed Self-supervised cut-and-paste GAN (SS-CPGAN) +• Singapore Whole-sky IMaging CATegories +(SWIMCAT) contains 784 images of a total +of five categories: patterned clouds, clear sky, +thick dark clouds, veil clouds, and thick white +clouds. +We use the SWIMCAT dataset as a +background set for the FGCV dataset. +We chose background datasets similar to the back- +ground of the images from the foreground dataset. +For the foreground datasets, we use Caltech-UCSD +Birds (CUB) 200-2011 [19], Oxford 102 Flowers +[20], and FGCV Aircraft (Airplanes) [21]. During +the training, we do not utilize the masks available +with datasets, Caltech-UCSD Birds and Flowers- +102. +For the background datasets, we use MIT +Places2 [22], and SWIMCAT [23]. +4.2. Implementation Details +Our implementation is based on the PyTorch +framework. For training our models, we deploy a +batch size of 16 and the Adam optimizer with an +initial learning rate of 2.10−4. +We use the unsu- +pervised segmentation algorithm based on the Grab- +Cut technique [24] for the self-supervision task. +Then, we set the weighting parameters for the self- +supervised term in the loss function according to +the Intersection-Over-Union (IoU) score between +the pseudo label (mask) and the predicted mask +by the discriminator. +Initially, when IoU < 0.2, +the hyperparameter value is set to 0.5 to boost the +model’s ability to learn useful representations from +the pseudo label. When the 0.2 < IoU < 0.8, we +refine the predicted mask using the hyperparameter +value λ of 0.1. To avoid the pseudo labels compro- +mising the predicted masks, we restrict the value λ to +0 when the IoU > 0.8. +4.3. Results +We utilized the Fr´echet Inception Distance (FID) +score and mean Intersection over Union (mIoU) met- +ric for quantitative evaluation of our methods. +In +this work, we use the FID score on the datasets +CUB2011, Oxford 102 Flowers, and FGCV Aircraft +(see Table 3) to compare the SS-CPGAN model with +the CPGAN model images spatially scaled to 64×64, +128 × 128, and 256 × 256. And for the datasets with +available ground truth masks, including CUB2011, +5 + +Self- +Supervised +Loss (Eq. 4) +SS-CPGAN +Encodel +Foreground +U-Net +Generator +Predicted Mask +Composite +Real/Fake +Background +U-Net +Discriminator +Itask +Unsupervised +Self-Supervised +Segmentation +using +GrabCut +Pseudo Label +Predicted Mask +byDiscriminator +Similarity +Update Loss +Check +WeightTable 1: FID comparison of the proposed method with the base- +line CPGAN model +FID ↓ +Methods +Image size +Caltech +UCSD- +Bird 200 +FGCV- +Aircraft +Oxford +102 +Flowers +CPGAN +64 x 64 +27.724 +43.353 +81.724 +128 x 128 +23.125 +39.674 +44.825 +256 x 256 +22.846 +44.825 +51.218 +SS-CPGAN 64 x 64 +23.751 +39.578 +63.343 +128 x 128 +16.893 +37.756 +54.982 +256 x 256 +13.422 +33.149 +49.181 +Table 2: mIOU comparison of the proposed method with the +baseline CPGAN model +mIoU ↑ +Methods +Image size +Caltech +UCSD- +Bird 200 +Oxford +102 +Flowers +w/o Self-Supervision 64 x 64 +0.537 +0.632 +128 x 128 +0.492 +0.674 +256 x 256 +0.484 +0.779 +Self-Supervision +64 x 64 +0.571 +0.625 +128 x 128 +0.543 +0.719 +256 x 256 +0.518 +0.791 +Oxford 102 Flowers, we use the mIoU metric as +shown in Table 2. +In Figure 2, we report the FID scores over the +training iterations. We show that our method sta- +bilizes GAN training across all the datasets by al- +lowing GAN training to converge faster and con- +sistently improve performance throughout the train- +ing. According to Table 3 and Figure 2, our method, +SS-CPGAN, utilizing self-supervision outperforms +the baseline method, CPGAN, on each dataset used. +Furthermore, as shown in Fig. 3, the generated masks +and composite images of our proposed SS-CPGAN +are of superior quality. The standard classification +based discriminator of CPGAN does not provide ef- +fective guidance to the generator. During the train- +ing, the standard discriminator is not encouraged to +learn a more robust data representation. The classi- +fication task learns only the representation based on +the discriminative differences between real/fake im- +ages and fails to give information on why the synthe- +sized image looks fake. Notedly, our self-supervision +based task assigned to the U-net based discrimina- +tor provides the generator with global feedback (real +or fake) as well as per-pixel feedback of the masks +with the help of pseudo labels. The self-supervisory +signals prevent the two scenarios for the generator +which the standard discriminator fails to do, i.e., cre- +ating constant masks of only all-zeros pixel values or +all-ones pixel values. The enhanced discriminator of +SS-CPGAN influences the generator to create high +quality masks that are devoid of any such anoma- +lies. As shown in Figure 3, the qualitative analysis of +the proposed SS-CPGAN shows that the generated +masks and composite images are of superior quality. +4.4. Comparison with the state-of-the-art +We compare our self-supervision based Cut-and- +Paste GAN (SS-CPGAN) with state-of-the-art. As +shown in Table 3, we report and compare the +FID score on the Caltech UCSD-Bird 200 dataset. +Specifically, the FID scores of StackGANv2 [25], +OneGAN [26], LR-GAN [27], ELGAN [28], and +FineGAN [29] are listed. +The results in Table 3 +show that our method delivers better performance +and outperforms the existing methods. +LR-GAN +[27] performed the worst, followed by the other +methods. The low performance of layer-wise GANs +[27] [28] is attributed to the fact that these methods +are prone to degenerate during the training phase, +with all the pixels being assigned as one compo- +nent. In Table 4, we compare the performance of our +method to the recent methods using the mIoU met- +ric on Caltech UCSD-Bird 200 and Oxford flowers- +102 respectively. In comparison to PerturbGAN [5], +ContraCAM [30], ReDO [4], UISB [31] and IIC- +seg [32], our method outperforms by a large mar- +gin on Caltech UCSD-Bird 200 dataset. On the Ox- +ford flowers-102 dataset, we perform better than the +methods, ReDO [4], Kyriazi et. al [33] and Voynov +et. al. [34]. Here, ReDO and Kyriazi et. al (2021) +are unsupervised approaches whereas Voynov et. al +(2021) is a weakly supervised approach to create seg- +mentation maps. The ability to leverage pseudo la- +bels in the training of Cut-and-Paste GAN assists in +creating foreground masks of superior quality. +6 + +Figure 3: Visualization results with the proposed SS-CPGAN on the datasets: Oxford 102 Flowers (left), FGVC Aircraft (center), +and Caltech-UCSD Birds (CUB) 200-2011 (right). +Table 3: FID comparison of our proposed method SS-CPGAN +with the state-of-art on Caltech UCSD-Bird 200 dataset +Method +FID +StackGANv2 +21.4 +FineGAN +23.0 +OneGAN +20.5 +LR-GAN +34.91 +ELGAN +15.7 +SS-CPGAN +13.42 +Table 4: Quantitative comparison of the segmentation perfor- +mance of our method SS-CPGAN with the state-of-art +Dataset +Method +mIoU +Caltech UCSD-Bird 200 +PerturbGAN +0.380 +ContraCAM +0.460 +ReDO +0.426 +UISB +0.442 +IIC-seg +0.365 +SS-CPGAN +0.571 +Oxford 102 flowers +ReDO +0.764 +Kyriazi et. al. +0.541 +Voynov et al. +0.540 +SS-CPGAN +0.791 +5. Conclusion +In this work, we proposed a novel Self-Supervised +Cut-and-Paste GAN method to learn object seg- +mentation. Specifically, we unify the cut-and-paste +adversarial training with the proposed segmenta- +tion based self-supervision learning. +Unlike the +existing transformation based self-supervised meth- +ods, our method improves the discriminator’s rep- +resentation ability by enhancing structure learning +with global and local feedback from the synthesized +masks. Furthermore, SS-CPGAN overcomes the is- +sue of unwanted trivial solutions (generating con- +stant masks of only all-zeros or all-ones pixel values) +that plagues the generator. The experimental results +show that our approach generates superior quality +images and achieves promising results on the bench- +mark datasets. +References +[1] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, +D. Warde-Farley, S. Ozair, A. Courville, Y. 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Vedaldi, +Finding an unsupervised image segmenter in each of your +deep generative models, arXiv preprint arXiv:2105.08127 +(2021). +[34] A. Voynov, S. Morozov, A. Babenko, Object segmenta- +tion without labels with large-scale generative models, in: +International Conference on Machine Learning, PMLR, +2021, pp. 10596–10606. +9 + diff --git a/ONAyT4oBgHgl3EQfgvjr/content/tmp_files/load_file.txt b/ONAyT4oBgHgl3EQfgvjr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60e0d3f6b3ca0f017c53755845cfd85dee7a0f40 --- /dev/null +++ b/ONAyT4oBgHgl3EQfgvjr/content/tmp_files/load_file.txt @@ -0,0 +1,464 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf,len=463 +page_content='Self-Supervised Object Segmentation with a Cut-and-Pasting GAN Kunal Chaturvedi, Ali Braytee, Jun Li, Mukesh Prasad aSchool of Computer Science, University of Technology Sydney, Ultimo, 2007, NSW, Australia Abstract This paper proposes a novel self-supervised based Cut-and-Paste GAN to perform foreground object seg- mentation and generate realistic composite images without manual annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We accomplish this goal by a simple yet effective self-supervised approach coupled with the U-Net based discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The proposed method extends the ability of the standard discriminators to learn not only the global data representations via classification (real/fake) but also learn semantic and structural information through pseudo labels created using the self-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The proposed method empowers the generator to create meaningful masks by forcing it to learn informative per-pixel as well as global image feedback from the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Our experiments demonstrate that our proposed method significantly outperforms the state-of-the-art methods on the standard benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Keywords: Generative adversarial networks, Self-supervised learning, Cut-and-Paste, Segmentation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Introduction Generative adversarial networks (GANs) [1] have become a popular class of image synthesis meth- ods due to their demonstrated ability to create high- dimensional samples with desired data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The primary objective of GANs is to generate di- verse, high-quality images while also ensuring the stability of GAN training [2] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' GAN consists of generator and discriminator networks trained in an adversarial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The generator attempts to syn- thesize the real data distribution to fool the discrim- inator, whereas the discriminator’s goal is to distin- guish between the generator’s real and fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In image segmentation, several compositional genera- tive models have been proposed [4, 5, 6, 7] , where the generator creates a synthesized composite image by copying the object from one image and pasting it in another to fool the discriminator about think- ing the synthesized composite image is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' But, the generator may not perform any segmentation, and the background may look realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Therefore, Email address: Kunal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='Chaturvedi, Ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='Braytee, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='Li, Mukesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='Prasad@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='au (Mukesh Prasad) for effective training, the discriminator needs to pro- vide the generator with informative learning signals by learning relevant semantics and structures of the data that may result in more effective generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' However, the current state-of-the-art GANs employ discriminators based on the classification network, which learn only a single discriminative signal such as the difference between real and fake images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In such a non-stationary environment, the generator be- comes prone to catastrophic forgetting and may lead to training instability or mode collapse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' To address the aforementioned issues, additional discriminatory signals are required to guide the train- ing mechanism and assist the generator in producing high-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' This can be accomplished by increasing the capacity of the discriminator with aux- iliary tasks and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' These auxiliary tasks on the labeled datasets resist the forgetting issues and im- prove the training stability of GANs, but it suffers with unlabeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Recently, self-supervised learning has been explored on numerous GANs methods [8],[9],[10],[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The self-supervised tasks provide the learning environment with additional guidance to the standard training mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Most of the recent self-supervision methods based GANs Preprint submitted to Nuclear Physics B January 3, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='00366v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='CV] 1 Jan 2023 use auxiliary tasks based on transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' For ex- ample, SS-GAN developed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' [8] uses rotation prediction as an auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In FX-GAN, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' [10] use the pretext task of predic- tion on corrupted real images, and in LT-GAN [9], the authors use distinguishing GAN-induced trans- formation as a pretext task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' However, the goals of these transformation-based self-supervised tasks are inconsistent with the GAN’s goal of mimicking the real data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Moreover, this problem ampli- fies when the generator’s task is to construct segmen- tation masks from the foreground images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In order to maintain an enriched real data repre- sentation and improve the quality of generated seg- mentation masks, we propose a Self-Supervised Cut- and-Paste GAN (SS-CPGAN) based on U-net archi- tecture [12], which unifies cut-and-paste adversar- ial training with a self-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' It allows the discriminator to learn both local and global dif- ferences between real and fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In contrast to the existing transformation based self-supervision methods, our self-supervision learning method cre- ates pseudo labels using unsupervised segmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Then, it simultaneously forces the discrim- inator to provide the generator with global feedback (real or fake) and the per-pixel feedback of the syn- thesized images with the help of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' To sum up, our main contributions are: We propose a novel Self-Supervised Cut-and- Paste GAN (SS-CPGAN), that unifies cut-and- paste adversarial training with a segmentation- based self-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' SS-CPGAN lever- age unlabeled data to maximize segmentation performance and generate highly realistic com- posite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The proposed self-supervised task in SS- CPGAN improves the discriminator’s represen- tation ability by enhancing structure learning with global and local feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' This enables the generator with additional discriminatory sig- nals to achieve superior results and stabilize the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We perform a comprehensive analysis on the benchmark datasets and compare our proposed method with the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Unsupervised Object Segmentation via GANs Unsupervised segmentation using GANs is an im- portant topic in research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Several works [4, 5, 6, 7] investigate the use of compositional generative mod- els to obtain high quality segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Copy- pasting GAN [7] performs unsupervised object dis- covery by extracting foreground objects and then copying and pasting them onto the different back- grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Similarly, PerturbGAN [5] generates a foreground mask along with a background and fore- ground image in an adversarial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Recently, Abdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' (2021) [6] propose a method to use an alpha network that includes two pretrained gen- erators and a discriminator based on the StyleGAN to generate high quality masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' These methods learn object segmentation without needing to use annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' However, they are prone to degener- ate solutions or other trivial cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' For example, the generator may not perform any segmentation, and the background looks realistic or the generator may segment foreground masks consisting of all- ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' To avoid such problems, special care needs to be taken while training the compositional genera- tive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Copy-pasting GAN uses anti-shortcut, border-zeroing, blur, and grounded fakes to prevent trivial solutions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' PerturbGAN avoids such so- lutions by randomly shifting object segments rel- ative to the background [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' However, Abdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' (2021) [6] make several changes to the original StyleGAN and use truncation trick along with regu- larization to avoid degenerate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Self-supervised learning Self-supervised learning learns useful feature rep- resentations from data with the help of pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Recently, many pretext tasks coupled with adversar- ial training have been introduced [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The motiva- tion for using self-supervised learning is to: (1) pre- vent discriminator forgetting [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' (2) improve train- ing stability [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' (3) and ensure high quality of im- ages generated [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The self-supervision techniques rely on pretext tasks on geometric transformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=', prediction on rotated images[8], corrupted im- ages [10], GAN-induced transformations [9], or a deshuffling task that predicts the shuffled orders [16]) 2 to increase the discriminator’s representation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Unlike the aforementioned methods, we incorporate segmentation-based self-supervised learning coupled with the Cut-and-Paste GAN to obtain high-quality segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Most importantly, with our self-supervised approach, no extra care is needed to deal with the trivial solutions prevalent in composi- tional generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Method In this section, we first present the standard ter- minology of adversarial training and the encoder- decoder based discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We then introduce our Self-Supervised Cut-and-Paste GAN built upon the cut-and-paste adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The uni- fied framework with the segmentation-based self- supervised task encourages the generator to empha- size local and global structures while synthesizing masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Adversarial Training As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 2, we build a generative model in which the generator takes the foreground image as the input and generates a composite image using a combination of the predicted mask, source fore- ground image and the background image to fool the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Formally, we define the input fore- ground source image as If ∈ Pdata and background image as Ib ∈ Pdata where Pdata denotes the set of input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Now, we define a generator (G) that is trained in an adversarial manner against the dis- criminator (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' During the training process, the generator predicts a segmentation mask defined by mg(If) = G(I f) where mg(I f) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Then, using the predicted mask mg(If), foreground source image If, and background image Ib, we define composite image as follows IC = mg(I f)I f + (1 − mg(I f))Ib (1) The discriminator’s objective is to classify the composite image as real or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' As a result, the stan- dard objective of the discriminator and the generator of the CPGAN is defined as follows LD = max D E � log D(If) + log(1 − D(IC)) � (2) LG = min G E �log D(IC)� (3) The discriminator works as a classification network that is restricted to learn only through the discrim- inative differences between the real and fake sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Thus, the discriminator fails to provide any useful information to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Therefore, we use an encoder-decoder-based discriminator network with self-supervised learning to mitigate this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Encoder-Decoder based Discriminator In this work, we replace standard classification- based discriminator with the U-net based discrimi- nator [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The U-net is an encoder-decoder-based architecture that consists of a network of convolu- tional layers, skip connections for semantic segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' It was initially proposed for biomedical im- age segmentation, which achieved precise segmen- tation results with few training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Further, it demonstrates good results in other applications, in- cluding ArcGIS [17], remote sensing [18], and oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Its architecture (see Figure 1) is symmetric and consists of two paths, an Encoder that extracts spatial features from the input image (downscaling process), and a Decoder that constructs the segmentation map from the extracted feature maps (upscaling process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We use the encoder part of the U-net as the standard classification-based discriminator that performs the binary decision on real/fake composite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' And the decoder part of the U-net architecture is utilized by the self-supervised task to give per-pixel feedback of the synthesized images with the help of pseudo la- bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' This allows the discriminator to learn both rel- evant local and global differences between real/fake images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Self-Supervised Cut-and-Paste GAN (SS- CPGAN) To improve the representation learning ability of the CPGAN, the discriminator must be able to learn semantic as well as structural information from the synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Therefore, we focus our ap- proach on using self-supervised learning to build comprehensive representations for the CPGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In this work, we employ a segmentation based self- supervised task, with the primary goal of enabling 3 Figure 1: An overview of U-net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The different ar- rows denote the different operations used in the encode-decoder based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' the discriminator with enhanced learned features that ultimately empower the generator to create consis- tent and structurally coherent masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The pseudo segmentation masks mUS(I f) ∈ [0, 1] are created using graph-based unsupervised segmentation algo- rithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' These masks obtained by the GrabCut technique acts as a good prior for the U-net based discriminator (see, Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Here, the discriminator performs two important tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=', 1) classification of real/fake compositing images and 2) performing per-pixel based classification on I f ∈ Pdata to gener- ate segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Given the self-supervised pseudo labels, we train the discriminator for accu- rate pixel-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The introduction of self- supervisory signals empowers the discriminator by enhancing its localization ability and forces the dis- criminator it to learn useful semantic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' This mechanism enables the generator to achieve op- timized results and makes the training process more stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Formally, we define I f ∈ Pdata as the source image containing foreground object, and Pdata de- notes the set of input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Given a source fore- ground image I f, we create pseudo label denoted by mUS(I f) ∈ [0, 1], using an unsupervised segmenta- tion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Then, we define mw(IC) ∈ [0, 1] as the pixel-wise segmentation mask produced by the decoder of the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Hereafter, we optimize the overall discriminator loss function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 5) by augmenting a new self-supervision based loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4) Lsel f−supervised = L � mw(IC), 1 − mUS(If) � (4) L′ D = LD + λLsel f−supervised (5) where L is the cross-entropy loss, and λ denotes the loss weight for the self-supervision based loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' This hyperparameter is updated based on the compari- son between mUS(I f) and mw(IC), using intersection- over-union (IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The details of the hyperparameter chosen are explained in the implementation details section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The framework of the self-supervised learn- ing is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Experimentation This section discusses the implementation details of the proposed method and an extensive set of ex- periments on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Datasets We utilize five different datasets for the foreground and background set to train our SS-CPGAN as de- scribed below: Caltech-UCSD Birds (CUB) 200-2011 is a fre- quently used benchmark for unsupervised im- age segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' It consists of 11,788 images from 200 birds species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Oxford 102 Flowers consists of 8,189 images from 102 flower classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' FGVC Aircraft (Airplanes) contains 102 dif- ferent aircraft model variants with 100 images of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' This dataset was originally used for the purpose of fine-grained visual categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' MIT Places2 is a scene-centric dataset with more than 10 million images consisting of over 400 unique scene classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' However, in the ex- periments, we use the classes: rainforest, for- est, sky, and swamp as a background set for the Caltech-UCSD Birds dataset, and the Ox- ford 102 Flowers, we use the class: herb garden as a background set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4 64 64 128 t9 64 Input indno Image abe 128 128 256 128 256 256 512 256 Conv 3 X 3 Max pool2x 2 Up-conv 2 × 2 512 512 1024512 Concatenation 1024 Finalconv 1x 1Figure 2: The proposed Self-supervised cut-and-paste GAN (SS-CPGAN) Singapore Whole-sky IMaging CATegories (SWIMCAT) contains 784 images of a total of five categories: patterned clouds, clear sky, thick dark clouds, veil clouds, and thick white clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We use the SWIMCAT dataset as a background set for the FGCV dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We chose background datasets similar to the back- ground of the images from the foreground dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' For the foreground datasets, we use Caltech-UCSD Birds (CUB) 200-2011 [19], Oxford 102 Flowers [20], and FGCV Aircraft (Airplanes) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' During the training, we do not utilize the masks available with datasets, Caltech-UCSD Birds and Flowers- 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' For the background datasets, we use MIT Places2 [22], and SWIMCAT [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Implementation Details Our implementation is based on the PyTorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' For training our models, we deploy a batch size of 16 and the Adam optimizer with an initial learning rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We use the unsu- pervised segmentation algorithm based on the Grab- Cut technique [24] for the self-supervision task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Then, we set the weighting parameters for the self- supervised term in the loss function according to the Intersection-Over-Union (IoU) score between the pseudo label (mask) and the predicted mask by the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Initially, when IoU < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='2, the hyperparameter value is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='5 to boost the model’s ability to learn useful representations from the pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' When the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='2 < IoU < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='8, we refine the predicted mask using the hyperparameter value λ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' To avoid the pseudo labels compro- mising the predicted masks, we restrict the value λ to 0 when the IoU > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Results We utilized the Fr´echet Inception Distance (FID) score and mean Intersection over Union (mIoU) met- ric for quantitative evaluation of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In this work, we use the FID score on the datasets CUB2011, Oxford 102 Flowers, and FGCV Aircraft (see Table 3) to compare the SS-CPGAN model with the CPGAN model images spatially scaled to 64×64, 128 × 128, and 256 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' And for the datasets with available ground truth masks, including CUB2011, 5 Self- Supervised Loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4) SS-CPGAN Encodel Foreground U-Net Generator Predicted Mask Composite Real/Fake Background U-Net Discriminator Itask Unsupervised Self-Supervised Segmentation using GrabCut Pseudo Label Predicted Mask byDiscriminator Similarity Update Loss Check WeightTable 1: FID comparison of the proposed method with the base- line CPGAN model FID ↓ Methods Image size Caltech UCSD- Bird 200 FGCV- Aircraft Oxford 102 Flowers CPGAN 64 x 64 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='724 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='353 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='724 128 x 128 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='125 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='674 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='825 256 x 256 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='846 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='825 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='218 SS-CPGAN 64 x 64 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='751 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='578 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='343 128 x 128 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='893 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='756 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='982 256 x 256 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='422 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='149 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='181 Table 2: mIOU comparison of the proposed method with the baseline CPGAN model mIoU ↑ Methods Image size Caltech UCSD- Bird 200 Oxford 102 Flowers w/o Self-Supervision 64 x 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='632 128 x 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='674 256 x 256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='779 Self-Supervision 64 x 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='625 128 x 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='719 256 x 256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='791 Oxford 102 Flowers, we use the mIoU metric as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In Figure 2, we report the FID scores over the training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' We show that our method sta- bilizes GAN training across all the datasets by al- lowing GAN training to converge faster and con- sistently improve performance throughout the train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' According to Table 3 and Figure 2, our method, SS-CPGAN, utilizing self-supervision outperforms the baseline method, CPGAN, on each dataset used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 3, the generated masks and composite images of our proposed SS-CPGAN are of superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The standard classification based discriminator of CPGAN does not provide ef- fective guidance to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' During the train- ing, the standard discriminator is not encouraged to learn a more robust data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The classi- fication task learns only the representation based on the discriminative differences between real/fake im- ages and fails to give information on why the synthe- sized image looks fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Notedly, our self-supervision based task assigned to the U-net based discrimina- tor provides the generator with global feedback (real or fake) as well as per-pixel feedback of the masks with the help of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The self-supervisory signals prevent the two scenarios for the generator which the standard discriminator fails to do, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=', cre- ating constant masks of only all-zeros pixel values or all-ones pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The enhanced discriminator of SS-CPGAN influences the generator to create high quality masks that are devoid of any such anoma- lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' As shown in Figure 3, the qualitative analysis of the proposed SS-CPGAN shows that the generated masks and composite images are of superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Comparison with the state-of-the-art We compare our self-supervision based Cut-and- Paste GAN (SS-CPGAN) with state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' As shown in Table 3, we report and compare the FID score on the Caltech UCSD-Bird 200 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Specifically, the FID scores of StackGANv2 [25], OneGAN [26], LR-GAN [27], ELGAN [28], and FineGAN [29] are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The results in Table 3 show that our method delivers better performance and outperforms the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' LR-GAN [27] performed the worst, followed by the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The low performance of layer-wise GANs [27] [28] is attributed to the fact that these methods are prone to degenerate during the training phase, with all the pixels being assigned as one compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In Table 4, we compare the performance of our method to the recent methods using the mIoU met- ric on Caltech UCSD-Bird 200 and Oxford flowers- 102 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' In comparison to PerturbGAN [5], ContraCAM [30], ReDO [4], UISB [31] and IIC- seg [32], our method outperforms by a large mar- gin on Caltech UCSD-Bird 200 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' On the Ox- ford flowers-102 dataset, we perform better than the methods, ReDO [4], Kyriazi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' al [33] and Voynov et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Here, ReDO and Kyriazi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' al (2021) are unsupervised approaches whereas Voynov et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' al (2021) is a weakly supervised approach to create seg- mentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The ability to leverage pseudo la- bels in the training of Cut-and-Paste GAN assists in creating foreground masks of superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 6 Figure 3: Visualization results with the proposed SS-CPGAN on the datasets: Oxford 102 Flowers (left), FGVC Aircraft (center), and Caltech-UCSD Birds (CUB) 200-2011 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Table 3: FID comparison of our proposed method SS-CPGAN with the state-of-art on Caltech UCSD-Bird 200 dataset Method FID StackGANv2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='4 FineGAN 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='0 OneGAN 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='5 LR-GAN 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='91 ELGAN 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='7 SS-CPGAN 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='42 Table 4: Quantitative comparison of the segmentation perfor- mance of our method SS-CPGAN with the state-of-art Dataset Method mIoU Caltech UCSD-Bird 200 PerturbGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='380 ContraCAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='460 ReDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='426 UISB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='442 IIC-seg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='365 SS-CPGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='571 Oxford 102 flowers ReDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='764 Kyriazi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='541 Voynov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='540 SS-CPGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content='791 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Conclusion In this work, we proposed a novel Self-Supervised Cut-and-Paste GAN method to learn object seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Specifically, we unify the cut-and-paste adversarial training with the proposed segmenta- tion based self-supervision learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Unlike the existing transformation based self-supervised meth- ods, our method improves the discriminator’s rep- resentation ability by enhancing structure learning with global and local feedback from the synthesized masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Furthermore, SS-CPGAN overcomes the is- sue of unwanted trivial solutions (generating con- stant masks of only all-zeros or all-ones pixel values) that plagues the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' The experimental results show that our approach generates superior quality images and achieves promising results on the bench- mark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' References [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Pouget-Abadie, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' Babenko, Object segmenta- tion without labels with large-scale generative models, in: International Conference on Machine Learning, PMLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 10596–10606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'} diff --git a/OdFOT4oBgHgl3EQf3TT6/content/tmp_files/2301.12946v1.pdf.txt b/OdFOT4oBgHgl3EQf3TT6/content/tmp_files/2301.12946v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e885cdb2a96139dc6c11d99af91294f04b12856 --- /dev/null +++ b/OdFOT4oBgHgl3EQf3TT6/content/tmp_files/2301.12946v1.pdf.txt @@ -0,0 +1,3015 @@ +Efficient learning of ground & thermal states within phases of matter +Emilio Onorati,1, ∗ Cambyse Rouz´e ,1, † Daniel Stilck Fran¸ca,2 and James D. Watson3 +1Zentrum Mathematik, Technische Universit¨at M¨unchen, 85748 Garching, Germany +2Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France‡ +3University of Maryland, College Park, QuICS 3353 Atlantic Building, MD 20742-2420, USA § +We consider two related tasks: (a) estimating a parameterisation of an unknown Gibbs +state and expectation values of Lipschitz observables on this state; and (b) learning the +expectation values of local observables within a thermal or quantum phase of matter. In +both cases, we wish to minimise the number of samples we use to learn these properties to +a given precision. +For the first task, we develop new techniques to learn parameterisations of classes of sys- +tems, including quantum Gibbs states of non-commuting Hamiltonians under the condition +of exponential decay of correlations and the approximate Markov property, thus improving +on work by [RF21]. We show that it is possible to infer the expectation values of all extensive +properties of the state from a number of copies that not only scales polylogarithmically with +the system size, but polynomially in the observable’s locality — an exponential improvement +— hence partially answering conjectures stated in [RF21] and [AAKS21] in the positive. This +class of properties includes expected values of quasi-local observables and entropic quantities +of the state. +For the second task, we turn our tomography tools into efficient algorithms for learning +observables in a phase of matter of a quantum system. By exploiting the locality of the +Hamiltonian, we show that M local observables can be learned with probability 1−δ and up +to precision ε with access to only N = O +� +log +� M +δ +� +epolylog(ε−1)� +samples — an exponential +improvement in the precision over the best previously known bounds [HKT+22]. Our results +apply to both families of ground states of Hamiltonians displaying local topological quantum +order, and thermal phases of matter displaying exponential decay of correlations. In addition, +our sample complexity applies to the worse case setting whereas previous results only applied +to the average case setting. +To prove our results, we develop new tools of independent interest, such as robust shadow +tomography algorithms for ground and Gibbs states, Gibbs approximations of locally indis- +tinguishable ground states, and generalisations of transportation cost inequalities for Gibbs +states of non-commuting Hamiltonians. +∗ Emilio Onorati emilio.onorati@tum.de +† Cambyse Rouz´e cambyse.rouze@tum.de +‡ Daniel Stilck Fran¸ca daniel.stilck franca@ens-lyon.fr +§ James D. Watson jdwatson@umd.edu +arXiv:2301.12946v1 [quant-ph] 30 Jan 2023 + +2 +I. +INTRODUCTION +Tomography of quantum states is among the most important tasks in quantum information +science. In quantum tomography, we have access to one or more copies of a quantum state and wish +to understand the structure of the state. However, for a general quantum state, all tomographic +methods inevitably require resources that scale exponentially in the size of the system [HHJ+17, +OW16]. This is due to the curse of dimensionality: the number of parameters needed to fully +describe a quantum system scales exponentially with the number of its constituents. Obtaining +these parameters often necessitates the preparation and destructive measurement of exponentially +many copies of the quantum system, as well as their storage in a classical memory. In particular, as +the size of quantum devices continues to increase beyond what can be easily simulated classically, +the community faces new challenges to characterise their output states in a robust and efficient +manner. +Thankfully, only a few physically relevant observables are often needed to describe the physics +of a system, e.g. its entanglement or energy. Recently, new methods of tomography have been +proposed which precisely leverage this important simplification to develop efficient state learning +algorithms. One highly relevant development in this direction is that of classical shadows [HKP20]. +This new set of protocols allows for estimating physical observables of quantum spin systems that +only depend on local properties from a number of measurements that scales logarithmically with the +total number of qubits. However, the number of required measurements still faces an exponential +growth with respect to the size of the observables that we want to estimate. Thus, using such +protocols to learn the expectation values of physical observables that depend on more than a few +qubits quickly becomes unfeasible. +Gibbs State Tomography. Some simplification can be achieved from the fact that physically +relevant quantum states, such as ground and Gibbs states of a locally interacting spin system, are +themselves often described by a number of parameters which scales only polynomially with the +number of qubits. From this observation, another direction in the characterisation of large quantum +systems that has received considerable attention is that of Hamiltonian learning and many-body +tomography, where it was recently shown that it is possible to robustly characterise the interactions +of a Gibbs state with a few samples [Ans, HKT21]. However, even for many-body states, recovery +in terms of the trace distance requires a number of samples that scales polynomially in the number +of qubits, in contrast to shadows for which the scaling is logarithmic. +These considerations naturally lead to the question of identifying settings where it is possible +to combine the strengths of shadows and many-body tomography. In [RF21], some of the authors +proposed a first solution by combining these with new insights from the emerging field of quantum +optimal transport. They obtained a tomography algorithm that only requires a number of samples +that scales logarithmically in the system’s size and learns all quasi-local properties of a state. +These properties are characterised by so-called “Lipschitz observables”. However, that first step +was confined to topologically trivial states such as high-temperature Gibbs states of commuting +Hamiltonians or outputs of shallow circuits. Here, we significantly extend these results to all states +exhibiting exponential decay of correlations and the approximate Markov property. +Learning Phases of Matter. Tomographical techniques by themselves are somewhat limited +in that they tell us nothing about nearby related states – often states belong to a phase of matter +in which the properties of the states vary smoothly and are in some sense “well behaved”, and we +wish to learn properties of this entire phase of matter. A recent line of research in this direction + +3 +that has gained significant attention from the quantum community is that of combining machine +learning methods with the ability to sample complex quantum states from a phase of matter to +efficiently characterise the entire phase [BWP+17, CM17]. A landmark result in this direction +is [HKT+22]. There the authors showed how to use machine learning methods combined with +classical shadows to learn local linear and nonlinear functions of states belonging to a gapped +phase of matter with a number of samples that only grows logarithmically with the system’s size. +That is, given states from that phase drawn from a distribution and the corresponding parameters +of the Hamiltonian, one can train a classical algorithm that would predict local properties of other +points of the phase. However, there are some caveats to this scheme: (i) the scaling of the number +of samples in terms of the precision is exponential, (ii) it does not immediately apply to phases of +matter beyond gapped ground states, (iii) the results only come with guarantees on the errors in +the prediction in expectation. That is, given another state sampled from the same distribution as +the one used to train, only on average is the error made by the ML algorithm proven to be small. +In this work, we address all of these shortcomings. First, our result extends to thermal phases +of matter which exhibit exponential decay of correlations, which includes all thermal systems away +from criticality/poles in the partition function [HMS20, Section 5]. Our result also extends to +gapped phases that satisfy local topological quantum order [MZ13, BHM10, NSY22]. Furthermore, +the sample complexity of our algorithm is quasi-polynomial in the desired precision, which is an +exponential improvement over previous work [HKT+22]. And, importantly, it comes with point- +wise guarantees on the quality of the recovery, as opposed to average guarantees. +Interestingly, our results are easier to grasp through the lens of the concentration of measure +phenomenon rather than machine learning: we show that local expectation values of quantum +states are quite smooth under perturbations in the same class of states. And, as is showcased +by the concentration of measure phenomenon, smooth functions on high-dimensional spaces do +not show a lot of variability. Thus, it suffices to collect a few examples to be able to predict +what happens in the whole space, while the price we pay for these stronger recovery guarantees +is that our algorithm does not work for any distribution over states, but needs some form of +anti-concentration which holds e.g. for the uniform distribution (see Appendix D for a technical +discussion). In other words, our algorithm necessitates to “see” enough of the space to work and +struggles if there are large, low-probability corners. +II. +SUMMARY OF MAIN RESULTS +In this paper, we consider a quantum system defined over a D-dimensional finite regular lattice +Λ = [−L, L]D, where n = (2L + 1)D denotes the total number of qubits constituting the system. +We assume for simplicity that each site of the lattice hosts a qubit, so that the total system’s +Hilbert space is HΛ := � +j∈Λ C2. All of the results presented here easily extend to qudits, but we +will focus on qubits for simplicity. +Our focus in this work are nontrivial statements about what can be learned about many-body +states of n qubits in the setting where we are only given Θ(polylog(n)) copies. +The common +theme is that we will assume exponential decay of correlations for our class of states, but will show +results in two different regimes. In Section II A we summarise our results on how to estimate all +quasi-local properties of a given state given identical copies of it. This is the traditional setting +of quantum tomography. In contrast, in Section II B we summarise our findings on how to learn +local properties of a class of states given samples from different states from that class. This is the + +4 +setting of [HKT+22] where ground states of gapped quantum phases of matter were studied. Here +we consider (a) thermal phases of matter with exponentially decaying correlations and (b) gapped +ground states with local topological quantum order. +A. +Optimal Tomography of Many-Body Quantum States +We first consider the task of obtaining a good approximation of expected values of extensive +properties of a fixed unknown n-qubit state over Λ. The state is assumed to be a Gibbs state +of an unknown local Hamiltonian H(x) := � +j∈Λ hj(xj), x = {xj} ∈ [−1, 1]m, defined through +interactions hj(xj), each depending on parameters xj ∈ [−1, 1]ℓ for some fixed integer ℓ and +supported on a ball Aj around site j ∈ Λ of radius r0. We also assume that the matrix-valued +functions xj �→ hj(xj) as well as their derivatives are uniformly bounded: ∥hj∥∞, ∥∇hj∥∞ ≤ h. +The corresponding Gibbs state at inverse temperature β > 0, and the ground state as β → ∞ take +the form +σ(β, x) := +e−βH(x) +tr +� +e−βH(x)� +and +ψg(x) := lim +β→∞ σ(β, x) . +(II.1) +In the case when [hj(xj), hj′(xj′)] = 0 for all j, j′ ∈ Λ, the Hamiltonian H(x) and its associated +Gibbs states σ(β, x) are said to be commuting. +1. +Preliminaries on Lipschitz observables +Extensive properties of a state are well-captured by the recently introduced class of Lipschitz +observables [RD19, DPMTL21]. +Definition II.1 (Lipschitz Observable [DPMTL21] ). An observable L on HΛ is said to be +Lipschitz if ∥L∥Lip := maxi∈Λ minLic 2∥L − Lic ⊗ Ii∥∞ = O(1), where ic is the complement of +the site i in Λ and the scaling is in terms of the number of qubits in the system. +In words, ∥L∥Lip quantifies the amount by which the expectation value of L changes for states +that are equal when tracing out one site. By a simple triangle inequality together with [DPMTL21, +Proposition 15], one can easily see that ∥L∥∞ ≤ n∥L∥Lip. Given the definition of the Lipschitz +constant, we can also define the quantum Wasserstein distance of order 1 by duality [DPMTL21]. +Definition II.2 (Wasserstein Distance [DPMTL21]). The Wasserstein distance between two n +qubit quantum states ρ1, ρ2 is defined as W1(ρ0, ρ1) := sup∥L∥Lip≤1 tr +� +L(ρ0 − ρ1) +� +≤ n∥ρ − σ∥1. +Having W1(ρ, σ) = O(εn) is sufficient to guarantee that the expectation value of ρ and σ on +extensive, quasi-local observables is the same up to a multiplicative error εn. This justifies why we +focus on learning states up to an error O(εn) in Wasserstein distance instead of the usual trace +distance bound of order O(ε): although a trace distance guarantee of order O(ε) gives the same +error estimate, it requires exponentially more samples even for product states, as shown in [RF21, +Appendix G]. In Appendix B, we argue that Lipschitz observables and the induced Wasserstein +distance capture linear and nonlinear extensive properties of many-body quantum states. + +5 +2. +Gibbs state tomography +In this section, we turn our attention to the problem of obtaining approximations of linear +functionals of the form fL(β, x) := tr[Lσ(β, x)] for all Lipschitz observables L from the measure- +ment and classical post-processing of as few copies of the associated unknown Gibbs state σ(β, x) +as possible. We will further require that the state satisfies the property of exponential decay of +correlations: for any two observables XA, resp. XB, supported on region A, resp. B, +Covσ(β,x)(XA, XB) ≤ C min{|A|, |B|} ∥XA∥∞ ∥XB∥∞ e−ν dist(A,B) , +(II.2) +for some constants C, ν > 0, where dist(A, B) denotes the distance between regions A and B, and +where the covariance is defined by +Covσ(X, Y ) := 1 +2 tr +� +σ +� +X − tr[σX], Y − tr[σY ] +�� +. +(II.3) +Our first main result is a method to learn Gibbs states with few copies of the unknown state: +Theorem II.3 (Tomography algorithm for decaying Gibbs states (informal)). For any unknown +commuting Gibbs state σ(β, x) satisfying Equation (II.2), there exists an algorithm that provides +the description of parameters x′ such that the state σ(β, x′) approximates σ(β, x) to precision nε in +Wasserstein distance with probability 1−δ with access to N = O +� +log(δ−1) polylog(n) ε−2� +samples +of the state (see Appendix C 3 a). The result extends to non-commuting Hamiltonians whenever +one of the following two assumptions is satisfied: +(i) the high-temperature regime, β < βc (see Appendix C 3 b). +(ii) uniform clustering/Markov conditions (see Corollary C.12). +In case (ii), we find good approximation guarantees under the following slightly worst scaling in +the precision ε: N = O(ε−4 polylog(nδ−1)). +The results for commuting Hamiltonians and in the high-temperature regime proceed directly +from the following continuity bound on the Wasserstein distance between two Gibbs states, whose +proof requires the notion of quantum belief propagation in the non-commuting case (see Corol- +lary C.4): for any x, y ∈ [−1, 1]m, +W1(σ(β, x), σ(β, y)) = ∥x − y∥ℓ1 O(polylog(n)) . +(II.4) +Furthermore, this inequality is tight up to a polylog(n) factor for β = Θ(1). Equation (II.4) reduces +the problem of recovery in Wasserstein distance to that of recovering the parameters x up to an +error εn/ polylog(n) in ℓ1 distance. This is a variation of the Hamiltonian learning problem for +Gibbs states [AAKS21, HKT21] which relies on lower bounding the ℓ2 strong convexity constant +for the log-partition function. +In [Ans], the authors give an algorithm estimating x with eO(βkD)O(log(δ−1n)ε−2) copies of +σ(β, x) up to ε in ℓ∞ distance when σ(β, x) belongs to a family of commuting, k-local Hamiltonians +on a D-dimensional lattice. If we assume m = O(n), this translates to an algorithm with sample +complexity eO(βkD)O(ε−2polylog(δ−1n)) to learn x up to εn in ℓ1 distance. +It should also be +noted that the time complexity of the algorithm in [Ans] is O(neO(βkD)ε−2polylog(δ−1n)). Thus, + +6 +any commuting model at constant temperature satisfying exponential decay of correlations can +be efficiently learned with polylog(n) samples. We refer the reader to Appendix C 3 for more +information and classes of commuting states that satisfy exponential decay of correlations. In the +high-temperature regime, we rely on a result of [HKT21] where the authors give a computationally +efficient algorithm to learn x up to error ε in ℓ∞ norm from O(ε−2polylog(δ−1n)) samples. This +again translates to a O(εn) error in ℓ1 norm thanks to (II.4). +Furthermore, in Appendix C 3 c we more directly extend the strategy of [AAKS21] by introdu- +cing the notion of a W1 strong convexity constant for the log-partition function and showing that +it scales linearly with the system size under (a) uniform clustering of correlations and (b) uniform +Markov condition. This result also generalises the strategy of [RF21] which relied on the exist- +ence of a so-called transportation cost inequality previously shown to be satisfied for commuting +models at high-temperature. For the larger class of states satisfying conditions (a) and (b), we are +able to find x′ s.t. W1(σ(β, x), σ(β, x′)) = O(εn) with O(ε−4polylog(δ−1n)) samples. Note that +the uniform Markov condition is expected to hold for a large class of models that goes beyond +high-temperature Gibbs states [KB19, KKBa20]. +3. +Beyond linear functionals +So far, we considered properties of the quantum system which could be related to local linear +functionals of the unknown state. In [HKP20, HKT+22], the authors propose a simple trick in +order to learn non-linear functionals of many-body quantum systems, e.g. their entropy over a small +subregion. However, such methods require a number of samples scaling exponentially with the size +of the subregion, and thus very quickly become inefficient as the size of the region increases. Here +instead, we make use of the continuity of the entropy functional with respect to the Wasserstein +distance, mentioned in Equation (B.6), together with the following Wasserstein continuity bound +in order to estimate the entropic quantities of Gibbs states over regions of arbitrary size (see +Corollary C.6): assuming Equation (II.2), for any region S of the lattice and any two x, y ∈ [−1, 1]m +W1(trSc(σ(β, x)), trSc(σ(β, y))) ≤ ∥x|S(rS) − y|S(rS)∥ℓ1 polylog(|S(rS)|) , +(II.5) +where rS = max +� +r0, 2ξ log +� +2|S|C1∥x|S(r0) − y|S(r0)∥−1 +ℓ1 +�� +with r0 being the smallest integer such +that x|S(r0) ̸= y|S(r0), S(rS) := {xj| supp(hj(xj)) ∩ S(rS) ̸= ∅}, S(rS) := {i ∈ Λ : dist(i, S) ≤ rS}, +and C1, ξ > 0 are constants introduced in Lemma C.5. +Let us recall a few definitions: denoting by ρR := trRc(ρ) the marginal of a state ρ ∈ D(HΛ) on +a region R ⊂ Λ, and given separated regions A, B, C ⊂ Λ of the lattice: S(A)ρ := − tr[ρA log ρA] +is the von Neumann entropy of ρ on A, S(A|B)ρ := S(AB)ρ − S(B)ρ is the conditional entropy +on region A conditioned on region B, I(A : B)ρ := S(A)ρ + S(B)ρ − S(AB)ρ is the mutual +information between regions A and B, and I(A : B|C)ρ := S(AC)ρ +S(BC)ρ −S(C)ρ −S(ABC)ρ +is the conditional mutual information between regions A and B conditioned on region C. The +following corollary is a direct consequence of Equation (B.6) together with Equation (II.5): +Corollary II.4. Assume the decay of correlations holds uniformly, as specified in Equation (II.2), +for all {σ(β, x)}x∈[−1,1]m, m = O(n). Then, in the notations of the above paragraph, for any two +Gibbs states σ(β, x) and σ(β, y), x, y ∈ [−1, 1]m, and any region A ⊂ Λ: +|S(A)σ(β,x) − S(A)σ(β,y)| = ∥x|S(rS) − y|S(rS)∥ℓ1O(polylog(|S(rS)|)) , + +7 +for S ≡ A. +The same conclusion holds for |S(A|B)σ(β,x) − S(A|B)σ(β,y)| (S ≡ AB), |I(A : +B)σ(β,x) − I(A : B)σ(β,y)| (S ≡ AB), and |I(A : B|C)σ(β,x) − I(A : B|C)σ(β,y)| (S ≡ ABC). +Thus, given an an estimate y of x satisfying ∥x − y∥ℓ∞ = O(ε/polylog(n)), we can also ap- +proximate entropic quantities of the Gibbs state to a multiplicative error. More generally, entropic +continuity bounds can be directly used together with Theorem II.3(ii) in order to estimate entropic +properties of Gibbs states satisfying both uniform clustering of correlations and the approximate +Markov condition (see Appendix C 3 c for details). +B. +Learning Expectation Values of Parametrised Families of Many-Body Quantum Systems +Next, we turn our attention to the task of learning Gibbs or ground states of a parameterised +Hamiltonian H(x) known to the learner and sampled according to the uniform distribution U +over x ∈ [−1, 1]m. More general distributions can also be dealt with under a condition of anti- +concentration, see Appendix D. Here we restrict our results to local observables of the form O = +�M +i=1 Oi where Si := supp(Oi) is contained in a ball of diameter independent of the system size. +The setup in this section is similar to [HKT+22]. The idea is that we have access to some samples +of a state chosen from different values of the parameterised Hamiltonian, and we want to use these +to learn observables everywhere in the parameter space with high precision. We then want to +know: what is the minimum number of samples drawn from this distribution which allows us to +accurately predict expectation values of local observables for all choices of parameters? +1. +Learning Expectation Values in Thermal Phases of Matter +The learner is given samples {(xi, σ(β, xi))}N +i=1, where the parameters xi ∼ U, and their task +is to learn fO(x) := tr[σ(β, x)O] for an arbitrary value of x ∈ [−1, 1]m and an arbitrary local +observable O. We assume that everywhere in the parameter space x ∈ [−1, 1]m the Gibbs states +are in the same phase of exponentially decaying correlations. Then we have: +Theorem II.5 (Learning algorithm for quantum Gibbs states). With the conditions of the previ- +ous paragraph, given a set of N samples {xi, ˜σ(β, xi)}N +i=1, where ˜σ(β, xi) can be stored efficiently +classically, and N = O +� +log +� M +δ +� +log +� n +δ +� +epolylog(ε−1)� +, there exists an algorithm that, on input +x ∈ [−1, 1]m and a local observable O = �M +i=1 Oi, produces an estimator ˆfO such that, with prob- +ability (1 − δ), +sup +x∈[−1,1]m |fO(x) − ˆfO(x)| ≤ ε +M +� +i=1 +∥Oi∥∞ . +Moreover, the samples ˜σ(β, xi) are efficiently generated from measurements of the Gibbs states +{σ(β, xi)}N +i=1 followed by classical post-processing. +Our estimator ˆfO is constructed as follows: +during a training stage, we pick N points +Y1, . . . , YN ∼ U and estimate the reduced Gibbs states over large enough enlargements Si∂ of +the supports Si := {xj| supp(hj(xj)) ∩ Si∂ ̸= ∅} ∩ [x − ε, x + ε]m of the observables Oi. Due to +the anti-concentration property of the uniform distribution, the probability that a small region +Si∂ in parameter space contains t variables Yi1, . . . , Yit becomes large for N ≈ log(M). We then + +8 +run the classical shadow tomography protocol on those states in order to construct efficiently +describable and computable product matrices �σ(β, Y1), . . . , �σ(β, YN). Then for any region Si, we +select the shadows �σ(β, Yi1), . . . �σ(β, Yit) whose local parameters are close to that of the target +state and construct the empirical average �σSi(x) := 1 +t +�t +j=1 trSc +i +� +�σ(β, Yij) +� +. Using belief propaga- +tion methods (see Proposition D.2), it is possible to show that exponential decay of correlations +ensures that the estimator is a good approximation to local observables. Thus such operators can +be well approximated using the reduced state trSc +i σ(β, x) for t ≈ log(n). The estimator ˆfO is then +naturally chosen as ˆfO(x) := �M +i=1 tr[Oi �σSi(x)]. A key part of the proof is demonstrating that +exponential decay of correlations implies that fO(x) does not change too much as x varies. +2. +Learning ground states under local indistinguishability +We now move our attention to the problem of learning ground states. Again, the learner is +given samples {xi, ψg(xi)}N +i=1, xi ∼ U, and their task is to learn fO,g(x) := tr[ψg(x)O]. In fact, the +previous argument for Gibbs states can be extended to the present setting as long as the condition +of exponentially decaying correlations in the Gibbs state is replaced by the following condition +of local topological quantum order (LTQO) [MZ13, BHM10, NSY22]: A quantum system satisfies +LTQO if for any two regions A ⊂ B ⊂ Λ and all x ∈ [−1, 1]m, +�� trAc(ψg(x, B) − ψg(x, Λ)) +�� +1 ≤ CT |A| e− dist(A,Bc) +ξ0 +(II.6) +for some constants CT , ξ0 > 0, and where, given a region R ⊂ Λ we denote by ψg(x, R) the +ground state corresponding to the Hamiltonian HR(x) = � +j∈R hj(xj). In words, LTQO states +that observables localised away from the boundary of the volume B cannot distinguish between +different ground states. Many systems of practical interest are known to satisfy Equation (II.6), +including frustration-free spin chains with a unique translation-invariant matrix product ground +state [AKLT88] and quantum double models, which include Kitaev’s toric code [Kit06, Kit03, +CDH+20]. For more details on LTQO, we refer to [NSY22] and the references therein. +Theorem II.6 (Learning algorithm for quantum ground states). With the conditions of the pre- +vious paragraph, given a set of N samples {xi, ˜ψ(xi)}N +i=1, where ˜ψ(xi) can be stored efficiently +classically, and N = O +� +log +� M +δ +� +log +� n +δ +� +epolylog(ε−1)� +, there exists an algorithm that, on input +x ∈ [−1, 1]m and a local observable O = �M +i=1 Oi, produces an estimator ˆfO such that, with prob- +ability (1 − δ), +sup +x∈[−1,1]m |fO(x) − ˆfO(x)| ≤ ε +M +� +i=1 +∥Oi∥∞ . +Moreover, the samples ˜ψ(xi) are efficiently generated from measurements of the ground states +{ψg(xi)}N +i=1 followed by classical post-processing. +To prove this statement, we reduce it to the problem of learning Gibbs states of the previous +section. The LTQO condition permits approximating the expectation of the local observable Oi +in the state ψg(x) by the one in the state ψg(x, Si∂). The latter is approximated by the local +Gibbs state σ(β, x, Si∂) ∝ e−βHSi∂(x) for large but constant β (see Lemma E.5). By a continuity +argument, these states are approached by σ(β, Yit, Si∂), which in turn are close to ψg(Yit). This + +9 +chain of approximation steps together with a robust version of the shadow tomography protocol for +ground states, stated in Proposition E.7, allows us to conclude. We expect that the assumption of +LTQO is not the only assumption that can be made to achieve similar scaling. Indeed, we expect +that a lower bound on the spectral gap in the parameterized region would achieve similar results. +III. +COMPARISON TO PREVIOUS WORK +A. +Classical literature +The problem of Hamiltonian learning for classical models has attracted a lot of attention in +the last years by the computer science community [Bre15, PSBR20, LVMC18, ZKKW20] which +traditionally refers to it as Ising model — or Markov field — learning. The question of what can be +inferred from very few samples was also asked classically [DDDK20]. Our work sheds further light +on this question and is of interest even when restricting to classicaapproximatingbservables. Indeed, +to the best of our knowledge, the statements of Corollary C.4 and Corollary C.6 are new even for +classical Gibbs distributions. Previous work by the authors of [RF21] already established similar +learning results for measures satisfying a so-called transportation cost inequality (TC) [BG99, +Tal96], although the present condition of exponential decay of correlations is more standard. +It should be noted that if a Gibbs measure satisfies TC, then any Lipschitz function of a random +variable distributed according to it satisfies a Gaussian concentration bound [Led01]. This can +easily be seen to imply that we can estimate the expectation value of M Lipschitz functions up to +an error ε with probability of success δ from O(ε−2 log(Mδ−1)) samples by taking the empirical +average. At first sight this might look comparable with the sample complexity we obtain with +our learning algorithm. +However, this only holds for one basis, whereas our result holds for +any basis. Furthermore, if the number of Lipschitz observables satisfies M = ec Ω(n), then the +number of samples required to obtain a good estimate through the empirical average becomes +polynomial. On the other hand, given that W1(σ(β, x), σ(β, x′)) ≤ εn, we can evaluate as many +Lipschitz observables as we wish from σ(β, x′) without requiring any further samples. Thus, even +for observables in a fixed basis our result has advantages. +B. +Previous work on many-body quantum state tomography +As mentioned before, one striking advantage of our Gibbs tomography algorithm when es- +timating expectation values of local observables compared to state-agnostic methods like classical +shadows is the exponential speedup in the size of the support of the observable. In fact, our method +gives good guarantees on the larger class of Lipschitz observables, which includes non-local ob- +servables. This advantage is even more visible when it comes to estimating entropic quantities: +whereas the polynomial approximation proposed in [HKP20] works universally for any n-qubit +state, it only gives good approximation guarantees for reduced states on very few qubits. Here +instead, we avoid this issue by leveraging the Wasserstein continuity bounds offered in [DPMTL21]. +Our framework also differs from the one of Hamiltonian learning algorithms tackled in [Ans, +AAKS21, HKT21]: in these papers, the authors were interested in estimating the parameter +x of a given Hamiltonian H(x) given access to copies of the state σ(β, x), in ℓ2 or ℓ∞. +Here +instead, we argue that a good recovery in W1 distance is implied by the weaker condition of +recovery in ℓ1. Clearly, one can leverage these previous results to further control our ℓ1 bound, + +10 +as we argue in Section II A 2. It should be noted however that our bound only requires that the +Gibbs state σ(β, x) satisfies an exponential decay of correlations, whereas these learning algorithms +provide very efficient ℓ∞ or ℓ2 recovery either for (i) commuting Hamiltonians or (ii) in the high- +temperature regime. It remains an important question whether the condition of exponential decay +of correlations is enough to get good ℓ1 recovery. Furthermore, in Appendix C 3 c we show that +under the additional assumptions of uniform Markovianity and clustering of correlations, it is +possible to learn in W1 through the maximum entropy method, without resorting directly to +learning the parameters x. +C. +Previous work on learning observables in phases of matter +In [HKT+22], the authors found a machine learning algorithm which, for any smoothly para- +meterised family of local Hamiltonians {H(x)}x∈[−1,1]m in a finite spatial dimension with a constant +spectral gap, can be trained to predict expected values of sums of local observables in the associated +ground state ψg(x). More precisely, given a local observable O = �M +i=1 Oi with supp(Oi) = O(1), +they construct an estimator ˆfO(x) of the expectation value of the observable such that +Ex∼U([−1,1]m) +��� tr[Oψg(x)] − ˆfO(x) +��2� +≤ ε2� M +� +i=1 +∥Oi∥∞ +�2 +, +(III.1) +as long as the training size (i.e. +the number of sampled points within the phase) is N = +� �M +i=1 ∥Oi∥∞ +�2 +mO(1/ε2). +In Theorem II.6, we improve this result for ground states in three ways, up to further imposing +the LTQO condition: first, we can assume that the parameters x are distributed according to a +much larger class of distributions than the uniform distribution. This extension does not carry +so easily in the proof of [HKT+22] which uses Fourier analysis techniques involving integration +over the Lebesgue measure to derive Equation (III.1). Second, theirs is a result in expectation, +that is in ∥.∥L2, whereas our bound in Theorem II.6 works in the worst-case setting associated to +the stronger ∥.∥∞-norm topology. Third and most importantly, the dependence of the number of +training data points scales exponentially in the precision parameter ε in Equation (III.1), whereas +ours scales only quasi-polynomially. +Finally, we extend the learning result beyond ground states to finite temperature phases of +matter with exponential decay of correlations. This not only includes all high-temperature phases +of matter (regardless of the Hamiltonian), but also low-temperature phases with the relevant +correlation functions [DCGR19]. This is a particularly relevant result since zero temperature is +never achieved in practice, so in reality we are always working with low-temperature thermal states. +We also recognise independent, concurrent work by [LTL+23]. Here the authors consider the +same setup of gapped ground states as [HKT+22] and also improved over Equation (III.1) to achieve +the same sample complexity as Theorem II.6. However, their result is not directly comparable to +ours. We emphasise [LTL+23] consider gapped, ground state phases, whereas our work focuses on +thermal phases and ground states with LTQO. We also note they remove all conditions on the prior +distribution over the samples x, whereas we still need to assume a type of mild anti-concentration +over the local marginals. However, their result is still stated as an ∥.∥L2-bound due to the use of +machine learning machinery, whereas our more straightforward Gibbs approximation tools allow +us to get stronger bounds in ∥.∥∞. Conceptually speaking, our methods for approximating local + +11 +expectation values requires no knowledge of machine learning techniques. Our work also shows that +it is possible to go beyond gapped quantum phases and learn thermal phases with exponentially +decaying correlations, as well as ground states with LTQO. +IV. +DISCUSSION AND CONCLUSIONS +We have contributed to the tasks of tomography and learnability of quantum many-body states +by combining previous techniques with approaches not considered so far in this field to obtain +novel and powerful features. +Tomography. +First, we extended the results of [RF21] on the efficient tomography of high- +temperature commuting Gibbs states to Gibbs states with exponentially decaying correlations. +This result permits to significantly enlarge the class of states for which we know how to learn +all quasi-local properties with a number of samples that scales polylogarithmically with the sys- +tem’s size. In particular, our results now also hold for classes of Gibbs states of non-commuting +Hamiltonians. As we require exponentially fewer samples to learn in the Wasserstein metric when +compared with the usual trace distance and still recover essentially all physically relevant quantit- +ies associated to the states, we hope that our results motivate the community to consider various +tomography problems in the Wasserstein instead of trace distance. +As we achieved this result by reducing the problem of learning the states to learning the para- +meters of the Hamiltonian in ℓ1, we hope our work further motivates the study of the Hamiltonian +learning problem in ℓ1-norm with polylog samples. 1D Gibbs states are a natural place to start, +but obtaining Hamiltonian learning algorithms just departing from exponential decay of correla- +tions would provide us with a complete picture. In Appendix C 3 c we also partially decoupled +the Hamiltonian learning problem from the W1 learning one by resorting to the uniform Markov +condition. Thus, it would be important to establish the latter for a larger number of systems. +It would be interesting to investigate the sharpness of our bounds, and to understand if expo- +nential decay of correlations is really necessary. One way of settling this question would be to prove +polynomial lower bounds for learning in Wasserstein distance for states at critical temperatures. +Learning Phases of Matter. +Second, we improved the results of [HKT+22] for learning +a class of states in several directions, including the scaling in precision, the classes of states it +applies to and the form of the recovery guarantee. In particular, the results now apply to Gibbs +states, which are the states of matter commonly encountered experimentally. Interestingly, we did +not need to resort to machine learning techniques to achieve an exponentially better scaling in +precision by making arguably mild assumptions on the distributions the states are drawn from. +Although the results proved here push the state-of-the-art of learning quantum states, we believe +that our methods, for instance the novel continuity bounds for various local properties of quantum +many-body states, will find applications in other areas of quantum information. +Beyond the thermal phases and LTQO ground states studied here, it would be interesting +to find other families of states which can be efficiently learned, and indeed if more restrictive +assumptions on the parameterization of Hamiltonians can result in more efficient learning. One +interesting open problem that goes beyond the present paper’s scope is finding families of states +satisfying LTQO without belonging to a common gapped phase of matter. If such a family existed, +it would clarify the differences between our framework and that of [HKT+22]. Finally, we realise +that although the results proved here are for lattice systems, they almost certainly generalise to +non-lattice configurations of particles. + +12 +V. +ACKNOWLEDGMENTS +The authors gratefully recognise useful discussions with +Hsin-Yuan (Robert) Huang and +Haonan Zhang. We thank Laura Lewis, Viet T. Tran, Sebastian Lehner, Richard Kueng, Hsin- +Yuan (Robert) Huang, and John Preskill for sharing a preliminary of the manuscript [LTL+23], +which was discussed in Section III C. +EO is supported by the Munich Quantum Valley and the Bavarian state government, with +funds from the Hightech Agenda Bayern Plus. CR acknowledges financial support from a Ju- +nior Researcher START Fellowship from the DFG cluster of excellence 2111 (Munich Center for +Quantum Science and Technology), from the ANR project QTraj (ANR-20-CE40-0024-01) of the +French National Research Agency (ANR), as well as from the Humboldt Foundation. +DSF is +supported by France 2030 under the French National Research Agency award number “ANR-22- +PNCQ-0002”. JDW acknowledges support from the United States Department of Energy, Office +of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum +Computing program, and also NSF QLCI grant OMA-2120757. +[AAKS21] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar. Sample- +efficient learning of interacting quantum systems. Nature Physics, 17(8):931–935, 2021. +[ABF23] Guillaume Aubrun, Emily Beatty, and Daniel Stilck Fran¸ca. 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PMLR, 13–18 Jul 2020. + +16 +SUPPLEMENTAL MATERIAL +Appendix A: Preliminaries +Given a finite dimensional Hilbert space H, we denote by B(H) the algebra of bounded operators +on H, whereas Bsa(H) denotes the subspace of self-adjoint operators. We denote by D(H) the set +of positive operators on H of unit trace, and by D+(H) the subset of positive, full-rank operators +on H. Schatten norms are denoted by ∥.∥p for p ≥ 1. The identity matrix in B(H) is denoted by +I. Given a bipartite system AB, the normalised partial trace over a subsystem A is written τA, +i.e. τA := 2−|A| trA. +In this work, we consider a family of local qubit interactions {hj(xj)}xj∈[−1,1]ℓ, j = 1, . . . , n over +the D-dimensional lattice Λ = [−L, L]D, for some fixed integer ℓ, where n = (2L+1)D denotes the +total number of qubits constituting the system. For each j and all xj ∈ [−1, 1]ℓ, hj(xj) is supported +on a ball Aj around site j ∈ Λ of radius r0. We also assume that the matrix-valued functions xj �→ +hj(xj) as well as their derivatives are uniformly bounded: ∥hj∥∞, ∥∇xhj(x)∥∞ ≤ h. For sake of +simplicity, we assume that the interactions are linear functions of their parameters, that is hj(xj) = +xjVj for some fixed operator Vj. However this assumption is not necessary in any of our proofs, +as commented in Appendix F. Concatenating the vectors xj into x = (x1, . . . , xn) = (x′ +1, . . . , x′ +m), +m = nℓ, the local interactions induce the following family of Hamiltonians {H(x)}x∈[−1,1]m, with: +H(x) = +m +� +j=1 +hj(xj) . +(A.1) +More generally, given a region B ⊂ Λ of the lattice, we denote by HB(x) := � +j|Aj⊂B hj(x) the +Hamiltonian restricted to B. We denote by x|S(r) the concatenation of vectors xj corresponding +to interactions hj supported on regions intersecting S(r) := {l ∈ Λ| dist(l, S) ≤ r}. +For much of the following, we will be concerned with Gibbs states, defined as +σ(β, x) := +e−βH(x) +tr[e−βH(x)]. +In particular, we will be interested in systems satisfying the following type of correlation decay: +Condition A.1 (Exponential Decay of Correlations). For a state σ and any operator XA, +resp. XB, supported on region A, resp. B, we say the state satisfies exponential decay of correla- +tions if +Covσ(XA, XB) ≤ C min{|A|, |B|} ∥XA∥∞ ∥XB∥∞ e−ν dist(A,B) , +(A.2) +for any choice of XA,XB, and for some parameters C, ν > 0 which we assume independent of x +and of the lattice size n, and where +Covσ(A, B) := 1 +2 tr +� +σ +� +A − tr[σA], B − tr[σB] +�� +. +Condition A.1 is satisfied by many classes of Gibbs states, including high-temperature Gibbs +states [HMS20, KKBa20] and 1D Gibbs states at any constant temperature [HMS20, BCPH22]. It +is also known to hold for ground states of gapped Hamiltonians [HK06]. In fact, the class of Gibbs + +17 +states for which Condition A.1 holds is larger than that for which polylog algorithms to learn the +parameters of the Hamiltonian are known. In Appendix C 3 we will discuss several examples for +which it is known how to learn the parameters efficiently. In Appendix C 3 c we will also consider +the case when we have the additional assumption of uniform Markovianity to show that then it is +possible to bypass having to learn the parameters. +Appendix B: Lipschitz observables +In this appendix, we argue that Lipschitz observables and the induced Wasserstein distance +capture most observables of physical interest, such as local and quasi-local observables, as well +as quasi-local polynomials of the state and entropic quantities of subsystems. +They can even +capture global properties, including some of physical interest like global entropies. These classes +of examples justify the claim that Lipschitz observables and the Wasserstein distance capture well +both linear and nonlinear extensive properties of quantum states. +Let us illustrate our previous claims. An important class of Lipschitz observables are those of +the form +M +� +i=1 +Oi, +M = O(n), +∥Oi∥ = O(1), +max +1≤j≤n |{i : supp(Oi) ∩ {j} ̸= ∅}| = O(1). +(B.1) +Observables like those defined in Equation (B.1) include local observables w.r.t. to a regular +lattice. However, it is also not difficult to see that the expectation values of such observables are +characterised by the marginals of the states on a few qubits. But Lipschitz observables capture +more than strictly local properties. Indeed, as shown in [RF21], the time evolution of local observ- +ables like those in Equation (B.1) by a shallow quantum circuit or a short continuous-time evolution +satisfying a Lieb-Robinson bound are Lipschitz. These include evolutions by Hamiltonians with +algebraically decaying interactions, which will map strictly local Hamiltonians to quasi-local ob- +servables. +In fact, recent results [ABF23] show that Lipschitz observables can distinguish two +random quantum states almost optimally. As such states are locally indistinguishable [BHH16, +Corollary 15], this fact shows that Lipschitz observables capture much more than just quasi-local +properties of quantum states. +Although so far we only discussed how to use the Wasserstein distance to control linear func- +tionals of the state, the fact that the Wasserstein distance behaves well under tensor products +means that it is also easy to control the error for non-linear functions. Indeed, in [DPMTL21, +Propostion 4], the authors show that the Wasserstein distance is additive under tensor products. +i.e. for all states ρ, σ and integer k we have +W1(ρ⊗k, σ⊗k) = kW1(ρ, σ). +(B.2) +We can then combine this additivity with the standard trick that a polynomial of degree k on a +quantum state can be expressed as the expectation value of a certain observable O on ρ⊗k. In +particular, if this polynomial is an average over polynomials in reduced density matrices of constant +size, it is not difficult to see that the corresponding observable on ρ⊗k will be Lipschitz as well. +Let us exemplify this in the case of the average purity of a state. For a subset A ⊂ [n] of the +qubits of size l, let FA ∈ +� +C2�⊗2l be the flip operator acting on two copies of those qubits: +FA(|ψ⟩ ⊗ |ϕ⟩) = |ϕ⟩ ⊗ |ψ⟩ . +(B.3) + +18 +It can be shown in a few lines that tr +� +FAρ⊗2� += tr +� +ρ2 +A +� +. Furthermore, observables of the form +O = +M +� +i=1 +FAi, +M = O(n), +max +1≤j≤n |{i : Ai ∩ {j} ̸= ∅}| = O(1). +(B.4) +satisfy ∥O∥Lip = O(1). Then +M +� +i=1 +tr +� +ρ2 +Ai − σ2 +Ai +� += tr +� +O(ρ⊗2 − σ⊗2) +� +≤ ∥O∥LipW1(ρ⊗2, σ⊗2) = 2∥O∥LipW1(ρ, σ). +(B.5) +By a direct generalisation of the above, we see that W1(ρ, σ) = O(εn/k) is sufficient to ensure that +degree k polynomials of the states are approximated to a multiplicative error. As we will see later +in Section II A 3, this polynomial trick can be used to ensure that averages of various subsystem +entropies, mutual informations and conditional mutual informations are well-approximated given +a Wasserstein bound. +Once again it should be emphasised that a Wasserstein bound can be used to control global +properties, even non-linear ones. +A good example of that is the entropy of a quantum state. +In [DPMTL21, Theorem 1], the authors show the continuity bound: +|S(ρ) − S(σ)| ≤ g(W1(ρ, σ)) + W1(ρ, σ) log(4n), +(B.6) +where g(t) = (t + 1) log(t + 1) − t log(t). In this case, it turns out that a Wasserstein distance of +W1(ρ, σ) = O(εn/ log(n)) suffices to obtain a multiplicative error for the entropy. Finally, it is also +worth mentioning observables that are not Lipschitz. Simple examples include linear combinations +of high-weight Paulis. +Appendix C: Gibbs states tomography +In this section, our main goal is to devise an efficient tomography algorithm for Gibbs states +σ(β, x). In particular, we wish to learn the parameters x to high precision. We prove the following +lemma: +Theorem C.1 (Tomography algorithm for decaying Gibbs states ). Let H(x) = � +i hi(xi) be +a Hamiltonian such that each hi(xi), xi ∈ [−1, 1]ℓ, is not more than k-local, for k = O(1), +and all terms commute. For some unknown x, let σ(β, x) be its associated Gibbs state satisfying +exponential decay of correlations as per Condition A.1. Then there exists an algorithm that provides +the description of parameters x′ such that the state σ(β, x′) satisfies: +W1(σ(β, x), σ(β, x′)) ≤ εn +(C.1) +with probability greater than 1 − δ, such that the algorithm requires access to no more than N = +O +� +log(δ−1) polylog(n) ε−2� +samples of the state (see Appendix C 3 a). +The result extends to the case where {hi(xi)}i do not commute whenever one of the following +two assumptions is satisfied: +(i) the high-temperature regime, β < βc (see Appendix C 3 b). + +19 +(ii) uniform clustering/Markov conditions (see Corollary C.12). +In case (ii), we find good approximation guarantees under the following slightly worst scaling in +the precision ε: N = O(ε−4 polylog(nδ−1)). +Proof Outline. The full proof is laid out in sections C 1, C 2 and C 3. +The fundamental part of the result uses the continuity estimate of the Wasserstein distance +between two Gibbs states that is of interest on its own. In Corollary C.4 we will show that under +exponential decay of correlations we have: +W1(σ(β, x), σ(β, y)) ≤ ∥x − y∥ℓ1 polylog(n) . +(C.2) +The significance of the bound in Equation (C.2) is that it reduces the problem of obtaining a good +estimate of σ(β, x) in W1 to estimating the parameters x in ℓ1 distance. This is a variation of +the Hamiltonian learning problem [AAKS21, GCC22, HKT21], and we can then directly import +results from the literature for our tomography algorithm. +As we argued before in Section II A 1, the recovery guarantee in Equation (C.1) suffices to ensure +that σ(β, x′) mirrors all the quasi-local properties of σ(β, x). Furthermore, the polylog complexity +in system size is exponentially better than what is required to obtain a recovery guarantee in trace +distance [RF21, Appendix G], even for product states. +1. +Quantum belief propagation +We start by recalling a well-known tool in the analysis of quantum Gibbs states known as +quantum belief propagation [Has07, Kim17, KB19]. We assume a parameterisation of the Hamilto- +nian as H(x) = �m +j=1 xjVj for appropriate operators Vj (we will generalise this to other paramet- +erisation later) and for some observable L we define the function fL(β, x) = tr [σ(β, x)L]. The +belief propagation method then states that we have that for any k ∈ [m], +∂x′ +kfL(β, x) = −β +2 tr +� +L +� +ΦH(x)(∂x′ +kH(x)), σ(β, x) +�� ++ β tr(∂x′ +kH(x)σ(β, x)) tr(Lσ(β, x)) . +where the quantum belief propagation operator ΦH(x) is defined as +ΦH(x)(V ) := +� ∞ +−∞ +dt κβ(t) e−iH(x)tV eiH(x)t , +for some smooth, fast-decaying probability density function κβ(t) := +1 +2π +� +�κβ(ω)eiωtdω of Fourier +transform +�κβ(ω) := tanh(βω/2) +βω/2 +. +The function κβ was in fact computed in [AAKS21, Appendix B]: for t ∈ R\{0}: +κβ(t) := 2 +πβ log eπ|t|/β + 1 +eπ|t|/β − 1 ≤ 4 +πβ +1 +eπ|t|/β − 1 +(C.3) + +20 +Rewriting the above derivative, and using the notations ⟨O⟩β,x ≡ tr(σ(β, x)O) for the expected +value of an observable O in the Gibbs state σ(β, x), we have that +∂x′ +kfL(β, x) = −β +2 ⟨ +� +L, �Hk(x) − ⟨ �Hk(x)⟩β,x +� +⟩β,x +(C.4) +where �Hk(x) := ΦH(x)(∂x′ +kH(x)). We define the covariance between two observables A and B in +the state σ as +Covσ(A, B) := 1 +2 tr +� +σ +� +A − tr[σA], B − tr[σB] +�� +. +Therefore +∂x′ +kfL(β, x) = −β Covσ(β,x) (L, �Hk(x)) . +(C.5) +In what follows, we will need to approximate �Hk(x) by observables supported on bounded regions. +For this, we make use of Lieb-Robinson bounds for Hamiltonians of finite-range interactions [LR72, +Pou10, KGE14, BHV06, Has10, Sid09, CLMPG15]. Here we choose a version proven in [CLMPG15, +Lemma 5.5]: for any observable OA supported on a region A of the lattice, and any B ⊃ A, we +denote by αt, resp. by αB +t , the unitary evolution generated by H(x), resp. by HB(x), up to time +t, i.e. +αt(O) := e−iH(x)tOeiH(x)t , +αB +t (O) := e−iHB(x)tOeiHB(x)t . +which then satisfy +∥αt(OA) − αB +t (OA)∥∞ ≤ c |A| ∥OA∥∞ evt−µ dist(A,Bc) , +(C.6) +for some parameters c, v, µ > 0 which depend on the interactions hj but can be chosen independent +of n and x. +Lemma C.2. For any region A ⊂ B ⊂ Λ and operator OA supported in A and all x, +∥ΦH(x)(OA) − ΦHB(x)(OA)∥∞ ≤ c′ |A| ∥OA∥∞ e−µ′ dist(A,Bc) +for some parameters c′ and µ′ depending on H(x) and β but independent of n. +Proof. We make use of the exponential decay of κβ provided in Equation (C.3) together with the +Lieb-Robinson bound Equation (C.6): +∥ΦH(x)(OA) − ΦHB(x)(OA)∥∞ ≤ +� ∞ +−∞ +|κβ(t)| ∥αt(OA) − αB +t (OA)∥∞ dt +≤ c |A| ∥OA∥∞e−µ dist(A,Bc) +� δ +−δ +|κβ(t)| evt dt ++ 2 ∥OA∥∞ +� +[−δ,δ]c |κβ(t)| dt . +For the first integral above, we use that |κβ(t)| ∝ log(1/t) for t small. More precisely, +� δ +−δ +|κβ(t)| evt dt ≤ 4evδ +πβ +� δ +0 +log +� +eπt/β + 1 +tπ/β +� +dt ≤ 4e(v+π/β)δ +π2 + +21 +For the other integral, we use the exponential decay of κβ: +� +[−δ,δ]c |κβ(t)| dt ≤ 8 +πβ +� ∞ +δ +1 +eπt/β − 1 dt ≤ 8 +πβ +� ∞ +δ +e− πt +2β dt = 16 +π2 e− πδ +2β , +where the second inequality holds for δ ≥ 2β +π sh−1 � 1 +2 +� +≡ δ1. Choosing δ := δ1+µ dist(A, Bc)/(2 +� +v+ +π/β +� +), we get +∥ΦH(x)(OA) − ΦHB(x)(OA)∥∞ ≤ c′ |A|∥OA∥∞ e−µ′d(A,Bc) , +for some constant c′ ≡ c′(β, v), where µ′ = µ min +� 1 +2, +π +4β(v+π/β) +� +. +2. +Continuity estimate for W1 distance on Gibbs states +In this subsection, we will prove Equation (C.2). First, we use the bound derived in Lemma C.2 +together with the assumption that σ(β, x) has exponential decay of correlations in order to control +the derivatives ∂x′ +kfL: +Lemma C.3. Assume that σ(β, x) satisfies the condition of decay of correlations, Condition A.1. +Then for any k ∈ [m], +|∂x′ +kfL(β, x)| ≤ ∥L∥Lip polylog(n) , +(C.7) +for some polynomial of log(n) of degree D with coefficients depending on β, r0, D, h, c′, ν, µ′ and C. +Proof. Denoting by jk the index of the interaction hjk which depends on variable x′ +k, we have that, +given ΦH(x)(∂x′ +khj) = δj,jkΦH(x)(∂x′ +khjk), and denoting �hk = ΦH(x)(∂x′ +khjk), from Equation (C.5) +we have: +|∂x′ +kfL(β, x)| = β Covσ(β,x)(L, �Hk(x)) = β Covσ(β,x)(L, �hk) . +Next, given a region Bk ⊃ Ajk, define the observable +OBk := ΦHBk(x)(∂x′ +khjk) − ⟨ΦHBk(x)(∂x′ +khjk)⟩β,x . +(C.8) +Then by Lemma C.2 we have that +Covσ(β,x)(L, �hk(x)) = Covσ(β,x)(L, �hk(x) − OBk) + Covσ(β,x)(L, OBk) +≤ 2∥L∥∞ ∥ΦH(x)(∂x′ +khjk) − ΦHBk(x)(∂x′ +khjk)∥∞ + Covσ(β,x)(L, OBk) +≤ 2nc′(2r0)D h ∥L∥Lip e−µ′ dist(Ajk,Bc +k) + Covσ(β,x)(L, OBk) . +Next, we estimate the last covariance above. Denoting Bk(r) := {i ∈ Λ : dist(i, Bk) ≤ r}, we get +Covσ(β,x)(L, OBk) = Covσ(β,x)(L − τBk(r)(L), OBk) + Covσ(β,x)(τBk(r)(L), OBk) +≤ 2h∥L − τBk(r)(L)∥∞ + 2C|Bk| h∥L∥∞ e−νr +≤ 2h|Bk(r)| ∥L∥Lip + 2C|Bk| h n∥L∥Lip e−νr , + +22 +where the second line above follows from the condition of decay of correlations Condition A.1. +Choosing Bk = Ajk(⌊log(n)/µ′⌋), so that dist(Ajk, Bc +k) = ⌊log(n)/µ′⌋, and r = ⌊log(n)/ν⌋, we +have shown that, given 1/ν′ := 1/µ′ + 1/ν, +|∂x′ +kfL(β, x)| ≤ 2β h ∥L∥Lip +� +c′(2r0)D h + (2(r0 + log(n)/ν′))D(1 + C) +� +The result follows. +With the bound of Lemma C.3, we show that for Gibbs states belonging to a phase with +exponentially decaying correlations, the difference of expected values of Lipschitz observables in +two such states is controlled by the ℓ1-norm of their associated parameters. +Corollary C.4. With the conditions of Lemma C.3, for any x, y ∈ [−1, 1]m, +W1(σ(β, x), σ(β, y)) ≤ ∥x − y∥ℓ1 polylog(n) . +(C.9) +Furthermore, this inequality is tight up to a polylog(n) factor for β = Θ(1). +Proof. To get the upper bound Equation (C.9), it suffices to interpolate between the two states as +follows: for any Lipschitz observable L, and a path x(s) = (1 − s)x + sy, +| tr [L(σ(β, x) − σ(β, y))] | ≤ +m +� +k=1 +|x′ +k − y′ +k| +� 1 +0 +|∂kfL(β, x)| ds . +The result follows from using Equation (C.7) above, and using the resulting inequality in the +definition of Wasserstein distance, definition II.2. +To see that the inequality is tight up to the polylog(n) factor, consider the family of Hamilto- +nians H(x) = � +i xiZi, which gives rise to diagonal, product Gibbs states that clearly satisfy +exponential decay of correlations. We then have: +W1(σ(β, x), σ(β, y)) ≥ 1 +2 tr +�� +i +Zi(σ(β, x) − σ(β, y)) +� +, +(C.10) +as � +i Zi has Lipschitz constant 2. A simple computation shows that: +1 +2 tr +�� +i +Zi(σ(β, x) − σ(β, y)) +� += 1 +2 +� +i +� +e−βxi +e−βxi + e+βxi − +e−βyi +e−βyi + e+βyi +� +. +(C.11) +We will assume without loss of generality that xi < yi (as otherwise we can consider the observable +with −Zi instead). Under this condition, the summands are all positive and thus: +1 +2 tr +�� +i +Ziσ(β, x) − σ(β, y)) +� += 1 +2 +� +i +���� +e−βxi +e−βxi + e+βxi − +e−βyi +e−βyi + e+βyi +���� . +(C.12) +Yet another simple computation shows that the derivative of the function y �→ +e−βy +e−βy+e+βy is given +by +−β +2 sech(βy). +(C.13) + +23 +Let cβ denote the minimum of the function in Equation (C.13) for a fixed β = Θ(1) over y ∈ [−1, 1]. +Then, by the mean value theorem: +1 +2 +� +i +���� +e−βxi +e−βxi + e+βxi − +e−βyi +e−βyi + e+βyi +���� ≥ cβ +2 +� +i +|xi − yi| , +(C.14) +from which we conclude that: +W1(σ(β, x), σ(β, y)) ≥ cβ +2 ∥x − y∥ℓ1. +(C.15) +We next prove that when given a local observable O supported on a ball S ⊂ Λ of diameter +at most k0 around site i of the lattice, to study its behaviour as H(x) varies for Gibbs states, +it is sufficient to only consider the components of x which parameterise local terms which are +geometrically close to the observable O (up to some small error). +Before we prove this, we remember that we denote by x|S(r) the concatenation of vectors xj +corresponding to interactions hj supported on regions intersecting S(r) := {i ∈ Λ| dist(i, S) ≤ r}. +Lemma C.5 (Gibbs local indistinguishability). Assuming the exponential decay of correlations in +Condition A.1, then for any observable O supported on region S, any r ∈ N, denoting fO(x) := +tr[O σ(β, x)] and identify x|S(r) with the vector (x|S(r), 0S(r)c) ∈ [−1, 1]m, then the following bound +holds: +sup +x∈[−1,1]m |fO(x) − fO(x|S(r))| ≤ C1 e− r +2ξ ∥O∥∞ , +for O(1) constants C1, ξ > 0 independent of n. In other words: +sup +x∈[−1,1]m ∥ trSc(σ(β, x) − σ(β, x|S(r)))∥1 ≤ C1 e− r +2ξ . +(C.16) +Proof. We identify x|S(r) with the vector (x|S(r), 0S(r)c) ∈ [−1, 1]m. Given the path x(s) = (1 − +s)x + sx|S(r) with components {x′ +l(s)}m +l=1, we get +|fO(x) − fO(x|S(r))| ≤ +� +l∈S(r)c +|x′ +l(0)| +� 1 +0 +��∂l tr +� +Oσ(β, x(s)) +��� ds +(C.17) += β +� +l∈S(r)c +|x′ +l(0)| +� 1 +0 +�� Covσ(β,x(s)) +� +O, �Hl(x(s)) +� +| ds , +for �Hl(x) := ΦH(x)(∂lH(x)), where the second line comes from Equation (C.5). Next, we call +jl ∈ Λ the unique site such that x′ +l is a coordinate of xjl, and denote Ajl be the support of hjl. +Now, the above covariance is small if r is large enough, since �Hj(x(s)) can be well approximated +by an observable on Sc. Indeed, +∂lH(x) = ∂lhjℓ , + +24 +where jl denotes the index of interaction hjl which depends on variable x′ +l. Therefore, whenever +Ajl ∩ S = ∅, we proceed similarly to Lemma C.3: given a region Bl ⊃ Ajl such that Bl ∩ S = ∅, +denoting the observable +OBl := ΦHBl(x)(∂x′ +lhjl) − ⟨ΦHBl(x)(∂x′ +lhjl)⟩β,x , +we have by Lemma C.2 as well as the assumption that the state σ(β, x) has exponential decay of +correlations we have the following (refer to Figure 1 for a diagram of the regions): +Covσ(β,x)(O, �Hl(x)) = Covσ(β,x)(O, �Hl(x) − OBl) + Covσ(β,x)(O, OBl) +≤ 2∥O∥∞ ∥ΦH(x)(∂x′ +lhjl) − ΦHBl(x)(∂x′ +lhjl)∥∞ + Covσ(β,x)(O, OBl) +≤ 2∥O∥∞c′|Ajl| h e−µ′ dist(Ajl,Bc +l ) + 2C|S| ∥O∥∞ h e−ν dist(S,Bl) +≤ 2(C + c′) ∥O∥∞ (2r0 + k0)D h +� +e−µ′ dist(Ajl,Bc +l ) + e−ν dist(S,Bl)� +Figure 1. Diagram showing the regions involved in the proof Lemma C.5. +By construction, for r > 2r0, the condition that Ajl ∩ S = ∅ is met, and therefore the bound +holds. We recall that i ∈ Λ is defined as the center of S. Since dist(i, jl) = k0/2 + dist(S, Bl) + +dist(Ajl, Bl)+r0, we can choose Bl so that dist(S, Bl), dist(Ajl, Bl) ≥ dist(i, jl)/2−k0/4−r0/2−1. +Therefore, +Covσ(β,x)(O, �Hl(x)) ≤ 4(C + c′)C′′∥O∥∞(2r0 + k0)D he− dist(i,jl)/ξ + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +Be +S(r) +0 +0 +6 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +dist(B, Aje) +0 +6 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +r +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +s +Aje +dist(Be, S) +Q +0 +0 +0 +0 +0 +0 +Q +0 +0 +0 +0 +0 +0 +0 +Q +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +9 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +D +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +025 +where 1/ξ = min{µ′, ν} and C′′ := emax{µ′,ν}(k0/4+r0/2+1). Therefore +|fO(x) − fO(x|Si(r))| ≤ 4β(C + c′) h (2r0 + k0)D ∥O∥∞ +� +l∈S(r)c +e− dist(i,jl)/ξ . +Upon shifting the center of the lattice at site i, we get +|fO(x) − fO(x|S(r))| ≤ 4β(C + c′)C′′ h (2r0 + k0)D ∥O∥∞ +� +|l|≥r+k0/2 +e−|l|/ξ += 4β(C + c′)C′′ h (2r0 + k0)D ∥O∥∞ +� +a>r+k0/2 +�a + D − 1 +D − 1 +� +e−a/ξ +≤ 4β(C + c′)C′′ h (2r0 + k0)D DD−1∥O∥∞ +� +a>r+k0/2 +aD−1 e−a/ξ +≤ 4β(C + c′)C′′ h (2r0 + k0)D(D − 1)!(2ξ)D−1 DD−1∥O∥∞ +� +a>r+k0/2 +e− a +2ξ +≤ 4β(C + c′)C′′ h (2r0 + k0)D(D − 1)!(2ξ)D−1 DD−1∥O∥∞ +e− r+k0/2+1 +2ξ +1 − e− 1 +2ξ +≡ C1 e− r +2ξ ∥O∥∞ , +where C1 depends upon all the parameters of the problem. +In the case when we are interested in distinguishing two Gibbs states with Lipschitz observables, +over extended subregions of the lattice, the following extension of Corollary C.4 can be easily shown +to hold: +Corollary C.6. Assume that the states σ(β, x) satisfy the condition of decay of correlations Con- +dition A.1. Then for any region S of the lattice and any two x, y ∈ [−1, 1]m +W1(trSc(σ(β, x)), trSc(σ(β, y))) ≤ ∥x|S(r) − y|S(r)∥ℓ1 polylog(|S(r)|) , +where r = max +� +r0, 2ξ log +� +2|S|C1 +∥x|S(r0)−y|S(r0)∥ℓ1 +�� +with r0 being the smallest integer such that +x|S(r0) ̸= y|S(r0), and C1, ξ are the same constants as in Lemma C.5. +Proof. Given LS a Lipchitz observable supported on region S of the lattice, we have for any r ∈ N: +��fLS(x) − fLS(y) +�� ≤ ∥LS∥∞ +��� trSc � +σ(β, x) − σ(β, x|S(r)) +��� +1 + +�� trSc � +σ(β, y) − σ(β, y|S(r)) +��� +1 +� ++ W1 +� +σ(β, x|S(r)), σ(β, y|S(r)) +� +≤ 2 |S| ∥LS∥Lip C1 e− r +2ξ + W1 +� +σ(β, x|S(r)), σ(β, y|S(r)) +� +, +where the second line follows from Equation (C.16). By Corollary C.4, we conclude that +W1 +� +trSc(σ(β, x)), trSc(σ(β, y)) +� +≤ 2 |S| C1 e− r +2ξ + W1 +� +σ(β, x|S(r)), σ(β, y|S(r)) +� +≤ 2 |S| C1 e− r +2ξ + ∥x|S(r) − y|S(r)∥ℓ1 polylog(|S(r)|) . +Next, we choose r = 2ξ log +� +2|S|C1 +∥x|S(r0)−y|S(r0)∥ℓ1 +� +, where r0 is the smallest integer such that x|S(r0) ̸= +y|S(r0). + +26 +3. +Hamiltonian estimation and optimal Gibbs state tomography +From Corollary C.4 it is immediate that we reduced the problem of obtaining a good estimate +in W1 to the problem of estimating the parameters of the Gibbs state σ(β, x). Indeed, it is clear +that if we can obtain an estimate x′ of x satisfying +∥x − x′∥ℓ1 = O(εn/polylog(n)), +(C.18) +then it suffices to ensure that W1(σ(β, x), σ(β, x′)) = εn. Let us discuss some examples where we +can obtain this efficiently with O(ε−2polylog(n)) samples. +a. +Commuting Hamiltonians +In [Ans], the authors give an algorithm which with +eO(βkD)O(log(δ−1n)ε−2) +(C.19) +copies of σ(β, x) learns x up to ε in ℓ∞ distance when σ(β, x) belongs to a family of commuting, +k-local Hamiltonians on a D-dimensional lattice. As we assumed that the number of parameters +m = O(n), this translates to an algorithm with sample complexity eO(βkD)O(ε−2polylog(δ−1n)) +to learn x up to εn in ℓ1 distance. It should be noted that the time complexity of their algorithm +is O(neO(βkD)ε−2polylog(δ−1n)). +Thus, any commuting model at constant temperature satisfying exponential decay of correla- +tions can be efficiently learned with polylog(n) samples. Examples of classes of commuting states +that satisfy exponential decay of correlations include: +1. 1D translation-invariant Hamiltonians at any positive temperature [Ara69]. +2. Commuting Gibbs states of Hamiltonians on regular lattices below a threshold temperat- +ure [KGK+14, HMS20]. +3. Classical spin models away from criticality [DS87, LSS19, HMS20]. +4. Ground states of uniformly gapped systems [BHM10, BH11, MZ13, NSY22]. +b. +High-temperature Gibbs states +Another class of states for which the conditions of our results hold are local Gibbs states on +a lattice above a threshold temperature that depends on the locality of the Hamiltonian and +the dimension of the lattice. +These systems are known to have exponential decay of correla- +tions [KGK+14, HMS20]. Furthermore, in [HKT21] the authors give an algorithm to learn x up +to error ε in ℓ∞ norm from O(ε−2polylog(δ−1n)) samples. This again translates to a O(εn) error +in ℓ1 norm. Note that their algorithm also is computationally efficient. +We note that in [AAKS21] the authors give an algorithm to learn the Hamiltonian of any Gibbs +state of positive temperature through the maximum entropy method. However, their results require +a polynomial number of samples to recover the parameters in ℓ1 distance. Thus, their results do +not work for the polylog regime investigated in this work. + +27 +c. +Gibbs state of exponentially decaying correlations and conditional mutual information +In the previous section, we extracted two regimes for which there exist efficient Gibbs tomo- +graphy algorithms from previous works, namely the commuting and the high-temperature regimes. +As said before, depending on the Hamiltonian, exponential decay of correlations can also occur in +the low-temperature regime, and it is an interesting open question whether our strategy can be +adapted to that setting for non-commuting interactions. +Here, we show that the Gibbs state σ(β, x) of a possibly non-commuting Hamiltonian H(x) can +also be estimated in Wasserstein distance up to multiplicative error εn given polylog(n) copies of +it as long as the latter has exponentially decaying correlations and is close to a quantum Markov +chain, hence partially answering an open problem previously raised in [AAKS21]. +To be more precise, in this section we will require a stronger notion of decay of correlations: +Definition C.7 (Uniform clustering). The Gibbs state σ(β, x) is said to be uniformly ζ(ℓ)- +clustering if for any X ⊂ Λ and any A ⊂ X and B ⊂ X such that dist(A, B) ≥ ℓ, +Covσ(β,x,X)(XA, XB) ≤ ∥XA∥∞ ∥XB∥∞ ζ(ℓ) +for any XA supported on A and XB supported on B. +As pointed out in [BK18], this property is called uniform clustering to contrast with regular +clustering property that usually only refers to properties of the state σ(β, x). +Definition C.8 (Uniform Markov condition). The Gibbs state σβ(x) is said to satisfy the uniform +δ(ℓ)-Markov condition if for any ABC = X ⊂ Λ with B shielding A away from C and such that +dist(i, j) ≥ ℓ for any i ∈ A and j ∈ C, we have +I(A : C|B)σ(β,x,X) ≤ δ(ℓ) . +This property always holds for commuting Gibbs states for a function δ(ℓ) = 0 as soon as +ℓ is larger than twice the interaction range. +Although not proven yet, it is believed that the +approximate Markov property holds with some generality for non-commuting Gibbs states. The +1D and high-temperature settings were investigated in [KB19] and [KKBa20], respectively. The +decay of the conditional mutual information was also shown for finite temperature Gibbs states +of free fermions, free bosons, conformal field theories, and holographic models [SM16], as well as +more recently for purely generated finitely correlated states in [SK22]. +We will now show how to learn states that satisfy both the uniform Markov condition and the +uniform clustering of correlations. Our strategy consists in using the maximum entropy estimation +[Jay57b, Jay57a, Jay82, BKL+17], already appearing in [AAKS21], to construct an estimator ˆx +of the parameter x ∈ [−1, 1]m. The condition of exponential decay of correlations and that of +approximate Markov chain will ensure that W1(σ(β, ˆx), σ(β, x)) = o(n). +Thus, we once again +emphasise that our goal is to obtain a good recovery of the state, not of the parameter x. +For sake of clarity and simplicity of presentation, we only consider the 1D setting, although our +method easily extends to arbitrary dimension. We assume that each interaction hj(xj) is of the +form +hj(xj) := +ℓ +� +l=1 +xj,l hj,l + +28 +for some self-adjoint operators hj,l supported in Aj := {k ∈ Λ| dist(k, j) ≤ r0} with ∥hj,l∥ ≤ h, +where we denoted by xj,l the entries of xk. We also recall that given a region R of the lattice, we +denote HR(x) := � +k∈R hk(xk). In what follows, with a slight abuse of notations, we denote by +the same symbol a vector y = {yk,l}k∈Nj and its embedding (y, 0) onto [−1, 1]m. Then, given an +inverse temperature β > 0, we define the partition function as +Zβ(x) = tr +� +e−βH(x)� +. +The maximum entropy problem consists in the following strongly convex optimisation problem. +Theorem C.9 ([AAKS21]). Given an unknown Hamiltonian H(x) = � hj(xj), define ek,l = +tr[hk,ℓσ(β, x)]. Solving the following optimisation problem: +ˆx := arg min +y∈[−1,1]m L(y) , +where +L(y) := log Zβ(y) + β +� +k∈Λ +ℓ +� +l=1 +yk,l ek,l +(C.20) +gives ˆx such that σ(β, ˆx) = σ(β, x). +In an experimental setting, we will not have access to the exact {ek,l}k,l, but instead may be +able to approximate them using by having access to the state. However, we want to be sure that +having a reasonably good approximation to ek,l is sufficient to approximate x. To do so one can +make use of the fact that +log Zβ(ˆx) ≤ log Zβ(x) + β +� +k∈Λ +ℓ +� +l=1 +(xk,l − ˆxk,l) �ek,l . +(C.21) +Further assuming α2 is a lower bound on the strong convexity constant associated to the function +x �→ log Zβ(x), that is ∇2Zβ ≥ α2 I, we have by Taylor expansion and since ∂xk,l log Zβ(x) = +−βek,l(x): +log Zβ(ˆx) ≥ log Zβ(x) − β +� +k∈Λ +ℓ +� +l=1 +(ˆxk,l − xk,l) ek,l(x) + α2 +2 ∥x − ˆx∥2 +ℓ2 . +(C.22) +Combining the two bounds above, we find that +∥x − ˆx∥2 +ℓ2 ≤ 2β +α2 +� +k,l +(xk,l − ˆxk,l)(�ek,l − ek,l(x)) ≤ 2β +α2 +∥x − ˆx∥ℓ2 ∥e − �e∥ℓ2 , +and hence ∥x − ˆx∥ℓ2 ≤ +2β√ +ℓ|Λ| η +α2 +, thus giving the following theorem: +Theorem C.10 ([AAKS21]). Suppose �ek,l is an approximation of ek,l(x) := tr +� +hk,l σ(β, x) +� +with +∥�e − e(x)∥ℓ∞ ≤ η. Assume that the following inequality is satisfied for some α2: ∇2Zβ ≥ αI. +Solving the following optimisation problem: +ˆx := arg min +y∈[−1,1]m L(y) , +where +L(y) := log Zβ(y) + β +� +k∈Λ +ℓ +� +l=1 +yk,l �ek,l +(C.23) +gives an output ˆx satisfying: +∥ˆx − x∥ℓ2 ≤ 2βη +√ +ℓΛ +α2 +. + +29 +Using the bound on ˆx from Theorem C.10 the equivalence between ℓ1 and ℓ2-norms, we have +that +∥x − ˆx∥ℓ1 ≤ 2βℓnη +α2 +, +which provides us with the right scaling for our ℓ1 approximation problem as long as η = o(1) +and α2 = Ω(1). Unfortunately, the constant α2 could only be proved to scale inverse polynomially +with n in [AAKS21]. A first idea from there is to try and find a constant α1 = Ω(n−1) such that +the following strong convexity bound with respect to the ℓ1-norm holds. As per eq. (C.22), this +would imply: +log Zβ(ˆx) ≥ log Zβ(x) − β +� +k,l +(ˆxk,l − xk,l) ek,l(x) + α1 +2 ∥x − ˆx∥2 +ℓ1 . +(C.24) +If such a bound held, we would conclude similarly to the previous setting that +∥x − ˆx∥ℓ1 ≤ 2βη +α1 += o(ηn) . +Which together with the continuity bound Equation (C.9) would allow us to get the desired +recovery estimate in Wasserstein distance. Now, it can be seen that Equation (C.24) is equivalent +to +∥x − ˆx∥2 +ℓ1 ≤ +2 +α1 +D(σ(β, x)∥σ(β, ˆx)) . +(C.25) +Here we recall that the relative entropy between two quantum states ρ and σ with supp(ρ) ⊆ +supp(σ) is D(ρ∥σ) := tr ρ log ρ − tr ρ log σ. This together with Equation (C.9) would lead to the +following local version of the transportation cost inequality +W1(σ(β, x), σ(β, ˆx))2 ≤ O(n polylog(n)) D(σ(β, x)∥σ(β, ˆx)) . +(C.26) +In [PR22], such inequality was shown to hold in the high-temperature regime only for commuting +H, albeit when σ(β, x) can be replaced by an arbitrary state ρ on the lattice. The latter is referred +to as a transportation-cost inequality for the state σ(β, ˆx). Since Equation (C.24) consists in a +strengthening of Equation (C.26), proving it directly appears difficult. Here instead, we want to +show the following weakening of (C.26): +W1(σ(β, x), σ(β, ˆx))2 ≤ O(n polylog(n)) D(σ(β, x)∥σ(β, ˆx)) + o(εn) , +for some constant δ which depends on the approximate Markov as well as the correlation decay +properties of the Gibbs state σ(β, ˆx). More precisely, we show the following extension of [PR22, +Theorem 4] to Gibbs states of non-commuting Hamiltonians. +Proposition C.11 (Generalised transportation-cost inequality). With the notations of the above +paragraph, for all states ρ: +W1(ρ, σ(β, x)) ≤ inf +ℓ∈N O(ℓ√n) +� +D(ρ∥σ(β, x)) + n2� +δ(O(ℓ)) + ζ(O(ℓ)) + e−O(ℓ)� +. +In particular, if both ζ(l), δ(l) = O(e−ξl), then for l = O(ξ−1 log(nε−1)) we have +W1(ρ, σ(β, x)) ≤ O(log(nε−1)√n) +� +D(ρ∥σ(β, x)) + o(εn). +(C.27) + +30 +Proof. The proof is adapted from that of [PR22, Theorem 4]. We first consider a bipartite quantum +subsystem AB ⊂ Λ and a joint quantum state ωAB of AB. We then define the so-called quantum +recovery map [SBT16, JRS+18] by its action on a quantum state ρA on region A: +ΦA→AB(ρA) = +� +R +ω +1−it +2 +AB ω +it−1 +2 +A +ρA ω +− 1+it +2 +A +ω +1+it +2 +AB dµ0(t) , +(C.28) +where µ0 is the probability distribution on R with density +dµ0(t) = +π dt +2 (cosh(πt) + 1) . +(C.29) +If A is in the state ωA, the recovery map ΦA→AB recovers the joint state ωAB, i.e., ΦA→AB(ωA) = +ωAB. The relevance of the recovery map comes from the recoverability theorem [SBT16], which +states that ΦA→AB can recover a generic joint state ρAB from its marginal ρA if removing the +subsystem B does not significantly decrease the relative entropy between ρ and ω. More precisely, +for any quantum state σAB of AB we have +D(σAB∥ωAB) − D(σA∥ωA) ≥ DM(σAB∥ΦA→AB(σA)) , +(C.30) +where DM denotes the measured relative entropy [Don86, Pet86, HP91, BFT17] +DM(σ∥ω) := sup +(X,M) +D(Pσ,M∥Pω,M) , +(C.31) +where the supremum above is over all positive operator valued measures M that map the input +quantum state to a probability distribution on a finite set X with probability mass function given +by Pρ,M(x) = tr ρM(x). +Next, we split region A into regions A1 and A2 such that A1 shields A2 away from B, and take +σAB := tr(AB)c(σ(β, x)) and ωAB = σA1B ⊗ σA2 In that case, (C.30) becomes +I(B : A2|A1)σ ≥ DM +� +σ∥ΦA1→A1B(σA) +� +, +(C.32) +where we also used that the state ω is a tensor product in the cut A1B − A2, so that ΦA→B = +ΦA1→B. +Next, we pave the chain Λ into unions of intervals A = ∪M +i=1Ai and B = ∪M +i=1Bi such that +Ai ∩ Bi ̸= ∅ and Bi ∩ Ai+1 ̸= ∅. As in [BK18], we then define the channel F := FA ◦ FB where +FB := � +i σ(β, x, Bi)⊗trBi and FA := � +j ΦAi\B→Ai ◦trAi. In words, the channel FB first prepares +the Gibbs state in the region B, whereas FA prepares the remaining of the Gibbs state onto region +A\B. Then, we have, for any state ρ +W1(ρ, σ(β, x)) ≤ W1(ρ, FB(ρ)) + W1(FB(ρ), FA ◦ FB(ρ)) + W1(F(ρ), σ(β, x)) +≤ +� +i +W1(σ(A, i), σ(A, i + 1)) + +� +i +W1(σ(B, i), σ(B, i + 1)) + n ∥F(ρ) − σ(β, x)∥1 +(1) +≤ R +� +i +∥σ(A, i) − σ(A, i + 1)∥1 + ∥σ(B, i) − σ(B, i + 1)∥1 + n∥F(ρ) − σ(β, x)∥1 , + +31 +where σ(A, i) := � +j 0, and restrict ourselves to the subset of parameters x|Si(r). The number +of parameters in that subset is bounded by the volume V (r + r0 + k0) of the ball Si(r + r0) times +the number ℓ of parameters per interaction. +We denote it by mr := V (r + r0 + k0)ℓ. +Next, +we partition the parameter space [−1, 1]mr onto cubes of side-size γ ∈ (0, 1). +By the coupon +collector’s problem, we have that the probability that none of the sub-vectors Yj|Si(r) is within one +of those cubes is upper bounded by e−N(γ/2)mr+mr log(2/γ). By union bound, the probability that +for any i ∈ [M], any cube is visited by at least one sub-vector Yj|Si(r) is lower bounded by 1 − δ, +δ := Me−N(γ/2)mr+mr log(2/γ). In other words, with probability 1 − δ there is a ˆYi(x)|Si(r) in the N +samples satisfying +∥x|Si(r) − ˆYi(x)|Si(r)∥ℓ∞ ≤ γ +(D.2) +for all i ∈ [M]. +Denoting ˆfOi(x) := tr +� +Oi σ(β, ˆYi(x)|Si(r)) +� +, we next control |fOi(x|Si(r))− ˆfOi(x)|. This is easily +done in terms of the ℓ∞ norm distance between ˆYi(x)|Si(r) and x|Si(r) by Lipschitz estimate: writing +H(x|Si(r)) ≡ H1 and H( ˆYi(x)|Si(r)) ≡ H2, +��fOi(x|Si(r)) − ˆfOi(x) +�� ≤ ∥Oi∥∞ ∥σ(β, x|Si(r)) − σ(β, ˆYi(x)|Si(r))∥1 +≤ ∥Oi∥∞ +���� +e−βH1 +tr[e−βH1] − +e−βH2 +tr[e−βH2] +���� +1 +≤ ∥Oi∥∞ +�∥e−βH1 − e−βH2∥1 +tr[e−βH1] ++ ∥e−βH2∥1 +| tr[e−βH1] − tr[e−βH2]| +tr[e−βH1] tr[e−βH2] +� +≤ ∥Oi∥∞ ∥e−βH1 − e−βH2∥1 +tr +� +e−βH1 + e−βH2� +tr +� +e−βH1� +tr +� +e−βH2� +≤ 22V (r+k0)+1∥Oi∥∞ ∥e−βH1 − e−βH2∥∞ e3βV (r+k0)h . + +34 +Next, we use the integral perturbation bound for the exponential in order to bound +∥e−βH1 − e−βH2∥∞ ≤ +� 1 +0 +∥e−(1−s)βH1(H1 − H2)e−βsH2∥∞ ds +≤ e2βV (r+k0)h ∥H1 − H2∥∞ +≤ e2βV (r+k0)h +� +j∈Si(r) +∥hj(xj − ˆYi(x)j)∥∞ +≤ e2βV (r+k0)h V (r + k0)hℓγ , +where the last line comes from Equation (D.2). Combining this with Lemma C.5, from which +we have that +| tr +� +Oi (σ(β, x) − σ(β, x|Si(r))) +� +|, | tr +� +Oi (σ(β, ˆYi(x)) − σ(β, ˆYi(x)|Si(r))) +� +| ≤ C1 e− r +2ξ ∥Oi∥∞, +we have proven that for all x ∈ [−1, 1]m, +|fOi(x) − ˆfOi(x)| ≤ +� +2C1e− r +2ξ + C2(r)γ +� +∥Oi∥∞ , +(D.3) +where C2(r) := 22V (r+k0)+1e5βV (r+k0)hV (r + k0)hℓ. Now, the volume V (s) of a ball of radius s in +Λ is equal to +V (s) = +� +a≤s +�a + D − 1 +D − 1 +� +≤ (2s)D . +We then fix r so that 2C1e−r/2ξ ≤ ε/2, γ so that C2(r)γ ≤ 22D+1(r+k0)D+1e5β2D(r+k0)Dh2D(r + +k0)Dhℓ ≤ ε/2, and therefore a lower bound on N arises from the constraint +δ := Me−N(γ/2)mr+mr log(2/γ) . +Namely: +r = +� +��� +2ξ log +� +�16β(C + c′) h (2r0 + k0)D(D − 1)!(2ξ)D−1 DD−1 +ε e +k0+1 +2ξ (1 − e− 1 +2ξ ) +� +� +� +��� +, +γ = ε e−[2(r+k0)]D(3 log 2+5βh) +2[2(r + k0)]Dhℓ +. +Therefore, +N = +�γ +2 +�−[2(r+r0+k0)]Dℓ +log +�M +δ +� ++ [2(r + r0 + k0)]Dℓ log +�2 +γ +��γ +2 +�−[2(r+r0+k0)]Dℓ +copies suffice for the approximation claimed to hold with probability 1 − δ. +At this stage, we use the shadow tomography protocol to get classical shadows �σ(β, ˆYi(x)) for +each of the states σ(β, ˆYi(x)). Since only one copy of each �σ(β, ˆYi(x)) is available, its reconstruc- +tion is likely going to be too noisy. Instead, we will use several non-identical copies σ(β, ˆYi(x)) + +35 +which almost coincide on large enough regions surrounding the supports of observables Oi in order +to improve the precision of the estimation of ˆfO. We first develop the following Gibbs shadow +tomography protocol which we believe to be of independent interest. +Consider a Gibbs state σ(β, x) and a family σ(β, x1), . . . , σ(β, xN) of Gibbs states with the +promise that for any i ∈ [M] there exist t vectors xi1, . . . , xit such that maxj∈[t] ∥x|Si(r) − +xij|Si(r)∥∞ ≤ γ. We run the shadow protocol and construct product operators �σ(β, x1), . . . , �σ(β, xN). +Then for any ball B of radius k0, we select the shadows �σ(β, xi1), . . . �σ(β, xit) and construct the +empirical average +�σB(x) := 1 +t +t +� +j=1 +trBc � +�σ(β, xij) +� +. +Proposition D.2 (Robust shadow tomography for Gibbs states). Fix ε, δ ∈ (0, 1). In the notations +of Proposition D.1, with probability 1 − δ′, for any ball B of radius k0, the shadow �σB satisfies +∥�σB − trBc[σ(β, x)]∥1 ≤ 2C1 e− r +2ξ + C2(r)γ + ε as long as +t ≥ 8.12k0 +3.ε2 log +�nk02k0+1 +δ′ +� +. +(D.4) +Proof. In view of Proposition G.1, it is enough to show that the reduced states trBc[σ(β, xij)] +are close to trBc[σ(β, x)]. This is done by simply adapting some of the estimates in the proof of +Proposition D.1. In particular, we have shown that +∥ trBc � +σ(β, x) − σ(β, xij) +� +∥1 ≤ 2C1 e− r +2ξ + C2(r)γ . +The result follows directly from Proposition G.1. +We are now ready to state and proof the main result of this section. We denote �fOi(x) = +tr +� +Oi �σSi(x) +� +the function constructed from the Gibbs shadow tomography protocol of Proposi- +tion D.2, and write �fO := �M +i=1 �fOi. +Theorem D.3 (Learning algorithm for quantum Gibbs states). In the notation of the previous +paragraph, consider a set of N shadows {�σ(β, xi)}N +i=1. Given an arbitrary local observable O, we +fix +r := +� +��� +2ξ log +� +�24β(C + c′) h (2r0 + k0)D(D − 1)!(2ξ)D−1 DD−1 +ε e +k0+1 +2ξ (1 − e− 1 +2ξ ) +� +� +� +��� +, +γ = ε e−[2(r+k0)]D(3 log 2+5βh) +3[2(r + k0)]Dhℓ +, +t := +�24.12k0 +ε2 +log +�nk02k0+1 +δ′ +�� +. +Then, we have that with probability (1 − δ).(1 − δ′), +|fO(x) − �fO(x)| ≤ ε +� +i +∥Oi∥∞ , +as long as +N = t +�γ +2 +�−[2(r+r0+k0)]Dℓ +log +�M +δ +� ++ t log +� +t +�2 +γ +�[2(r+r0+k0)]Dℓ��γ +2 +�−[2(r+r0+k0)]Dℓ +. + +36 +Once again, upon careful checking of the bounds, we have found +N = Θ +� +log +�M +δ +� +log +� n +δ′ +� +epolylog(ε−1) +� +. +Proof. Adapting the proof of Proposition D.1, it is clear that with probability +1 − δ := 1 − Me−N 1 +t (γ/2)mr+mr log(2/γ)+log t +each cube is visited at least t times. Conditioned on that event, and choosing t such that Equa- +tion (D.4) holds, we have that with probability 1 − δ′ +|fOi(x) − �fOi(x)| ≤ +� +2C1e− r +2ξ + C2(r)γ + ε +� +∥Oi∥∞ . +Remark D.4. We emphasise that the classical data {�σ(β, xi)}N +i=1 are fixed, and then any local +observable O can be chosen after the data has been taken which will satisfy the bounds in the- +orem D.3. +Remark D.5. Our proof readily extends to distributions µ that satisfy the following anti-concentration +property: for any x0 ∈ [−1, 1]mr and for all π permutations of the coordinates of [−1, 1]n we have: +µ(A(x0, ε, π)) > 0 =⇒ µ(A(x0, ε, π)) = Ω((2ε)mr/polylog(n)) , +(D.5) +where A(x0, ε, π) := π((x0 + [−ε, ε]mr) × [−1, 1]n−mr). To see this, notice that the condition in +Equation (D.5) implies that we can discretise the number of cubes with size ε and positive weight +into at most O((2ε)−mrpolylog(n)) cubes for any choice of mr coordinates. +By e.g. +rejection +sampling we can then generate a sample from the uniform distribution on those cubes by taking +at most O(polylog(n)) samples from the distribution µ. Once given uniform samples over those +cubes we can argue as in the proof above. +One distribution that satisfies the condition in Equation (D.5) but is far from uniform over +the whole space is e.g. a Dirac measure on a single state. It is also satisfied for various natural +distributions, such as i.i.d. distributions on each coordinate. +Remark D.6. It is clear that our stronger L∞ recovery guarantee cannot hold for arbitrary dis- +tributions and requires some sort of anti-concentration. To see this, consider a distribution over +parameters that outputs a state ρ0 with probability 1 − p and a different, locally distinguishable +state ρ1 with probability p. Before we have drawn Ω(p−1) samples it is unlikely that we gained +access to even a single sample of ρ1. But algorithms like ours with L∞ guarantees also need to +perform well on such rare outputs. Thus, we see that the sample complexity for L∞ guarantees +will have to depend on the parameter p and will blow-up as p → 0. In contrast, if we wish to +obtain good recovery in L2 for this simple example as p → 0, we can always output the expectation +value w.r.t. ρ0. + +37 +1. +Learning the High-Temperature Phase +Theorem D.7 (Learnability of the High-Temperature Phase). Let H(x) be a geometrically local +Hamiltonian. Then there exists a temperature range β ∈ [βc, 0], such for all x ∈ [−1, 1]m then the +parameters of Theorem D.3 are sufficient to learn +|fO(x) − �fO(x)| ≤ ε +� +i +∥Oi∥∞ , +with probabilities and parameters as given in Theorem D.3. +Proof. From [KGK+14], we see that for sufficiently high-temperatures (low β), then the Gibbs +states must satisfy exponentially decaying correlations. Thus we can utilise Theorem D.3 directly +to get the parameters required to learn the high-temperature phase. +Remark D.8. For 1D, translationally invariant Hamiltonians, the Gibbs state has exponential +decay of correlations for all temperatures [BCPH22] and hence the phase can be learned efficiently +everywhere. +Remark D.9. For commuting and 1D Hamiltonians we can relate the learnability of the phase to +the analyticity of the free energy, and thus to a more rigorous notion of phase, defined as regions +of parameter space where the free energy is analytic (assuming the Hamiltonian is parameterised +in an analytic fashion). The free energy is defined as F(β, x) = − log(tr[e−βH(x)]). This is done +using [HMS20, Theorem 32] which demonstrates that for commuting and 1D geometrically local +Hamiltonians, exponential decay of correlations holds in the sense of Equation (II.2). Thus we can +utilise Theorem D.3 directly to get the parameters. +One might attempt to relate this to the free energy of non-commuting Hamiltonians, however, +as per [HMS20, Theorem 31], exponential decay of correlations in regions with analytic free energy +is only known for observables O1, O2 whose supports are distance Ω(log(n)) away from each other. +Although one would be able to prove learnability with more samples scaling with n, [HMS20, +Theorem 31] is not strong enough to give the scaling we desire. +Remark D.10. Although we do not prove it here, it is likely we can flip the above remark on its +head. If we consider a region of parameter space in which the free energy is analytic, we expect +all local observables to be analytic in x in this region. As such, we should be able to approximate +the local observable using polynomial interpolation (or some other technique) and learning the +polynomial of this observable everywhere in the phase with small error. +It is worth noting that high-temperature Gibbs states of commuting Hamiltonians as well as +those of 1D Hamiltonians can be efficiently prepared and hence desired observables could be meas- +ured directly [BK18]. Hence Theorem D.7 becomes most useful in the setting where parameters +are a priori unknown to us and we wish to extract useful information. +Appendix E: Learning ground states +The previous results can be adapted to ground states with exponentially decaying correlations +through the following tricks. Here by ground state we mean the limit +ψg(x) := lim +β→∞ σ(β, x) . + +38 +1. +Commuting models +We first consider the same family of Hamiltonians {H(x)}x∈[−1,1]m as the one defined in Equa- +tion (A.1), and assume the interactions hj(xj) to be commuting for all x. The ground state H(x) is +denoted by ψg(x). It was shown in [Ans] that the reduced Gibbs states trRc σ(β, x) of commuting +Hamiltonians can be written as +trRc σ(β, x) = +e−β(HR(x)+ΦR(x)) +tr +� +e−β(HR(x)+ΦR(x))� ≡ σ′(x, β, R), +(E.1) +where we recall that HR(x) corresponds to the Hamiltonian restricted to the regions R, and where +ΦR(x) is an effective interaction term supported on the inner boundary ∂−R of R, which commutes +with HR and satisfies ∥ΦR(x)∥∞ ≤ 2|∂R|, where ∂R denotes the boundary of R. +Lemma E.1. For any region R of the lattice, we denote by λR(x) the spectral gap of HR(x) + +ΦR(x). Then for any observable XA, resp. XB, supported on region A, resp. B, and any β ≥ +2 +λA∪B(x) log +� 8.2|A∪B| +ε +� +, +�� Covσ(β,x)(XA, XB) − Covψg(x)(XA, XB) +�� ≤ ε ∥XA∥∞ ∥XB∥∞ . +Proof. The proof follows by rudimentary estimates: given an arbitrary region R ⊆ Λ, we denote +σR, resp. ψR, the reduced state trRc(σ(β, x)), resp. trRc(ψg(x)). By the triangle inequality and +H¨older’s inequality, we have +�� Covσ(β,x)(XA, XB) − Covψg(x)(XA, XB) +�� +≤ +� +4 ∥σAB − ψAB∥1 + 2 +� +∥σA − ψA∥1 + ∥σB − ψB∥1 +�� +∥XA∥∞ ∥XB∥∞ +≤ 8 ∥σAB − ψAB∥1 ∥XA∥∞ ∥XB∥∞ , +where the third line follows from the monotonicity of the trace distance under partial tracing. +Next, for any region R, we denote ψ′ +g(x, R) the limit limβ→∞ σ′(x, β, R). This is a projection +onto the ground eigenspace of HR(x) + ΦR(x). By Pinsker inequality, we have, denoting E0(x) < +E1(x) < . . . the ordered energies of HR(x) + ΦR(x) with corresponding multiplicities mj(x), and + +39 +λR(x) its gap: +∥σAB − ψAB∥2 +1 = ∥σ′(x, β, AB) − ψ′(x, AB)∥2 +1 +≤ 2 D(ψ′ +g(x, AB)∥σ′(x, β, AB)) +≤ −2 log(m0(x)) − 2 tr +� +ψ′ +g(x, AB) log(σ′(x, β, AB)) +� +≤ −2 log(m0(x)) − 2 log +� +e−βE0(x) +tr +� +e−β(HR(x)+ΦR(x))� +� += −2 log(m0(x)) + 2 +� +βE0(x) + log +� +j≥0 +e−βEj(x)mj(x) +� += −2 log(m0(x)) + 2 log +� � +j≥0 +e−β(Ej(x)−E0(x))mj(x) +� += 2 log +� +�1 + +� +j≥1 +e−β(Ej(x)−E0(x)) mj(x) +m0(x) +� +� +≤ 2 e−βλAB(x) � +j≥1 +mj(x) +m0(x) +≤ 22|A∪B| e−βλA∪B(x) . +(E.2) +The result follows after imposing the above bound to be smaller than ε2/8. +Remark E.2. Under suitable assumptions on the density of states, it is possible to improve the +dependency of the temperature to β ≥ Ω +� +1 +λA∪B(x) log +� +|A ∪ B|ε−1�� +. +2. +Non-commuting models and LTQO +To go beyond the commuting case, we make use of the notion of local topological quantum order +(LTQO) [MZ13, BHM10, NSY22]: in words, the latter states that observables localised away from +the boundary of the volume cannot distinguish between different ground states. From now on, +given a region R ⊂ Λ we denote by σ(β, x, R), resp. ψg(x, R), the Gibbs state corresponding to +the Hamiltonian HR(x) at inverse temperature β, resp. the ground state +ψg(x, R) := lim +β→∞ σ(β, x, R) . +Definition E.3 (Local Topological Quantum Order). A quantum system satisfies LTQO if for +any region A ⊂ B ⊂ Λ, all x ∈ [−1, 1]m, +�� trAc(ψg(x, B) − ψg(x, Λ)) +�� +1 ≤ CT |A| e− dist(A,Bc)/ξ0 +(LTQO) +for some constants CT , ξ0 > 0. +Remark E.4. The LTQO property is generally defined with respect to a fast-enough decaying +function of dist(A, Bc). Here, we only considered exponential decay for simplicity, but it is not +difficult to extend our proof to the general settings. It is also worth mentioning that the notion of + +40 +LTQO was also extended to models with multiple distinguishable ground states, such as the Ising +model, by replacing the above trace distance by an optimisation over observables XA that satisfy +a symmetry condition in [NSY22]. +With this notion at hand, we can extend Lemma E.1 to the non-commutative setting: +Lemma E.5. Assume that the quantum system is uniformly gapped, infR infx HR(x) ≥ λ0 > 0, +and satisfies LTQO. Then for any balls A, B of radius r0, and all β ≥ +2 +λ0 log +� +25(2(r+r0))D +ε +� +, +�� Covψg(x)(XA, XB) − Covσ(β,x,A∪B(r))(XA, XB) +�� ≤ ε ∥XA∥∞ ∥XB∥∞ , +where r := ξ0 log +� +16|A∪B|CT +ε +� +. Similarly, for any observable XA supported on ball A of radius r0, +�� tr(XAψg(x)) − tr(XAσ(β, x, A(r))) +�� ≤ ε ∥XA∥∞ . +(E.3) +for all β ≥ +2 +λ0 log +� +22(2(r+r0))D +ε +� +, where r = ξ0 log +� +2CT |A| +ε +� +. +Proof. We take a region R that includes A ∪ B. By LTQO, we have +�� Covψg(x)(XA, XB) − Covψg(x,R)(XA, XB) +�� ≤ 8 ∥ tr(AB)c(ψg(x) − ψg(x, R))∥1 ∥XA∥∞ ∥XB∥∞ +≤ 8 CT |A ∪ B| e− dist(A∪B,Rc)/ξ0 ∥XA∥∞ ∥XB∥∞ . +(E.4) +Next, by the same computation to the one that leads to Equation (E.2), we have +�� Covψg(x,R)(XA, XB) − Covσ(β,x,R)(XA, XB) +�� ≤ 8 ∥ψg(x, R) − σ(β, x, R)∥1 ∥XA∥∞ ∥XB∥∞ +≤ 24|R| e− β +2 λR(x) ∥XA∥∞ ∥XB∥∞ . +(E.5) +We now choose R and β so that the right-hand sides of Equation (E.4) and Equation (E.5) are +each bounded by ε +2∥XA∥∞∥XB∥∞. The proof of Equation (E.3) follows the same lines. +Both Lemma E.1 and Lemma E.5 can be used to learn ground states of local Hamiltonians. +We illustrate this in the next theorem. We first need a replacement for Proposition D.1. +Proposition E.6. Denote fOi,g(x) := tr(ψg(x)Oi) and fO,g(x) := � +i fOi,g(x). With the assump- +tions of Lemma E.5, the estimator ˆfO,g(x) := � +i tr[Oi ψg( ˆYi(x))] satisfies the bound +sup +x∈[−1,1]m |fO,g(x) − ˆfO,g(x)| ≤ ε +M +� +i=1 +∥Oi∥∞ , +with probability at least 1 − δ, whenever +N = +�γ +2 +�−[2(r+r0+k0)]Dℓ +log +�M +δ +� ++ [2(r + r0 + k0)]Dℓ log +�2 +γ +��γ +2 +�−[2(r+r0+k0)]Dℓ + +41 +with +r = ξ0 log +�3|Si|Ct +ε +� +, +β = 2 +λ0 +log +� +3.2(2r+k0)D +ε +� +, +γ = ε e−[2(r+k0)]D(3 log 2+5βh) +3[2(r + k0)]Dhℓ +. +Proof. We simply need to adapt the proof of Proposition D.1. Using the same notations as there, +with probability +1 − δ := 1 − Me−N(γ/2)mr+mr log(2/γ) +each cube is visited at least one time. By Lemma E.5, we have that +�� tr +� +Oi(ψg(x) − σ(β, x|Si(r))) +��� ≤ +� +CT e− r +ξ0 |Si| + 2|Si(r)|e− βλ0 +2 +� +∥Oi∥∞ ≡ �C1(r, β) ∥Oi∥∞ . +Similarly +�� tr +� +Oi(ψg( ˆYi(x)) − σ(β, ˆYi(x)|Si(r))) +��� ≤ �C1(r, β) ∥Oi∥∞ . +Next, we can reuse the bound on |fi(x|Si(r))− ˆfOi(x)| = +�� tr[Oiσ(β, x|Si(r))]−tr +� +Oiσ(β, ˆYi(x)|Si(r)) +��� +found in the proof of Proposition D.1. This leads us to the following adaptation of the bound in +Equation (D.3): +��fOi,g(x) − ˆfOi,g(x) +�� ≤ +� +CT e− r +ξ0 |Si| + 2|Si(r)|e +−βλ0 +2 ++ C2(r)γ +� +∥Oi∥∞ , +where ˆfOi,g(x) := tr +� +Oiψg( ˆYi(x)|Si(r)) +� +. We conclude by choosing r so that CT e− r +ξ0 |Si| ≤ ε +3, β +such that 2|Si(r)|e− βλ0 +2 +≤ ε +3 and γ such that C2(r)γ ≤ ε +3. +Next, in complete analogy with Proposition D.2, we develop a robust classical shadow tomo- +graphy algorithm for ground states of quantum systems with LTQO: consider a ground state ψg(x) +and a family ψg(x1), . . . , ψg(xN) of groups states with the promise that for any i ∈ [M] there exist +t vectors xi1, . . . , xit such that maxj∈[t] ∥x|Si(r) −xij|Si(r)∥∞ ≤ γ. We run the shadow protocol and +construct product operators �ψg(x1), . . . , �ψg(xN). Then for any ball B of radius k0, we select the +shadows �ψg(xi1), . . . �ψg(xit) and construct the empirical average +�ψB(x) := 1 +t +t +� +j=1 +trBc � �ψg(xij) +� +. +Proposition E.7 (Robust shadow tomography for ground states). Fix ε, δ ∈ (0, 1). In the nota- +tions of Proposition E.6, with probability 1−δ′, for any ball B of radius k0, the shadow �ψB satisfies +∥ �ψB − trBc[ψg(x)]∥1 ≤ CT e− r +ξ0 |B| + 2|B(r)|e +−βλ0 +2 ++ C2(r)γ + ε as long as +t ≥ 8.12k0 +3.ε2 log +�nk02k0+1 +δ′ +� +. +(E.6) + +42 +Proof. In view of Proposition G.1, it is enough to show that the reduced states trBc[ψg(xij)] +are close to trBc[ψg(x)]. This is done by simply adapting some of the estimates in the proof of +Proposition E.7. In particular, we have shown that +∥ trBc � +ψg(x) − ψ(xij) +� +∥1 ≤ CT e− r +ξ0 |B| + 2|B(r)|e +−βλ0 +2 ++ C2(r)γ . +The result follows directly from Proposition G.1. +We are now ready to state and prove the main result of this section. We denote �fOi(x) = +tr +� +Oi �ψSi(x) +� +the function constructed from the Gibbs shadow tomography protocol of Proposi- +tion D.2, and write �fO := �M +i=1 �fOi. +Theorem E.8 (Learning algorithm for ground states). With the assumptions of Proposition E.6, +we fix +r = ξ0 log +�4|Si|Ct +ε +� +, +β = 2 +λ0 +log +� +4.2(2r+k0)D +ε +� +, +γ = ε e−[2(r+k0)]D(4 log 2+5βh) +3[2(r + k0)]Dhℓ +. +Then, we have that with probability (1 − δ).(1 − δ′), +|fO(x) − �fO(x)| ≤ ε +� +i +∥Oi∥∞ , +as long as +N = t +�γ +2 +�−[2(r+r0+k0)]Dℓ +log +�M +δ +� ++ t log +� +t +�2 +γ +�[2(r+r0+k0)]Dℓ��γ +2 +�−[2(r+r0+k0)]Dℓ +. +In other words, once again, +N = Θ +� +log +�M +δ +� +log +� n +δ′ +� +epolylog(ε−1) +� +. +Proof. Adapting the proof of Proposition D.1, it is clear that with probability +1 − δ := 1 − Me−N 1 +t (γ/2)mr+mr log(2/γ)+log t +each cube is visited at least t times. Conditioned on that event, and choosing t such that Equa- +tion (E.6) holds, we have that with probability 1 − δ′ +|fOi(x) − �fOi(x)| ≤ +� +CT e− r +ξ0 |B| + 2|B(r)|e +−βλ0 +2 ++ C2(r)γ + ε +� +∥Oi∥∞ . +3. +Examples +In this section, we gather examples of lattice quantum systems satisfying LTQO. For more +details, we refer the interested reader to [NSY22] and references therein. + +43 +i. Frustration-free spin chains with a unique translation-invariant matrix product ground state, +such as the famous AKLT chain [AKLT88], provide a class of families of ground states +satisfying LTQO [CGLW13, OT19, Tas18]. +ii. Quantum double models, among which the well-known Toric code introduced by Kitaev +[Kit06, Kit03], models of commuting Hamiltonians and were recently shown to satisfy LTQO +in [CDH+20]. +iii. Levin-Wen models are another class of two-dimensional commuting Hamiltonians studied +for their good properties as quantum error correcting codes. Their LTQO property was +shown in [QW20]. +iv. The stability of LTQO was proved in [NSY22, Theorem 7.2] for a large class of local per- +turbations under the condition that an unperturbed family of Hamiltonians is uniformly +gapped and frustration-free. If the frustration-free condition is removed, stability no longer +holds in general, and there exist families of Hamiltonians with uncomputable spectral gaps +[CPGW15]. +Appendix F: Non-Linear Parameterisations of the Hamiltonian +Here we show that taking a parameterisation of the Hamiltonian that is not a sum of Paulis +does not change the results of Appendix D. +Lemma F.1. Consider a Hamiltonian parameterised in terms of Pauli strings: +H(x) = +� +j +xjPj +where Pj is a Pauli string. Consider an alternative parameterisation of the same Hamiltonian in +terms of the local terms: +H(y) = +� +j +hj(y(j)) +where y(j) ∈ [−1, 1]b, b = O(1), and hj only depends on the coordinates in y(j). We will also +assume that each hj is k-local and ∥∂uhj(y)∥ ≤ 1. +Then, assuming all the non-zero elements of Jacobian are bounded as 1/C′ ≤ |∂ym/∂xk| ≤ C +for C, C′ = O(1), the following holds: +����� +∂hj(y(j)) +∂xm +����� ≤ b4kC, +and +|∂ymfL(y)| ≤ C′′ max +m |∂xifL(β, x)|, +where C′ = O(1). + +44 +Proof. We see that: +∂hj(y(j)) +∂xm += +� +i +∂yi +∂xm +∂hj(y(j)) +∂yi +We note that ∂hj(y(j)) +∂yi +is only non-zero for a b terms. Furthermore, since hj is k-local, then it can +be written as a sum of ≤ 4k Pauli strings. Hence: +����� +∂hj(y(j)) +∂xm +����� ≤ +� +i +��� ∂yi +∂xm +��� +����� +∂hj(y(j)) +∂yi +����� +≤ b4kC max +����� +∂hj(y(j)) +∂yi +����� +≤ b4kC = O(1). +We now consider the functions fL(y) = tr[Lρ(y)], fL(β, x) = tr[Lρ(x)]. +∂ymfL(y) = +� +i +∂xi +∂ym +∂xifL(β, x) +Using that a given ym can depend on at most 4k xm coordinates, we see that for a given ym, at +most poly(4k) = O(1) many +∂xi +∂ym can be non-zero. Thus +|∂ymfL(y)| ≤ poly(4k) +��� ∂xi +∂ym +���|∂xifL(β, x)| +≤ poly(4k)C′|∂xifL(β, x)|. +The lemma statement then follows for C′′ = poly(4k)C′. +This lemma allows us to prove up bounds on the derivative of fL(y), and thus an equivalent to +Lemma C.2 holds for local observables. The rest of the results in appendix D follow similarly. +Appendix G: Shadow tomography for non-identical copies +In this appendix, we extend the shadow tomography protocol to the case of non-identical copies. +Consider a state σ and a family σ1, . . . , σN of states over n qubits with the promise that for any +subset A of qubits of size |A| ≤ r there exists a subfamily of states σi1 . . . σit, flagged in advance, +with the promise that maxj∈[t] ∥ trAc(σij − σ)∥1 ≤ η. +We run the shadow protocol and construct product operators �σ1, . . . , �σN. Then for any region +A, we select the shadows �σi1, . . . �σit and construct the empirical average +�σA := 1 +t +t +� +j=1 +trAc(�σij) . +Proposition G.1 (Shadows for non-identical copies). Fix ε, δ ∈ (0, 1). With probability 1 − δ, the +shadow �σA satisfies ∥�σA − σA∥1 ≤ ε + η as long as +t ≥ 8.12r +3.ε2 log +�nr2r+1 +δ +� +. + +45 +In order to prove the above proposition, we need an extension of the matrix Bernstein inequal- +ity used in proving the convergence guarantee of the standard shadow protocol to the case of +independent, non-identically distributed random matrices: +Lemma G.2 (Matrix Bernstein for non-i.i.d. random matrices [T+15]). Let S1, . . . , St be inde- +pendent, centered random matrices with common dimension d1 × d2, and assume that each one is +uniformly bounded: +E[Sj] = 0 +and +∥Sj∥∞ ≤ L +for all j = 1, . . . , t. +Denote the sum Z = �t +j=1 Sj and let ν(Z) := max +� +∥E[ZZ∗]∥∞, ∥E[Z∗Z]∥∞ +� +. Then, +P +� +∥Z∥∞ ≥ s +� +≤ (d1 + d2) exp +� +−s2/2 +ν(Z) + Ls/3 +� +for all s ≥ 0 . +Proof of Proposition G.1. In the notations of the previous paragraph and of Lemma G.2, we take +Sj := trAc(�σij − σij), j = 1 . . . t, so that Z/t = �σA − E[�σA]. Adapting the proof for the standard +shadow tomography protocol (see e.g. [HKT+22]), we have +∥ trAc(�σij − σij)∥∞ ≤ 2r + 1 +and +ν(Z) +t += 1 +t +��� +t +� +j=1 +E[S2 +j ] +��� +∞ ≤ 3r . +Since ∥X∥∞ ≤ ∥X∥1 ≤ 2r ∥X∥∞, we have +P +� +∥�σA − E[�σA]∥1 ≥ s +� +≤ 2r+1 exp +� +−ts2/22r+1 +3r + (2r + 1)s/(3.2r) +� +≤ 2r+1 exp +�−3ts2 +8.12r +� +. +Next, we observe that under the assumption maxj∈[t] ∥ trAc(σij − σ)∥1 ≤ η, then: +E[�σA] = 1 +t +t +� +j=1 +trAc(σij) +⇒ +∥E[�σA] − trAc(σ)]∥1 ≤ η . +Hence, +P +� +∥�σA − trAc(σ)∥1 ≥ η + ε +� +≤ P +� +∥�σA − E[�σA]∥1 ≥ s +� +≤ 2r+1 exp +�−3tε2 +8.12r +� +. +By union bound, the result follows after choosing δ := nr2r+1 exp +� +−3tε2 +8.12r +� +. + diff --git a/OdFOT4oBgHgl3EQf3TT6/content/tmp_files/load_file.txt b/OdFOT4oBgHgl3EQf3TT6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d69d5dcb0c2ae6f18bd4695bb44da0949dd9bfd9 --- /dev/null +++ b/OdFOT4oBgHgl3EQf3TT6/content/tmp_files/load_file.txt @@ -0,0 +1,1369 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf,len=1368 +page_content='Efficient learning of ground & thermal states within phases of matter Emilio Onorati,1, ∗ Cambyse Rouz´e ,1, † Daniel Stilck Fran¸ca,2 and James D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Watson3 1Zentrum Mathematik, Technische Universit¨at M¨unchen, 85748 Garching, Germany 2Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France‡ 3University of Maryland, College Park, QuICS 3353 Atlantic Building, MD 20742-2420, USA § We consider two related tasks: (a) estimating a parameterisation of an unknown Gibbs state and expectation values of Lipschitz observables on this state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' and (b) learning the expectation values of local observables within a thermal or quantum phase of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In both cases, we wish to minimise the number of samples we use to learn these properties to a given precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For the first task, we develop new techniques to learn parameterisations of classes of sys- tems, including quantum Gibbs states of non-commuting Hamiltonians under the condition of exponential decay of correlations and the approximate Markov property, thus improving on work by [RF21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We show that it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable’s locality — an exponential improvement — hence partially answering conjectures stated in [RF21] and [AAKS21] in the positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This class of properties includes expected values of quasi-local observables and entropic quantities of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For the second task, we turn our tomography tools into efficient algorithms for learning observables in a phase of matter of a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' By exploiting the locality of the Hamiltonian, we show that M local observables can be learned with probability 1−δ and up to precision ε with access to only N = O � log � M δ � epolylog(ε−1)� samples — an exponential improvement in the precision over the best previously known bounds [HKT+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter displaying exponential decay of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In addition, our sample complexity applies to the worse case setting whereas previous results only applied to the average case setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' To prove our results, we develop new tools of independent interest, such as robust shadow tomography algorithms for ground and Gibbs states, Gibbs approximations of locally indis- tinguishable ground states, and generalisations of transportation cost inequalities for Gibbs states of non-commuting Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' ∗ Emilio Onorati emilio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='onorati@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='de † Cambyse Rouz´e cambyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='rouze@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='de ‡ Daniel Stilck Fran¸ca daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='stilck franca@ens-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='fr § James D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Watson jdwatson@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='12946v1 [quant-ph] 30 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' INTRODUCTION Tomography of quantum states is among the most important tasks in quantum information science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In quantum tomography, we have access to one or more copies of a quantum state and wish to understand the structure of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, for a general quantum state, all tomographic methods inevitably require resources that scale exponentially in the size of the system [HHJ+17, OW16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This is due to the curse of dimensionality: the number of parameters needed to fully describe a quantum system scales exponentially with the number of its constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Obtaining these parameters often necessitates the preparation and destructive measurement of exponentially many copies of the quantum system, as well as their storage in a classical memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, as the size of quantum devices continues to increase beyond what can be easily simulated classically, the community faces new challenges to characterise their output states in a robust and efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thankfully, only a few physically relevant observables are often needed to describe the physics of a system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' its entanglement or energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Recently, new methods of tomography have been proposed which precisely leverage this important simplification to develop efficient state learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' One highly relevant development in this direction is that of classical shadows [HKP20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This new set of protocols allows for estimating physical observables of quantum spin systems that only depend on local properties from a number of measurements that scales logarithmically with the total number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, the number of required measurements still faces an exponential growth with respect to the size of the observables that we want to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, using such protocols to learn the expectation values of physical observables that depend on more than a few qubits quickly becomes unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Gibbs State Tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Some simplification can be achieved from the fact that physically relevant quantum states, such as ground and Gibbs states of a locally interacting spin system, are themselves often described by a number of parameters which scales only polynomially with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' From this observation, another direction in the characterisation of large quantum systems that has received considerable attention is that of Hamiltonian learning and many-body tomography, where it was recently shown that it is possible to robustly characterise the interactions of a Gibbs state with a few samples [Ans, HKT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, even for many-body states, recovery in terms of the trace distance requires a number of samples that scales polynomially in the number of qubits, in contrast to shadows for which the scaling is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' These considerations naturally lead to the question of identifying settings where it is possible to combine the strengths of shadows and many-body tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In [RF21], some of the authors proposed a first solution by combining these with new insights from the emerging field of quantum optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' They obtained a tomography algorithm that only requires a number of samples that scales logarithmically in the system’s size and learns all quasi-local properties of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' These properties are characterised by so-called “Lipschitz observables”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, that first step was confined to topologically trivial states such as high-temperature Gibbs states of commuting Hamiltonians or outputs of shallow circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here, we significantly extend these results to all states exhibiting exponential decay of correlations and the approximate Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Learning Phases of Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Tomographical techniques by themselves are somewhat limited in that they tell us nothing about nearby related states – often states belong to a phase of matter in which the properties of the states vary smoothly and are in some sense “well behaved”, and we wish to learn properties of this entire phase of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A recent line of research in this direction 3 that has gained significant attention from the quantum community is that of combining machine learning methods with the ability to sample complex quantum states from a phase of matter to efficiently characterise the entire phase [BWP+17, CM17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A landmark result in this direction is [HKT+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' There the authors showed how to use machine learning methods combined with classical shadows to learn local linear and nonlinear functions of states belonging to a gapped phase of matter with a number of samples that only grows logarithmically with the system’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' That is, given states from that phase drawn from a distribution and the corresponding parameters of the Hamiltonian, one can train a classical algorithm that would predict local properties of other points of the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, there are some caveats to this scheme: (i) the scaling of the number of samples in terms of the precision is exponential, (ii) it does not immediately apply to phases of matter beyond gapped ground states, (iii) the results only come with guarantees on the errors in the prediction in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' That is, given another state sampled from the same distribution as the one used to train, only on average is the error made by the ML algorithm proven to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In this work, we address all of these shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' First, our result extends to thermal phases of matter which exhibit exponential decay of correlations, which includes all thermal systems away from criticality/poles in the partition function [HMS20, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our result also extends to gapped phases that satisfy local topological quantum order [MZ13, BHM10, NSY22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, the sample complexity of our algorithm is quasi-polynomial in the desired precision, which is an exponential improvement over previous work [HKT+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' And, importantly, it comes with point- wise guarantees on the quality of the recovery, as opposed to average guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Interestingly, our results are easier to grasp through the lens of the concentration of measure phenomenon rather than machine learning: we show that local expectation values of quantum states are quite smooth under perturbations in the same class of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' And, as is showcased by the concentration of measure phenomenon, smooth functions on high-dimensional spaces do not show a lot of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, it suffices to collect a few examples to be able to predict what happens in the whole space, while the price we pay for these stronger recovery guarantees is that our algorithm does not work for any distribution over states, but needs some form of anti-concentration which holds e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' for the uniform distribution (see Appendix D for a technical discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In other words, our algorithm necessitates to “see” enough of the space to work and struggles if there are large, low-probability corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' SUMMARY OF MAIN RESULTS In this paper, we consider a quantum system defined over a D-dimensional finite regular lattice Λ = [−L, L]D, where n = (2L + 1)D denotes the total number of qubits constituting the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We assume for simplicity that each site of the lattice hosts a qubit, so that the total system’s Hilbert space is HΛ := � j∈Λ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' All of the results presented here easily extend to qudits, but we will focus on qubits for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our focus in this work are nontrivial statements about what can be learned about many-body states of n qubits in the setting where we are only given Θ(polylog(n)) copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The common theme is that we will assume exponential decay of correlations for our class of states, but will show results in two different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Section II A we summarise our results on how to estimate all quasi-local properties of a given state given identical copies of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This is the traditional setting of quantum tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In contrast, in Section II B we summarise our findings on how to learn local properties of a class of states given samples from different states from that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This is the 4 setting of [HKT+22] where ground states of gapped quantum phases of matter were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here we consider (a) thermal phases of matter with exponentially decaying correlations and (b) gapped ground states with local topological quantum order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Optimal Tomography of Many-Body Quantum States We first consider the task of obtaining a good approximation of expected values of extensive properties of a fixed unknown n-qubit state over Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The state is assumed to be a Gibbs state of an unknown local Hamiltonian H(x) := � j∈Λ hj(xj), x = {xj} ∈ [−1, 1]m, defined through interactions hj(xj), each depending on parameters xj ∈ [−1, 1]ℓ for some fixed integer ℓ and supported on a ball Aj around site j ∈ Λ of radius r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We also assume that the matrix-valued functions xj �→ hj(xj) as well as their derivatives are uniformly bounded: ∥hj∥∞, ∥∇hj∥∞ ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The corresponding Gibbs state at inverse temperature β > 0, and the ground state as β → ∞ take the form σ(β, x) := e−βH(x) tr � e−βH(x)� and ψg(x) := lim β→∞ σ(β, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) In the case when [hj(xj), hj′(xj′)] = 0 for all j, j′ ∈ Λ, the Hamiltonian H(x) and its associated Gibbs states σ(β, x) are said to be commuting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Preliminaries on Lipschitz observables Extensive properties of a state are well-captured by the recently introduced class of Lipschitz observables [RD19, DPMTL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1 (Lipschitz Observable [DPMTL21] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' An observable L on HΛ is said to be Lipschitz if ∥L∥Lip := maxi∈Λ minLic 2∥L − Lic ⊗ Ii∥∞ = O(1), where ic is the complement of the site i in Λ and the scaling is in terms of the number of qubits in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In words, ∥L∥Lip quantifies the amount by which the expectation value of L changes for states that are equal when tracing out one site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' By a simple triangle inequality together with [DPMTL21, Proposition 15], one can easily see that ∥L∥∞ ≤ n∥L∥Lip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Given the definition of the Lipschitz constant, we can also define the quantum Wasserstein distance of order 1 by duality [DPMTL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2 (Wasserstein Distance [DPMTL21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The Wasserstein distance between two n qubit quantum states ρ1, ρ2 is defined as W1(ρ0, ρ1) := sup∥L∥Lip≤1 tr � L(ρ0 − ρ1) � ≤ n∥ρ − σ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Having W1(ρ, σ) = O(εn) is sufficient to guarantee that the expectation value of ρ and σ on extensive, quasi-local observables is the same up to a multiplicative error εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This justifies why we focus on learning states up to an error O(εn) in Wasserstein distance instead of the usual trace distance bound of order O(ε): although a trace distance guarantee of order O(ε) gives the same error estimate, it requires exponentially more samples even for product states, as shown in [RF21, Appendix G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Appendix B, we argue that Lipschitz observables and the induced Wasserstein distance capture linear and nonlinear extensive properties of many-body quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Gibbs state tomography In this section, we turn our attention to the problem of obtaining approximations of linear functionals of the form fL(β, x) := tr[Lσ(β, x)] for all Lipschitz observables L from the measure- ment and classical post-processing of as few copies of the associated unknown Gibbs state σ(β, x) as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We will further require that the state satisfies the property of exponential decay of correlations: for any two observables XA, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' XB, supported on region A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' B, Covσ(β,x)(XA, XB) ≤ C min{|A|, |B|} ∥XA∥∞ ∥XB∥∞ e−ν dist(A,B) , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2) for some constants C, ν > 0, where dist(A, B) denotes the distance between regions A and B, and where the covariance is defined by Covσ(X, Y ) := 1 2 tr � σ � X − tr[σX], Y − tr[σY ] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3) Our first main result is a method to learn Gibbs states with few copies of the unknown state: Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3 (Tomography algorithm for decaying Gibbs states (informal)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For any unknown commuting Gibbs state σ(β, x) satisfying Equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2), there exists an algorithm that provides the description of parameters x′ such that the state σ(β, x′) approximates σ(β, x) to precision nε in Wasserstein distance with probability 1−δ with access to N = O � log(δ−1) polylog(n) ε−2� samples of the state (see Appendix C 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The result extends to non-commuting Hamiltonians whenever one of the following two assumptions is satisfied: (i) the high-temperature regime, β < βc (see Appendix C 3 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (ii) uniform clustering/Markov conditions (see Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In case (ii), we find good approximation guarantees under the following slightly worst scaling in the precision ε: N = O(ε−4 polylog(nδ−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The results for commuting Hamiltonians and in the high-temperature regime proceed directly from the following continuity bound on the Wasserstein distance between two Gibbs states, whose proof requires the notion of quantum belief propagation in the non-commuting case (see Corol- lary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4): for any x, y ∈ [−1, 1]m, W1(σ(β, x), σ(β, y)) = ∥x − y∥ℓ1 O(polylog(n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4) Furthermore, this inequality is tight up to a polylog(n) factor for β = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4) reduces the problem of recovery in Wasserstein distance to that of recovering the parameters x up to an error εn/ polylog(n) in ℓ1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This is a variation of the Hamiltonian learning problem for Gibbs states [AAKS21, HKT21] which relies on lower bounding the ℓ2 strong convexity constant for the log-partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In [Ans], the authors give an algorithm estimating x with eO(βkD)O(log(δ−1n)ε−2) copies of σ(β, x) up to ε in ℓ∞ distance when σ(β, x) belongs to a family of commuting, k-local Hamiltonians on a D-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' If we assume m = O(n), this translates to an algorithm with sample complexity eO(βkD)O(ε−2polylog(δ−1n)) to learn x up to εn in ℓ1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It should also be noted that the time complexity of the algorithm in [Ans] is O(neO(βkD)ε−2polylog(δ−1n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, 6 any commuting model at constant temperature satisfying exponential decay of correlations can be efficiently learned with polylog(n) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We refer the reader to Appendix C 3 for more information and classes of commuting states that satisfy exponential decay of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In the high-temperature regime, we rely on a result of [HKT21] where the authors give a computationally efficient algorithm to learn x up to error ε in ℓ∞ norm from O(ε−2polylog(δ−1n)) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This again translates to a O(εn) error in ℓ1 norm thanks to (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, in Appendix C 3 c we more directly extend the strategy of [AAKS21] by introdu- cing the notion of a W1 strong convexity constant for the log-partition function and showing that it scales linearly with the system size under (a) uniform clustering of correlations and (b) uniform Markov condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This result also generalises the strategy of [RF21] which relied on the exist- ence of a so-called transportation cost inequality previously shown to be satisfied for commuting models at high-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For the larger class of states satisfying conditions (a) and (b), we are able to find x′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' W1(σ(β, x), σ(β, x′)) = O(εn) with O(ε−4polylog(δ−1n)) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Note that the uniform Markov condition is expected to hold for a large class of models that goes beyond high-temperature Gibbs states [KB19, KKBa20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Beyond linear functionals So far, we considered properties of the quantum system which could be related to local linear functionals of the unknown state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In [HKP20, HKT+22], the authors propose a simple trick in order to learn non-linear functionals of many-body quantum systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' their entropy over a small subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, such methods require a number of samples scaling exponentially with the size of the subregion, and thus very quickly become inefficient as the size of the region increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here instead, we make use of the continuity of the entropy functional with respect to the Wasserstein distance, mentioned in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6), together with the following Wasserstein continuity bound in order to estimate the entropic quantities of Gibbs states over regions of arbitrary size (see Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6): assuming Equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2), for any region S of the lattice and any two x, y ∈ [−1, 1]m W1(trSc(σ(β, x)), trSc(σ(β, y))) ≤ ∥x|S(rS) − y|S(rS)∥ℓ1 polylog(|S(rS)|) , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5) where rS = max � r0, 2ξ log � 2|S|C1∥x|S(r0) − y|S(r0)∥−1 ℓ1 �� with r0 being the smallest integer such that x|S(r0) ̸= y|S(r0), S(rS) := {xj| supp(hj(xj)) ∩ S(rS) ̸= ∅}, S(rS) := {i ∈ Λ : dist(i, S) ≤ rS}, and C1, ξ > 0 are constants introduced in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Let us recall a few definitions: denoting by ρR := trRc(ρ) the marginal of a state ρ ∈ D(HΛ) on a region R ⊂ Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' and given separated regions A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' C ⊂ Λ of the lattice: S(A)ρ := − tr[ρA log ρA] is the von Neumann entropy of ρ on A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' S(A|B)ρ := S(AB)ρ − S(B)ρ is the conditional entropy on region A conditioned on region B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' I(A : B)ρ := S(A)ρ + S(B)ρ − S(AB)ρ is the mutual information between regions A and B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' and I(A : B|C)ρ := S(AC)ρ +S(BC)ρ −S(C)ρ −S(ABC)ρ is the conditional mutual information between regions A and B conditioned on region C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The following corollary is a direct consequence of Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6) together with Equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5): Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Assume the decay of correlations holds uniformly, as specified in Equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2), for all {σ(β, x)}x∈[−1,1]m, m = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then, in the notations of the above paragraph, for any two Gibbs states σ(β, x) and σ(β, y), x, y ∈ [−1, 1]m, and any region A ⊂ Λ: |S(A)σ(β,x) − S(A)σ(β,y)| = ∥x|S(rS) − y|S(rS)∥ℓ1O(polylog(|S(rS)|)) , 7 for S ≡ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The same conclusion holds for |S(A|B)σ(β,x) − S(A|B)σ(β,y)| (S ≡ AB), |I(A : B)σ(β,x) − I(A : B)σ(β,y)| (S ≡ AB), and |I(A : B|C)σ(β,x) − I(A : B|C)σ(β,y)| (S ≡ ABC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, given an an estimate y of x satisfying ∥x − y∥ℓ∞ = O(ε/polylog(n)), we can also ap- proximate entropic quantities of the Gibbs state to a multiplicative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' More generally, entropic continuity bounds can be directly used together with Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3(ii) in order to estimate entropic properties of Gibbs states satisfying both uniform clustering of correlations and the approximate Markov condition (see Appendix C 3 c for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Learning Expectation Values of Parametrised Families of Many-Body Quantum Systems Next, we turn our attention to the task of learning Gibbs or ground states of a parameterised Hamiltonian H(x) known to the learner and sampled according to the uniform distribution U over x ∈ [−1, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' More general distributions can also be dealt with under a condition of anti- concentration, see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here we restrict our results to local observables of the form O = �M i=1 Oi where Si := supp(Oi) is contained in a ball of diameter independent of the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The setup in this section is similar to [HKT+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The idea is that we have access to some samples of a state chosen from different values of the parameterised Hamiltonian, and we want to use these to learn observables everywhere in the parameter space with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We then want to know: what is the minimum number of samples drawn from this distribution which allows us to accurately predict expectation values of local observables for all choices of parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Learning Expectation Values in Thermal Phases of Matter The learner is given samples {(xi, σ(β, xi))}N i=1, where the parameters xi ∼ U, and their task is to learn fO(x) := tr[σ(β, x)O] for an arbitrary value of x ∈ [−1, 1]m and an arbitrary local observable O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We assume that everywhere in the parameter space x ∈ [−1, 1]m the Gibbs states are in the same phase of exponentially decaying correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then we have: Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5 (Learning algorithm for quantum Gibbs states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' With the conditions of the previ- ous paragraph, given a set of N samples {xi, ˜σ(β, xi)}N i=1, where ˜σ(β, xi) can be stored efficiently classically, and N = O � log � M δ � log � n δ � epolylog(ε−1)� , there exists an algorithm that, on input x ∈ [−1, 1]m and a local observable O = �M i=1 Oi, produces an estimator ˆfO such that, with prob- ability (1 − δ), sup x∈[−1,1]m |fO(x) − ˆfO(x)| ≤ ε M � i=1 ∥Oi∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Moreover, the samples ˜σ(β, xi) are efficiently generated from measurements of the Gibbs states {σ(β, xi)}N i=1 followed by classical post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our estimator ˆfO is constructed as follows: during a training stage, we pick N points Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' , YN ∼ U and estimate the reduced Gibbs states over large enough enlargements Si∂ of the supports Si := {xj| supp(hj(xj)) ∩ Si∂ ̸= ∅} ∩ [x − ε, x + ε]m of the observables Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Due to the anti-concentration property of the uniform distribution, the probability that a small region Si∂ in parameter space contains t variables Yi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' , Yit becomes large for N ≈ log(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We then 8 run the classical shadow tomography protocol on those states in order to construct efficiently describable and computable product matrices �σ(β, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' , �σ(β, YN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then for any region Si, we select the shadows �σ(β, Yi1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' �σ(β, Yit) whose local parameters are close to that of the target state and construct the empirical average �σSi(x) := 1 t �t j=1 trSc i � �σ(β, Yij) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Using belief propaga- tion methods (see Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2), it is possible to show that exponential decay of correlations ensures that the estimator is a good approximation to local observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus such operators can be well approximated using the reduced state trSc i σ(β, x) for t ≈ log(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The estimator ˆfO is then naturally chosen as ˆfO(x) := �M i=1 tr[Oi �σSi(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A key part of the proof is demonstrating that exponential decay of correlations implies that fO(x) does not change too much as x varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Learning ground states under local indistinguishability We now move our attention to the problem of learning ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Again, the learner is given samples {xi, ψg(xi)}N i=1, xi ∼ U, and their task is to learn fO,g(x) := tr[ψg(x)O].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In fact, the previous argument for Gibbs states can be extended to the present setting as long as the condition of exponentially decaying correlations in the Gibbs state is replaced by the following condition of local topological quantum order (LTQO) [MZ13, BHM10, NSY22]: A quantum system satisfies LTQO if for any two regions A ⊂ B ⊂ Λ and all x ∈ [−1, 1]m, �� trAc(ψg(x, B) − ψg(x, Λ)) �� 1 ≤ CT |A| e− dist(A,Bc) ξ0 (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6) for some constants CT , ξ0 > 0, and where, given a region R ⊂ Λ we denote by ψg(x, R) the ground state corresponding to the Hamiltonian HR(x) = � j∈R hj(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In words, LTQO states that observables localised away from the boundary of the volume B cannot distinguish between different ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Many systems of practical interest are known to satisfy Equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6), including frustration-free spin chains with a unique translation-invariant matrix product ground state [AKLT88] and quantum double models, which include Kitaev’s toric code [Kit06, Kit03, CDH+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For more details on LTQO, we refer to [NSY22] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6 (Learning algorithm for quantum ground states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' With the conditions of the pre- vious paragraph, given a set of N samples {xi, ˜ψ(xi)}N i=1, where ˜ψ(xi) can be stored efficiently classically, and N = O � log � M δ � log � n δ � epolylog(ε−1)� , there exists an algorithm that, on input x ∈ [−1, 1]m and a local observable O = �M i=1 Oi, produces an estimator ˆfO such that, with prob- ability (1 − δ), sup x∈[−1,1]m |fO(x) − ˆfO(x)| ≤ ε M � i=1 ∥Oi∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Moreover, the samples ˜ψ(xi) are efficiently generated from measurements of the ground states {ψg(xi)}N i=1 followed by classical post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' To prove this statement, we reduce it to the problem of learning Gibbs states of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The LTQO condition permits approximating the expectation of the local observable Oi in the state ψg(x) by the one in the state ψg(x, Si∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The latter is approximated by the local Gibbs state σ(β, x, Si∂) ∝ e−βHSi∂(x) for large but constant β (see Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' By a continuity argument, these states are approached by σ(β, Yit, Si∂), which in turn are close to ψg(Yit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This 9 chain of approximation steps together with a robust version of the shadow tomography protocol for ground states, stated in Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='7, allows us to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We expect that the assumption of LTQO is not the only assumption that can be made to achieve similar scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Indeed, we expect that a lower bound on the spectral gap in the parameterized region would achieve similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' COMPARISON TO PREVIOUS WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Classical literature The problem of Hamiltonian learning for classical models has attracted a lot of attention in the last years by the computer science community [Bre15, PSBR20, LVMC18, ZKKW20] which traditionally refers to it as Ising model — or Markov field — learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The question of what can be inferred from very few samples was also asked classically [DDDK20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our work sheds further light on this question and is of interest even when restricting to classicaapproximatingbservables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Indeed, to the best of our knowledge, the statements of Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4 and Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6 are new even for classical Gibbs distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Previous work by the authors of [RF21] already established similar learning results for measures satisfying a so-called transportation cost inequality (TC) [BG99, Tal96], although the present condition of exponential decay of correlations is more standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It should be noted that if a Gibbs measure satisfies TC, then any Lipschitz function of a random variable distributed according to it satisfies a Gaussian concentration bound [Led01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This can easily be seen to imply that we can estimate the expectation value of M Lipschitz functions up to an error ε with probability of success δ from O(ε−2 log(Mδ−1)) samples by taking the empirical average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' At first sight this might look comparable with the sample complexity we obtain with our learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, this only holds for one basis, whereas our result holds for any basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, if the number of Lipschitz observables satisfies M = ec Ω(n), then the number of samples required to obtain a good estimate through the empirical average becomes polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' On the other hand, given that W1(σ(β, x), σ(β, x′)) ≤ εn, we can evaluate as many Lipschitz observables as we wish from σ(β, x′) without requiring any further samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, even for observables in a fixed basis our result has advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Previous work on many-body quantum state tomography As mentioned before, one striking advantage of our Gibbs tomography algorithm when es- timating expectation values of local observables compared to state-agnostic methods like classical shadows is the exponential speedup in the size of the support of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In fact, our method gives good guarantees on the larger class of Lipschitz observables, which includes non-local ob- servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This advantage is even more visible when it comes to estimating entropic quantities: whereas the polynomial approximation proposed in [HKP20] works universally for any n-qubit state, it only gives good approximation guarantees for reduced states on very few qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here instead, we avoid this issue by leveraging the Wasserstein continuity bounds offered in [DPMTL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our framework also differs from the one of Hamiltonian learning algorithms tackled in [Ans, AAKS21, HKT21]: in these papers, the authors were interested in estimating the parameter x of a given Hamiltonian H(x) given access to copies of the state σ(β, x), in ℓ2 or ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here instead, we argue that a good recovery in W1 distance is implied by the weaker condition of recovery in ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Clearly, one can leverage these previous results to further control our ℓ1 bound, 10 as we argue in Section II A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It should be noted however that our bound only requires that the Gibbs state σ(β, x) satisfies an exponential decay of correlations, whereas these learning algorithms provide very efficient ℓ∞ or ℓ2 recovery either for (i) commuting Hamiltonians or (ii) in the high- temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It remains an important question whether the condition of exponential decay of correlations is enough to get good ℓ1 recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, in Appendix C 3 c we show that under the additional assumptions of uniform Markovianity and clustering of correlations, it is possible to learn in W1 through the maximum entropy method, without resorting directly to learning the parameters x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Previous work on learning observables in phases of matter In [HKT+22], the authors found a machine learning algorithm which, for any smoothly para- meterised family of local Hamiltonians {H(x)}x∈[−1,1]m in a finite spatial dimension with a constant spectral gap, can be trained to predict expected values of sums of local observables in the associated ground state ψg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' More precisely, given a local observable O = �M i=1 Oi with supp(Oi) = O(1), they construct an estimator ˆfO(x) of the expectation value of the observable such that Ex∼U([−1,1]m) ��� tr[Oψg(x)] − ˆfO(x) ��2� ≤ ε2� M � i=1 ∥Oi∥∞ �2 , (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) as long as the training size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' the number of sampled points within the phase) is N = � �M i=1 ∥Oi∥∞ �2 mO(1/ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6, we improve this result for ground states in three ways, up to further imposing the LTQO condition: first, we can assume that the parameters x are distributed according to a much larger class of distributions than the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This extension does not carry so easily in the proof of [HKT+22] which uses Fourier analysis techniques involving integration over the Lebesgue measure to derive Equation (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Second, theirs is a result in expectation, that is in ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='∥L2, whereas our bound in Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6 works in the worst-case setting associated to the stronger ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='∥∞-norm topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Third and most importantly, the dependence of the number of training data points scales exponentially in the precision parameter ε in Equation (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1), whereas ours scales only quasi-polynomially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Finally, we extend the learning result beyond ground states to finite temperature phases of matter with exponential decay of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This not only includes all high-temperature phases of matter (regardless of the Hamiltonian), but also low-temperature phases with the relevant correlation functions [DCGR19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This is a particularly relevant result since zero temperature is never achieved in practice, so in reality we are always working with low-temperature thermal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We also recognise independent, concurrent work by [LTL+23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here the authors consider the same setup of gapped ground states as [HKT+22] and also improved over Equation (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) to achieve the same sample complexity as Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, their result is not directly comparable to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We emphasise [LTL+23] consider gapped, ground state phases, whereas our work focuses on thermal phases and ground states with LTQO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We also note they remove all conditions on the prior distribution over the samples x, whereas we still need to assume a type of mild anti-concentration over the local marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, their result is still stated as an ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='∥L2-bound due to the use of machine learning machinery, whereas our more straightforward Gibbs approximation tools allow us to get stronger bounds in ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Conceptually speaking, our methods for approximating local 11 expectation values requires no knowledge of machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our work also shows that it is possible to go beyond gapped quantum phases and learn thermal phases with exponentially decaying correlations, as well as ground states with LTQO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS We have contributed to the tasks of tomography and learnability of quantum many-body states by combining previous techniques with approaches not considered so far in this field to obtain novel and powerful features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' First, we extended the results of [RF21] on the efficient tomography of high- temperature commuting Gibbs states to Gibbs states with exponentially decaying correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This result permits to significantly enlarge the class of states for which we know how to learn all quasi-local properties with a number of samples that scales polylogarithmically with the sys- tem’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, our results now also hold for classes of Gibbs states of non-commuting Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As we require exponentially fewer samples to learn in the Wasserstein metric when compared with the usual trace distance and still recover essentially all physically relevant quantit- ies associated to the states, we hope that our results motivate the community to consider various tomography problems in the Wasserstein instead of trace distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As we achieved this result by reducing the problem of learning the states to learning the para- meters of the Hamiltonian in ℓ1, we hope our work further motivates the study of the Hamiltonian learning problem in ℓ1-norm with polylog samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 1D Gibbs states are a natural place to start, but obtaining Hamiltonian learning algorithms just departing from exponential decay of correla- tions would provide us with a complete picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Appendix C 3 c we also partially decoupled the Hamiltonian learning problem from the W1 learning one by resorting to the uniform Markov condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, it would be important to establish the latter for a larger number of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It would be interesting to investigate the sharpness of our bounds, and to understand if expo- nential decay of correlations is really necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' One way of settling this question would be to prove polynomial lower bounds for learning in Wasserstein distance for states at critical temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Learning Phases of Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Second, we improved the results of [HKT+22] for learning a class of states in several directions, including the scaling in precision, the classes of states it applies to and the form of the recovery guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, the results now apply to Gibbs states, which are the states of matter commonly encountered experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Interestingly, we did not need to resort to machine learning techniques to achieve an exponentially better scaling in precision by making arguably mild assumptions on the distributions the states are drawn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Although the results proved here push the state-of-the-art of learning quantum states, we believe that our methods, for instance the novel continuity bounds for various local properties of quantum many-body states, will find applications in other areas of quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Beyond the thermal phases and LTQO ground states studied here, it would be interesting to find other families of states which can be efficiently learned, and indeed if more restrictive assumptions on the parameterization of Hamiltonians can result in more efficient learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' One interesting open problem that goes beyond the present paper’s scope is finding families of states satisfying LTQO without belonging to a common gapped phase of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' If such a family existed, it would clarify the differences between our framework and that of [HKT+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Finally, we realise that although the results proved here are for lattice systems, they almost certainly generalise to non-lattice configurations of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors gratefully recognise useful discussions with Hsin-Yuan (Robert) Huang and Haonan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We thank Laura Lewis, Viet T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Tran, Sebastian Lehner, Richard Kueng, Hsin- Yuan (Robert) Huang, and John Preskill for sharing a preliminary of the manuscript [LTL+23], which was discussed in Section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' EO is supported by the Munich Quantum Valley and the Bavarian state government, with funds from the Hightech Agenda Bayern Plus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' CR acknowledges financial support from a Ju- nior Researcher START Fellowship from the DFG cluster of excellence 2111 (Munich Center for Quantum Science and Technology), from the ANR project QTraj (ANR-20-CE40-0024-01) of the French National Research Agency (ANR), as well as from the Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' DSF is supported by France 2030 under the French National Research Agency award number “ANR-22- PNCQ-0002”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' JDW acknowledges support from the United States Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing program, and also NSF QLCI grant OMA-2120757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [AAKS21] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Sample- efficient learning of interacting quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Nature Physics, 17(8):931–935, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [ABF23] Guillaume Aubrun, Emily Beatty, and Daniel Stilck Fran¸ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In preparation, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [AKLT88] Ian Affleck, Tom Kennedy, Elliott H Lieb, and Hal Tasaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Valence bond ground states in isotropic quantum antiferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Condensed matter physics and exactly soluble models, pages 253–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Springer, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [Ans] Anurag Anshu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Efficient learning of commuting Hamiltonians on lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Electronic notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [Ara69] Huzihiro Araki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Gibbs states of a one dimensional quantum lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Communications in Mathem- atical Physics, 14(2):120–157, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [BCPH22] Andreas Bluhm, ´Angela Capel, and Antonio P´erez-Hern´andez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Exponential decay of mutual information for Gibbs states of local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Quantum, 6:650, February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [BFT17] Mario Berta, Omar Fawzi, and Marco Tomamichel.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS’20, Red Hook, NY, USA, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [QW20] Yang Qiu and Zhenghan Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Ground subspaces of topological phases of matter as error cor- recting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Annals of Physics, 422:168318, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [RD19] Cambyse Rouz´e and Nilanjana Datta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Concentration of quantum states from quantum functional and transportation cost inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Journal of Mathematical Physics, 60(1):012202, January 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [RF21] Cambyse Rouz´e and Daniel Stilck Fran¸ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Learning quantum many-body systems from a few copies.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Talagrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Transportation cost for gaussian and other product measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Geometric and Functional Analysis, 6(3):587–600, May 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [Tas18] Hal Tasaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Topological phase transition and Z2 index for s = 1 quantum spin chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Physical Review Letters, 121(14):140604, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' [ZKKW20] Huanyu Zhang, Gautam Kamath, Janardhan Kulkarni, and Steven Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Privately learning Markov random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Hal Daum´e III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 11129–11140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' PMLR, 13–18 Jul 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 16 SUPPLEMENTAL MATERIAL Appendix A: Preliminaries Given a finite dimensional Hilbert space H, we denote by B(H) the algebra of bounded operators on H, whereas Bsa(H) denotes the subspace of self-adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We denote by D(H) the set of positive operators on H of unit trace, and by D+(H) the subset of positive, full-rank operators on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Schatten norms are denoted by ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='∥p for p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The identity matrix in B(H) is denoted by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Given a bipartite system AB, the normalised partial trace over a subsystem A is written τA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' τA := 2−|A| trA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In this work, we consider a family of local qubit interactions {hj(xj)}xj∈[−1,1]ℓ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' , n over the D-dimensional lattice Λ = [−L, L]D, for some fixed integer ℓ, where n = (2L+1)D denotes the total number of qubits constituting the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For each j and all xj ∈ [−1, 1]ℓ, hj(xj) is supported on a ball Aj around site j ∈ Λ of radius r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We also assume that the matrix-valued functions xj �→ hj(xj) as well as their derivatives are uniformly bounded: ∥hj∥∞, ∥∇xhj(x)∥∞ ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For sake of simplicity, we assume that the interactions are linear functions of their parameters, that is hj(xj) = xjVj for some fixed operator Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However this assumption is not necessary in any of our proofs, as commented in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Concatenating the vectors xj into x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' , xn) = (x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' , x′ m), m = nℓ, the local interactions induce the following family of Hamiltonians {H(x)}x∈[−1,1]m, with: H(x) = m � j=1 hj(xj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) More generally, given a region B ⊂ Λ of the lattice, we denote by HB(x) := � j|Aj⊂B hj(x) the Hamiltonian restricted to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We denote by x|S(r) the concatenation of vectors xj corresponding to interactions hj supported on regions intersecting S(r) := {l ∈ Λ| dist(l, S) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For much of the following, we will be concerned with Gibbs states, defined as σ(β, x) := e−βH(x) tr[e−βH(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, we will be interested in systems satisfying the following type of correlation decay: Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1 (Exponential Decay of Correlations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For a state σ and any operator XA, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' XB, supported on region A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' B, we say the state satisfies exponential decay of correla- tions if Covσ(XA, XB) ≤ C min{|A|, |B|} ∥XA∥∞ ∥XB∥∞ e−ν dist(A,B) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2) for any choice of XA,XB, and for some parameters C, ν > 0 which we assume independent of x and of the lattice size n, and where Covσ(A, B) := 1 2 tr � σ � A − tr[σA], B − tr[σB] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1 is satisfied by many classes of Gibbs states, including high-temperature Gibbs states [HMS20, KKBa20] and 1D Gibbs states at any constant temperature [HMS20, BCPH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It is also known to hold for ground states of gapped Hamiltonians [HK06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In fact, the class of Gibbs 17 states for which Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1 holds is larger than that for which polylog algorithms to learn the parameters of the Hamiltonian are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Appendix C 3 we will discuss several examples for which it is known how to learn the parameters efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Appendix C 3 c we will also consider the case when we have the additional assumption of uniform Markovianity to show that then it is possible to bypass having to learn the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Appendix B: Lipschitz observables In this appendix, we argue that Lipschitz observables and the induced Wasserstein distance capture most observables of physical interest, such as local and quasi-local observables, as well as quasi-local polynomials of the state and entropic quantities of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' They can even capture global properties, including some of physical interest like global entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' These classes of examples justify the claim that Lipschitz observables and the Wasserstein distance capture well both linear and nonlinear extensive properties of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Let us illustrate our previous claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' An important class of Lipschitz observables are those of the form M � i=1 Oi, M = O(n), ∥Oi∥ = O(1), max 1≤j≤n |{i : supp(Oi) ∩ {j} ̸= ∅}| = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) Observables like those defined in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) include local observables w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' to a regular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, it is also not difficult to see that the expectation values of such observables are characterised by the marginals of the states on a few qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' But Lipschitz observables capture more than strictly local properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Indeed, as shown in [RF21], the time evolution of local observ- ables like those in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) by a shallow quantum circuit or a short continuous-time evolution satisfying a Lieb-Robinson bound are Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' These include evolutions by Hamiltonians with algebraically decaying interactions, which will map strictly local Hamiltonians to quasi-local ob- servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In fact, recent results [ABF23] show that Lipschitz observables can distinguish two random quantum states almost optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As such states are locally indistinguishable [BHH16, Corollary 15], this fact shows that Lipschitz observables capture much more than just quasi-local properties of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Although so far we only discussed how to use the Wasserstein distance to control linear func- tionals of the state, the fact that the Wasserstein distance behaves well under tensor products means that it is also easy to control the error for non-linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Indeed, in [DPMTL21, Propostion 4], the authors show that the Wasserstein distance is additive under tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' for all states ρ, σ and integer k we have W1(ρ⊗k, σ⊗k) = kW1(ρ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2) We can then combine this additivity with the standard trick that a polynomial of degree k on a quantum state can be expressed as the expectation value of a certain observable O on ρ⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, if this polynomial is an average over polynomials in reduced density matrices of constant size, it is not difficult to see that the corresponding observable on ρ⊗k will be Lipschitz as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Let us exemplify this in the case of the average purity of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For a subset A ⊂ [n] of the qubits of size l, let FA ∈ � C2�⊗2l be the flip operator acting on two copies of those qubits: FA(|ψ⟩ ⊗ |ϕ⟩) = |ϕ⟩ ⊗ |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3) 18 It can be shown in a few lines that tr � FAρ⊗2� = tr � ρ2 A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, observables of the form O = M � i=1 FAi, M = O(n), max 1≤j≤n |{i : Ai ∩ {j} ̸= ∅}| = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4) satisfy ∥O∥Lip = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then M � i=1 tr � ρ2 Ai − σ2 Ai � = tr � O(ρ⊗2 − σ⊗2) � ≤ ∥O∥LipW1(ρ⊗2, σ⊗2) = 2∥O∥LipW1(ρ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5) By a direct generalisation of the above, we see that W1(ρ, σ) = O(εn/k) is sufficient to ensure that degree k polynomials of the states are approximated to a multiplicative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As we will see later in Section II A 3, this polynomial trick can be used to ensure that averages of various subsystem entropies, mutual informations and conditional mutual informations are well-approximated given a Wasserstein bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Once again it should be emphasised that a Wasserstein bound can be used to control global properties, even non-linear ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A good example of that is the entropy of a quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In [DPMTL21, Theorem 1], the authors show the continuity bound: |S(ρ) − S(σ)| ≤ g(W1(ρ, σ)) + W1(ρ, σ) log(4n), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6) where g(t) = (t + 1) log(t + 1) − t log(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In this case, it turns out that a Wasserstein distance of W1(ρ, σ) = O(εn/ log(n)) suffices to obtain a multiplicative error for the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Finally, it is also worth mentioning observables that are not Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Simple examples include linear combinations of high-weight Paulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Appendix C: Gibbs states tomography In this section, our main goal is to devise an efficient tomography algorithm for Gibbs states σ(β, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, we wish to learn the parameters x to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We prove the following lemma: Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1 (Tomography algorithm for decaying Gibbs states ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Let H(x) = � i hi(xi) be a Hamiltonian such that each hi(xi), xi ∈ [−1, 1]ℓ, is not more than k-local, for k = O(1), and all terms commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For some unknown x, let σ(β, x) be its associated Gibbs state satisfying exponential decay of correlations as per Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then there exists an algorithm that provides the description of parameters x′ such that the state σ(β, x′) satisfies: W1(σ(β, x), σ(β, x′)) ≤ εn (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) with probability greater than 1 − δ, such that the algorithm requires access to no more than N = O � log(δ−1) polylog(n) ε−2� samples of the state (see Appendix C 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The result extends to the case where {hi(xi)}i do not commute whenever one of the following two assumptions is satisfied: (i) the high-temperature regime, β < βc (see Appendix C 3 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 19 (ii) uniform clustering/Markov conditions (see Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In case (ii), we find good approximation guarantees under the following slightly worst scaling in the precision ε: N = O(ε−4 polylog(nδ−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Proof Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The full proof is laid out in sections C 1, C 2 and C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The fundamental part of the result uses the continuity estimate of the Wasserstein distance between two Gibbs states that is of interest on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4 we will show that under exponential decay of correlations we have: W1(σ(β, x), σ(β, y)) ≤ ∥x − y∥ℓ1 polylog(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2) The significance of the bound in Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2) is that it reduces the problem of obtaining a good estimate of σ(β, x) in W1 to estimating the parameters x in ℓ1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This is a variation of the Hamiltonian learning problem [AAKS21, GCC22, HKT21], and we can then directly import results from the literature for our tomography algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As we argued before in Section II A 1, the recovery guarantee in Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1) suffices to ensure that σ(β, x′) mirrors all the quasi-local properties of σ(β, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, the polylog complexity in system size is exponentially better than what is required to obtain a recovery guarantee in trace distance [RF21, Appendix G], even for product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Quantum belief propagation We start by recalling a well-known tool in the analysis of quantum Gibbs states known as quantum belief propagation [Has07, Kim17, KB19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We assume a parameterisation of the Hamilto- nian as H(x) = �m j=1 xjVj for appropriate operators Vj (we will generalise this to other paramet- erisation later) and for some observable L we define the function fL(β, x) = tr [σ(β, x)L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The belief propagation method then states that we have that for any k ∈ [m], ∂x′ kfL(β, x) = −β 2 tr � L � ΦH(x)(∂x′ kH(x)), σ(β, x) �� + β tr(∂x′ kH(x)σ(β, x)) tr(Lσ(β, x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' where the quantum belief propagation operator ΦH(x) is defined as ΦH(x)(V ) := � ∞ −∞ dt κβ(t) e−iH(x)tV eiH(x)t , for some smooth, fast-decaying probability density function κβ(t) := 1 2π � �κβ(ω)eiωtdω of Fourier transform �κβ(ω) := tanh(βω/2) βω/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The function κβ was in fact computed in [AAKS21, Appendix B]: for t ∈ R\\{0}: κβ(t) := 2 πβ log eπ|t|/β + 1 eπ|t|/β − 1 ≤ 4 πβ 1 eπ|t|/β − 1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3) 20 Rewriting the above derivative, and using the notations ⟨O⟩β,x ≡ tr(σ(β, x)O) for the expected value of an observable O in the Gibbs state σ(β, x), we have that ∂x′ kfL(β, x) = −β 2 ⟨ � L, �Hk(x) − ⟨ �Hk(x)⟩β,x � ⟩β,x (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4) where �Hk(x) := ΦH(x)(∂x′ kH(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We define the covariance between two observables A and B in the state σ as Covσ(A, B) := 1 2 tr � σ � A − tr[σA], B − tr[σB] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Therefore ∂x′ kfL(β, x) = −β Covσ(β,x) (L, �Hk(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5) In what follows, we will need to approximate �Hk(x) by observables supported on bounded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For this, we make use of Lieb-Robinson bounds for Hamiltonians of finite-range interactions [LR72, Pou10, KGE14, BHV06, Has10, Sid09, CLMPG15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here we choose a version proven in [CLMPG15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5]: for any observable OA supported on a region A of the lattice, and any B ⊃ A, we denote by αt, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' by αB t , the unitary evolution generated by H(x), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' by HB(x), up to time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' αt(O) := e−iH(x)tOeiH(x)t , αB t (O) := e−iHB(x)tOeiHB(x)t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' which then satisfy ∥αt(OA) − αB t (OA)∥∞ ≤ c |A| ∥OA∥∞ evt−µ dist(A,Bc) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6) for some parameters c, v, µ > 0 which depend on the interactions hj but can be chosen independent of n and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For any region A ⊂ B ⊂ Λ and operator OA supported in A and all x, ∥ΦH(x)(OA) − ΦHB(x)(OA)∥∞ ≤ c′ |A| ∥OA∥∞ e−µ′ dist(A,Bc) for some parameters c′ and µ′ depending on H(x) and β but independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We make use of the exponential decay of κβ provided in Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3) together with the Lieb-Robinson bound Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6): ∥ΦH(x)(OA) − ΦHB(x)(OA)∥∞ ≤ � ∞ −∞ |κβ(t)| ∥αt(OA) − αB t (OA)∥∞ dt ≤ c |A| ∥OA∥∞e−µ dist(A,Bc) � δ −δ |κβ(t)| evt dt + 2 ∥OA∥∞ � [−δ,δ]c |κβ(t)| dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For the first integral above, we use that |κβ(t)| ∝ log(1/t) for t small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' More precisely, � δ −δ |κβ(t)| evt dt ≤ 4evδ πβ � δ 0 log � eπt/β + 1 tπ/β � dt ≤ 4e(v+π/β)δ π2 21 For the other integral, we use the exponential decay of κβ: � [−δ,δ]c |κβ(t)| dt ≤ 8 πβ � ∞ δ 1 eπt/β − 1 dt ≤ 8 πβ � ∞ δ e− πt 2β dt = 16 π2 e− πδ 2β , where the second inequality holds for δ ≥ 2β π sh−1 � 1 2 � ≡ δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Choosing δ := δ1+µ dist(A, Bc)/(2 � v+ π/β � ), we get ∥ΦH(x)(OA) − ΦHB(x)(OA)∥∞ ≤ c′ |A|∥OA∥∞ e−µ′d(A,Bc) , for some constant c′ ≡ c′(β, v), where µ′ = µ min � 1 2, π 4β(v+π/β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Continuity estimate for W1 distance on Gibbs states In this subsection, we will prove Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' First, we use the bound derived in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2 together with the assumption that σ(β, x) has exponential decay of correlations in order to control the derivatives ∂x′ kfL: Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Assume that σ(β, x) satisfies the condition of decay of correlations, Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then for any k ∈ [m], |∂x′ kfL(β, x)| ≤ ∥L∥Lip polylog(n) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='7) for some polynomial of log(n) of degree D with coefficients depending on β, r0, D, h, c′, ν, µ′ and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Denoting by jk the index of the interaction hjk which depends on variable x′ k, we have that, given ΦH(x)(∂x′ khj) = δj,jkΦH(x)(∂x′ khjk), and denoting �hk = ΦH(x)(∂x′ khjk), from Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5) we have: |∂x′ kfL(β, x)| = β Covσ(β,x)(L, �Hk(x)) = β Covσ(β,x)(L, �hk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Next, given a region Bk ⊃ Ajk, define the observable OBk := ΦHBk(x)(∂x′ khjk) − ⟨ΦHBk(x)(∂x′ khjk)⟩β,x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='8) Then by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2 we have that Covσ(β,x)(L, �hk(x)) = Covσ(β,x)(L, �hk(x) − OBk) + Covσ(β,x)(L, OBk) ≤ 2∥L∥∞ ∥ΦH(x)(∂x′ khjk) − ΦHBk(x)(∂x′ khjk)∥∞ + Covσ(β,x)(L, OBk) ≤ 2nc′(2r0)D h ∥L∥Lip e−µ′ dist(Ajk,Bc k) + Covσ(β,x)(L, OBk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Next, we estimate the last covariance above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Denoting Bk(r) := {i ∈ Λ : dist(i, Bk) ≤ r}, we get Covσ(β,x)(L, OBk) = Covσ(β,x)(L − τBk(r)(L), OBk) + Covσ(β,x)(τBk(r)(L), OBk) ≤ 2h∥L − τBk(r)(L)∥∞ + 2C|Bk| h∥L∥∞ e−νr ≤ 2h|Bk(r)| ∥L∥Lip + 2C|Bk| h n∥L∥Lip e−νr , 22 where the second line above follows from the condition of decay of correlations Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Choosing Bk = Ajk(⌊log(n)/µ′⌋), so that dist(Ajk, Bc k) = ⌊log(n)/µ′⌋, and r = ⌊log(n)/ν⌋, we have shown that, given 1/ν′ := 1/µ′ + 1/ν, |∂x′ kfL(β, x)| ≤ 2β h ∥L∥Lip � c′(2r0)D h + (2(r0 + log(n)/ν′))D(1 + C) � The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' With the bound of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3, we show that for Gibbs states belonging to a phase with exponentially decaying correlations, the difference of expected values of Lipschitz observables in two such states is controlled by the ℓ1-norm of their associated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' With the conditions of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3, for any x, y ∈ [−1, 1]m, W1(σ(β, x), σ(β, y)) ≤ ∥x − y∥ℓ1 polylog(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='9) Furthermore, this inequality is tight up to a polylog(n) factor for β = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' To get the upper bound Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='9), it suffices to interpolate between the two states as follows: for any Lipschitz observable L, and a path x(s) = (1 − s)x + sy, | tr [L(σ(β, x) − σ(β, y))] | ≤ m � k=1 |x′ k − y′ k| � 1 0 |∂kfL(β, x)| ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The result follows from using Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='7) above, and using the resulting inequality in the definition of Wasserstein distance, definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' To see that the inequality is tight up to the polylog(n) factor, consider the family of Hamilto- nians H(x) = � i xiZi, which gives rise to diagonal, product Gibbs states that clearly satisfy exponential decay of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We then have: W1(σ(β, x), σ(β, y)) ≥ 1 2 tr �� i Zi(σ(β, x) − σ(β, y)) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='10) as � i Zi has Lipschitz constant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A simple computation shows that: 1 2 tr �� i Zi(σ(β, x) − σ(β, y)) � = 1 2 � i � e−βxi e−βxi + e+βxi − e−βyi e−βyi + e+βyi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='11) We will assume without loss of generality that xi < yi (as otherwise we can consider the observable with −Zi instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Under this condition, the summands are all positive and thus: 1 2 tr �� i Ziσ(β, x) − σ(β, y)) � = 1 2 � i ���� e−βxi e−βxi + e+βxi − e−βyi e−βyi + e+βyi ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='12) Yet another simple computation shows that the derivative of the function y �→ e−βy e−βy+e+βy is given by −β 2 sech(βy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='13) 23 Let cβ denote the minimum of the function in Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='13) for a fixed β = Θ(1) over y ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then, by the mean value theorem: 1 2 � i ���� e−βxi e−βxi + e+βxi − e−βyi e−βyi + e+βyi ���� ≥ cβ 2 � i |xi − yi| , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='14) from which we conclude that: W1(σ(β, x), σ(β, y)) ≥ cβ 2 ∥x − y∥ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='15) We next prove that when given a local observable O supported on a ball S ⊂ Λ of diameter at most k0 around site i of the lattice, to study its behaviour as H(x) varies for Gibbs states, it is sufficient to only consider the components of x which parameterise local terms which are geometrically close to the observable O (up to some small error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Before we prove this, we remember that we denote by x|S(r) the concatenation of vectors xj corresponding to interactions hj supported on regions intersecting S(r) := {i ∈ Λ| dist(i, S) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5 (Gibbs local indistinguishability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Assuming the exponential decay of correlations in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1, then for any observable O supported on region S, any r ∈ N, denoting fO(x) := tr[O σ(β, x)] and identify x|S(r) with the vector (x|S(r), 0S(r)c) ∈ [−1, 1]m, then the following bound holds: sup x∈[−1,1]m |fO(x) − fO(x|S(r))| ≤ C1 e− r 2ξ ∥O∥∞ , for O(1) constants C1, ξ > 0 independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In other words: sup x∈[−1,1]m ∥ trSc(σ(β, x) − σ(β, x|S(r)))∥1 ≤ C1 e− r 2ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We identify x|S(r) with the vector (x|S(r), 0S(r)c) ∈ [−1, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Given the path x(s) = (1 − s)x + sx|S(r) with components {x′ l(s)}m l=1, we get |fO(x) − fO(x|S(r))| ≤ � l∈S(r)c |x′ l(0)| � 1 0 ��∂l tr � Oσ(β, x(s)) ��� ds (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='17) = β � l∈S(r)c |x′ l(0)| � 1 0 �� Covσ(β,x(s)) � O, �Hl(x(s)) � | ds , for �Hl(x) := ΦH(x)(∂lH(x)), where the second line comes from Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Next, we call jl ∈ Λ the unique site such that x′ l is a coordinate of xjl, and denote Ajl be the support of hjl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Now, the above covariance is small if r is large enough, since �Hj(x(s)) can be well approximated by an observable on Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Indeed, ∂lH(x) = ∂lhjℓ , 24 where jl denotes the index of interaction hjl which depends on variable x′ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Therefore, whenever Ajl ∩ S = ∅, we proceed similarly to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='3: given a region Bl ⊃ Ajl such that Bl ∩ S = ∅, denoting the observable OBl := ΦHBl(x)(∂x′ lhjl) − ⟨ΦHBl(x)(∂x′ lhjl)⟩β,x , we have by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='2 as well as the assumption that the state σ(β, x) has exponential decay of correlations we have the following (refer to Figure 1 for a diagram of the regions): Covσ(β,x)(O, �Hl(x)) = Covσ(β,x)(O, �Hl(x) − OBl) + Covσ(β,x)(O, OBl) ≤ 2∥O∥∞ ∥ΦH(x)(∂x′ lhjl) − ΦHBl(x)(∂x′ lhjl)∥∞ + Covσ(β,x)(O, OBl) ≤ 2∥O∥∞c′|Ajl| h e−µ′ dist(Ajl,Bc l ) + 2C|S| ∥O∥∞ h e−ν dist(S,Bl) ≤ 2(C + c′) ∥O∥∞ (2r0 + k0)D h � e−µ′ dist(Ajl,Bc l ) + e−ν dist(S,Bl)� Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Diagram showing the regions involved in the proof Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' By construction, for r > 2r0, the condition that Ajl ∩ S = ∅ is met, and therefore the bound holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We recall that i ∈ Λ is defined as the center of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Since dist(i, jl) = k0/2 + dist(S, Bl) + dist(Ajl, Bl)+r0, we can choose Bl so that dist(S, Bl), dist(Ajl, Bl) ≥ dist(i, jl)/2−k0/4−r0/2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Covσ(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='x)(O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' �Hl(x)) ≤ 4(C + c′)C′′∥O∥∞(2r0 + k0)D he− dist(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='jl)/ξ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Be S(r) 0 0 6 0 0 0 0 0 0 0 0 0 0 dist(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Aje) 0 6 0 0 0 0 0 0 0 0 0 0 0 r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 s Aje dist(Be,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' S) Q 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 025 where 1/ξ = min{µ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' ν} and C′′ := emax{µ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='ν}(k0/4+r0/2+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Therefore |fO(x) − fO(x|Si(r))| ≤ 4β(C + c′) h (2r0 + k0)D ∥O∥∞ � l∈S(r)c e− dist(i,jl)/ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Upon shifting the center of the lattice at site i, we get |fO(x) − fO(x|S(r))| ≤ 4β(C + c′)C′′ h (2r0 + k0)D ∥O∥∞ � |l|≥r+k0/2 e−|l|/ξ = 4β(C + c′)C′′ h (2r0 + k0)D ∥O∥∞ � a>r+k0/2 �a + D − 1 D − 1 � e−a/ξ ≤ 4β(C + c′)C′′ h (2r0 + k0)D DD−1∥O∥∞ � a>r+k0/2 aD−1 e−a/ξ ≤ 4β(C + c′)C′′ h (2r0 + k0)D(D − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (2ξ)D−1 DD−1∥O∥∞ � a>r+k0/2 e− a 2ξ ≤ 4β(C + c′)C′′ h (2r0 + k0)D(D − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (2ξ)D−1 DD−1∥O∥∞ e− r+k0/2+1 2ξ 1 − e− 1 2ξ ≡ C1 e− r 2ξ ∥O∥∞ , where C1 depends upon all the parameters of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In the case when we are interested in distinguishing two Gibbs states with Lipschitz observables, over extended subregions of the lattice, the following extension of Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4 can be easily shown to hold: Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Assume that the states σ(β, x) satisfy the condition of decay of correlations Con- dition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then for any region S of the lattice and any two x, y ∈ [−1, 1]m W1(trSc(σ(β, x)), trSc(σ(β, y))) ≤ ∥x|S(r) − y|S(r)∥ℓ1 polylog(|S(r)|) , where r = max � r0, 2ξ log � 2|S|C1 ∥x|S(r0)−y|S(r0)∥ℓ1 �� with r0 being the smallest integer such that x|S(r0) ̸= y|S(r0), and C1, ξ are the same constants as in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Given LS a Lipchitz observable supported on region S of the lattice, we have for any r ∈ N: ��fLS(x) − fLS(y) �� ≤ ∥LS∥∞ ��� trSc � σ(β, x) − σ(β, x|S(r)) ��� 1 + �� trSc � σ(β, y) − σ(β, y|S(r)) ��� 1 � + W1 � σ(β, x|S(r)), σ(β, y|S(r)) � ≤ 2 |S| ∥LS∥Lip C1 e− r 2ξ + W1 � σ(β, x|S(r)), σ(β, y|S(r)) � , where the second line follows from Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' By Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4, we conclude that W1 � trSc(σ(β, x)), trSc(σ(β, y)) � ≤ 2 |S| C1 e− r 2ξ + W1 � σ(β, x|S(r)), σ(β, y|S(r)) � ≤ 2 |S| C1 e− r 2ξ + ∥x|S(r) − y|S(r)∥ℓ1 polylog(|S(r)|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Next, we choose r = 2ξ log � 2|S|C1 ∥x|S(r0)−y|S(r0)∥ℓ1 � , where r0 is the smallest integer such that x|S(r0) ̸= y|S(r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Hamiltonian estimation and optimal Gibbs state tomography From Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='4 it is immediate that we reduced the problem of obtaining a good estimate in W1 to the problem of estimating the parameters of the Gibbs state σ(β, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Indeed, it is clear that if we can obtain an estimate x′ of x satisfying ∥x − x′∥ℓ1 = O(εn/polylog(n)), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='18) then it suffices to ensure that W1(σ(β, x), σ(β, x′)) = εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Let us discuss some examples where we can obtain this efficiently with O(ε−2polylog(n)) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Commuting Hamiltonians In [Ans], the authors give an algorithm which with eO(βkD)O(log(δ−1n)ε−2) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='19) copies of σ(β, x) learns x up to ε in ℓ∞ distance when σ(β, x) belongs to a family of commuting, k-local Hamiltonians on a D-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As we assumed that the number of parameters m = O(n), this translates to an algorithm with sample complexity eO(βkD)O(ε−2polylog(δ−1n)) to learn x up to εn in ℓ1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' It should be noted that the time complexity of their algorithm is O(neO(βkD)ε−2polylog(δ−1n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, any commuting model at constant temperature satisfying exponential decay of correla- tions can be efficiently learned with polylog(n) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Examples of classes of commuting states that satisfy exponential decay of correlations include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 1D translation-invariant Hamiltonians at any positive temperature [Ara69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Commuting Gibbs states of Hamiltonians on regular lattices below a threshold temperat- ure [KGK+14, HMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Classical spin models away from criticality [DS87, LSS19, HMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Ground states of uniformly gapped systems [BHM10, BH11, MZ13, NSY22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' High-temperature Gibbs states Another class of states for which the conditions of our results hold are local Gibbs states on a lattice above a threshold temperature that depends on the locality of the Hamiltonian and the dimension of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' These systems are known to have exponential decay of correla- tions [KGK+14, HMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Furthermore, in [HKT21] the authors give an algorithm to learn x up to error ε in ℓ∞ norm from O(ε−2polylog(δ−1n)) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This again translates to a O(εn) error in ℓ1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Note that their algorithm also is computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We note that in [AAKS21] the authors give an algorithm to learn the Hamiltonian of any Gibbs state of positive temperature through the maximum entropy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, their results require a polynomial number of samples to recover the parameters in ℓ1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, their results do not work for the polylog regime investigated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 27 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Gibbs state of exponentially decaying correlations and conditional mutual information In the previous section, we extracted two regimes for which there exist efficient Gibbs tomo- graphy algorithms from previous works, namely the commuting and the high-temperature regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As said before, depending on the Hamiltonian, exponential decay of correlations can also occur in the low-temperature regime, and it is an interesting open question whether our strategy can be adapted to that setting for non-commuting interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here, we show that the Gibbs state σ(β, x) of a possibly non-commuting Hamiltonian H(x) can also be estimated in Wasserstein distance up to multiplicative error εn given polylog(n) copies of it as long as the latter has exponentially decaying correlations and is close to a quantum Markov chain, hence partially answering an open problem previously raised in [AAKS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' To be more precise, in this section we will require a stronger notion of decay of correlations: Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='7 (Uniform clustering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The Gibbs state σ(β, x) is said to be uniformly ζ(ℓ)- clustering if for any X ⊂ Λ and any A ⊂ X and B ⊂ X such that dist(A, B) ≥ ℓ, Covσ(β,x,X)(XA, XB) ≤ ∥XA∥∞ ∥XB∥∞ ζ(ℓ) for any XA supported on A and XB supported on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As pointed out in [BK18], this property is called uniform clustering to contrast with regular clustering property that usually only refers to properties of the state σ(β, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='8 (Uniform Markov condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The Gibbs state σβ(x) is said to satisfy the uniform δ(ℓ)-Markov condition if for any ABC = X ⊂ Λ with B shielding A away from C and such that dist(i, j) ≥ ℓ for any i ∈ A and j ∈ C, we have I(A : C|B)σ(β,x,X) ≤ δ(ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This property always holds for commuting Gibbs states for a function δ(ℓ) = 0 as soon as ℓ is larger than twice the interaction range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Although not proven yet, it is believed that the approximate Markov property holds with some generality for non-commuting Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The 1D and high-temperature settings were investigated in [KB19] and [KKBa20], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The decay of the conditional mutual information was also shown for finite temperature Gibbs states of free fermions, free bosons, conformal field theories, and holographic models [SM16], as well as more recently for purely generated finitely correlated states in [SK22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We will now show how to learn states that satisfy both the uniform Markov condition and the uniform clustering of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Our strategy consists in using the maximum entropy estimation [Jay57b, Jay57a, Jay82, BKL+17], already appearing in [AAKS21], to construct an estimator ˆx of the parameter x ∈ [−1, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The condition of exponential decay of correlations and that of approximate Markov chain will ensure that W1(σ(β, ˆx), σ(β, x)) = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Thus, we once again emphasise that our goal is to obtain a good recovery of the state, not of the parameter x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' For sake of clarity and simplicity of presentation, we only consider the 1D setting, although our method easily extends to arbitrary dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We assume that each interaction hj(xj) is of the form hj(xj) := ℓ � l=1 xj,l hj,l 28 for some self-adjoint operators hj,l supported in Aj := {k ∈ Λ| dist(k, j) ≤ r0} with ∥hj,l∥ ≤ h, where we denoted by xj,l the entries of xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We also recall that given a region R of the lattice, we denote HR(x) := � k∈R hk(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In what follows, with a slight abuse of notations, we denote by the same symbol a vector y = {yk,l}k∈Nj and its embedding (y, 0) onto [−1, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then, given an inverse temperature β > 0, we define the partition function as Zβ(x) = tr � e−βH(x)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The maximum entropy problem consists in the following strongly convex optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='9 ([AAKS21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Given an unknown Hamiltonian H(x) = � hj(xj), define ek,l = tr[hk,ℓσ(β, x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Solving the following optimisation problem: ˆx := arg min y∈[−1,1]m L(y) , where L(y) := log Zβ(y) + β � k∈Λ ℓ � l=1 yk,l ek,l (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='20) gives ˆx such that σ(β, ˆx) = σ(β, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In an experimental setting, we will not have access to the exact {ek,l}k,l, but instead may be able to approximate them using by having access to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' However, we want to be sure that having a reasonably good approximation to ek,l is sufficient to approximate x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' To do so one can make use of the fact that log Zβ(ˆx) ≤ log Zβ(x) + β � k∈Λ ℓ � l=1 (xk,l − ˆxk,l) �ek,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='21) Further assuming α2 is a lower bound on the strong convexity constant associated to the function x �→ log Zβ(x), that is ∇2Zβ ≥ α2 I, we have by Taylor expansion and since ∂xk,l log Zβ(x) = −βek,l(x): log Zβ(ˆx) ≥ log Zβ(x) − β � k∈Λ ℓ � l=1 (ˆxk,l − xk,l) ek,l(x) + α2 2 ∥x − ˆx∥2 ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='22) Combining the two bounds above, we find that ∥x − ˆx∥2 ℓ2 ≤ 2β α2 � k,l (xk,l − ˆxk,l)(�ek,l − ek,l(x)) ≤ 2β α2 ∥x − ˆx∥ℓ2 ∥e − �e∥ℓ2 , and hence ∥x − ˆx∥ℓ2 ≤ 2β√ ℓ|Λ| η α2 , thus giving the following theorem: Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='10 ([AAKS21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Suppose �ek,l is an approximation of ek,l(x) := tr � hk,l σ(β, x) � with ∥�e − e(x)∥ℓ∞ ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Assume that the following inequality is satisfied for some α2: ∇2Zβ ≥ αI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Solving the following optimisation problem: ˆx := arg min y∈[−1,1]m L(y) , where L(y) := log Zβ(y) + β � k∈Λ ℓ � l=1 yk,l �ek,l (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='23) gives an output ˆx satisfying: ∥ˆx − x∥ℓ2 ≤ 2βη √ ℓΛ α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' 29 Using the bound on ˆx from Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='10 the equivalence between ℓ1 and ℓ2-norms, we have that ∥x − ˆx∥ℓ1 ≤ 2βℓnη α2 , which provides us with the right scaling for our ℓ1 approximation problem as long as η = o(1) and α2 = Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Unfortunately, the constant α2 could only be proved to scale inverse polynomially with n in [AAKS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' A first idea from there is to try and find a constant α1 = Ω(n−1) such that the following strong convexity bound with respect to the ℓ1-norm holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='22), this would imply: log Zβ(ˆx) ≥ log Zβ(x) − β � k,l (ˆxk,l − xk,l) ek,l(x) + α1 2 ∥x − ˆx∥2 ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='24) If such a bound held, we would conclude similarly to the previous setting that ∥x − ˆx∥ℓ1 ≤ 2βη α1 = o(ηn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Which together with the continuity bound Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='9) would allow us to get the desired recovery estimate in Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Now, it can be seen that Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='24) is equivalent to ∥x − ˆx∥2 ℓ1 ≤ 2 α1 D(σ(β, x)∥σ(β, ˆx)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='25) Here we recall that the relative entropy between two quantum states ρ and σ with supp(ρ) ⊆ supp(σ) is D(ρ∥σ) := tr ρ log ρ − tr ρ log σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' This together with Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='9) would lead to the following local version of the transportation cost inequality W1(σ(β, x), σ(β, ˆx))2 ≤ O(n polylog(n)) D(σ(β, x)∥σ(β, ˆx)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='26) In [PR22], such inequality was shown to hold in the high-temperature regime only for commuting H, albeit when σ(β, x) can be replaced by an arbitrary state ρ on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The latter is referred to as a transportation-cost inequality for the state σ(β, ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Since Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='24) consists in a strengthening of Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='26), proving it directly appears difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Here instead, we want to show the following weakening of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='26): W1(σ(β, x), σ(β, ˆx))2 ≤ O(n polylog(n)) D(σ(β, x)∥σ(β, ˆx)) + o(εn) , for some constant δ which depends on the approximate Markov as well as the correlation decay properties of the Gibbs state σ(β, ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' More precisely, we show the following extension of [PR22, Theorem 4] to Gibbs states of non-commuting Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='11 (Generalised transportation-cost inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' With the notations of the above paragraph, for all states ρ: W1(ρ, σ(β, x)) ≤ inf ℓ∈N O(ℓ√n) � D(ρ∥σ(β, x)) + n2� δ(O(ℓ)) + ζ(O(ℓ)) + e−O(ℓ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In particular, if both ζ(l), δ(l) = O(e−ξl), then for l = O(ξ−1 log(nε−1)) we have W1(ρ, σ(β, x)) ≤ O(log(nε−1)√n) � D(ρ∥σ(β, x)) + o(εn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='27) 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The proof is adapted from that of [PR22, Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We first consider a bipartite quantum subsystem AB ⊂ Λ and a joint quantum state ωAB of AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' We then define the so-called quantum recovery map [SBT16, JRS+18] by its action on a quantum state ρA on region A: ΦA→AB(ρA) = � R ω 1−it 2 AB ω it−1 2 A ρA ω − 1+it 2 A ω 1+it 2 AB dµ0(t) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='28) where µ0 is the probability distribution on R with density dµ0(t) = π dt 2 (cosh(πt) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='29) If A is in the state ωA, the recovery map ΦA→AB recovers the joint state ωAB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=', ΦA→AB(ωA) = ωAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' The relevance of the recovery map comes from the recoverability theorem [SBT16], which states that ΦA→AB can recover a generic joint state ρAB from its marginal ρA if removing the subsystem B does not significantly decrease the relative entropy between ρ and ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' More precisely, for any quantum state σAB of AB we have D(σAB∥ωAB) − D(σA∥ωA) ≥ DM(σAB∥ΦA→AB(σA)) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='30) where DM denotes the measured relative entropy [Don86, Pet86, HP91, BFT17] DM(σ∥ω) := sup (X,M) D(Pσ,M∥Pω,M) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='31) where the supremum above is over all positive operator valued measures M that map the input quantum state to a probability distribution on a finite set X with probability mass function given by Pρ,M(x) = tr ρM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Next, we split region A into regions A1 and A2 such that A1 shields A2 away from B, and take σAB := tr(AB)c(σ(β, x)) and ωAB = σA1B ⊗ σA2 In that case, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='30) becomes I(B : A2|A1)σ ≥ DM � σ∥ΦA1→A1B(σA) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content='32) where we also used that the state ω is a tensor product in the cut A1B − A2, so that ΦA→B = ΦA1→B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Next, we pave the chain Λ into unions of intervals A = ∪M i=1Ai and B = ∪M i=1Bi such that Ai ∩ Bi ̸= ∅ and Bi ∩ Ai+1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' As in [BK18], we then define the channel F := FA ◦ FB where FB := � i σ(β, x, Bi)⊗trBi and FA := � j ΦAi\\B→Ai ◦trAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' In words, the channel FB first prepares the Gibbs state in the region B, whereas FA prepares the remaining of the Gibbs state onto region A\\B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFOT4oBgHgl3EQf3TT6/content/2301.12946v1.pdf'} +page_content=' Then, we have, for any state ρ W1(ρ, σ(β, x)) ≤ W1(ρ, FB(ρ)) + W1(FB(ρ), FA ◦ FB(ρ)) + W1(F(ρ), σ(β, x)) ≤ � i W1(σ(A, i), σ(A, i + 1)) + � i W1(σ(B, i), σ(B, i + 1)) + n ∥F(ρ) − σ(β, x)∥1 (1) ≤ R � i ∥σ(A, i) − σ(A, i + 1)∥1 + ∥σ(B, i) − σ(B, i + 1)∥1 + n∥F(ρ) − σ(β, x)∥1 , 31 where σ(A, i) := � j ⟨U A⟩), although +not statistically incompatible with the assumption that +the data is sampling a stationary mean along the plateau. +We can therefore conclude that this data is consistent +with a plateau along the VEXIT axis, flat at the 0.1 ppm +level, but significantly offset from ef by 0.22 ppm. Fig- +ure 4 (b) shows the same data re-analysed with the first +700 data points rejected from the beginning of each 1000- +point data segment instead of the standard 300. This was +to test for the presence of a time constant in the current, +as discussed in section V. +Only one precision run was performed along the VENT +axis before the interruption, illustrated by the heavy +filled points in figure 2 (b). +One data point, marked +with a ∗, failed the stationary-mean test. It is not clear +from this single run whether this data point indicates +real structure to the plateau at level of ∼ 5 ppm, or if it +is the result of a drift in the device state. The remain- +ing 4 data points mark a plateau region which, combined + +(a) +-0.4- +8 +^/ +-0.8 +ENT +EXIT +/ nA V-1 +-1.2 +-1.6 +-1.2 +V +V +-0.8 +EXIT +1 +(q) +2 +(c) +3 +0 +5 +/ef +4 +-2 +4 +Log +-8 +-1.0 +-0.5 +-1.6 +-1.4 +-1.2 +V +ENT +EXIT4 +FIG. 3. +Results of precision measurements for runs 1-4 as a +function of VEXIT, expressed as ∆IP = (IP − ef)/ef. The data +have been analysed with (a): 300 and (b): 700 data points +rejected from the start of each 1000-point data segment. Error +bars show the combined standard uncertainty UT The horizontal +dashed lines show the weighted means of each data set. Arrows +highlight run 1, measurement 4 and run 4, measurement 4. A +breakdown of the uncertainty for these measurements is given in +table I. +with the stability of the pump map from runs 1-5, gives +confidence that the fixed value of of VENT selected for +runs 1-4 is in the middle of an experimentally-determined +plateau. The mean of the 4 measurements from run 5 is +∆IP = 0.15 ppm, with a standard deviation of 0.09 ppm. +This is consistent with the deviation measured in runs +1-4 given the much smaller sample size. +B. +Uncertainty +In table I, the breakdown of the uncertainty is given +for two measurements indicated by arrows in figure 3a. +The uncertainties due to the two stages of the ULCA cal- +ibration are presented as separate components, with the +uncertainty due to the drift of the ULCA gains in between +calibrations included in these two terms. This was sig- +nificantly reduced by performing frequent ULCA calibra- +tions, with more detail given in supplementary sections +D and E. Run 1, measurement 4 is a typical represen- +tative measurement, and run 4, measurement 4 had the +lowest combined uncertainty of the campaign. As in pre- +vious measurement campaigns, the type A uncertainty +of the pump measurement is the largest single contribu- +tion, but the larger pump current achieved in this study +has reduced UA to below 0.1 ppm and the uncertainty in +the ULCA calibration is now a significant contribution. +Specifically, the uncertainty in the output stage gain RIV +(nominal value 1 MΩ) is limited by the 0.04 ppm uncer- +tainty in the 100 kΩ reference resistor traceable to the +TABLE I. Uncertainty breakdown for run 1, measurement 4, +and run 4, measurement 4. All entries in the table are dimen- +sionless relative uncertainties (k = 1) in parts per million. +Contribution +Meas. 1.4 +Meas. 4.4 +ULCA GI Cal. +0.024 +0.024 +ULCA RIV Cal. +0.062 +0.043 +ULCA Temp. corr. +0.023 +0.023 +DVM Cal. +0.014 +0.016 +Pump UA +0.088 +0.061 +Total UT +0.111 +0.084 +QHR via a chain of 3 intermediate measurements18,19. +C. +stability of the pump +The measurement campaign was divided into two parts +by an instrument issue which forced a period of 8 days’ +down-time between runs 5 and 6. During this time, the +pump was thermally cycled to room temperature and +back to 4 K twice. From examination of the pump maps +in supplementary section H, it is clear that the pump +became less stable after this interruption, although even +before the interruption, small changes in the ‘nose’ (the +onset of pumped current as VEXIT is made less negative) +of the pump map are visible. This contrasts with the data +of Ref. 13 showing this sample of pump to be extremely +stable over multiple cool-downs in different laboratories, +when driven with a sine wave at ∼ 1 GHz. We conjecture +that at least some of the changes visible in the pump +maps during the present campaign may be due to changes +in the transmission of the cryogenic microwave line at +frequencies ≫ 1 GHz. +This could plausibly arise due +to changes in the temperature gradient along the line, +and would affect the high frequency components of the +AWG waveform, causing distortion of the waveform at +the pump entrance gate. +V. +DISCUSSION +The study was complicated by instability in the pump +map, which made it difficult in the later parts of the mea- +surement campaign to interpret the results as sampling a +stable state of the device. However, enough results were +obtained from runs 1-5 to establish that the pump cur- +rent is invariant in the exit gate voltage at the level of 1 +part in 107. Averages over these data points presented +in the previous section give ∆IP ∼ 2×10−7, a significant +offset from the ideal current IP = ef. The flatness of +the plateau suggests that the offset is due to an error in +the measurement system which applies a constant offset +to all the measurements, rather than an error due to the +physics of the pump itself. +One possible cause of error is a time constant in the +pump current. This was discussed in Ref. 13, and could + +4.4 +1.4 +(a) +0.5 +(udd) d/v +---在----0.22 ppm +0.0 +王 +-0.5 +-1.40 +-1.35 +-1.30 +Run number +1 +3 +2 +4 +(b) +0.5 +(wdd) +0.13 ppm +△/p ( +0.0 +工 +-0.5 +-1.40 +-1.35 +-1.30 +VEXIT (V)5 +plausibly arise from heating due to the relatively large +RF powers applied to the device gate. +Repeating the +data analysis of runs 1-4 with 700 data points rejected +from the start of each segment instead of 300 did indeed +yield an average pump current closer to ef, as shown +in figure 3b. However, the larger type A uncertainties +in this analysis make it difficult to draw a firm conclu- +sion regarding possible time constants. +Measurements +with much longer on-off cycle times could potentially +resolve this question, but require the 1/f noise corner +of the ULCA current measurement to be at frequencies +well below 1 mHz. ULCA units have demonstrated this +performance in bench tests16, but the cryogenic wiring +involved in a pump measurement introduces additional +sources of noise and drift. Another possible cause of er- +ror is a non-linearity in the gain of the ULCA. The ULCA +input stage gain GI is calibrated at a current of 6 nA, and +the pump current is 320 pA. Comparisons of the gains of +two ULCA units with different input stage designs, de- +tailed in supplementary section F, set an upper limit to +possible non-linearity of a few parts in 108, so this is +unlikely to cause errors of a part in 107. +Possibly the most important cause of error could arise +from the CCC calibration of the ULCA GI. This could +result from rectification of noise by the CCC’s SQUID +detector leading to different SQUID offsets for the two +polarities of current used in the ULCA calibration20. +One study on CCC performance in the low-flux regime21 +concluded that noise pickup might lead to this type of +error at SQUID flux levels below 1 µΦ0, although this +number was based on a limited number of measurements +and is specific to a particular CCC design22, different +in detail to the CCC used to calibrate the ULCA in +our experiments. We calibrated the ULCA GI using a +CCC18 with a 10000 : 10 turns ratio, and a sensitiv- +ity of 6 µA turns/Φ0. The current in the large winding +was approximately ±5 nA, giving a full-signal ampere- +turns product of 100 µA turns, corresponding to a flux +of 16.7 Φ0. +A flux of 1 µΦ0 therefore corresponds to +0.06 ppm of the full signal in the ULCA GI calibra- +tion, three times smaller than the observed discrepancy +in the electron pump current. +However, no investiga- +tions have yet been carried out on the performance of +our CCC in the low-flux regime, so the size of possible +noise-rectification errors is not known. Low flux ratio ac- +curacy tests such as those presented in Ref. 21 should +provide useful information on the scale of possible errors. +We note that if these errors are affecting the ULCA cali- +brations in our experiment, they are remarkably constant +in time, as shown by the ∼ 5 × 10−8 relative stability of +the ULCA input gain over the duration of the measure- +ment campaign illustrated in supplementary section E. +This indicates that if noise is affecting the SQUID, its +most likely source is the CCC bridge electronics, rather +than external sources. +The upper frequency limit for accurate pumping with +tunable-barrier pumps has previously been empirically +established at around 1 GHz4. We have shown that this +can be increased, albeit in a rather exceptional sample +of pump. In this study, the practical upper frequency +limit was determined by a combination of plateau round- +ing, and increased incidence of switching events which +shifted the pump operating point in the VENT − VEXIT +plane. +This hints at device-physics factors which may +limit the practical upper operation frequency, possibly +charge traps which are activated by high frequency com- +ponents present in the drive signal. Further investigation +of more samples of pump could shed fruitful light on this +question. +VI. +CONCLUSIONS +Precision measurements have been made of the cur- +rent from a silicon electron pump driven at a frequency +of 2 GHz using a custom drive waveform applied to the +entrance gate. The pump current is invariant in exit gate +voltage with a precision of 0.1 ppm (32 aA), but offset by +roughly 0.2 ppm from the expected current corresponding +to one electron for each pump cycle. The application of +a blind measurement protocol provides added confidence +that this result is not affected by experimenter bias. At +this accuracy level, the measurement of the pump cur- +rent challenges the state of the art in existing electrical +metrology methods, with scaling of small currents using +CCCs at low flux levels posing a particularly interesting +problem. The recent demonstration of current plateaus +due to the dual Josephson effect23 raises the possibility +of a metrological investigation of the dual Josephson ef- +fect in the near future, providing added motivation for a +better understanding of low current scaling. +ACKNOWLEDGMENTS +The authors would like to thank Colin Porter and Scott +Wilkins for making the NPL primary Josephson voltage +standard available, and for assistance with setting up the +voltmeter calibration. This research was supported by +the UK department for Business, Energy and Industrial +Strategy. 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A. +Ritchie, “Tunable nonadiabatic excitation in a single-electron +quantum dot,” Physical Review Letters 106, 126801 (2011). + +7 +FIG. S1. (a): Grey-scale derivative pump map using sine wave +drive at 1.05 GHz, PRF = 11.6 dBm. (b): Pump map using a +waveform from an AWG at repetition rate f = 1.04 GHz. One +cycle of the AWG waveform is shown in the inset. Note that this +waveform is subsequently amplified by an inverting amplifier to +yield the correct polarity of gate voltage, whereby the negative +voltage pulse on the entrance gate raises the entrance barrier +to pump an electron. (c): Log-scale plots of the pump current +along the horizontal dashed lines in plots (a) and (b). +VII. +SUPPLEMENTARY INFORMATION +A. +AWG waveform at 1 GHz +The silicon pump in this study has already exhibited +robust quantisation at 1.05 GHz with sine wave drive13, +as illustrated in the pump map and log plot of figure +S1 (a) and (c). As an initial part of the setup process, +we tested the pump operation using an AWG waveform +at a similar frequency of 1.04 GHz. This resulted in a +substantially wider plateau, seen by comparing the log +plots with sine wave and AWG drive in figure S1 (c). Note +that the AWG waveform leads to substantial distortion +of the pump map (figure S1 (b)), due to the electron +capture occurring at different rates dVENT/dt as VENT is +scanned. This data was an important motivator towards +the main study because it showed for the first time that +the type of waveform first used on GaAs pumps in Ref. +10 could also yield a substantial improvement in plateau +flatness with Si pumps. +B. +Exploration of higher frequencies +During the setup of the experiments reported in the +main text, frequencies above 2 GHz were explored using +custom waveforms (figure S2). The data at 4 GHz shows +a feature which may be attributable to non-adiabatic +excitation24 resulting from the rapid deformation of the +FIG. S2. Exit gate scans of pump current using AWG waveforms. +The waveforms are illustrated as insets at the top of the plot on +a common time axis. Dashed horizontal lines indicate the cur- +rent ef at each frequency. Entrance gate voltages are −0.79 V, +−1.08 V, and −1.35 V at frequencies of 2 GHz, 2.574 GHz and +4 GHz respectively. The AWG output amplitude was 0.47 V pp +for all measurements prior to amplification by a 15 dB wide-band +inverting amplifier. The arrow indicates a feature possibly due +to non-adiabatic excitation. +confining potential formed by the entrance and exit gates. +Although the 1ef plateau at 4 GHz looks superficially +flat on this expanded current scale, its slope could easily +be resolved by zooming the data and no precision mea- +surements were attempted. The plateau at 2.574 GHz +was sufficiently flat for metrological investigation, but +the stability of the pump map was degraded compared +to 2 GHz, with sudden shifts along the entrance and +exit gate axes becoming common on time-scales of a few +hours. +Switches in the pump state generally occurred +more frequently as f was increased, and we speculate that +high frequency components in the drive signal may acti- +vate charge traps in the device structure. Consequently, +all the precision measurements reported in the main text +used f = 2 GHz, with the waveform shown in the inset +of figure S2, and also the inset of figure 1 (b) of the main +text. +C. +Raw data +The measurement apparatus and procedure, with two +exceptions, are the same as described in Ref. 13 and its +supplementary information. The exceptions are firstly, +the use of a blind protocol as discussed in the main text, +and secondly, the use of a noise-optimised ULCA16 in- +stead of a standard ULCA17. All measurements are per- +formed as on-off cycles. +For pump measurements, the +‘on’ and ‘off’ states correspond to the entrance gate drive +waveform from the arbitrary waveform generator (AWG) +being turned on and off respectively. For calibrations of +the digital voltmeter (DVM) used to read out the ULCA, + +(a) Sine 1.05 GHz +Sine 1.05 GHz +-1.2- +AWG 1.04 GHz +VENT +0 +lp/ef +(c) +/ V +-1.8 +-2 +11-1 +-1.6 +-1.2 +4 +Log 1 +6 +(b) AWG 1.04 GHz +-1.6 +-1.4 +-1.2 +-0.8 +VEXIT / V +VENT +/ V +8 +plots a, b: +-1.6 +-1.2 +-1.6 +VEXIT / V2 GHz +1000 +2.574 GHz +4 GHz +lp / pA +500 +e1 +0 +-1.6 +-1.4 +-1.2 +-1.0 +-0.8 +VEXIT / V8 +FIG. S3. Raw data, as viewed in a LabView program used to visualise the data during the measurement campaign. The top pair +of plots show the raw voltmeter data from one measurement - run 16, measurement 3. The plots are zoomed to highlight the ‘on’ +(upper plot, yellow points) and ‘off’ (lower plot, blue points) pump data. The voltmeter calibration data are off the scale of these +plots. The lower pair of plots show the beginning of the measurement on an expanded y-axis, and a zoomed x-axis. The first 800 +data points are voltmeter calibrations. The x-axis is simply the sequential data point number. This does not quite map linearly onto +time, because the cal cycles used a voltmeter auto zero with every data point, whereas an auto zero was performed after every 25th +data point for the measure cycles. All data points were integrated over 10 power line cycles. On both plots, vertical bars indicate +the y-scale in raw voltmeter units (ULCA input current). +the ‘on’ and ‘off’ states correspond to the Josephson volt- +age standard programmed to output 0.32 V and 0 V re- +spectively. +In figure S3 we illustrate some raw data, and explain +the nomenclature used to describe the data files. The +illustrated data is measurement 3 from run 16. The data +are the blind-scaled readings of the Agilent 3458A DVM, +connected to either the Josephson voltage standard for +the calibration cycles, or the ULCA for the measure cy- +cles. For the calibration cycles, the data are the com- +pletely raw readings from the voltmeter, and for the mea- +sure cycles the raw readings have been multiplied by the +blind scaling factor β = 1.00000387. The particular mea- +surement illustrated here consisted of 7 ‘sequences’. Each +sequence starts with 8 voltmeter calibration cycles. The +calibration cycles were done with the DVM auto zero +turned on, and 50 data points for each on or off segment. +After the calibration cycles, the voltmeter was connected +to the ULCA output, and a set of pump measurement +cycles were done with 1000 data points for each segment, +auto zero off, and an auto zero operation every 25 data +points (optimisation of the DVM auto zero interval in the +context of single-electron pump measurements was first +discussed in Ref. 6). For the illustrated measurement, +there were 8 measurement cycles in one sequence. Other +measurements in the campaign used from 7 to 11 cycles +per sequence. After the 7 cal-measure sequences, a final +set of 8 calibration cycles was performed, so that each +set of measure cycles had a calibration cycle before and +after, for evaluating the calibration factor to apply to the +measurement data as described in the supplementary in- +formation to Ref. 13. The data analysis evaluated the +pump current separately for each sequence, and the sta- +tistical properties of this data was used as a pass / fail +criteria for the measurement, as described in supplemen- +tary section G. The current reported for the measurement +was the weighted mean over the sequences. +Two points are worth remarking in the data. +The +first is that the hysteretic Josephson voltage standard +does not always yield the same step number (it was pro- +grammed to switch between nominal values of 320 mV +and 0 V). As discussed in the supplementary informa- +tion to Ref. 13, this is not an issue as long as the DVM +is linear over the narrow range of voltages sampled by +the different calibration steps. The second is the remark- +able stability of the ULCA offset. By eye, it does not +appear to drift by more than about 1 fA over the course +of the measurement. We will examine the stability of the +ULCA gain and offset in more detail in supplementary +section E. + +measurement +0.32048- +20UV(20fA) +0.3204- +5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +45000 +50000 +55000 +60000 +65000 +70000 +75000 +8000 +85000 +90000 +95000 +100000 +105000 +110000 +115000 +12000 +Data point number +20 +(20 fA +I reading (M) +sequence +2E-5 +: +-2E-5 +-4E-5- + 5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +45000 + 50000 +55000 +60000 +65000 +7000 +75000 +80000 +85000 +90000 +95000 +10000010500011000 +115000120000 +cal cycle +0.323 +pump cycle +VM reading +0.32 +2 mV (2 pA) + 0.318 +0.317- +100　200　300　400　500 +1400 1500 1600 1700 1800 +700 +900 +10001100 +1200 +1300 +Data point number +0.003 +M 0.002- +2mV(2pA +-0.003- +100 200 300 400 500 600 700 800 +1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 40009 +FIG. S4. Plot (a): filled circles, left axis: Calibration factor kDVM +of the DVM. line, right axis: laboratory temperature. Coloured +blocks at the bottom of the plot, and events labeled ‘E1’ and +‘E2’ are explained in the supplementary text. Inset: Log-scale +histogram of the difference between adjacent measurements of +kDVM. The black square in the main plot shows a range of DVM +calibration data plotted on expanded axes in plot (b). +D. +Voltmeter calibrations and measurement time-line +In figure S4 we have combined several pieces of infor- +mation pertinent to the measurement campaign. +The +main graph of plot (a) shows, on the left axis, the cali- +bration factors, kDVM, of the DVM recorded during the +measurement campaign. We define the calibration factor +as kDVM∆VIND = ∆VREF, where ∆VIND is the change +in indicated voltage and ∆VREF is the change in applied +reference voltage evaluated from an on-off cal cycle. Each +plotted point is averaged from a set of 8 calibration cycles +directly against the Josephson array at a nominal volt- +age of 0.32 V. No data points have been omitted from +this plot, and some outlying data points with large er- +ror bars are the result of failure of the frequency lock to +the Josephson array control electronics. The pink line +plotted on the right axis shows the laboratory tempera- +ture, as measured by a sensor integrated into the ceiling. +Periods when the experiment was not running are vis- +ible as gaps in the voltmeter calibration data, and to +clarify the experimental time-line, shaded blocks at the +bottom of the plot indicate what was happening. Four +types of activity are indicated: The experimental runs, +numbered 1-17; the weekend calibrations of the ULCA +input stage gain GI; The short calibrations of the ULCA +output stage RIV, and finally a period of down-time indi- +cated by a cross-hatched block when the experiment was +stopped due to a fault in the AWG used to generate the +pump drive signal. +Two events marked E1 and E2 are indicated. E1 marks +when an un-used instrument in the experimental rack +(a sine wave generator) was switched off. +The reduc- +tion of heat produced in the rack caused a noticeable +change in the calibration factor of the voltmeter, which +was mounted directly above the sine wave generator. The +fact that this is visible in the kDVM data illustrates the +sensitivity of the direct calibrations of the DVM against +the Josephson array. The event E1 also lowered the tem- +perature of the ULCA, mounted higher up in the rack, +reducing ATR by roughly 0.15 ppm. Event E2 marks a +dramatic excursion of the laboratory temperature caused +by planned maintenance of the air conditioning. This re- +sulted in a larger uncertainty assigned to some of the +measurements of run 15 because of rapid changes in the +ULCA temperature. The transition from stable to fluc- +tuating temperature roughly half-way through the mea- +surement campaign was co-incident with a transfer of +liquid helium into the experimental dewar. It may also +be related to increased activity in adjacent laboratories +as activities were re-started and staff returned following +relaxation of covid-19 control measures. +One important contribution to the uncertainty of the +current measurement is the stability of the DVM on the +1-hour time taken for a cal-measure sequence. The in- +set to figure S4 (a) shows a histogram of the difference +in kDVM between adjacent calibrations during measure- +ments, denoted ∆kDVM. Generally, the DVM is stable +to better than 0.2 ppm on time-scales of an hour, but +jumps in kDVM of up to 0.5 ppm sometimes occur. As in +our previous study13, the uncertainty due to the drift in +kDVM was evaluated using a rectangular distribution as +∆kDVM/2 +√ +3, so a jump in kDVM of 0.2 ppm contributes +0.057 ppm to the combined uncertainty in the pump cur- +rent. The 1-hour DVM calibration interval is therefore +consistent with achieving a combined uncertainty in the +pump measurement of 0.1 ppm. To visualise the short- +term stability of the DVM in the time domain, plot (b) +shows a portion of the main plot on an expanded time +axis. Over this 3-day period, the voltmeter calibration +did not drift by more than 0.3 ppm. The voltmeter cali- +bration data are of general interest for electrical metrol- +ogy, where voltmeters such as the 3458A are commonly +used as transfer standards. From the general perspec- +tive of evaluating the DVM performance in metrological +applications, this data set shows the DVM comfortably +exceeding its manufacturer’s 24-hour accuracy specifica- +tion of 1.5 ppm on the 1 V range. Calibrations over longer +time-scales (not shown) show that the 90-day specifica- +tion of 4.6 ppm is also exceeded by typically a factor 5. +E. +ULCA calibrations +The noise-optimised ULCA was calibrated using a +cryogenic current comparator (CCC) bridge, as described +in Ref. 18. For the calibrations, the ULCA was hand- +carried to an adjacent laboratory. It was specifically car- + +100 +Counts +10 +(a) +0.999983 +E1 +0.0 △kDVM0.5 +/ ppm +(00) +0.999982 +KDVM +20 +Lab Temp. +0.999981 +E2 +0.999980 +16, +G +6,7G. +G. +1-3 +G. +4.5 +8-11 +12-15 +15 +17 +0.999979 +15 Jan +29 Jan +12 Feb +26 Feb +Date (2021) +(b) +0.9999814 +KDVM +HH +0.9999812 +0.9999810 +19 Feb +21 Feb +Date10 +FIG. S5. (a): Deviations from nominal of (upper plot): ULCA +input current gain and (lower plot): ULCA output stage gain, +corrected to a standard temperature. +The plot shows all the +calibrations performed on this ULCA since its delivery to NPL. +The time period covered by the measurement campaign is shown +as a purple shaded box, and the fixed value adopted for the +input stage gain during the measurement campaign is shown as +a horizontal dashed line. (b): Deviation from nominal of the +ULCA transresistance gain calculated from the data in plot (a), +for use during the measurement campaign. +The boxes above +each data point show the run numbers covered by the 6 values +of trans-resistance gain. +ried by hand rather than on a trolley to minimise the +possibility of mechanical shocks. +As illustrated in the +time-line of figure S4 (a), a total of 4 calibrations of the +input stage gain GI (nominal value 1000), and 6 calibra- +tions of the output gain RIV (nominal value 1 MΩ) were +preformed during the measurement campaign. The over- +all trans-resistance gain of the ULCA is ATR = GIRIV +(nominal value 1 GΩ)17. The results of all calibrations +of this ULCA unit since its delivery to NPL are shown in +figure S5 (a). The historical behaviour of the input and +output gains is different, and resulted in different statis- +tical treatments. The input stage gain does not show any +significant drift over the measurement campaign, and fur- +thermore, the limited number of additional calibrations +before and after the campaign did not give any evidence +for long-term drift. Consequently it was assumed to be +constant during the measurement campaign. +Its value +was taken to be the weighted mean of the four calibra- +tions during the campaign, shown as a horizontal dashed +FIG. S6. Averaged DVM readings with the pump (a): on, and +(b): off. Each data point is averaged from one measurement, so +is the mean of ∼ 60, 000 DVM readings. A scale bar indicates 1 +ppm of the 320 pA pump current. +line in figure S5 (a). On the other hand, the output stage +gain shows some drift over time. +Values of RIV were +chosen half way between ‘before’ and ‘after’ calibration +values, with uncertainties which included a drift term de- +rived from a rectangular distribution. In this way, five +values of ATR were calculated to cover runs 4-17. Runs +1-3 were not preceded immediately by any ULCA calibra- +tions, so the value of RIV was taken to be the first RIV +calibration, in between runs 3 and 4, with an uncertainty +derived from a rectangular distribution bounded by the +highest and lowest RIV calibrations during the measure- +ment campaign. In other words, we assumed that the +drift behaviour of RIV for the few days covering runs 1-3 +was similar to the behaviour during the rest of the mea- +surement runs. The 6 values of ATR with their combined +standard uncertainties used to analyse the measurements +are shown in figure S5 (b). +The remarkable stability of the ULCA offset current +is already visible in the raw data of figure S3 (a), and +in figure S6 we go further and show the averaged values +of the ‘ON’ and ‘OFF’ signals measured by the DVM. +Each data point in this graph is the average of all the +ON (plot (a)) or OFF (plot (b)) DVM readings after +rejecting the first 300 readings in each segment. +The +offset current does not change by more than 2 fA over +the 2-month period covered by the measurements. The +drift in offset current may be partially attributable to +changes in ULCA temperature, but there may also be +contributions due to changes in leakage currents through + +(a) +7.8- +SG, / ppm +7.7 +-- 1 +7.6 +7.5 +7.4 +7.3 +27.0 +TI +27.2 +-27.4 +01/10/2019 +01/10/2020 +Date +(b) +8-11 +12-15 +1-3 +4,5 +6,7 +ppm +-19.4 +16,17 +T +-19.5 +-19.6 +15 Jan +29 Jan +12 Feb +26 Feb +Date in 2021(a) +0.320433 +0.320432 +1 ppm = 0.32 fA +0.320431 +15 Jan +29 Jan +12 Feb +26 Feb +(b) +-0.000003 +V +-> +1 ppm = 0.32 fA +0.000004 +-0.000005 +15 Jan +29 Jan +12 Feb +26 Feb11 +FIG. S7. (a): Difference in input current gains GI of two ULCA +units for two series of measurements in self-test configuration, in +which the test current was alternated between a ‘high’ current of +4.8 nA and a ‘Low’ current, either 320 pA or 640 pA. (b): Differ- +ence in trans-resistance gains ATR, alternating the test current +between 4.8 nA and 320 pA. The data plot legend refers to both +panels (a) and (b). +the electron pump control gates. The possible leakage +current paths through the device gates were discussed in +the supplementary information to Ref. 13. +F. +ULCA linearity +The linearity of the ULCA gain is a key assumption in +this experiment, because the calibration of GI is done at +an input current of ∼ 5 nA and the pump current during +the measurement is 320 pA. One previous investigation +set an upper bound on the non-linearity of the overall +UCLA transresistance gain ATR at around the 0.1 ppm +level16. We attempted to reduce this upper bound, using +two test methods previously demonstrated for the ULCA. +First, we compared the input stage current gains of two +ULCA units, as was first demonstrated in Ref. 17. This +is called the ‘self-test’ configuration. A standard ULCA +unit, not otherwise used in our experiment, was used as a +source to generate a test current for comparing its input +stage gain GI,source with the input stage gain of the noise- +optimised experimental ULCA GI,measure. This self-test +configuration is quite straightforward to implement, be- +cause the readout DVM measures a small signal derived +from the difference in the input gains of the two ULCAs, +denoted αGI = GI,source − GI,measure. We alternated sets +of forward-reverse cycles with test currents of ±4.8 nA, +±320 pA and ±640 pA to obtain the data of figure S7 +(a). The forward-reverse cycle time was 60 s, and the +data points are averaged from 100 and 1000 cycles for the +±4.8 nA and ±320 pA currents respectively. The back- +ground drift of αGI visible in the high current data is due +to temperature variation of the ULCAs, but by evaluat- +ing the difference between each low-current data points +(orange triangles) and the mean of the two adjacent high +current data points (green circles), we can extract the +current dependence as a mean over 6 cycles of high-low- +high current. We obtain the current dependence in αGI +between 4.8 nA and 320 pA as 0.002 ± 0.029 ppm. An +additional run examined the current dependence between +4.8 nA and 640 pA (blue diamonds). This data was not +evaluated, but clearly the current dependence is around +a part in 108 or less. +For the second test, we measured the current depen- +dence of the difference in the overall trans-resistance +gains of the two ULCAs, again with the standard ULCA +in ‘source’ mode, and the noise-optimised experimental +ULCA in ‘measure’ mode. This test configuration is il- +lustrated in figure 6 of Ref. 16. It is less straightforward +to implement than the self-test configuration, because +the voltage outputs of the source and measure ULCAs +have opposite signs. We implemented a protocol equiv- +alent to figure 7b of Ref. 16. A single DVM could be +connected to either the source or measure ULCA us- +ing an automated switch - the same switch that was +used in the main experiment to connect the DVM ei- +ther to the ULCA output or the JVS. One cycle con- +sisted of four segments of data: +the test current was +applied with both polarities with the DVM connected +to the source ULCA, recording a forward-reverse differ- +ence voltage ∆Vsource and then the test current was ap- +plied with both polarities with the DVM connected to the +measure ULCA, recording a difference voltage ∆Vmeasure. +Acquiring one cycle took 2 minutes. Assuming that the +DVM calibration factor does not change on this time- +scale, The ratio of ULCA transresistance gains is given by +αATR = ATR,source/ATR,measure = ∆Vmeasure/∆Vsource. +We are interested in whether the ratio of gains depends +on current, so as in the tests of GI linearity, we alternated +1000 cycles at ±320 pA test current, with 100 cycles at +±4.8 nA test current to yield the averaged data points +in figure S7 (b). Similarly to the data of figure S7 (a), +we averaged the high-low-high differences, to obtain the +current dependence of αATR as 0.006 ± 0.023 ppm. +Of course, this data does not conclusively rule out non- +linearity in the ULCA unit used for the measurements. +It only gives information on the linearity of the differ- +ence in the gains of the two ULCA units. It is a slightly +stronger test than the one published in Ref. 16, how- +ever. While that measurement used two nominally iden- +tical noise-optimised ULCAs, our measurement used a +standard ULCA in the ‘source’ role. The different values +of resistors used in the current scaling networks make it +less likely that both ULCA units would have the same + +(a) +8.8×106 +αG, / ppm +-8.9×10-6 +-9.0×10-6 +-9.1×106 +-9.2×10-6 +Measurement number +(b) +± 4.8 nA +-35.6 +± 320 pA +± 640 pA +-35.7 +-35.8 +-35.9 +Measurement number12 +current-dependence to the gain. +G. +Statistical tests and data set rejection +As mentioned in the main text, the pump state, as doc- +umented by the ‘pump maps’, changed during the mea- +surement campaign, with some obvious dramatic changes +occurring during some measurements, and more subtle +changes during other measurements. Even if the pump +map was stable, some of the measurements close to the +edges of the current plateaus could be affected by small +fluctuations in offset charge, leading to relatively large +changes in pump current as the operating point drifted +on and off the plateau. +It could not generally be as- +sumed that the pump current sampled by a measure- +ment lasting more than 10 hours represented a station- +ary mean. Each measurement was therefore subjected to +a statistical test. Recall from supplementary section S3, +that the pump current from each sequence was evaluated +separately. +This yielded m values of IP, denoted IP,m +with uncertainties U(IP,m), where m is the number of se- +quences in the measurement. If all the IP,m are sampling +the same value of pump current, on average the stan- +dard deviation of the IP,m, σ(IP,m) will be equal to the +mean of the uncertainties, ⟨U(IP,m)⟩. We propose the ra- +tio σ(IP,m)/⟨U(IP,m)⟩ = R as a statistical measure of the +stationarity of the data, and in figure S8, we plot a his- +togram of this quantity (grey bars, right axis) for the 64 +measurements performed during our campaign. We also +plot (red bars, left axis) a histogram of the same quantity +obtained from 1000 simulated measurements, in which +the simulated raw data, both for the measurement and +calibration cycles, was generated from a stationary mean +multiplied by Gaussian white noise with the same stan- +dard deviation as the real data. As expected, the most +probable value of R for this simulated stationary data is +1, and the probability of obtaining a measurement with +R > 2 from a set of 1000 measurements becomes neg- +ligible. Since we only performed 64 measurements, we +assigned a cutoff of R = 1.7, and rejected measurements +with R > 1.7. Comparing the histogram of the measured +data with the simulation, it is clear that a significant +number of data sets have an R value which would be +improbably high if the pump current was constant dur- +ing the measurement. This is actually expected, for the +reason that some of the precision measurements were se- +lected with control parameter values close to the edges of +the current plateau. For these measurements, small fluc- +tuations in offset charge during the measurement (equiv- +alent to a drift in the control parameters) would cause +the pump current to drift away from ef. +To see the accept / reject criteria in action, two exam- +ple measurements from run 10 are plotted in figures S8 +(b) and (c), with the corresponding R values marked with +red and green arrows on the x-axis of panel (a). The data +of panel (b) clearly shows a decrease in the pump current, +and it would be tempting to reject this data set based just +FIG. S8. (a): Histograms of the quantity R, defined in the sup- +plementary text as the ratio of the standard deviation of the m +values of IP calculated for each measurement (m is sequence +number) to the average uncertainty of IP,m. The grey bars re- +ferred to the right axis are for the measured data, and the red +cross-hatched bars referred to the left axis are for 1000 sim- +ulated measurements assuming a stationary mean. +A vertical +dashed line shows the R = 1.7 rejection threshold derived from +the probability of the simulated data having an R greater than +this value. Panels (b) and (c) show IP,m for two example mea- +surements from run 10, with (b): R > 1.7 and (c): R < 1.7. +The R values for these data sets are indicated with red and green +arrows respectively on panel (a). +on this time-domain visualisation of IP,m. However, the +definition of the R parameter makes this otherwise sub- +jective process more quantitative. Altogether, 14 mea- +surements during the entire measurement campaign had +R > 1.7. +H. +full data set +Due to instability of the pump after run 5, only the +data from runs 1-5 are analysed in the main text. The +increasing instability is visible in the pump maps of figure +S10, and also in the increasing number of runs which +failed the stationary mean test. In figure S9, we present +all of the precision data on linear axes. +Plots (a,b,c) +show all of the measurements on expanded y-axes, and +plots (d,e,f) show the sub-set of the measurements which +passed the stationary-mean test. Figure S10 shows the +full set of ‘fingerprint’ pump maps obtained before and +after each precision measurement run. For data integrity + +(a) +200 +15 +Simulated data +measured data +Count (Simulated) +150. +Count (Measured) +REJECT +10 +100 +5 +50. +0 +0 +0 +2 +4 +5 +6 +7 +8 +R +(b) +(c) +Run 10, V. += -0.835 V +Run 10, V += -0.775 V +ENT +ENT +wdd +REJECT +0 +ACCEPT +p,m +-2 +12 hours +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +seguence m +seguence m13 +FIG. S9. a-c: Deviation of the pump current from its nominal value as a function of (a): Exit gate voltage, (b): Entrance gate +voltage and (c): AWG output amplitude. All of the measurements from the 17 runs are shown in these plots. One data point in panel +(a) is off the y-axis scale, and is indicated by an arrow. (d), (e) and (f): the sub-set of data in plots (a), (b) and (c) respectively, +which passed the stationary mean test, on expanded axes. In each plot, vertical dotted lines indicate fixed values of the scanned +parameter for runs in the other plots. Error bars indicate combined standard uncertainties UT. +purposes, this figure also includes the 4-digit hexadecimal +file identifier for the precision raw data. + +(a) +(d) +1.0 +9,105,7,15 +8 +1 +4 +11 +2 +12 +14 +17 +6 +6 +run 6 +0.5 +3 +8 +16 ++27 +udd +0.0 +△/p +4 +11 +-0.5 +2 +6 +12 +0. +3 +8 +16 +-2 +-1.0 +-1.40 +-1.35 +-1.30 +-1.25 +-1.40 +-1.35 +-1.30 +-1.25 +VeXIT / V +VEXIT / V +5 +13 +(b) +(c) +(e) +(f) +6 +7 +14 +1.0 +16,17 +1,2,3,4,6,8,11,12 +1-16 +9 +15 +4 +0.5 +10 +17 +udd +udd +2 +0.0 +N/p +0 +△/p / +0.5 +-2 +5 +14 +17 +7 +10 +15 +-1.0 +4 +0.90 -0.85 -0.80 -0.75 -0.70 +0.460.470.48 +0.85-0.80-0.75-0.70 +0.46 +0.47 +0.48 +VENT / V +VAc / V +VENT / V +VAc / V14 +FIG. S10. Thumbnail pump maps measured before and after each precision scan. Each pump map is an inverted grey-scale derivitive +plot of the current similar to the one shown in figures 1(b) and 2(a) of the main text. The axis limits are the same for each thumbnail: +The x-axis is VEXIT, from −1.7 V to −0.8 V, and the y-axis is VENT, from −1.4 V to −0.4 V. Each horizontal row of the table represents +a precision run. The middle cell contains some text data describing the run, including the 4-digit hexadecimal file number identifying +the raw data set for the precision run. The left-most cell shows the pump map recorded before the run, and the right-most cell shows +the pump map recorded after the run. Missing pump maps for runs 9,10 and 11 were due to software crashes. Red arrows highlight +runs in which the pump map changed dramatically during the run. + +8 Days down-time due to AWG issue \ No newline at end of file diff --git a/SdE3T4oBgHgl3EQfZwoa/content/tmp_files/load_file.txt b/SdE3T4oBgHgl3EQfZwoa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88881859e7c9e373dd72670068b5835bf093c477 --- /dev/null +++ b/SdE3T4oBgHgl3EQfZwoa/content/tmp_files/load_file.txt @@ -0,0 +1,879 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf,len=878 +page_content='Precision measurement of an electron pump at 2 GHz Stephen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Giblin,1 Gento Yamahata,2 Akira Fujiwara,2 and Masaya Kataoka1 1)National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, United Kingdom 2)NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan (Dated: 12 January 2023) A well-characterised sample of silicon tunable-barrier electron pump has been operated at a frequency of 2 GHz using a custom drive waveform, generating a pump current of 320 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Precision measurements of the current were made as a function of pump control parameters, using a blind protocol, over a 7-week campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The combined standard uncertainty for each ∼ 10 hour measurement was ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 parts per million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The pump current exhibits a plateau along the exit gate voltage flat to approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 parts per million, but offset from ef by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 parts per million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This offset may be a sign of errors in the current traceability chain, indicating a limit to the accuracy of small current scaling using existing methods based on cryogenic current comparators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' PACS numbers: 1234 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' INTRODUCTION Electron pumps are devices that aim to generate a ref- erence DC electric current by moving electrons one at a time in response to a periodic control signal at frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' They potentially offer a simple and elegant traceabil- ity route for small currents, direct to the SI definition of the ampere1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A class of pumps fabricated from semicon- ductor materials3 has demonstrated accurate and robust current generation at roughly the part-per-million (ppm) accuracy level, for currents IP up to 160 pA4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' However, important questions must be answered before electron pumps can confidently be adopted as reference current standards at the uncertainty levels of primary electrical metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Most significantly, the robustness and device independence of the current needs to be demonstrated at at least the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm level, over a range of device designs and operating parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To date, two studies have fo- cused on the robustness of the current from GaAs pumps, at current levels of ∼ 100 pA, at uncertainty levels for each data point of ∼ 2 ppm5 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 ppm6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' However, blind measurement techniques which have been implemented in other metrology areas to remove bias7 have not yet been applied to the study of electron pumps where the pump current is treated as an unknown and compared to a known reference current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Address- ing unconscious experimenter bias is particularly impor- tant in experiments where the expectation of the result is strongly constrained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' in this case, we expect IP = ef, and there is a possibility that in a non-blind measure- ment, the experimenter may unconsciously favour pump control parameters that yield this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Particularly important in the electron pump context is the lack of reproducibility in attempts to realise a capacitance stan- dard based on pumping a known number of electrons onto a cryogenic capacitor8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 9 were un- able to reproduce the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 8, and identified components in the capacitance measurement uncertainty which had previously been under-estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Evaluating the robustness of the pump current presents a challenge due to the time-scales involved: the small currents require many hours of averaging to resolve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm for a single data point, and the time-scale of the whole measurement campaign challenges the stability of the measurement system and the electron pump itself4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To reduce the measurement time, or equivalently, to al- low more data points to be measured within the timescale of a measurement campaign, the pump current should be increased as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Custom gate drive waveforms which slow down the electron capture pro- cess have been used to operate GaAs pumps accurately at much higher frequencies than were possible with sine wave drive6,10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' With these pumps the upper frequency limit for accurate pumping was f ∼ 1 GHz even with the custom waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Silicon pumps, on the other hand, have demonstrated accurate pumping at f = 1 GHz with sine wave drive12,13, and the possibility of increasing the frequency further while maintaining sub-ppm pumping accuracy using custom waveforms has not yet been ex- plored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' EXPERIMENTAL METHOD AND BLIND PROTOCOL We investigate a single well-characterised sample of silicon pump which has previously been the subject of two precision measurement campaigns13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The pump is a silicon nanowire-MOSFET, in which charge carri- ers are induced by a positive voltage applied to a global top gate14,15 which was set to 4 V for all the measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Two finger gates, denoted the entrance gate and exit gate, define the region of the nanowire where a sin- gle electron can be trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Negative DC voltages VENT and VEXIT applied to these gates define the pump operat- ing point, and the periodic pump drive signal is added to VENT using a room-temperature bias-tee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A 50 Giga sam- ples/s arbitrary waveform generator (AWG, Tektronix arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='04499v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='mes-hall] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a): Grey-scale derivative pump map using sine wave drive at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='062 GHz, PRF = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (b): Pump map using a waveform from an AWG at repetition rate f = 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One cycle of the AWG waveform is shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Note that this waveform is subsequently amplified by an inverting amplifier to yield the correct polarity of gate voltage, whereby the negative voltage pulse on the entrance gate raises the entrance barrier to pump an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (c): Log-scale plots of the pump current along the horizontal dashed lines in plots (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 70001A) was used to generate a custom waveform for the pump drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Because the AWG output had a max- imum peak-peak amplitude of VAC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 V, the output was amplified by a wide-band inverting RF amplifier with +15 dB gain before the bias-tee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The AWG is referenced to a 10 MHz frequency reference derived from a hydrogen maser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Figure 1 shows characterisation data using both sine wave drive and the custom AWG waveform at a repeti- tion frequency of 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It is clear from the log-scale plots of figure 1 (c) that there is a substantial plateau along the exit gate axis when using the AWG drive wave- form, but not when using sine wave drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The inset to figure 1(b) shows the AWG waveform used for all the measurements reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Characterisation data at other frequencies is inlcuded in supplementary sections A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The experimental apparatus and methods used for this study are in many respects identical to that used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As in those experiments, the pump is cooled to a temperature close to 4 K by suspending it above a liquid helium surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The pump current IP is measured us- ing a noise-optimised ultrastable low-noise current ampli- fier (ULCA)16, with a precision digital voltmeter (DVM) recording the ULCA output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13, the DVM was calibrated roughly once every hour by switching its input to a Josephson voltage standard (JVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A single precision measurement typically lasted between 8 and 10 hours and included between 7 and 11 voltmeter calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To re- move offset drifts in the measurement system during pre- cision measurements, the pump drive signal was toggled on and off with a cycle time of 228 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Roughly the first 34 seconds of each data segment (300 out of 1000 data points) following each on or off switch was rejected from the analysis to remove transient effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' More details of the measurement protocol are given in supplementary section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The pump current IP is calculated from the on-off difference in the DVM voltages ∆V using the equation IP = ∆V/ATR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Here, ATR is the trans-resistance gain of the ULCA, nominally equal to 109 V/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This gain is calibrated against the quantum Hall resistance (QHR) in 2 stages17 and via some intermediate transfer standards, using a cryogenic current comparator (CCC) with rel- ative uncertainty less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Detailed calibra- tion results are reported in supplementary section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The measurement of the pump current was therefore traceable to the SI unit ampere via the JVS, the QHR, and the re- lationship I = V/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For characterisation measurements such as those reported in figures 1, 2a, and small filled points in figures 2b and 2c, no offset subtraction was per- formed: the pump drive signal was left on, and each data point is a single 20 power line cycle DVM measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A blind protocol was implemented so that the lead ex- perimenter could not see the true value of IP while the measurements and data analysis were in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This is achieved by multiplying all the DVM readings by a hidden scaling factor β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='00000387, at a low level in the measurement software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' An exception occurs when when the DVM is connected to the JVS for calibration, in which case β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' While tuning the pump and per- forming measurements, the experimenter does not know the scaling factor and can only access the scaled pump current IP,B = β∆V/ATR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Therefore, the tuning of the pump operating parameters and the choice of parame- ters for the precision measurements can only be made with reference to the flatness of the current plateau, not the deviation of the current from ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The scaling factor was programmed and password-protected by a member of the team who was not otherwise involved in the exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The experimenter knew that it was constrained such that |1 − β| < 5 × 10−6 so that gross failures of the pump or apparatus would be apparent during character- ization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The scaling factor was revealed after the experiments were finished and data analysis, including analysis of the ULCA calibrations, completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' PRECISION MEASUREMENT CAMPAIGN The aim of the measurements was to study the pump current as a function of control parameters VENT, VEXIT and VAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To this end, a total of 67 precision measure- ments were made during a 7-week campaign, employing the apparatus and blind protocol described in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The measurements were divided into 17 ‘runs’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For most of the runs, several measurements were made while vary- ing one control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Runs 11-13 consisted of sin- (a) Sine 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='062 GHz Sine 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='062 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 VENT AWG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='000 GHz 0 / V lp/ef (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 2 Log I 1 - / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 VEXIT / V 4 6 (b) AWG 2 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 VEXIT / V VENT 10 / V plots a, b: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 d/p / dVExIT / nAV-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 0 VEXIT / V3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a): Derivative pump map measured after precision run 4 and before precision run 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (b) and (c): Line-scans of the pump current measured along the gate voltage axes indicated by solid colored lines in (a), and plotted on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The vertical dashed lines indicate the fixed value of entrance (exit) gate voltage used for the exit (entrance) gate scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The diagonal dashed lines are guides to the eye extrapolating the exponential edges of the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Larger filled points are the precision measurement data for runs 1-5, with the run number indicated in the plot legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Data points indicated with a star (∗) failed the stationary-mean test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' gle measurements without varying a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Further detail of the measurement chronology is given in supple- mentary section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To monitor the stability of the pump, a ‘fingerprint’ pump map was obtained before and after each run, apart from a few occasions when it was pre- vented by an experimental difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For completeness, all of these pump maps are shown in supplementary sec- tion H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Additional line scans of current as a function of one or more control parameters were also measured to assess the optimal values of fixed control parameters for the next precision run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Typically, these scans were used to find the value of the control parameter that maximised the plateau width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' They used a single 20 PLC measure- ment for each data point, with a relative uncertainty of approximately 10 ppm per data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A pass / fail sta- tionary mean statistical test, described in supplementary section G, was applied at the data analysis stage to each precision measurement to evaluate whether the current was stable during the measurement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' RESULTS OF PRECISION MEASUREMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Precision results After run 5, the pump became less stable, (supple- mentary section H), making it difficult to establish the flatness of plateaus along VENT and VEXIT axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For this reason, we concentrate here on the data from runs 1-5, although the full precision data set is presented in sup- plementary figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In figure 2, we present the data from the first 5 precision runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Panel (a) shows a pump map recorded between runs 4 and 5, and panels (b) and (c) show line-scans on a log scale which highlight the de- viation of the current from the ideal value on the 1ef plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The fixed value of VENT (VEXIT) for the VEXIT (VENT) line-scan was adjusted in order to maximise the width of the plateau in the log-scale plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The results of precision runs 1-5 are plotted as solid points in figures 2 (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Runs 1-4 were VEXIT scans, plotted in figure 2 (c), and run 5 was a VENT scan, plotted in figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The 18 data points along the VEXIT axis (figure 2 (c)) define a plateau in agreement with an extrapolation of the standard-accuracy measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The precision data point marked with a star (∗), failed the stationary-mean test, presumably because it was close to the edge of the plateau, and small fluctuations in offset charge, equiva- lent to shifts in VEXIT, caused fluctuations in the pumped current to be resolved on the time-scale of the precision measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The precision IP(VEXIT) data for runs 1-4 (apart from the point that failed the stationary mean test) are re- plotted on a linear y-axis in figure 3a as ∆IP = (IP − ef)/ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The mean of these 17 points is ∆IP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='22 ppm, with a standard deviation σ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='14 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The individ- ual data points have a mean combined uncertainty ⟨U T ⟩ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='102 ppm, although the uncorrelated random uncer- tainty, UA, for each data point is smaller, in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='08 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='09 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The scatter of the points is therefore slightly larger than what would be expected from the type A uncertainty of each point (σ > ⟨U A⟩), although not statistically incompatible with the assumption that the data is sampling a stationary mean along the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We can therefore conclude that this data is consistent with a plateau along the VEXIT axis, flat at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm level, but significantly offset from ef by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='22 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Fig- ure 4 (b) shows the same data re-analysed with the first 700 data points rejected from the beginning of each 1000- point data segment instead of the standard 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This was to test for the presence of a time constant in the current, as discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Only one precision run was performed along the VENT axis before the interruption, illustrated by the heavy filled points in figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One data point, marked with a ∗, failed the stationary-mean test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It is not clear from this single run whether this data point indicates real structure to the plateau at level of ∼ 5 ppm, or if it is the result of a drift in the device state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The remain- ing 4 data points mark a plateau region which, combined (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4- 8 ^/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 ENT EXIT / nA V-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 V V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 EXIT 1 (q) 2 (c) 3 0 5 /ef 4 2 4 Log 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 V ENT EXIT4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Results of precision measurements for runs 1-4 as a function of VEXIT, expressed as ∆IP = (IP − ef)/ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The data have been analysed with (a): 300 and (b): 700 data points rejected from the start of each 1000-point data segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Error bars show the combined standard uncertainty UT The horizontal dashed lines show the weighted means of each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Arrows highlight run 1, measurement 4 and run 4, measurement 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A breakdown of the uncertainty for these measurements is given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' with the stability of the pump map from runs 1-5, gives confidence that the fixed value of of VENT selected for runs 1-4 is in the middle of an experimentally-determined plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The mean of the 4 measurements from run 5 is ∆IP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='15 ppm, with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='09 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This is consistent with the deviation measured in runs 1-4 given the much smaller sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Uncertainty In table I, the breakdown of the uncertainty is given for two measurements indicated by arrows in figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The uncertainties due to the two stages of the ULCA cal- ibration are presented as separate components, with the uncertainty due to the drift of the ULCA gains in between calibrations included in these two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This was sig- nificantly reduced by performing frequent ULCA calibra- tions, with more detail given in supplementary sections D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Run 1, measurement 4 is a typical represen- tative measurement, and run 4, measurement 4 had the lowest combined uncertainty of the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As in pre- vious measurement campaigns, the type A uncertainty of the pump measurement is the largest single contribu- tion, but the larger pump current achieved in this study has reduced UA to below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm and the uncertainty in the ULCA calibration is now a significant contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Specifically, the uncertainty in the output stage gain RIV (nominal value 1 MΩ) is limited by the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='04 ppm uncer- tainty in the 100 kΩ reference resistor traceable to the TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Uncertainty breakdown for run 1, measurement 4, and run 4, measurement 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' All entries in the table are dimen- sionless relative uncertainties (k = 1) in parts per million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Contribution Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 ULCA GI Cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='024 ULCA RIV Cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='043 ULCA Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='023 DVM Cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='016 Pump UA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='061 Total UT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='084 QHR via a chain of 3 intermediate measurements18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' stability of the pump The measurement campaign was divided into two parts by an instrument issue which forced a period of 8 days’ down-time between runs 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' During this time, the pump was thermally cycled to room temperature and back to 4 K twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' From examination of the pump maps in supplementary section H, it is clear that the pump became less stable after this interruption, although even before the interruption, small changes in the ‘nose’ (the onset of pumped current as VEXIT is made less negative) of the pump map are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This contrasts with the data of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13 showing this sample of pump to be extremely stable over multiple cool-downs in different laboratories, when driven with a sine wave at ∼ 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We conjecture that at least some of the changes visible in the pump maps during the present campaign may be due to changes in the transmission of the cryogenic microwave line at frequencies ≫ 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This could plausibly arise due to changes in the temperature gradient along the line, and would affect the high frequency components of the AWG waveform, causing distortion of the waveform at the pump entrance gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' DISCUSSION The study was complicated by instability in the pump map, which made it difficult in the later parts of the mea- surement campaign to interpret the results as sampling a stable state of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' However, enough results were obtained from runs 1-5 to establish that the pump cur- rent is invariant in the exit gate voltage at the level of 1 part in 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Averages over these data points presented in the previous section give ∆IP ∼ 2×10−7, a significant offset from the ideal current IP = ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The flatness of the plateau suggests that the offset is due to an error in the measurement system which applies a constant offset to all the measurements, rather than an error due to the physics of the pump itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One possible cause of error is a time constant in the pump current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This was discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13, and could 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 (udd) d/v ---在----0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='22 ppm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 王 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='30 Run number 1 3 2 4 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 (wdd) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='13 ppm △/p ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 工 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='30 VEXIT (V)5 plausibly arise from heating due to the relatively large RF powers applied to the device gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Repeating the data analysis of runs 1-4 with 700 data points rejected from the start of each segment instead of 300 did indeed yield an average pump current closer to ef, as shown in figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' However, the larger type A uncertainties in this analysis make it difficult to draw a firm conclu- sion regarding possible time constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Measurements with much longer on-off cycle times could potentially resolve this question, but require the 1/f noise corner of the ULCA current measurement to be at frequencies well below 1 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' ULCA units have demonstrated this performance in bench tests16, but the cryogenic wiring involved in a pump measurement introduces additional sources of noise and drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Another possible cause of er- ror is a non-linearity in the gain of the ULCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The ULCA input stage gain GI is calibrated at a current of 6 nA, and the pump current is 320 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Comparisons of the gains of two ULCA units with different input stage designs, de- tailed in supplementary section F, set an upper limit to possible non-linearity of a few parts in 108, so this is unlikely to cause errors of a part in 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Possibly the most important cause of error could arise from the CCC calibration of the ULCA GI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This could result from rectification of noise by the CCC’s SQUID detector leading to different SQUID offsets for the two polarities of current used in the ULCA calibration20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One study on CCC performance in the low-flux regime21 concluded that noise pickup might lead to this type of error at SQUID flux levels below 1 µΦ0, although this number was based on a limited number of measurements and is specific to a particular CCC design22, different in detail to the CCC used to calibrate the ULCA in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We calibrated the ULCA GI using a CCC18 with a 10000 : 10 turns ratio, and a sensitiv- ity of 6 µA turns/Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The current in the large winding was approximately ±5 nA, giving a full-signal ampere- turns product of 100 µA turns, corresponding to a flux of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7 Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A flux of 1 µΦ0 therefore corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='06 ppm of the full signal in the ULCA GI calibra- tion, three times smaller than the observed discrepancy in the electron pump current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' However, no investiga- tions have yet been carried out on the performance of our CCC in the low-flux regime, so the size of possible noise-rectification errors is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Low flux ratio ac- curacy tests such as those presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 21 should provide useful information on the scale of possible errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We note that if these errors are affecting the ULCA cali- brations in our experiment, they are remarkably constant in time, as shown by the ∼ 5 × 10−8 relative stability of the ULCA input gain over the duration of the measure- ment campaign illustrated in supplementary section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This indicates that if noise is affecting the SQUID, its most likely source is the CCC bridge electronics, rather than external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The upper frequency limit for accurate pumping with tunable-barrier pumps has previously been empirically established at around 1 GHz4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We have shown that this can be increased, albeit in a rather exceptional sample of pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In this study, the practical upper frequency limit was determined by a combination of plateau round- ing, and increased incidence of switching events which shifted the pump operating point in the VENT − VEXIT plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This hints at device-physics factors which may limit the practical upper operation frequency, possibly charge traps which are activated by high frequency com- ponents present in the drive signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Further investigation of more samples of pump could shed fruitful light on this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' CONCLUSIONS Precision measurements have been made of the cur- rent from a silicon electron pump driven at a frequency of 2 GHz using a custom drive waveform applied to the entrance gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The pump current is invariant in exit gate voltage with a precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm (32 aA), but offset by roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 ppm from the expected current corresponding to one electron for each pump cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The application of a blind measurement protocol provides added confidence that this result is not affected by experimenter bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' At this accuracy level, the measurement of the pump cur- rent challenges the state of the art in existing electrical metrology methods, with scaling of small currents using CCCs at low flux levels posing a particularly interesting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The recent demonstration of current plateaus due to the dual Josephson effect23 raises the possibility of a metrological investigation of the dual Josephson ef- fect in the near future, providing added motivation for a better understanding of low current scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors would like to thank Colin Porter and Scott Wilkins for making the NPL primary Josephson voltage standard available, and for assistance with setting up the voltmeter calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This research was supported by the UK department for Business, Energy and Industrial Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' are supported by JSPS KAK- ENHI Grant Number JP18H05258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 1N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Kaneko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Nakamura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Okazaki, “A review of the quantum current standard,” Measurement Science and Technol- ogy 27, 032001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Scherer and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Schumacher, “Single-electron pumps and quantum current metrology in the revised SI,” Annalen der Physik , 1800371 (2019).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Ritchie, “Tunable nonadiabatic excitation in a single-electron quantum dot,” Physical Review Letters 106, 126801 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a): Grey-scale derivative pump map using sine wave drive at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='05 GHz, PRF = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (b): Pump map using a waveform from an AWG at repetition rate f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='04 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One cycle of the AWG waveform is shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Note that this waveform is subsequently amplified by an inverting amplifier to yield the correct polarity of gate voltage, whereby the negative voltage pulse on the entrance gate raises the entrance barrier to pump an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (c): Log-scale plots of the pump current along the horizontal dashed lines in plots (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' SUPPLEMENTARY INFORMATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' AWG waveform at 1 GHz The silicon pump in this study has already exhibited robust quantisation at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='05 GHz with sine wave drive13, as illustrated in the pump map and log plot of figure S1 (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As an initial part of the setup process, we tested the pump operation using an AWG waveform at a similar frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='04 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This resulted in a substantially wider plateau, seen by comparing the log plots with sine wave and AWG drive in figure S1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Note that the AWG waveform leads to substantial distortion of the pump map (figure S1 (b)), due to the electron capture occurring at different rates dVENT/dt as VENT is scanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This data was an important motivator towards the main study because it showed for the first time that the type of waveform first used on GaAs pumps in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 10 could also yield a substantial improvement in plateau flatness with Si pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Exploration of higher frequencies During the setup of the experiments reported in the main text, frequencies above 2 GHz were explored using custom waveforms (figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The data at 4 GHz shows a feature which may be attributable to non-adiabatic excitation24 resulting from the rapid deformation of the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Exit gate scans of pump current using AWG waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The waveforms are illustrated as insets at the top of the plot on a common time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Dashed horizontal lines indicate the cur- rent ef at each frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Entrance gate voltages are −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='79 V, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='08 V, and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='35 V at frequencies of 2 GHz, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='574 GHz and 4 GHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The AWG output amplitude was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='47 V pp for all measurements prior to amplification by a 15 dB wide-band inverting amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The arrow indicates a feature possibly due to non-adiabatic excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' confining potential formed by the entrance and exit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Although the 1ef plateau at 4 GHz looks superficially flat on this expanded current scale, its slope could easily be resolved by zooming the data and no precision mea- surements were attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The plateau at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='574 GHz was sufficiently flat for metrological investigation, but the stability of the pump map was degraded compared to 2 GHz, with sudden shifts along the entrance and exit gate axes becoming common on time-scales of a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Switches in the pump state generally occurred more frequently as f was increased, and we speculate that high frequency components in the drive signal may acti- vate charge traps in the device structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Consequently, all the precision measurements reported in the main text used f = 2 GHz, with the waveform shown in the inset of figure S2, and also the inset of figure 1 (b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Raw data The measurement apparatus and procedure, with two exceptions, are the same as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13 and its supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The exceptions are firstly, the use of a blind protocol as discussed in the main text, and secondly, the use of a noise-optimised ULCA16 in- stead of a standard ULCA17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' All measurements are per- formed as on-off cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For pump measurements, the ‘on’ and ‘off’ states correspond to the entrance gate drive waveform from the arbitrary waveform generator (AWG) being turned on and off respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For calibrations of the digital voltmeter (DVM) used to read out the ULCA, (a) Sine 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='05 GHz Sine 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='05 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2- AWG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='04 GHz VENT 0 lp/ef (c) / V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 2 11-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 4 Log 1 6 (b) AWG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='04 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 VEXIT / V VENT / V 8 plots a, b: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 VEXIT / V2 GHz 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='574 GHz 4 GHz lp / pA 500 e1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 VEXIT / V8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Raw data, as viewed in a LabView program used to visualise the data during the measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The top pair of plots show the raw voltmeter data from one measurement - run 16, measurement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The plots are zoomed to highlight the ‘on’ (upper plot, yellow points) and ‘off’ (lower plot, blue points) pump data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The voltmeter calibration data are off the scale of these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The lower pair of plots show the beginning of the measurement on an expanded y-axis, and a zoomed x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The first 800 data points are voltmeter calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The x-axis is simply the sequential data point number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This does not quite map linearly onto time, because the cal cycles used a voltmeter auto zero with every data point, whereas an auto zero was performed after every 25th data point for the measure cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' All data points were integrated over 10 power line cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' On both plots, vertical bars indicate the y-scale in raw voltmeter units (ULCA input current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' the ‘on’ and ‘off’ states correspond to the Josephson volt- age standard programmed to output 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='32 V and 0 V re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In figure S3 we illustrate some raw data, and explain the nomenclature used to describe the data files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The illustrated data is measurement 3 from run 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The data are the blind-scaled readings of the Agilent 3458A DVM, connected to either the Josephson voltage standard for the calibration cycles, or the ULCA for the measure cy- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For the calibration cycles, the data are the com- pletely raw readings from the voltmeter, and for the mea- sure cycles the raw readings have been multiplied by the blind scaling factor β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='00000387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The particular mea- surement illustrated here consisted of 7 ‘sequences’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each sequence starts with 8 voltmeter calibration cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The calibration cycles were done with the DVM auto zero turned on, and 50 data points for each on or off segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' After the calibration cycles, the voltmeter was connected to the ULCA output, and a set of pump measurement cycles were done with 1000 data points for each segment, auto zero off, and an auto zero operation every 25 data points (optimisation of the DVM auto zero interval in the context of single-electron pump measurements was first discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For the illustrated measurement, there were 8 measurement cycles in one sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Other measurements in the campaign used from 7 to 11 cycles per sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' After the 7 cal-measure sequences, a final set of 8 calibration cycles was performed, so that each set of measure cycles had a calibration cycle before and after, for evaluating the calibration factor to apply to the measurement data as described in the supplementary in- formation to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The data analysis evaluated the pump current separately for each sequence, and the sta- tistical properties of this data was used as a pass / fail criteria for the measurement, as described in supplemen- tary section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The current reported for the measurement was the weighted mean over the sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Two points are worth remarking in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The first is that the hysteretic Josephson voltage standard does not always yield the same step number (it was pro- grammed to switch between nominal values of 320 mV and 0 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As discussed in the supplementary informa- tion to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13, this is not an issue as long as the DVM is linear over the narrow range of voltages sampled by the different calibration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The second is the remark- able stability of the ULCA offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' By eye, it does not appear to drift by more than about 1 fA over the course of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We will examine the stability of the ULCA gain and offset in more detail in supplementary section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' measurement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='32048- 20UV(20fA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='3204- 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000 70000 75000 8000 85000 90000 95000 100000 105000 110000 115000 12000 Data point number 20 (20 fA I reading (M) sequence 2E-5 : 2E-5 4E-5- 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000 7000 75000 80000 85000 90000 95000 10000010500011000 115000120000 cal cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='323 pump cycle VM reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='32 2 mV (2 pA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='317- 100 200 300 400 500 1400 1500 1600 1700 1800 700 900 10001100 1200 1300 Data point number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='003 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='002- 2mV(2pA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='003- 100 200 300 400 500 600 700 800 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 40009 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Plot (a): filled circles, left axis: Calibration factor kDVM of the DVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' line, right axis: laboratory temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Coloured blocks at the bottom of the plot, and events labeled ‘E1’ and ‘E2’ are explained in the supplementary text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Inset: Log-scale histogram of the difference between adjacent measurements of kDVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The black square in the main plot shows a range of DVM calibration data plotted on expanded axes in plot (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Voltmeter calibrations and measurement time-line In figure S4 we have combined several pieces of infor- mation pertinent to the measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The main graph of plot (a) shows, on the left axis, the cali- bration factors, kDVM, of the DVM recorded during the measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We define the calibration factor as kDVM∆VIND = ∆VREF, where ∆VIND is the change in indicated voltage and ∆VREF is the change in applied reference voltage evaluated from an on-off cal cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each plotted point is averaged from a set of 8 calibration cycles directly against the Josephson array at a nominal volt- age of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='32 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' No data points have been omitted from this plot, and some outlying data points with large er- ror bars are the result of failure of the frequency lock to the Josephson array control electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The pink line plotted on the right axis shows the laboratory tempera- ture, as measured by a sensor integrated into the ceiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Periods when the experiment was not running are vis- ible as gaps in the voltmeter calibration data, and to clarify the experimental time-line, shaded blocks at the bottom of the plot indicate what was happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Four types of activity are indicated: The experimental runs, numbered 1-17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' the weekend calibrations of the ULCA input stage gain GI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The short calibrations of the ULCA output stage RIV, and finally a period of down-time indi- cated by a cross-hatched block when the experiment was stopped due to a fault in the AWG used to generate the pump drive signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Two events marked E1 and E2 are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' E1 marks when an un-used instrument in the experimental rack (a sine wave generator) was switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The reduc- tion of heat produced in the rack caused a noticeable change in the calibration factor of the voltmeter, which was mounted directly above the sine wave generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The fact that this is visible in the kDVM data illustrates the sensitivity of the direct calibrations of the DVM against the Josephson array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The event E1 also lowered the tem- perature of the ULCA, mounted higher up in the rack, reducing ATR by roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='15 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Event E2 marks a dramatic excursion of the laboratory temperature caused by planned maintenance of the air conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This re- sulted in a larger uncertainty assigned to some of the measurements of run 15 because of rapid changes in the ULCA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The transition from stable to fluc- tuating temperature roughly half-way through the mea- surement campaign was co-incident with a transfer of liquid helium into the experimental dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It may also be related to increased activity in adjacent laboratories as activities were re-started and staff returned following relaxation of covid-19 control measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One important contribution to the uncertainty of the current measurement is the stability of the DVM on the 1-hour time taken for a cal-measure sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The in- set to figure S4 (a) shows a histogram of the difference in kDVM between adjacent calibrations during measure- ments, denoted ∆kDVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Generally, the DVM is stable to better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 ppm on time-scales of an hour, but jumps in kDVM of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 ppm sometimes occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As in our previous study13, the uncertainty due to the drift in kDVM was evaluated using a rectangular distribution as ∆kDVM/2 √ 3, so a jump in kDVM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 ppm contributes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='057 ppm to the combined uncertainty in the pump cur- rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The 1-hour DVM calibration interval is therefore consistent with achieving a combined uncertainty in the pump measurement of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To visualise the short- term stability of the DVM in the time domain, plot (b) shows a portion of the main plot on an expanded time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Over this 3-day period, the voltmeter calibration did not drift by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='3 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The voltmeter cali- bration data are of general interest for electrical metrol- ogy, where voltmeters such as the 3458A are commonly used as transfer standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' From the general perspec- tive of evaluating the DVM performance in metrological applications, this data set shows the DVM comfortably exceeding its manufacturer’s 24-hour accuracy specifica- tion of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 ppm on the 1 V range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Calibrations over longer time-scales (not shown) show that the 90-day specifica- tion of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 ppm is also exceeded by typically a factor 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' ULCA calibrations The noise-optimised ULCA was calibrated using a cryogenic current comparator (CCC) bridge, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For the calibrations, the ULCA was hand- carried to an adjacent laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It was specifically car- 100 Counts 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='999983 E1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 △kDVM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 / ppm (00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='999982 KDVM 20 Lab Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='999981 E2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='999980 16, G 6,7G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 1-3 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 8-11 12-15 15 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='999979 15 Jan 29 Jan 12 Feb 26 Feb Date (2021) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='9999814 KDVM HH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='9999812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='9999810 19 Feb 21 Feb Date10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a): Deviations from nominal of (upper plot): ULCA input current gain and (lower plot): ULCA output stage gain, corrected to a standard temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The plot shows all the calibrations performed on this ULCA since its delivery to NPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The time period covered by the measurement campaign is shown as a purple shaded box, and the fixed value adopted for the input stage gain during the measurement campaign is shown as a horizontal dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (b): Deviation from nominal of the ULCA transresistance gain calculated from the data in plot (a), for use during the measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The boxes above each data point show the run numbers covered by the 6 values of trans-resistance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' ried by hand rather than on a trolley to minimise the possibility of mechanical shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As illustrated in the time-line of figure S4 (a), a total of 4 calibrations of the input stage gain GI (nominal value 1000), and 6 calibra- tions of the output gain RIV (nominal value 1 MΩ) were preformed during the measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The over- all trans-resistance gain of the ULCA is ATR = GIRIV (nominal value 1 GΩ)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The results of all calibrations of this ULCA unit since its delivery to NPL are shown in figure S5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The historical behaviour of the input and output gains is different, and resulted in different statis- tical treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The input stage gain does not show any significant drift over the measurement campaign, and fur- thermore, the limited number of additional calibrations before and after the campaign did not give any evidence for long-term drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Consequently it was assumed to be constant during the measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Its value was taken to be the weighted mean of the four calibra- tions during the campaign, shown as a horizontal dashed FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Averaged DVM readings with the pump (a): on, and (b): off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each data point is averaged from one measurement, so is the mean of ∼ 60, 000 DVM readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A scale bar indicates 1 ppm of the 320 pA pump current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' line in figure S5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' On the other hand, the output stage gain shows some drift over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Values of RIV were chosen half way between ‘before’ and ‘after’ calibration values, with uncertainties which included a drift term de- rived from a rectangular distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In this way, five values of ATR were calculated to cover runs 4-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Runs 1-3 were not preceded immediately by any ULCA calibra- tions, so the value of RIV was taken to be the first RIV calibration, in between runs 3 and 4, with an uncertainty derived from a rectangular distribution bounded by the highest and lowest RIV calibrations during the measure- ment campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In other words, we assumed that the drift behaviour of RIV for the few days covering runs 1-3 was similar to the behaviour during the rest of the mea- surement runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The 6 values of ATR with their combined standard uncertainties used to analyse the measurements are shown in figure S5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The remarkable stability of the ULCA offset current is already visible in the raw data of figure S3 (a), and in figure S6 we go further and show the averaged values of the ‘ON’ and ‘OFF’ signals measured by the DVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each data point in this graph is the average of all the ON (plot (a)) or OFF (plot (b)) DVM readings after rejecting the first 300 readings in each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The offset current does not change by more than 2 fA over the 2-month period covered by the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The drift in offset current may be partially attributable to changes in ULCA temperature, but there may also be contributions due to changes in leakage currents through (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8- SG, / ppm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7 -- 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 TI 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 01/10/2019 01/10/2020 Date (b) 8-11 12-15 1-3 4,5 6,7 ppm 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 16,17 T 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 15 Jan 29 Jan 12 Feb 26 Feb Date in 2021(a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='320433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='320432 1 ppm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='32 fA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='320431 15 Jan 29 Jan 12 Feb 26 Feb (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='000003 V > 1 ppm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='32 fA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='000004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='000005 15 Jan 29 Jan 12 Feb 26 Feb11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a): Difference in input current gains GI of two ULCA units for two series of measurements in self-test configuration, in which the test current was alternated between a ‘high’ current of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA and a ‘Low’ current, either 320 pA or 640 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (b): Differ- ence in trans-resistance gains ATR, alternating the test current between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA and 320 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The data plot legend refers to both panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' the electron pump control gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The possible leakage current paths through the device gates were discussed in the supplementary information to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' ULCA linearity The linearity of the ULCA gain is a key assumption in this experiment, because the calibration of GI is done at an input current of ∼ 5 nA and the pump current during the measurement is 320 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One previous investigation set an upper bound on the non-linearity of the overall UCLA transresistance gain ATR at around the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1 ppm level16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We attempted to reduce this upper bound, using two test methods previously demonstrated for the ULCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' First, we compared the input stage current gains of two ULCA units, as was first demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This is called the ‘self-test’ configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A standard ULCA unit, not otherwise used in our experiment, was used as a source to generate a test current for comparing its input stage gain GI,source with the input stage gain of the noise- optimised experimental ULCA GI,measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This self-test configuration is quite straightforward to implement, be- cause the readout DVM measures a small signal derived from the difference in the input gains of the two ULCAs, denoted αGI = GI,source − GI,measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We alternated sets of forward-reverse cycles with test currents of ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA, ±320 pA and ±640 pA to obtain the data of figure S7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The forward-reverse cycle time was 60 s, and the data points are averaged from 100 and 1000 cycles for the ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA and ±320 pA currents respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The back- ground drift of αGI visible in the high current data is due to temperature variation of the ULCAs, but by evaluat- ing the difference between each low-current data points (orange triangles) and the mean of the two adjacent high current data points (green circles), we can extract the current dependence as a mean over 6 cycles of high-low- high current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We obtain the current dependence in αGI between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA and 320 pA as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='029 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' An additional run examined the current dependence between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA and 640 pA (blue diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This data was not evaluated, but clearly the current dependence is around a part in 108 or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For the second test, we measured the current depen- dence of the difference in the overall trans-resistance gains of the two ULCAs, again with the standard ULCA in ‘source’ mode, and the noise-optimised experimental ULCA in ‘measure’ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This test configuration is il- lustrated in figure 6 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It is less straightforward to implement than the self-test configuration, because the voltage outputs of the source and measure ULCAs have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We implemented a protocol equiv- alent to figure 7b of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A single DVM could be connected to either the source or measure ULCA us- ing an automated switch - the same switch that was used in the main experiment to connect the DVM ei- ther to the ULCA output or the JVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One cycle con- sisted of four segments of data: the test current was applied with both polarities with the DVM connected to the source ULCA, recording a forward-reverse differ- ence voltage ∆Vsource and then the test current was ap- plied with both polarities with the DVM connected to the measure ULCA, recording a difference voltage ∆Vmeasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Acquiring one cycle took 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Assuming that the DVM calibration factor does not change on this time- scale, The ratio of ULCA transresistance gains is given by αATR = ATR,source/ATR,measure = ∆Vmeasure/∆Vsource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We are interested in whether the ratio of gains depends on current, so as in the tests of GI linearity, we alternated 1000 cycles at ±320 pA test current, with 100 cycles at ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA test current to yield the averaged data points in figure S7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Similarly to the data of figure S7 (a), we averaged the high-low-high differences, to obtain the current dependence of αATR as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='023 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Of course, this data does not conclusively rule out non- linearity in the ULCA unit used for the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It only gives information on the linearity of the differ- ence in the gains of the two ULCA units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It is a slightly stronger test than the one published in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 16, how- ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' While that measurement used two nominally iden- tical noise-optimised ULCAs, our measurement used a standard ULCA in the ‘source’ role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The different values of resistors used in the current scaling networks make it less likely that both ULCA units would have the same (a) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8×106 αG, / ppm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='9×10-6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0×10-6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='1×106 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='2×10-6 Measurement number (b) ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 nA 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='6 ± 320 pA ± 640 pA 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='9 Measurement number12 current-dependence to the gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Statistical tests and data set rejection As mentioned in the main text, the pump state, as doc- umented by the ‘pump maps’, changed during the mea- surement campaign, with some obvious dramatic changes occurring during some measurements, and more subtle changes during other measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Even if the pump map was stable, some of the measurements close to the edges of the current plateaus could be affected by small fluctuations in offset charge, leading to relatively large changes in pump current as the operating point drifted on and off the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' It could not generally be as- sumed that the pump current sampled by a measure- ment lasting more than 10 hours represented a station- ary mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each measurement was therefore subjected to a statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Recall from supplementary section S3, that the pump current from each sequence was evaluated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This yielded m values of IP, denoted IP,m with uncertainties U(IP,m), where m is the number of se- quences in the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' If all the IP,m are sampling the same value of pump current, on average the stan- dard deviation of the IP,m, σ(IP,m) will be equal to the mean of the uncertainties, ⟨U(IP,m)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We propose the ra- tio σ(IP,m)/⟨U(IP,m)⟩ = R as a statistical measure of the stationarity of the data, and in figure S8, we plot a his- togram of this quantity (grey bars, right axis) for the 64 measurements performed during our campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' We also plot (red bars, left axis) a histogram of the same quantity obtained from 1000 simulated measurements, in which the simulated raw data, both for the measurement and calibration cycles, was generated from a stationary mean multiplied by Gaussian white noise with the same stan- dard deviation as the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' As expected, the most probable value of R for this simulated stationary data is 1, and the probability of obtaining a measurement with R > 2 from a set of 1000 measurements becomes neg- ligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Since we only performed 64 measurements, we assigned a cutoff of R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7, and rejected measurements with R > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Comparing the histogram of the measured data with the simulation, it is clear that a significant number of data sets have an R value which would be improbably high if the pump current was constant dur- ing the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' This is actually expected, for the reason that some of the precision measurements were se- lected with control parameter values close to the edges of the current plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For these measurements, small fluc- tuations in offset charge during the measurement (equiv- alent to a drift in the control parameters) would cause the pump current to drift away from ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' To see the accept / reject criteria in action, two exam- ple measurements from run 10 are plotted in figures S8 (b) and (c), with the corresponding R values marked with red and green arrows on the x-axis of panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The data of panel (b) clearly shows a decrease in the pump current, and it would be tempting to reject this data set based just FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a): Histograms of the quantity R, defined in the sup- plementary text as the ratio of the standard deviation of the m values of IP calculated for each measurement (m is sequence number) to the average uncertainty of IP,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The grey bars re- ferred to the right axis are for the measured data, and the red cross-hatched bars referred to the left axis are for 1000 sim- ulated measurements assuming a stationary mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' A vertical dashed line shows the R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7 rejection threshold derived from the probability of the simulated data having an R greater than this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Panels (b) and (c) show IP,m for two example mea- surements from run 10, with (b): R > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7 and (c): R < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The R values for these data sets are indicated with red and green arrows respectively on panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' on this time-domain visualisation of IP,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' However, the definition of the R parameter makes this otherwise sub- jective process more quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Altogether, 14 mea- surements during the entire measurement campaign had R > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' full data set Due to instability of the pump after run 5, only the data from runs 1-5 are analysed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The increasing instability is visible in the pump maps of figure S10, and also in the increasing number of runs which failed the stationary mean test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In figure S9, we present all of the precision data on linear axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Plots (a,b,c) show all of the measurements on expanded y-axes, and plots (d,e,f) show the sub-set of the measurements which passed the stationary-mean test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Figure S10 shows the full set of ‘fingerprint’ pump maps obtained before and after each precision measurement run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' For data integrity (a) 200 15 Simulated data measured data Count (Simulated) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Count (Measured) REJECT 10 100 5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 0 0 0 2 4 5 6 7 8 R (b) (c) Run 10, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='835 V Run 10, V = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='775 V ENT ENT wdd REJECT 0 ACCEPT p,m 2 12 hours 0 2 4 6 8 10 0 2 4 6 8 10 seguence m seguence m13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' a-c: Deviation of the pump current from its nominal value as a function of (a): Exit gate voltage, (b): Entrance gate voltage and (c): AWG output amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' All of the measurements from the 17 runs are shown in these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' One data point in panel (a) is off the y-axis scale, and is indicated by an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (d), (e) and (f): the sub-set of data in plots (a), (b) and (c) respectively, which passed the stationary mean test, on expanded axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' In each plot, vertical dotted lines indicate fixed values of the scanned parameter for runs in the other plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Error bars indicate combined standard uncertainties UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' purposes, this figure also includes the 4-digit hexadecimal file identifier for the precision raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' (a) (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='0 9,105,7,15 8 1 4 11 2 12 14 17 6 6 run 6 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='85 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='80 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='75 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='48 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Thumbnail pump maps measured before and after each precision scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each pump map is an inverted grey-scale derivitive plot of the current similar to the one shown in figures 1(b) and 2(a) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The axis limits are the same for each thumbnail: The x-axis is VEXIT, from −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='7 V to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='8 V, and the y-axis is VENT, from −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 V to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content='4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Each horizontal row of the table represents a precision run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The middle cell contains some text data describing the run, including the 4-digit hexadecimal file number identifying the raw data set for the precision run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' The left-most cell shows the pump map recorded before the run, and the right-most cell shows the pump map recorded after the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Missing pump maps for runs 9,10 and 11 were due to software crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' Red arrows highlight runs in which the pump map changed dramatically during the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} +page_content=' 8 Days down-time due to AWG issue' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfZwoa/content/2301.04499v1.pdf'} diff --git a/VNAyT4oBgHgl3EQfhfi9/content/tmp_files/2301.00379v1.pdf.txt b/VNAyT4oBgHgl3EQfhfi9/content/tmp_files/2301.00379v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2d21b435536a1ac10b60b4b53f863a9318f8210 --- /dev/null +++ b/VNAyT4oBgHgl3EQfhfi9/content/tmp_files/2301.00379v1.pdf.txt @@ -0,0 +1,1294 @@ + +Review Article +A review of Implementation and Challenges of Unmanned Aerial Vehicles for Spraying +Applications and Crop Monitoring in Indonesia + +Authors: +Muhamad Rausyan Fikri1,2, Taufiq Candra3, Kushendarsyah Saptaji4, Ajeng Nindi Noviarini5, Dilla Ayu Wardani3 +1. +Automation Technology and Mechanical Engineering, Faculty of Engineering and Science, Tampere +University, Tampere, 33720, Finland +2. +Information Systems, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, +Indonesia +3. +Industrial Engineering, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, +Indonesia +4. +Mechanical Engineering, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, +Indonesia +5. +Computer Science, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, +Indonesia + +Abstract: +The rapid development of technology has brought unmanned aerial vehicles (UAVs) to become +widely known in the current era. The market of UAVs is also predicted to continue growing with +related technologies in the future. UAVs have been used in various sectors, including livestock, +forestry, and agriculture. In agricultural applications, UAVs are highly capable of increasing the +productivity of the farm and reducing farmers' workload. This paper discusses the application of +UAVs in agriculture, particularly in spraying and crop monitoring. This study examines the +urgency of UAV implementation in the agriculture sector. A short history of UAVs is provided in +this paper to portray the development of UAVs from time to time. The classification of UAVs is +also discussed to differentiate various types of UAVs. The application of UAVs in spraying and +crop monitoring is based on the previous studies that have been done by many scientific groups +and researchers who are working closely to propose solutions for agriculture-related issues. +Furthermore, the limitations of UAV applications are also identified. The challenges in +implementing agricultural UAVs in Indonesia are also presented. + +Keywords: +Unmanned aerial vehicle, agricultural UAV, spraying, crop monitoring. + + + +1. Introduction +According to the United Nations (UN), the world population is projected to reach 9.7 billion people +in 2050 (UN, 2015). This vast population would potentially double the food demand in the future +(Hunter et al., 2017). Consequently, the ever-growing population that would emerge could cause +food shortages in the future. This issue has become a severe problem since the Food and +Agriculture Organization (FAO) announced similar speculation in which the current agricultural +production must be increased by 70 percent by 2050 to meet the increasing demand for high- +quality food (Mundial, 2021). Many people suffering from hunger become a signal of how severe +the food shortage is, and it was reported that more than 820 million people in 2018 were considered +undernutrition (WHO, 2019). Surprisingly, the earlier data mentioned shows the increasing +tendency towards people suffering from hunger since only around 690 million people were +considered suffering from hunger in 2015. This kind of data indeed contradicts the second +Sustainable Development Goals (SDGs) approved by the United Nations (UN) in 2015 with the +aims to eradicate hunger and ensure access to food for all people (UN, 2015). On the other hand, +the labor shortages in the agricultural sector due to the aging population and the decreasing number +of workers have exacerbated the situation. The lack of laborers in the agricultural field would +expand the cultivation area per worker and increase the workload of workers (Seo & Umeda, +2021). +Alternative and innovative solutions to increase food production in dealing with those +issues are needed. One way to increase food production is to promote the internet of things (IoT), +robotics, and artificial intelligence (AI). By shifting the workforce to the technology's utilization, +it is expected to solve the labor shortages and improve farmers' skills. This transformation is +inseparable from revolutionary industries that constantly bring industrial innovations. Some +industrial innovations found in recent years, such as sensor technologies, big data, and artificial +intelligence (AI), have been considered as the beginning of the "Industry 5.0" era by the European +Commission (EC) (EC, 2021). The emergence of technologies characterized by advanced +digitalization is believed to play a significant role in increasing production flexibility and making +the value chain more robust so that technology could minimize the farmers' workload and improve +the speed and accuracy of the work. + + +Among the technologies mentioned earlier, unmanned aerial vehicles (UAVs) are one +viable way to increase food production. UAVs are less expensive and have contributed to many +areas in agriculture, including spraying, weed recognition, and crop monitoring (Mogili & Deepak, +2018). UAVs' timely and reliable information about the production, yield and crop management +would become beneficial to ensure food safety and security for stakeholders such as farmers and +sales units (Martos et al., 2021). UAV technologies in agriculture could also enable the complete +monitoring of crop conditions from the beginning of the growing season until the end of harvest +(Silver et al., 2017). +Some leading technologies are possible by implementing UAVs in the agriculture sector. +Therefore, this paper focuses on reviewing UAVs applications for spraying and crop monitoring +in the agricultural field. Some research results on the use of UAVs in spraying and crop monitoring +are discussed thoroughly to highlight the use of UAVs and the characteristics of the farming sector. +Some limitations exist during UAVs implementation are also reviewed to reveal the gap of UAV +implementation in the agriculture field. The rest of this paper is organized as follows. Section 2 +describes UAVs' history and the classification of UAVs. Section 3 describes the application of +UAVs focusing on spraying and crop monitoring. Section 4 provides some limitations in adopting +UAV technologies in the agriculture sector. In section 5, the challenge in implementing UAVs in +Indonesia is discussed. The last section provides the conclusion. + +2. Agriculture in Indonesia and Its Challenge +2.1. Current Condition +Agriculture programs in Indonesia have been a big agenda at the national level, such as +National Agenda 21, National Development Programs, and Agricultural and Forestry +Revitalization Strategies, encouraging Indonesia to adopt sustainable agriculture. The Central +Planning Authority (BAPPENAS), the Ministry of Agriculture, and the Environment Ministry +have implemented these ideas. Most of these plans include components suitable for effective +environmental management of Indonesian agricultural exports. +The motivations for using these tactics have shifted over time, and they seem to be +responding to a variety of distinct trends. First and foremost, Indonesian national plans have +prioritized socio-economic objectives above ecologically sustainable ones. Nonetheless, +environmental concerns have become more critical, as evidenced by recent reforms and the + + +increasing frequency of ecological issues in strategic documents. Second, strategy papers also +show a change in direction as the combination of means changes, with less focus on laws and +regulations and more attention to the means for market creation and voluntary methods over time. +The tensions between diverse skills and conservation goals and local revenue-generating needs +have led to different patterns of success in different states across the country. Significant +advancements have been achieved in modernizing agro-environmental rules, made possible by +increased information and worldwide best practices. The extent to which environmental hazards +pose local or global dangers, the degree of environmental degradation of a particular product, and +the availability of legal, enforcement, budgetary, and regulatory capacities for sub-national +governments all influence the choice of the policy tool. +For practical reasons, Indonesian policymakers have used a range of mechanisms to +minimize agriculture's environmental footprint, including direct regulation, market creation or +market modification incentives, voluntary and beneficial solutions, and market modification +incentives. Policies are implemented via legislative and regulatory mechanisms, which are +probably targeted at plantation states and large farms. It is essential to note the existence of +obligatory ISPO standards (in the section on local regulatory instruments), since they have just +recently been adopted as a result of voluntary standards being adopted as mandatory. Additional +factors that impact policymakers' choices to implement a particular instrument include the +potential efficacy of the instrument in comparison to its costs and the capacity of the policymaker +to enforce the instrument in the face of likely political opposition. In this respect, implementing +regulatory and legislative tools seems to be the most effective method of monitoring prominent +investments, such as planting restrictions and the demand for environmental impact assessments. +According to the findings of the Indonesian research, foreign pressure had a role in the spread of +planting restrictions throughout the country. In addition, the implementation of regulatory +instruments may be most effective when their administrative and monitoring costs are already +integrated into a current administration, such as indirect product charges for import limitations, +which are already embedded into an existing administration. + +2.2. The Challenge in Indonesia’s Agriculture +One of the factors is the limited availability of agricultural land in Indonesia due to land +reform, which is widespread in big provinces. As the population rose, so did the need for housing. + + +As a result, developers exploit a large portion of agricultural land to construct real estate. The +growth in the number of people also increased demand for trade and tourism, contributing to +increased demand for land. Farmers could not be faulted for selling their farms in this scenario. +Farmers were driven to sell their lands due to a lack of knowledge and technology, high agricultural +costs, and rising necessities. Farmers in Indonesia with low levels of education have little choice +except to work outside the agricultural industry; therefore, those who do not own land are tenant +farmers. Food price increases should be a dream come true for farmers, as their revenue would +almost certainly rise. Unfortunately, because most farmers in Indonesia are tenant farmers, it has +become a boomerang for their wellbeing. The rise in food prices has little effect on the well-being +of Indonesian farmers. Their income remains minimal, and they must continue to purchase their +basic necessities at market prices. Those who own land have benefited from the growing price. +Furthermore, the general public's perception of farmers is that they do not do a good job. The +younger generation is interested in non-agricultural jobs, such as parenting a farmer's child. +Farmers' regeneration is hampered as a result, and many opt to sell their land to be established as +capital or to work in the non-agricultural sector. +As the world's population rises at an alarming rate, agriculture must expand to supply the +growing demand for food against all odds. The agriculture sector is the most vulnerable to the +impact of integrating fresh, modern innovations in eradicating environmental-related challenges +and enhancing the current productivity rate. Now, the question is, how could we possibly do this? +Marking a third wave of the “Green Revolution”, the concept of precision agriculture with +technological help such as Unmanned Aerial Vehicle (UAV) has become popular nowadays in the +vast area of agriculture due to its tremendous benefits. Farmers and managers can boost operational +efficiency, cut expenses, minimize waste, and improve the quality of crops with the aid of accurate +data. Overall, technology has been a key component behind agricultural development and other +discoveries brought into the industry. + +3. Unmanned Aerial Vehicle (UAV) +3.1 UAVs History +An unmanned Aerial Vehicle (UAV) is an aircraft with no pilot on board; in other words, +it refers to auto-piloted aircraft (Ahmad et al., 2021). The unmanned type of aircraft can be +operated in two ways, either by a human operator or autonomously operated under the control of + + +an onboard computer (Pablo et al., 2020). According to the US Department of Defense (DOD), +UAV can be described as either a single air vehicle (with equipped surveillance sensors) or a UAV +system (UAS) that consists of three to six air vehicles, a ground control station, and support +equipment (Gertler, 2012). Furthermore, UAVs are often associated with remote sensing in +carrying out their task. This remote sensing is commonly known as UAV remote sensing, which +combines UAV and remote sensing technology that can quickly capture information about land, +environment, and resources for further data processing (Shi & Liu, 2011). In the US and other +developed countries, UAV remote sensing has been applied in many fields such as forestry, +environmental protection, land, and military (Xiang & Tian, 2011). UAV remote sensing are used +because it could be deployed quickly in repeated times. In addition, they are less costly, safer than +piloted aircraft, flexible in terms of flying height, and able to obtain very high-resolution imagery +(Yang et al., 2011). +The term unmanned aerial vehicles are also known as remotely piloted aircraft (RPA). +Even though the terms UAV and RPA are interchangeable, the term UAV is commonly used by +aviation organizations (Santos et al., 2019), while the term RPA is widely used in Europe +(Gallardo- Saavedra et al., 2018). Back then, in 1930, UAVs were also known as “Queen Bees” +(Vroegindeweij et al., 2014) and were initially used for military purposes (Muchiri & Kimathi, +2016). In 1986, UAVs that work specifically in agricultural contexts were introduced by launching +UAVs for Montana’s forest fires monitoring and followed by the capture of enhanced image +resolution using UAVs in 1994 (Muchiri & Kimathi, 2016). Then, a more complex UAV model +was finally developed by Yamaha through “Yamaha RMAX,” with the primary function for pest +control and crop monitoring application (Mogili & Deepak, 2018). This UAV model is used for +pesticide spraying in rice fields of Asia. As opposed to ground-based sprayers, the pesticides +deposition of this UAV model is quite similar, but this UAV model is used explicitly for a high- +value crop environment (Giles & Billing, 2015). + +3.2 Classification of UAVs +Generally, there are three types of UAV platforms: fixed-wing, rotary-wing UAVs, and +non-wing UAVs (Figure 1). A fixed-wing UAV resembles an airplane and requires a runway or +Modelsurface (meadow or road) for take-off and landing (Pederi & Cheporniuk, 2015). This kind +of UAV uses thrust and aerodynamic lifting forces to fly. It has a larger size than a rotary-wing + + +model and is mainly used for aerial mapping, spraying, and photography over a wide range of time +(Li & Yang, 2012). This UAV type typically lacks hovering while offering high-speed flights for +longer durations (Ahmad et al., 2021). The gliding capabilities possessed by fixed-wing aircraft +could enable greater flight endurance, allowing them to operate over longer distances (up to 15-20 +km) (Paneque-Gálvez et al., 2014). + + +D +Figure 1. Illustration of Basic UAVs (A) Fixed-Wing UAV (B) Rotary-Wing UAV (C) Combinational Concepts +(source: Ahmad et al., 2021), (D) Blimps (source: Tao et al., 2018) +On the other hand, rotary-wing UAVs is primarily categorized into the helicopter and +multi-rotor types. The helicopter type of rotary-wing UAV has a unique feature with a large +propeller atop the aircraft. It is widely used for spraying and aerial photography (see Figure 2) +(Swain et al., 2010). They can hover, vertical takeoff, and land with nimble maneuverability while +exhibiting low-speed flight for a shorter duration (Ahmad et al., 2021). In comparison, the multi- +rotor models are called according to the number of rotors (Kim et al., 2019). Quadcopter, +hexacopter, and octocopter are some multi-rotors UAVs that are widely known (Figure 3). These +UAVs are lifted and propelled according to the number of rotors (Mogili & Deepak, 2018). + + + + +Figure 2. Single Rotor/Helicopter UAV Type (source: Huang et al., 2009) + +Figure 3. Multirotor UAVs, (A) Quadcopter (source: Spoorthi et al., 2017) (B) Hexacopter (source: Yallappa et al., +2017) (C) Octocopter (source: Wallace et al., 2016) +For example, the rotor movement of a quadcopter is responsible for generating the lift of a +quadcopter. In a quadcopter, each of two rotors moves in an opposite way of which two rotors turn +in the clockwise direction and the other two turn in the anticlockwise direction. The movement of +the quadcopter around the axis consists of yaw (clockwise and anticlockwise), pitch (backward +and forward), and roll angles (right and left). The quadcopter uses a control system to balance the + +SR200 +SR20B +c +thrust of each rotor in order to support the UAVs' lift and yaw, pitch, and roll angles (Mogili & +Deepak, 2018). This control system turned out to be practical to produce a stable flight of the +UAVs (Patel et al., 2013). Moreover, two quadcopter configuration types include the plus (+) and +cross (X) models, as shown in Figure 4. The cross model is more popular between the two models +due to its stability (Kedari et al., 2016). + + +Figure 4. Quadcopter Configuration Model (A) Plus configuration (B) Cross Configuration (source: Mogili & +Deepak, 2018) +Furthermore, these multirotor UAVs have extended their functionalities by equipping appropriate +sensors such as vision, infrared, multispectral, and hyperspectral cameras. The expansion of UAV +features brings great influences, especially in adding the capabilities of the UAVs. Those sensors +are used to obtain data such as vegetation, reflectance indexes, and leaf areas in order to provide +information about the current state of crops. With this information, farmers can make possible +remedies or policies (weed control, fertilization, irrigation) according to the condition of the crops +(Gonzalez-De-Santos, 2016). +Further, non-wing UAVs have been developed to cope with the long endurance of flying robots +and are lighter than air (LTA) such as Blimp. A blimp is identically has a larger size than fixed- +wing and rotary-wing and cushioned with a helium-filled envelope, making the robot safe to fly +indoors, causing no threat to humans and the surroundings even with collisions (Tao et al., 2018). +With the lifting force provided by air buoyancy, the blimp has flight endurance for more than 2 +hours (Cho et al., 2017). Blimp is the one type of UAV lighter than the air UAVs (Krishna, 2021a; +Thusoo, 2021; Tsouros et al., 2019). It has a balloon-like body created from tough fabric and filled +with helium gas (Prisacariu et al., 2019). Blimp is notorious as the dirigible and was firstly + +PitchForward +PitchForward +Roll Left +Roll Left +Roll Right +Roll Right +A +B +PitchBackward +PitchBackward +designed in 1852 by Henri Giffard (Krishna, 2021a). This type of UAV has high endurance and +can flow longer than other types of UAV, approximately 1-3 weeks travel (Krishna, 2021a). Due +to its characteristics, the blimp is advantageous in numerous aspects of life, including the military +and agriculture. It was purposeful as the cargo transit and the sentinels between the missile site +and the military camp (Krishna, 2021a). It is also used to monitor the long-distance aspect, +especially in urban traffic and buildings. In PA, it monitors crop production, identifies the plant’s +disease, erosion, and detects either flood or drought conditions (Krishna, 2021a). Unlike the other +UAV, the blimp is also considered a safe technology since it remains in the air and did not collide +even if it loses its power (Krishna, 2021a; Tsouros et al., 2019). Besides, the University of Leeds +research reveals that blimp is chosen as the cheapest UAV to conduct a terrain survey (Krishna, +2021a). Thus, it could provide a detailed but further explanation of crops, surface minerals, +vegetation, and water quality. +4. Application of Blimp in Agriculture +Compared to other UAVs, the blimp has a pivotal function, including load capacity, safety, +quality, and environmental safes, which make it useful for everyday life. Researchers stated that +blimp could carry up to 400 tons of load with 110-160 km speed travel (Krishna, 2021a). Besides +its ability to fly longer, blimps could land on every land’s surface. In the case of environmental +footprint, blimp could reduce carbon dioxide emission (Krishna, 2021a). +Nowadays, there are several features of blimps with specific purposes. Those five types +include tethered, untethered, remote-controlled blimp, Giga blimps, and hybrid blimps (Krishna, +2021a). Tethered blimp (aerostats) ables for free flight and a steady anchored flight using solid +tethers (Mahmood & Ismail, 2020). A tethered blimp enables accurately obtaining a stereoscopic +image that might cover 35 m2 to 20,000 m3 (Krishna, 2021a). Meanwhile, the untethered is +commonly used in cargo transit, travel, and aerial surveillance. Thirdly, the remote-controlled +blimp uses the robotic that use the program flight plan to fly. There are two different types of +remote control blimps; small and large. The small blimp is commonly used for advertisement, and +the giant blimp (Giga blimp) is significant for military purposes. Lastly, the hybrid blimp, a new +modification of blimp that is prone to extreme conditions, can transport goods and civilian travel. +Unlike the other popular UAVs, multirotor and fixed-wing, the number of blimps applied +in PA is insignificant (Krishna, 2021b; Tsouros et al., 2019). However, considering its strength +characteristic and function, blimp could start considering the blimp as the priority to improve the + + +PA. (Mogili, Rao Deepak, 2021) stated that the integration of blimp with quadcopter aerial +automated pesticide sprayer (AAPS) is pivotal for pesticide spraying in lower altitudes by +following the GPS altitude. This technology is controlled with an android app to create an effective +cost-saving (Mogili, Rao Deepak, 2021). Besides, other researchers use the blimp with a Charge- +Couple Device to identify the Leaf Arena Index and biomass in soybean and paddy fields +(Chilonga & Kiswisch, 2016). The results showed that the technology is stable and provides high- +resolution images (Chilonga & Kiswisch, 2016). (Ponti et al., 2016) also stated that the blimp could +be practically used to monitor the bean crop dataset using the combination of 1/2.3 inch of CCD +sensor, 6.3 to 18.99 lens focal, and 10 Mega Pixel digital camera. The research found 29,556 +examples of the positive dataset and 11404 negative datasets in Brazil (Ponti et al., 2016). + The blimp could be significantly used in monitoring agriculture (Mahmood & +Ismail, 2020). For instance, the research conducted by (Bajoria et al., 2017) proposed a tethered +aerostat system that could be used to mitigate the vertebrate mammal and bird hazard, which is +positively contributed to 18%-43% of crop loss in India. The proposed design has been proven to +carry about a 50 kg payload and 25 m/s ambient wind speed (Bajoria et al., 2017). Other research +revealed that tethered aerostat combined with the electro-optical, acoustic, and laser-based sensors +could scare the bird and other pests (Krishna, 2021a). To mitigate the occurrence of pests, the other +researchers also create a Hawk Kite and Helikite aerostat hybrid that is purposeful to scare some +of the bird’s species, including the pigeons, seagulls, parrots, rooks, blackbird, etc. (Perigrine Ltd. +2018). + Besides, a tethered blimp (aerostat) could provide aerial images surrounding the +natural disaster zone. This image helps identify the cropped field due to the flood, large soil +erosion, drought, and crop loss due to the pest attack (Krishna, 2021a). It is also used to maintain +the field quality because the aerostat could lofty 24 hours surveillance above. Thus, it could help +the farmer control and watch the field without going directly to the farm. Besides the +aforementioned reasons, the aerostat could reduce the enormous cost of capturing the crop’s data. +The farmer could use the aerostat to pertain the crop data and send its digital data in a computer +program (Krishna, 2021a). Furthermore, the integration of aerostat with the sensor could help the +farmer obtain continuous data of Nutrients crop status. Thus, it could help the farmer evaluate the +number of nutrients placed for the crop (Krishna, 2021a). +5. Agricultural Unmanned Aerial Vehicle + + +In recent years, the application of more advanced technology in agriculture has gained +more attention. Several technologies such as satellites, UAVs, Geographic Information System +(GIS), Global Positioning System (GPS), and many other applications of technologies have been +able to pave their way into the agricultural field. The process modernization and industrial +revolution that brought many innovations in technology applications have opened the gate of +precision agriculture (Ahmad et al., 2021). Precision agriculture is defined as the utilization of +technology in the agricultural production system in order to determine, analyze, and manage the +farming factors to increase crop productivity, ensure environmental sustainability, and improve +business profitability (Unal & Topakci, 2013). This precision agriculture is seemingly possible to +increase food production due to its effective functionalities under pressure conditions such as the +ongoing reduction of arable land, the increase in global population, and the high cost of farming +due to wastage in the use of water and chemicals (Abdullahi et al., 2015). +UAVs have gained popularity as a pivotal part of precision agriculture to ensure +agricultural sustainability (Rani et al., 2019). The use of UAV, which plays a key role in reducing +the data acquisition time and processing cost, is considered as the main reason for its popularity +(Berni et al., 2009). The rapid development of UAVs that extend its functions to aerial photography +and video and weather forecasting, with the support of spatial data collection to help stakeholders +create policies and decisions, has attracted many parties to UAVs (Sylvester, 2018). +Moreover, the market of UAVs that is estimated to reach up to US$200 billion by the end +of 2020 has successfully described the popularity of UAVs as well (Puri et al., 2017). The huge +estimation of the total market of UAVs has shown that the market value of UAVs has doubled +within three years. PwC’s Drone Powered Solutions team quantified that the total market value of +UAVs is about US$127.3 billion in 2017 (Silver et al., 2017). Although the estimation of the +market value of UAVs in 2020 and the total market value of UAVs in 2017 are not exclusively +focused on agriculture sectors, this number was sufficient to portray the UAV market development. +In addition, the affordable cost of UAVs is another factor that influences its popularity +nowadays. This low-cost factor motivates many small companies to switch to using UAVs with +its simple and easy-to-understand operating systems in serving some activities in the agriculture +sector, including area measurement and crop monitoring (Hatfield & Prueger, 2010). + +6. Applications of Unmanned Aerial Vehicles in Precision Agriculture + + +Currently, there are numerous applications of UAVs in precision agriculture. They are used +in many areas of crops. This section introduces two applications of agricultural UAVs i.e., spraying +and crop monitoring. The summary is shown in Table 1. + +6.1 Spraying +Prior to the implementation of UAVs for spraying, the farmers used spraying bags to spray +pesticides all over the farm (Spoorthi et al., 2017). Manual spraying is very dangerous for the +workers because the measure of pesticides per hectare of agricultural land correlates to the risk of +worker ailments. The heavy bag carried by the farmers could also make them get strained. +Fortunately, the use of UAVs can reduce the usage of pesticides, maximize efficiency, and improve +the well-being of the workers (Luck et al., 2010; Pyo, 2006). + +Manual spraying is also considered ineffective for spraying the farmland because the +pesticides may not spread evenly in every area. The excessive use of chemicals or pesticides in +certain agricultural land is responsible for loss of soil fertility, soil degradation, and subsequent +degradation of water-related ecosystems. In addition, the chemicals or pesticides absorbed by the +crops and natural resources such as water and soil might cause pollution risk and severe health +impacts for the environment. Therefore, UAVs are required to minimize such dangers by helping +the spraying process specifically in the targeted area (Daponte et al., 2019). + +In addition, to pave the way towards sustainable agriculture, the employment of UAVs in +the agriculture sector also offers other benefits in terms of their operation. The implementation of +UAVs can make the process relatively faster and cheaper than other methods (Rani et al., 2019). +The efficient usage of UAVs was also widely reported in the literature. The use of 3WWDZ-10A, +XAG is successfully effective in controlling Spodoptera frugiperda, an invasive sugarcane crop +pest, by spraying pesticides (Song et al., 2020). In addition, the use of UAV (DJI Phantom 3) is +found to be effective in spraying pesticides in the nominated areas using electronic traps (E-traps), +which can count the insect and transmit the data to the server (Psirofonia et al., 2017). Studies have +also found that UAVs might improve the accuracy of control over crops by equipping the UAVs +with precision control algorithms (Faiçal et al., 2016). In summary, it is proved that the UAV +application offers several advantages in reducing the workload of the farmers and providing +efficient and low-cost service in the agriculture field. + + + +Some issues have been identified regarding the use of UAVs in crop areas overlapping and +outer edges, despite the advantages during UAVs implementation. These issues arise because some +crop fields are not fully covered properly during spraying, leading to reduced crop quality in +particular areas. To overcome this problem, the swarm of UAVs was introduced in a control loop +algorithm during UAV operation (Yao et al., 2016). Swarm control is considered a practical +technology since it could control multiple UAVs via one operator or program. In the swarm control +method, the operator can select an efficient shape based on the application so that the swarm can +be centralized, decentralized, and distributed according to the desired shape (Kim et al., 2019). In +addition, the spraying pesticides process on the crop is then organized by considering the feedback +from the Wireless Sensor Networks (WSNs) deployed in the field (Costa et al., 2012). The control +loop is responsible for the communication of each UAV in adjusting the UAVs’ route according +to the changes in wind speed and the number of messages exchanged in between (Faiçal et al., +2014). During this communication process, a short delay might exist in the control loop since the +UAVs need time to analyze the data from WSN to route further (Kale et al., 2015). An automatic +navigation spraying system of UAV was developed to direct the UAV in a particular area (Xue et +al., 2016). + +Another way in using swarm control is through task allocation technology. This technology +is currently used in mapping agricultural lands (Barrientos et al., 2011). In order to use swarm +technology, a route is assigned to each UAV. A route is built by dividing each region or area +among several UAVs. A map of the area is obtained by capturing a single picture through a camera +sensor attached to a UAV (Ju & Son, 2018). This kind of technique requires K-mean algorithms +in order to reduce complexity and prevent collision among UAVs. The most significant aspect of +this swarm technique is the combination of algorithms that come in handy in maintaining the +consistent distances between UAVs. These consistent distances allow linear and nonlinear control +that resist strong external influences (Kim et al., 2019). The implementation of swarm techniques +and task allocation in agriculture can be seen in Figure 5. This application most likely improves +the accuracy of agricultural operations, reduces operator control efforts, reduces work time, and +induces battery and payload shortages. + + + +Figure 5. (A) Swarm Control (source: Ju & Son, 2018) (B) Task allocation (source: Barrientos et al., 2017) +In the spraying using UAVs, the sprinkling system is mounted at the lower region of the +UAV, which has a nozzle under the pesticide tank in order to sprinkle the pesticide downstream in +the field. An appropriately selected nozzle is a significant part of pesticide application since it is a +significant factor in determining the amount of spray applied to an area, the coverage obtained on +the target surface, the amount of potential drift, and the uniformity of application (Ru et al., 2014). +Furthermore, the sprinkling system generally has two modules: the controller and the sprinkling +system. The sprinkling system consists of the spraying content, either pesticides or fertilizers. +Meanwhile, the controller is used to trigger the nozzle of the sprayer. The controller efficiency +could be increased by using a PWM controller in pesticide applications (Zhu et al., 2010; Huang +et al., 2009). +Another important component of the sprinkling system is a pressure pump used to put +pressure into the pesticide in the tank to flow through the nozzle (Tang et al., 2018). This pressure +pump works closely with the motor driver integrated circuit in completing their task in putting the +pressure to sprinkle the pesticide (Mogili & Deepak, 2018). The full spraying system can be seen +in Figure 6. The integration between UAV and the spraying system is expected to provide a +potential platform for pest management and vector control, an accurate site-specific application +for a large crop field. For this objective, a heavy lift of UAVs is required to cover many areas +(Sarghini & De Vivo, 2017). + +RemoteSensingTaskusingcameramountedonUAV +(MultipleUAVs) +contro +UAV +ensing +Agriculturalfield +Teleoperationcontrolwith +device +Hapticdevice(NovintFalcon) +A +B + +Figure 6. Spraying System Structure Diagram (source: Tang et al., 2018) + + + +6.2 Crop Monitoring +A crop monitoring is defined as predicting the yield or crop quality by analyzing the available crop +data (Kim et al., 2019). It is essential for optimizing crop production because it can assess crop +health and indicate bacterial or fungal infections. Furthermore, the crop scanning produced by +visible and near-infrared (NIR) light could reflect the different amounts of green light and NIR +light that are extremely essential in producing multispectral images that can track changes in crops +assess their health (Costa et al., 2012). The farmers can plan and apply remedies more precisely +according to the identified issues with such information. It makes the fast response to bacterial or +fungal infection, and infestation comes in handy and increases crop endurance into future issues. + +The use of UAVs for crop monitoring is also highlighted due to their ability to monitor a +large farm. By utilizing the UAVs, a large area of farmland can be fully monitored. It reduces the +significant time and labor required for monitoring large farm areas manually (Kim et al., 2019). +Aasen et al. (2015) reported that the UAVs application offers low crop monitoring costs. This is +due to the use of lightweight sensors and the implementation of low-flying UAVs (see Figure 7). + +Electronic speed control +Tank +Sprayboom +Nozzle1 +Nozzle2 +Waterpump + +Figure 7. UAV Platform (source: Aasen et al., 2015) +A camera-equipped UAV can also observe the crop with different indices (Simelli & +Tsagaris, 2015). Turner et al. (2011) used multispectral cameras mounted in UAVs to analyze the +vegetation index of grapes obtained from vineyards. These vegetation index data are considered +very important to emphasize the significant indicators to increase productivity and improve the +shortcoming from farming activity. Furthermore, the application of UAVs in crop monitoring +could also be seen through UAVs' capability to fly up to hectares of a field in one single flight. For +this purpose, multispectral and thermal cameras are mounted at the UAVs' downside to recording +the vegetation canopy's reflection (Bendig et al., 2012; Colomina & Molina, 2014). These cameras +can take one capture per second and store it in the memory. The images are captured in the visible +five bands with five different wavelengths (i) blues wavelength 440-510 nm (ii) green wavelength +520-590 nm, (iii) red wavelength 630-685 nm, (iv) red edge wavelength 690-730 nm, (v) near- +infrared wavelength 760-850 nm. Then, those images were sent to the ground station through +telemetry. The process of communication used the MAVLINK protocol. The data collected from +the multispectral camera was analyzed by the geographic indicator Normalized Difference +Vegetation Index (NDVI) (Reinecke & Prinsloo, 2017; Bhandari et al., 2012). +Moreover, the application of UAVs for crop monitoring has been implemented for +conducting several tasks including monitoring crop growth, chlorophyll, and phenology +measurement, and counting plants (Pino, 2019). These tasks are performed using SenseFly's e Bee +Ag that has NIR and NDVI sensors. These sensors can replace traditional farm scouting by +minimizing human error (Natu & Kulkarni, 2016). In addition, UAVs are involved in monitoring +crops in hilly areas that are considered to be difficult for traditional scouting (Rani et al., 2019). + +OktoXL +UHD185 +Gimbal +CP +ICS +SBC +Table 1. Applications of Agricultural UAVs +Task +UAV +Model +Indices +Crop +Flight +Altitude +(m) +Sensors +Task Period +Reference +Type +Model +Spraying +Fixed-wing +UAV +Normalized Difference Vegetation +Index (NDVI) +Maize Silage +150 +- +Canon s110 Throughout +the year +(Castaldi et +al., 2017) +Helicopter +Spray Work Rate +Vineyard +3-4 +Digital Camera +- +May +(Giles & +Billing, +2015) +Route Precision, Spraying +Uniformity +Wheat +5, 7, 9 +Image Transmitter +- +Summer +(Xue et al., +2016) +Droplet size, Flow rate +Field +6 +Proprietary Radio +Receiver +- +Throughout +the year +(Huang et +al., 2009) +Leaf Area Index (LAI), +Normalized Difference Vegetation +Index (NDVI) +Maize Silage +35 +Multi-Spectral Camera +Agrosenso +Throughout +the year +(Castaldi et +al., 2017) +- +Field +20 +Wireless Sensor Networks +- +- +(Faiçal et al., +2017) +Quadcopter + +Time of Communication between a +Sensor +Soy, Rice, Corn +Gapes, +Sugarcane +5, 10, 20 +RF Module +XBee-PRO +series 2 +Summer +(Faiçal et al., +2014) +Droplet coverage rate, Density, +Droplet size +Cocktail, +Grapefruit, +Citrus +3.5, 4, 4.5 +Digital Plant Canopy +Imager +Camas CI- +110 +Spring- +Summer +(Pan et al., +2016) +Observed Deposition Rate, Field +Work Rate +Field +Few +meters +Multi-Spectral camera, +Hyper-Spectral camera, +Near-Infrared, Color- +Infrared +- +Throughout +the year +(Meivel et +al., 2016) +Droplet Coverage Rate, Density, +Droplet size +Citrus +0.6, 1.2, +1.8 +Water-Sensitive Paper +Cards (WSPs) +- +One day +(Tang et al., +2018) +Hexacopter +Discharge and Pressure of Spray +Liquid, Spray Uniformity, Spray +Liquid Loss, Droplet Size and +Density. +Paddy and +groundnut +1 +HD FPV camera +- +Throughout +the year +(Yallappa et +al., 2017) + + + +Table 1. Cont. +Task +UAV Model +Indices +Crop +Flight +Altitude +(m) +Sensors +Task Period +Reference +Type +Model +Crop +Monitoring +Fixed-wing +UAV +Normalized Difference Vegetation +Index (NDVI) +Arable crops (corn, cotton, +sunflower) +120 +Multi- +Spectral +Camera +Parrot +Sequoia Plus June-October (Bollas et al., +2021) +Normalized Difference Vegetation +Index (NDVI) +Rice +20 +Multi- +Spectral +Camera +Tetracam +ADC +camera +95 days +(Swain et al., +2010) +Quadcopter +NDVI, Ontario Soil and Crop +Improvement Association +Soybean, Wheat, Barley, +Oat, Canola +120 +Digital +Camera +Aeryon +Photo3S +Spring- +Autmn +(Zhang et al., +2014) +Visible-Band Difference +Vegetation Index, Normalized +Green-Blue Difference Wheat +Index, Green-Red Ratio Index +Wheat +100 +Digital +Camera +SONY +ILCE-6000 +September- +July +(Du & +Noguchi, +2017) +Leaf Area Index (LAI), Total Dry +Weight (TDW), Plant Lenght (PL) +Three Rice Cultivars: +Nipponbae (Japonica), +IR64 (Indica), Basmati370 +(Indica) +30 +RGB +Camera +Zenmuse +X4s +Summer +(Peprah et al., +2021) +Vegetation Index (VI), Leaf Area +Index (LAI) +Coffee +30 +Digital +RGB +Camera +Sony +EXMOR +1/2.3" +Throughout +the year +(Barbosa et +al., 2021) +Soil-Adjusted Vegetation Index +(SAVI), Leaf Area Index (LAI), +Normalized Difference Vegetation +Index (NDVI) +Sunflower +75 +Digital +Camera +Tetracam +ADC Lite +four days +(Vega et al., +2015) +- +Field +- +RGB +Camera +- +- +(Doering et +a., 2014) +Hexacopter Normalized Difference Vegetation +Index (NDVI) +Vineyard +150 +ADC-Lite +Camera +Tetracam +ADC-lite +camera +One day +(Primicerio et +al., 2012) + + + +Table 1. Cont. +Task +UAV +Model +Indices +Crop +Flight +Altitude +(m) +Sensors +Task +Period +Reference +Type +Model +Crop +Monitoring +Hexacopter +Normalized Green-Red Difference +Index (NGRDI) +Pea, Oat +30 +RGB Camera +Panasonic Lumix +DMC-GF1 +April- +August +(Jannoura et al., +2015) +Blue Green Pigment Index 2 +(BGI2), Reformed Difference +Vegetation Index (RDVI) +Barley +30 +Hyper-Spectral +Camera +Firefly ultra-high +definition 185 +Summer +(Aasen et al., +2015) +Octocopter +NDVI, Soil Adjusted Vegetation +Index (SAVI), Optimized SAVI +(OSAVI) and Li +Barley +50 +RGB-Sensor +Panasonic Lumix +GXI +April- +July +(Bending et al., +2015) +Structure-from-Motion (SfM), +airborne laser scanning (ALS) +Eucalyptus +Pulchella +30 +RGB Camera +Canon55D +One day (Wallace et al., +2016 +Normalized Difference Vegetation +Index (NDVI), thermal temperature +Sugarbeet +55 +Multiple Camera +Array (MCA) +Camera +Tetracam mini +MCA +One day +(Bendig et al., +2012) +- +Sunflower +122 +Multi-Spectral +Camera +ADC Snap +- +(Noriega & +Anderson, +2016) + + + + + + + +7. Limitations in Adopting UAVs Technologies in Agriculture Sector +7.1 Technical Decisions + + +Various types of UAVs have been produced in the commercial market by many manufacturers and companies, starting from hobby-type +products up to industrial model aircraft. Since there is no specific standard about the UAV development for agricultural purposes, it is +hard to find a UAV built specifically for the agricultural context (Huang et al., 2013). Moreover, suppose the available commercial +software packages, which support the photogrammetric data processing, are not standardized for agricultural purposes. In that case, the +desired UAV images may not be appropriately captured by the sensor. Therefore, it can prevent the users from taking the right actions +if unexpected situations such as a collision with another flying object occur (Abdullahi, 2015). +Another major problem associated with technical decisions is the battery usage and flight time limitations. The lithium-ion +batteries currently used in UAVs have an advantage over conventional batteries, especially in their larger capacity. However, the larger +capacity affects the weight of the batteries that become heavier in return (Saha et al., 2011). Unfortunately, this issue is challenging to +be solved within this day. Another problem related to battery usages is battery management. Even though it is known that the batteries +of UAVs need to have constant maintenance, most UAV operators often forget and do not carefully pay attention to this issue. As a +result, it caused periodic replacement that required additional cost (Lee et al., 2012). Lastly, the possible time for UAVs to fly, which is +around 20-30 minutes with a fresh battery, can still provide enough time for complete crop monitoring (Baha et al., 2012). Researchers +try to develop optimized hybrid batteries as solutions in dealing with this issue. +7.2 Cost +The lack of awareness of the UAVs' cost, is one of the reasons for the slower adoption of this technology in the agriculture sector. For +a starter system, agricultural UAVs can range from US$1,000 that might go up to US$10,000 or US$20,000, depending upon the cameras +and the features (Stehr, 2015). This cost is not quite affordable and surely will be an impending stop to adopt UAVs technology for +smallholder farmers (Ahmad et al., 2021). The interested farmers who could not afford the cost of UAVs may need to contract as a +group to get UAV services to reduce the individual expenses. +Another possible solution to minimize the cost of UAVs by purchasing inexpensive airframes and low-cost cameras. However, +this solution could build up a short endurance of the UAV platform. Moreover, the low-cost UAVs are usually equipped with lightweight + + +engines that might limit the reachable altitude of the UAVs. The low cost of cameras also limited the sensor payload both in dimension +and weight, and reduced image quality (Abdullahi, 2015). In addition, the separate purchases of UAV components require highly skilled +engineers or technicians to integrate and assembly, which may increase the total expenses (Huang et al., 2013). + +Apart from the cost of vehicles equipped with cameras and software for aerial imagery processing, the farmers need to consider +the expenditures for the operator's license. The presence of this operator implies extra time and cost that need to be spent since not +everyone is allowed to operate the UAVs. Nevertheless, all these costs will constantly decrease over the years (Bollas et al., 2021). +7.3 Payload +Payload weight and size are critical factors for UAVs because they need to be carefully configured based on the specific +application of the UAV. When the UAV is ready to use, it needs to be configured by paying attention to payload design, mechanical +and electrical accommodation even though there is no specific engineering guideline to be followed (Huang et al., 2013). +7.4 Operation +In the UAV operation, most UAV types do not have the capability of automated take-off and landing (Huang et al., 2013). Furthermore, +the frequency of flying UAVs should be carefully selected because there are insufficient regulations about flying UAVs. Even certain +regions restrict the usage of UAVs as a security precaution (Eisenbeiss, 2009). Another challenge is the UAVs' inability to take readings +during extreme weather conditions like rain or storm (Abdullahi et al., 2015). Therefore, highly skilled operators for remote control are +required. However, the demand for skilled users to operate the UAVs is a problem for small and medium producers to adopt UAV +technology. Training issues and lack of demonstrated financial returns in the short and medium term are considered the reason for this +issue (Abdullahi et al., 2015). Thus, autonomous flight according to georeferenced coordinates has then become a highly desirable +component for practical use of UAVs in agriculture (Huang et al., 2013). +The swarm-control techniques can be applied to efficiently control multiple UAVs in performing a wide range of tasks. Although +swarm-control can provide practical techniques to lower the battery cost and operate more efficiently with shorter flight times, there is +a need for user interface improvement so that people who are older or unfamiliar with UAVs can easily control the UAVs. The user + + +interface improvement is made by considering multimodal feedback, including visual, auditory, and haptic feedback. Therefore, an +improvement that mainly focused on human-centered user interface and feedback are two ways that seem to be effective to deal with +multiple UAVs (Hong et al., 2017). + +8. Challenges in Implementing UAVs in Indonesia +Kavianand et al. (2016) have reported that agricultural development in Indonesia is critical since it has primarily contributed to +Indonesia's GDP. Roughly about 14.4 percent of Indonesia's total GDP comes from the agriculture sector and has reduced the +unemployment rate by absorbing 38.6 percent of the workforce (David & Ardiansyah, 2017). Despite its considerable contribution to +Indonesia's GDP, the contribution of agriculture to Indonesia's GDP has been remarkably decreasing for the last five decades due to low +productivity. Some natural phenomena, such as extreme weather changes, have also influenced Indonesia's agriculture (Syuaib, 2016). + +Many researchers have suggested implementing precision agriculture via UAVs to improve the productivity of the work in +agriculture. The application of UAVs offered many benefits that could grow the economic profit and provide a proper solution in +solving current issues in agriculture. However, as much as Indonesia depends upon agriculture, the application of UAVs in the +agriculture field is relatively far from adopting the latest technology into farms. Even though some developed countries have started to +use UAVs in their precision agriculture and proved that this technology is essential in reducing farmers' workload, Indonesia seems to +fall behind and keep using manual operating for farm activity. +One of the viable reasons for preventing the adoption of UAVs is the education level of farmers. The majority of farmers in +Indonesia do not complete their high school education in which 38 percent of local farmers have graduated from primary school (Haq +et al., 2016). Furthermore, only 6 percent of the local farmers can complete high school or university. These numbers significantly +describe the current state of Indonesia's farmers. The lack of education could cause a low understanding of the technology application. +Moreover, this low level of understanding can lead to the anxiety of relearning integrating agriculture and technology (Suryanegara et +al., 2019). + + +Despite the reasons mentioned above, researchers have tested the application of UAV in several agricultural sectors. For instance, +the UAV has been practically implemented or tested in Indonesia’s agriculture, including the sugar cane plantation in PTPN +(Perkebunan Nusantara Maospati East Java) and a paddy field near Menara Cigarette Factory, and the Teak Wood Forest in Madiun +(Rokhmana, 2015). Besides, the application of UAV has been significantly tested in one of the paddy fields in Parankasalak, Sukabumi, +West Java, to monitor the crop by mapping the paddy field through differentiating them based on their spectral characteristic +(Rokhmatuloh et al., 2019). However, the research found that the implementation of UAV poses several limitations and challenges that +become a prohibitive factor for broader use in Indonesia’s agriculture. +The application of UAV in PA requires a high investment cost to purchase the technology and the maintenance cost (Tsouros et +al., 2019). Furthermore, due to the limited space of agricultural land and the unstable market price for the crop yield, the implementation +of UAV might pose another operational cost to the farmers (Tsouros et al., 2019; Vera et al., 2021). Even though the market has +commercially offered an amateur and cheaper UAV, the product has several limitations related to stability, accuracy, and quality +(Norasma et al., 2019). The cheaper UAV has a low ability to reach a certain altitude due to its low power engine (Norasma et al., +2019). This notion is reinforced by (Rokhmana, 2015) who notes that amateur UAVs generally have an error in their camera lens. This +case happens because both the stability and accuracy of the non-metric lens are low. Besides, the UAV is relatively light, possesses +only 3 kg weight, which causes them to be easily disturbed by the wind and air turbulence when the weather is windy (> 40km/h) and +rainy days (Norasma et al., 2019; Rokhmana, 2015; Tsouros et al., 2019). This case requires huge attention because Indonesia is located +along the equatorial belt region to have periodic heavy rain. + Furthermore, the UAV requires data-intensive procedures and skilled personal for exploiting the acquainted imagery data. +(Tsouros et al., 2019). Hence, the farmers need to hire the expertise of UAV technology or do intensive training that may be costly. +This case requires intensive consideration because the average farmer in Indonesia is not in productive ages with low educational +background (Haq et al., 2016). It reported that 88% of the average farmer in Bantarkawung is on the 15-60 years and the remaining +farmer is on non-productive ages (Haq et al., 2016). Moreover, it discovered that only 6% of the majority graduated from high school + + +and university; the remaining only attended primary school, and 38% were in junior high school (Haq et al., 2016). As a result, most +farmers have a common understanding of technology, IoT and little comprehension of imagery data. Another viable reason for +preventing people from using the UAV technology is its limited flight time. (Tsouros et al., 2019) revealed that most commercial UAVs +only have 20 min to 1 hour flight time. Moreover, it only covers a small restricted area for each flight. Thus, the total cost expenses to +purchase the UAV technology for PA might not be advantageous. +9. Conclusion +The application of UAVs in current days has opened unlimited potential, especially in the agriculture sector. Two main UAVs +applications in agriculture sectors, such as spraying, and crop monitoring have been discussed. The urgency of UAVs and the +implementation of UAVs were necessary to be implemented in order to establish precision agriculture. Numerous issues and problems +that might occur in the future have also been highlighted to build awareness about the issues by providing various data and sources. The +application of UAVs in spraying and crop monitoring are the main parts of this paper since we were thoroughly investigating the +application of UAVs that includes the benefits obtained, various application forms of UAVs from several types of research, and the flow +of operating the UAVs. Moreover, the limitation found in the application of UAVs was also identified to reveal the gap of UAVs +implementation in the agriculture field. Lastly, the challenges in implementing UAVs are also being discussed, especially in Indonesia. + +REFERENCES +Aasen, H., Burkart, A., Bolten, A., & Bareth, G. (2015). Generating 3D hyperspectral information with lightweight UAV snapshot +cameras for vegetation monitoring: From camera calibration to quality assurance. 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(2010) “Development of a PWM precision spraying +controller for unmanned aerial vehicles.” Journal of Bionic Engineering, 7(3), pp.276-283. + + + + + diff --git a/VNAyT4oBgHgl3EQfhfi9/content/tmp_files/load_file.txt b/VNAyT4oBgHgl3EQfhfi9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..960dd6035bea16f11745de494a732aee2c7edee5 --- /dev/null +++ b/VNAyT4oBgHgl3EQfhfi9/content/tmp_files/load_file.txt @@ -0,0 +1,1554 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf,len=1553 +page_content='Review Article A review of Implementation and Challenges of Unmanned Aerial Vehicles for Spraying Applications and Crop Monitoring in Indonesia Authors: Muhamad Rausyan Fikri1,2, Taufiq Candra3, Kushendarsyah Saptaji4, Ajeng Nindi Noviarini5, Dilla Ayu Wardani3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Automation Technology and Mechanical Engineering, Faculty of Engineering and Science, Tampere University, Tampere, 33720, Finland 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Information Systems, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, Indonesia 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Industrial Engineering, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, Indonesia 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Mechanical Engineering, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, Indonesia 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Computer Science, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, Indonesia Abstract: The rapid development of technology has brought unmanned aerial vehicles (UAVs) to become widely known in the current era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The market of UAVs is also predicted to continue growing with related technologies in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' UAVs have been used in various sectors, including livestock, forestry, and agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" In agricultural applications, UAVs are highly capable of increasing the productivity of the farm and reducing farmers' workload." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This paper discusses the application of UAVs in agriculture, particularly in spraying and crop monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This study examines the urgency of UAV implementation in the agriculture sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' A short history of UAVs is provided in this paper to portray the development of UAVs from time to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The classification of UAVs is also discussed to differentiate various types of UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The application of UAVs in spraying and crop monitoring is based on the previous studies that have been done by many scientific groups and researchers who are working closely to propose solutions for agriculture-related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, the limitations of UAV applications are also identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The challenges in implementing agricultural UAVs in Indonesia are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Keywords: Unmanned aerial vehicle, agricultural UAV, spraying, crop monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Introduction According to the United Nations (UN), the world population is projected to reach 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='7 billion people in 2050 (UN, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This vast population would potentially double the food demand in the future (Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Consequently, the ever-growing population that would emerge could cause food shortages in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This issue has become a severe problem since the Food and Agriculture Organization (FAO) announced similar speculation in which the current agricultural production must be increased by 70 percent by 2050 to meet the increasing demand for high- quality food (Mundial, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Many people suffering from hunger become a signal of how severe the food shortage is, and it was reported that more than 820 million people in 2018 were considered undernutrition (WHO, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Surprisingly, the earlier data mentioned shows the increasing tendency towards people suffering from hunger since only around 690 million people were considered suffering from hunger in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This kind of data indeed contradicts the second Sustainable Development Goals (SDGs) approved by the United Nations (UN) in 2015 with the aims to eradicate hunger and ensure access to food for all people (UN, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' On the other hand, the labor shortages in the agricultural sector due to the aging population and the decreasing number of workers have exacerbated the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The lack of laborers in the agricultural field would expand the cultivation area per worker and increase the workload of workers (Seo & Umeda, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Alternative and innovative solutions to increase food production in dealing with those issues are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' One way to increase food production is to promote the internet of things (IoT), robotics, and artificial intelligence (AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" By shifting the workforce to the technology's utilization, it is expected to solve the labor shortages and improve farmers' skills." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This transformation is inseparable from revolutionary industries that constantly bring industrial innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Some industrial innovations found in recent years, such as sensor technologies, big data, and artificial intelligence (AI), have been considered as the beginning of the "Industry 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='0" era by the European Commission (EC) (EC, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" The emergence of technologies characterized by advanced digitalization is believed to play a significant role in increasing production flexibility and making the value chain more robust so that technology could minimize the farmers' workload and improve the speed and accuracy of the work." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Among the technologies mentioned earlier, unmanned aerial vehicles (UAVs) are one viable way to increase food production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' UAVs are less expensive and have contributed to many areas in agriculture, including spraying, weed recognition, and crop monitoring (Mogili & Deepak, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" UAVs' timely and reliable information about the production, yield and crop management would become beneficial to ensure food safety and security for stakeholders such as farmers and sales units (Martos et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' UAV technologies in agriculture could also enable the complete monitoring of crop conditions from the beginning of the growing season until the end of harvest (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Some leading technologies are possible by implementing UAVs in the agriculture sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Therefore, this paper focuses on reviewing UAVs applications for spraying and crop monitoring in the agricultural field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Some research results on the use of UAVs in spraying and crop monitoring are discussed thoroughly to highlight the use of UAVs and the characteristics of the farming sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Some limitations exist during UAVs implementation are also reviewed to reveal the gap of UAV implementation in the agriculture field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Section 2 describes UAVs' history and the classification of UAVs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Section 3 describes the application of UAVs focusing on spraying and crop monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Section 4 provides some limitations in adopting UAV technologies in the agriculture sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In section 5, the challenge in implementing UAVs in Indonesia is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The last section provides the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Agriculture in Indonesia and Its Challenge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Current Condition Agriculture programs in Indonesia have been a big agenda at the national level, such as National Agenda 21, National Development Programs, and Agricultural and Forestry Revitalization Strategies, encouraging Indonesia to adopt sustainable agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The Central Planning Authority (BAPPENAS), the Ministry of Agriculture, and the Environment Ministry have implemented these ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Most of these plans include components suitable for effective environmental management of Indonesian agricultural exports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The motivations for using these tactics have shifted over time, and they seem to be responding to a variety of distinct trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' First and foremost, Indonesian national plans have prioritized socio-economic objectives above ecologically sustainable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Nonetheless, environmental concerns have become more critical, as evidenced by recent reforms and the increasing frequency of ecological issues in strategic documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Second, strategy papers also show a change in direction as the combination of means changes, with less focus on laws and regulations and more attention to the means for market creation and voluntary methods over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The tensions between diverse skills and conservation goals and local revenue-generating needs have led to different patterns of success in different states across the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Significant advancements have been achieved in modernizing agro-environmental rules, made possible by increased information and worldwide best practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The extent to which environmental hazards pose local or global dangers, the degree of environmental degradation of a particular product, and the availability of legal, enforcement, budgetary, and regulatory capacities for sub-national governments all influence the choice of the policy tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" For practical reasons, Indonesian policymakers have used a range of mechanisms to minimize agriculture's environmental footprint, including direct regulation, market creation or market modification incentives, voluntary and beneficial solutions, and market modification incentives." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Policies are implemented via legislative and regulatory mechanisms, which are probably targeted at plantation states and large farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It is essential to note the existence of obligatory ISPO standards (in the section on local regulatory instruments), since they have just recently been adopted as a result of voluntary standards being adopted as mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Additional factors that impact policymakers' choices to implement a particular instrument include the potential efficacy of the instrument in comparison to its costs and the capacity of the policymaker to enforce the instrument in the face of likely political opposition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In this respect, implementing regulatory and legislative tools seems to be the most effective method of monitoring prominent investments, such as planting restrictions and the demand for environmental impact assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' According to the findings of the Indonesian research, foreign pressure had a role in the spread of planting restrictions throughout the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, the implementation of regulatory instruments may be most effective when their administrative and monitoring costs are already integrated into a current administration, such as indirect product charges for import limitations, which are already embedded into an existing administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The Challenge in Indonesia’s Agriculture One of the factors is the limited availability of agricultural land in Indonesia due to land reform, which is widespread in big provinces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' As the population rose, so did the need for housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' As a result, developers exploit a large portion of agricultural land to construct real estate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The growth in the number of people also increased demand for trade and tourism, contributing to increased demand for land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Farmers could not be faulted for selling their farms in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Farmers were driven to sell their lands due to a lack of knowledge and technology, high agricultural costs, and rising necessities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Farmers in Indonesia with low levels of education have little choice except to work outside the agricultural industry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' therefore, those who do not own land are tenant farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Food price increases should be a dream come true for farmers, as their revenue would almost certainly rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Unfortunately, because most farmers in Indonesia are tenant farmers, it has become a boomerang for their wellbeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The rise in food prices has little effect on the well-being of Indonesian farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Their income remains minimal, and they must continue to purchase their basic necessities at market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Those who own land have benefited from the growing price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Furthermore, the general public's perception of farmers is that they do not do a good job." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" The younger generation is interested in non-agricultural jobs, such as parenting a farmer's child." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Farmers' regeneration is hampered as a result, and many opt to sell their land to be established as capital or to work in the non-agricultural sector." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" As the world's population rises at an alarming rate, agriculture must expand to supply the growing demand for food against all odds." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The agriculture sector is the most vulnerable to the impact of integrating fresh, modern innovations in eradicating environmental-related challenges and enhancing the current productivity rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Now, the question is, how could we possibly do this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Marking a third wave of the “Green Revolution”, the concept of precision agriculture with technological help such as Unmanned Aerial Vehicle (UAV) has become popular nowadays in the vast area of agriculture due to its tremendous benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Farmers and managers can boost operational efficiency, cut expenses, minimize waste, and improve the quality of crops with the aid of accurate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Overall, technology has been a key component behind agricultural development and other discoveries brought into the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Unmanned Aerial Vehicle (UAV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='1 UAVs History An unmanned Aerial Vehicle (UAV) is an aircraft with no pilot on board;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' in other words, it refers to auto-piloted aircraft (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The unmanned type of aircraft can be operated in two ways, either by a human operator or autonomously operated under the control of an onboard computer (Pablo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' According to the US Department of Defense (DOD), UAV can be described as either a single air vehicle (with equipped surveillance sensors) or a UAV system (UAS) that consists of three to six air vehicles, a ground control station, and support equipment (Gertler, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, UAVs are often associated with remote sensing in carrying out their task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This remote sensing is commonly known as UAV remote sensing, which combines UAV and remote sensing technology that can quickly capture information about land, environment, and resources for further data processing (Shi & Liu, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In the US and other developed countries, UAV remote sensing has been applied in many fields such as forestry, environmental protection, land, and military (Xiang & Tian, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' UAV remote sensing are used because it could be deployed quickly in repeated times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, they are less costly, safer than piloted aircraft, flexible in terms of flying height, and able to obtain very high-resolution imagery (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The term unmanned aerial vehicles are also known as remotely piloted aircraft (RPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Even though the terms UAV and RPA are interchangeable, the term UAV is commonly used by aviation organizations (Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019), while the term RPA is widely used in Europe (Gallardo- Saavedra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Back then, in 1930, UAVs were also known as “Queen Bees” (Vroegindeweij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014) and were initially used for military purposes (Muchiri & Kimathi, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In 1986, UAVs that work specifically in agricultural contexts were introduced by launching UAVs for Montana’s forest fires monitoring and followed by the capture of enhanced image resolution using UAVs in 1994 (Muchiri & Kimathi, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Then, a more complex UAV model was finally developed by Yamaha through “Yamaha RMAX,” with the primary function for pest control and crop monitoring application (Mogili & Deepak, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This UAV model is used for pesticide spraying in rice fields of Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' As opposed to ground-based sprayers, the pesticides deposition of this UAV model is quite similar, but this UAV model is used explicitly for a high- value crop environment (Giles & Billing, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='2 Classification of UAVs Generally, there are three types of UAV platforms: fixed-wing, rotary-wing UAVs, and non-wing UAVs (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' A fixed-wing UAV resembles an airplane and requires a runway or Modelsurface (meadow or road) for take-off and landing (Pederi & Cheporniuk, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This kind of UAV uses thrust and aerodynamic lifting forces to fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It has a larger size than a rotary-wing model and is mainly used for aerial mapping, spraying, and photography over a wide range of time (Li & Yang, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This UAV type typically lacks hovering while offering high-speed flights for longer durations (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The gliding capabilities possessed by fixed-wing aircraft could enable greater flight endurance, allowing them to operate over longer distances (up to 15-20 km) (Paneque-Gálvez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' D Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Illustration of Basic UAVs (A) Fixed-Wing UAV (B) Rotary-Wing UAV (C) Combinational Concepts (source: Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021), (D) Blimps (source: Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2018) On the other hand, rotary-wing UAVs is primarily categorized into the helicopter and multi-rotor types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The helicopter type of rotary-wing UAV has a unique feature with a large propeller atop the aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It is widely used for spraying and aerial photography (see Figure 2) (Swain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' They can hover, vertical takeoff, and land with nimble maneuverability while exhibiting low-speed flight for a shorter duration (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In comparison, the multi- rotor models are called according to the number of rotors (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Quadcopter, hexacopter, and octocopter are some multi-rotors UAVs that are widely known (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' These UAVs are lifted and propelled according to the number of rotors (Mogili & Deepak, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Single Rotor/Helicopter UAV Type (source: Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2009) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Multirotor UAVs, (A) Quadcopter (source: Spoorthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) (B) Hexacopter (source: Yallappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) (C) Octocopter (source: Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016) For example, the rotor movement of a quadcopter is responsible for generating the lift of a quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In a quadcopter, each of two rotors moves in an opposite way of which two rotors turn in the clockwise direction and the other two turn in the anticlockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The movement of the quadcopter around the axis consists of yaw (clockwise and anticlockwise), pitch (backward and forward), and roll angles (right and left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" The quadcopter uses a control system to balance the SR200 SR20B c thrust of each rotor in order to support the UAVs' lift and yaw, pitch, and roll angles (Mogili & Deepak, 2018)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This control system turned out to be practical to produce a stable flight of the UAVs (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, two quadcopter configuration types include the plus (+) and cross (X) models, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The cross model is more popular between the two models due to its stability (Kedari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Quadcopter Configuration Model (A) Plus configuration (B) Cross Configuration (source: Mogili & Deepak, 2018) Furthermore, these multirotor UAVs have extended their functionalities by equipping appropriate sensors such as vision, infrared, multispectral, and hyperspectral cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The expansion of UAV features brings great influences, especially in adding the capabilities of the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Those sensors are used to obtain data such as vegetation, reflectance indexes, and leaf areas in order to provide information about the current state of crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' With this information, farmers can make possible remedies or policies (weed control, fertilization, irrigation) according to the condition of the crops (Gonzalez-De-Santos, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Further, non-wing UAVs have been developed to cope with the long endurance of flying robots and are lighter than air (LTA) such as Blimp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' A blimp is identically has a larger size than fixed- wing and rotary-wing and cushioned with a helium-filled envelope, making the robot safe to fly indoors, causing no threat to humans and the surroundings even with collisions (Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' With the lifting force provided by air buoyancy, the blimp has flight endurance for more than 2 hours (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Blimp is the one type of UAV lighter than the air UAVs (Krishna, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thusoo, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It has a balloon-like body created from tough fabric and filled with helium gas (Prisacariu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Blimp is notorious as the dirigible and was firstly PitchForward PitchForward Roll Left Roll Left Roll Right Roll Right A B PitchBackward PitchBackward designed in 1852 by Henri Giffard (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This type of UAV has high endurance and can flow longer than other types of UAV, approximately 1-3 weeks travel (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Due to its characteristics, the blimp is advantageous in numerous aspects of life, including the military and agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It was purposeful as the cargo transit and the sentinels between the missile site and the military camp (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It is also used to monitor the long-distance aspect, especially in urban traffic and buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In PA, it monitors crop production, identifies the plant’s disease, erosion, and detects either flood or drought conditions (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Unlike the other UAV, the blimp is also considered a safe technology since it remains in the air and did not collide even if it loses its power (Krishna, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides, the University of Leeds research reveals that blimp is chosen as the cheapest UAV to conduct a terrain survey (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thus, it could provide a detailed but further explanation of crops, surface minerals, vegetation, and water quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Application of Blimp in Agriculture Compared to other UAVs, the blimp has a pivotal function, including load capacity, safety, quality, and environmental safes, which make it useful for everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Researchers stated that blimp could carry up to 400 tons of load with 110-160 km speed travel (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides its ability to fly longer, blimps could land on every land’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In the case of environmental footprint, blimp could reduce carbon dioxide emission (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Nowadays, there are several features of blimps with specific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Those five types include tethered, untethered, remote-controlled blimp, Giga blimps, and hybrid blimps (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Tethered blimp (aerostats) ables for free flight and a steady anchored flight using solid tethers (Mahmood & Ismail, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' A tethered blimp enables accurately obtaining a stereoscopic image that might cover 35 m2 to 20,000 m3 (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Meanwhile, the untethered is commonly used in cargo transit, travel, and aerial surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thirdly, the remote-controlled blimp uses the robotic that use the program flight plan to fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' There are two different types of remote control blimps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' small and large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The small blimp is commonly used for advertisement, and the giant blimp (Giga blimp) is significant for military purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Lastly, the hybrid blimp, a new modification of blimp that is prone to extreme conditions, can transport goods and civilian travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Unlike the other popular UAVs, multirotor and fixed-wing, the number of blimps applied in PA is insignificant (Krishna, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' However, considering its strength characteristic and function, blimp could start considering the blimp as the priority to improve the PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (Mogili, Rao Deepak, 2021) stated that the integration of blimp with quadcopter aerial automated pesticide sprayer (AAPS) is pivotal for pesticide spraying in lower altitudes by following the GPS altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This technology is controlled with an android app to create an effective cost-saving (Mogili, Rao Deepak, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides, other researchers use the blimp with a Charge- Couple Device to identify the Leaf Arena Index and biomass in soybean and paddy fields (Chilonga & Kiswisch, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The results showed that the technology is stable and provides high- resolution images (Chilonga & Kiswisch, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016) also stated that the blimp could be practically used to monitor the bean crop dataset using the combination of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='3 inch of CCD sensor, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='3 to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='99 lens focal, and 10 Mega Pixel digital camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The research found 29,556 examples of the positive dataset and 11404 negative datasets in Brazil (Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The blimp could be significantly used in monitoring agriculture (Mahmood & Ismail, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' For instance, the research conducted by (Bajoria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) proposed a tethered aerostat system that could be used to mitigate the vertebrate mammal and bird hazard, which is positively contributed to 18%-43% of crop loss in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The proposed design has been proven to carry about a 50 kg payload and 25 m/s ambient wind speed (Bajoria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Other research revealed that tethered aerostat combined with the electro-optical, acoustic, and laser-based sensors could scare the bird and other pests (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' To mitigate the occurrence of pests, the other researchers also create a Hawk Kite and Helikite aerostat hybrid that is purposeful to scare some of the bird’s species, including the pigeons, seagulls, parrots, rooks, blackbird, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (Perigrine Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides, a tethered blimp (aerostat) could provide aerial images surrounding the natural disaster zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This image helps identify the cropped field due to the flood, large soil erosion, drought, and crop loss due to the pest attack (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It is also used to maintain the field quality because the aerostat could lofty 24 hours surveillance above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thus, it could help the farmer control and watch the field without going directly to the farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides the aforementioned reasons, the aerostat could reduce the enormous cost of capturing the crop’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The farmer could use the aerostat to pertain the crop data and send its digital data in a computer program (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, the integration of aerostat with the sensor could help the farmer obtain continuous data of Nutrients crop status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thus, it could help the farmer evaluate the number of nutrients placed for the crop (Krishna, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Agricultural Unmanned Aerial Vehicle In recent years, the application of more advanced technology in agriculture has gained more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Several technologies such as satellites, UAVs, Geographic Information System (GIS), Global Positioning System (GPS), and many other applications of technologies have been able to pave their way into the agricultural field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The process modernization and industrial revolution that brought many innovations in technology applications have opened the gate of precision agriculture (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Precision agriculture is defined as the utilization of technology in the agricultural production system in order to determine, analyze, and manage the farming factors to increase crop productivity, ensure environmental sustainability, and improve business profitability (Unal & Topakci, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This precision agriculture is seemingly possible to increase food production due to its effective functionalities under pressure conditions such as the ongoing reduction of arable land, the increase in global population, and the high cost of farming due to wastage in the use of water and chemicals (Abdullahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' UAVs have gained popularity as a pivotal part of precision agriculture to ensure agricultural sustainability (Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The use of UAV, which plays a key role in reducing the data acquisition time and processing cost, is considered as the main reason for its popularity (Berni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The rapid development of UAVs that extend its functions to aerial photography and video and weather forecasting, with the support of spatial data collection to help stakeholders create policies and decisions, has attracted many parties to UAVs (Sylvester, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, the market of UAVs that is estimated to reach up to US$200 billion by the end of 2020 has successfully described the popularity of UAVs as well (Puri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The huge estimation of the total market of UAVs has shown that the market value of UAVs has doubled within three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' PwC’s Drone Powered Solutions team quantified that the total market value of UAVs is about US$127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='3 billion in 2017 (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Although the estimation of the market value of UAVs in 2020 and the total market value of UAVs in 2017 are not exclusively focused on agriculture sectors, this number was sufficient to portray the UAV market development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, the affordable cost of UAVs is another factor that influences its popularity nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This low-cost factor motivates many small companies to switch to using UAVs with its simple and easy-to-understand operating systems in serving some activities in the agriculture sector, including area measurement and crop monitoring (Hatfield & Prueger, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Applications of Unmanned Aerial Vehicles in Precision Agriculture Currently, there are numerous applications of UAVs in precision agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' They are used in many areas of crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This section introduces two applications of agricultural UAVs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', spraying and crop monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The summary is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='1 Spraying Prior to the implementation of UAVs for spraying, the farmers used spraying bags to spray pesticides all over the farm (Spoorthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Manual spraying is very dangerous for the workers because the measure of pesticides per hectare of agricultural land correlates to the risk of worker ailments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The heavy bag carried by the farmers could also make them get strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Fortunately, the use of UAVs can reduce the usage of pesticides, maximize efficiency, and improve the well-being of the workers (Luck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Pyo, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Manual spraying is also considered ineffective for spraying the farmland because the pesticides may not spread evenly in every area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The excessive use of chemicals or pesticides in certain agricultural land is responsible for loss of soil fertility, soil degradation, and subsequent degradation of water-related ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, the chemicals or pesticides absorbed by the crops and natural resources such as water and soil might cause pollution risk and severe health impacts for the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Therefore, UAVs are required to minimize such dangers by helping the spraying process specifically in the targeted area (Daponte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, to pave the way towards sustainable agriculture, the employment of UAVs in the agriculture sector also offers other benefits in terms of their operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The implementation of UAVs can make the process relatively faster and cheaper than other methods (Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The efficient usage of UAVs was also widely reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The use of 3WWDZ-10A, XAG is successfully effective in controlling Spodoptera frugiperda, an invasive sugarcane crop pest, by spraying pesticides (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, the use of UAV (DJI Phantom 3) is found to be effective in spraying pesticides in the nominated areas using electronic traps (E-traps), which can count the insect and transmit the data to the server (Psirofonia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Studies have also found that UAVs might improve the accuracy of control over crops by equipping the UAVs with precision control algorithms (Faiçal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In summary, it is proved that the UAV application offers several advantages in reducing the workload of the farmers and providing efficient and low-cost service in the agriculture field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Some issues have been identified regarding the use of UAVs in crop areas overlapping and outer edges, despite the advantages during UAVs implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' These issues arise because some crop fields are not fully covered properly during spraying, leading to reduced crop quality in particular areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' To overcome this problem, the swarm of UAVs was introduced in a control loop algorithm during UAV operation (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Swarm control is considered a practical technology since it could control multiple UAVs via one operator or program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In the swarm control method, the operator can select an efficient shape based on the application so that the swarm can be centralized, decentralized, and distributed according to the desired shape (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, the spraying pesticides process on the crop is then organized by considering the feedback from the Wireless Sensor Networks (WSNs) deployed in the field (Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The control loop is responsible for the communication of each UAV in adjusting the UAVs’ route according to the changes in wind speed and the number of messages exchanged in between (Faiçal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' During this communication process, a short delay might exist in the control loop since the UAVs need time to analyze the data from WSN to route further (Kale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' An automatic navigation spraying system of UAV was developed to direct the UAV in a particular area (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Another way in using swarm control is through task allocation technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This technology is currently used in mapping agricultural lands (Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In order to use swarm technology, a route is assigned to each UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' A route is built by dividing each region or area among several UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' A map of the area is obtained by capturing a single picture through a camera sensor attached to a UAV (Ju & Son, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This kind of technique requires K-mean algorithms in order to reduce complexity and prevent collision among UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The most significant aspect of this swarm technique is the combination of algorithms that come in handy in maintaining the consistent distances between UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' These consistent distances allow linear and nonlinear control that resist strong external influences (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The implementation of swarm techniques and task allocation in agriculture can be seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This application most likely improves the accuracy of agricultural operations, reduces operator control efforts, reduces work time, and induces battery and payload shortages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (A) Swarm Control (source: Ju & Son, 2018) (B) Task allocation (source: Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) In the spraying using UAVs, the sprinkling system is mounted at the lower region of the UAV, which has a nozzle under the pesticide tank in order to sprinkle the pesticide downstream in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' An appropriately selected nozzle is a significant part of pesticide application since it is a significant factor in determining the amount of spray applied to an area, the coverage obtained on the target surface, the amount of potential drift, and the uniformity of application (Ru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, the sprinkling system generally has two modules: the controller and the sprinkling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The sprinkling system consists of the spraying content, either pesticides or fertilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Meanwhile, the controller is used to trigger the nozzle of the sprayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The controller efficiency could be increased by using a PWM controller in pesticide applications (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Another important component of the sprinkling system is a pressure pump used to put pressure into the pesticide in the tank to flow through the nozzle (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This pressure pump works closely with the motor driver integrated circuit in completing their task in putting the pressure to sprinkle the pesticide (Mogili & Deepak, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The full spraying system can be seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The integration between UAV and the spraying system is expected to provide a potential platform for pest management and vector control, an accurate site-specific application for a large crop field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' For this objective, a heavy lift of UAVs is required to cover many areas (Sarghini & De Vivo, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' RemoteSensingTaskusingcameramountedonUAV (MultipleUAVs) contro UAV ensing Agriculturalfield Teleoperationcontrolwith device Hapticdevice(NovintFalcon) A B Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Spraying System Structure Diagram (source: Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2018) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='2 Crop Monitoring A crop monitoring is defined as predicting the yield or crop quality by analyzing the available crop data (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It is essential for optimizing crop production because it can assess crop health and indicate bacterial or fungal infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, the crop scanning produced by visible and near-infrared (NIR) light could reflect the different amounts of green light and NIR light that are extremely essential in producing multispectral images that can track changes in crops assess their health (Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The farmers can plan and apply remedies more precisely according to the identified issues with such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It makes the fast response to bacterial or fungal infection, and infestation comes in handy and increases crop endurance into future issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The use of UAVs for crop monitoring is also highlighted due to their ability to monitor a large farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' By utilizing the UAVs, a large area of farmland can be fully monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It reduces the significant time and labor required for monitoring large farm areas manually (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Aasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (2015) reported that the UAVs application offers low crop monitoring costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This is due to the use of lightweight sensors and the implementation of low-flying UAVs (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Electronic speed control Tank Sprayboom Nozzle1 Nozzle2 Waterpump Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' UAV Platform (source: Aasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015) A camera-equipped UAV can also observe the crop with different indices (Simelli & Tsagaris, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (2011) used multispectral cameras mounted in UAVs to analyze the vegetation index of grapes obtained from vineyards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' These vegetation index data are considered very important to emphasize the significant indicators to increase productivity and improve the shortcoming from farming activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Furthermore, the application of UAVs in crop monitoring could also be seen through UAVs' capability to fly up to hectares of a field in one single flight." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" For this purpose, multispectral and thermal cameras are mounted at the UAVs' downside to recording the vegetation canopy's reflection (Bendig et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Colomina & Molina, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' These cameras can take one capture per second and store it in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The images are captured in the visible five bands with five different wavelengths (i) blues wavelength 440-510 nm (ii) green wavelength 520-590 nm, (iii) red wavelength 630-685 nm, (iv) red edge wavelength 690-730 nm, (v) near- infrared wavelength 760-850 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Then, those images were sent to the ground station through telemetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The process of communication used the MAVLINK protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The data collected from the multispectral camera was analyzed by the geographic indicator Normalized Difference Vegetation Index (NDVI) (Reinecke & Prinsloo, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Bhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, the application of UAVs for crop monitoring has been implemented for conducting several tasks including monitoring crop growth, chlorophyll, and phenology measurement, and counting plants (Pino, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" These tasks are performed using SenseFly's e Bee Ag that has NIR and NDVI sensors." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' These sensors can replace traditional farm scouting by minimizing human error (Natu & Kulkarni, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, UAVs are involved in monitoring crops in hilly areas that are considered to be difficult for traditional scouting (Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' OktoXL UHD185 Gimbal CP ICS SBC Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Applications of Agricultural UAVs Task UAV Model Indices Crop Flight Altitude (m) Sensors Task Period Reference Type Model Spraying Fixed-wing UAV Normalized Difference Vegetation Index (NDVI) Maize Silage 150 Canon s110 Throughout the year (Castaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) Helicopter Spray Work Rate Vineyard 3-4 Digital Camera May (Giles & Billing, 2015) Route Precision, Spraying Uniformity Wheat 5, 7, 9 Image Transmitter Summer (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016) Droplet size, Flow rate Field 6 Proprietary Radio Receiver Throughout the year (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2009) Leaf Area Index (LAI), Normalized Difference Vegetation Index (NDVI) Maize Silage 35 Multi-Spectral Camera Agrosenso Throughout the year (Castaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) Field 20 Wireless Sensor Networks (Faiçal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) Quadcopter Time of Communication between a Sensor Soy, Rice, Corn Gapes, Sugarcane 5, 10, 20 RF Module XBee-PRO series 2 Summer (Faiçal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014) Droplet coverage rate, Density, Droplet size Cocktail, Grapefruit, Citrus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='5, 4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='5 Digital Plant Canopy Imager Camas CI- 110 Spring- Summer (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016) Observed Deposition Rate, Field Work Rate Field Few meters Multi-Spectral camera, Hyper-Spectral camera, Near-Infrared, Color- Infrared Throughout the year (Meivel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016) Droplet Coverage Rate, Density, Droplet size Citrus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='8 Water-Sensitive Paper Cards (WSPs) One day (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2018) Hexacopter Discharge and Pressure of Spray Liquid, Spray Uniformity, Spray Liquid Loss, Droplet Size and Density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Paddy and groundnut 1 HD FPV camera Throughout the year (Yallappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Task UAV Model Indices Crop Flight Altitude (m) Sensors Task Period Reference Type Model Crop Monitoring Fixed-wing UAV Normalized Difference Vegetation Index (NDVI) Arable crops (corn, cotton, sunflower) 120 Multi- Spectral Camera Parrot Sequoia Plus June-October (Bollas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021) Normalized Difference Vegetation Index (NDVI) Rice 20 Multi- Spectral Camera Tetracam ADC camera 95 days (Swain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2010) Quadcopter NDVI, Ontario Soil and Crop Improvement Association Soybean, Wheat, Barley, Oat, Canola 120 Digital Camera Aeryon Photo3S Spring- Autmn (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014) Visible-Band Difference Vegetation Index, Normalized Green-Blue Difference Wheat Index, Green-Red Ratio Index Wheat 100 Digital Camera SONY ILCE-6000 September- July (Du & Noguchi, 2017) Leaf Area Index (LAI), Total Dry Weight (TDW), Plant Lenght (PL) Three Rice Cultivars: Nipponbae (Japonica), IR64 (Indica), Basmati370 (Indica) 30 RGB Camera Zenmuse X4s Summer (Peprah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021) Vegetation Index (VI), Leaf Area Index (LAI) Coffee 30 Digital RGB Camera Sony EXMOR 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='3" Throughout the year (Barbosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021) Soil-Adjusted Vegetation Index (SAVI), Leaf Area Index (LAI), Normalized Difference Vegetation Index (NDVI) Sunflower 75 Digital Camera Tetracam ADC Lite four days (Vega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015) Field RGB Camera (Doering et a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2014) Hexacopter Normalized Difference Vegetation Index (NDVI) Vineyard 150 ADC-Lite Camera Tetracam ADC-lite camera One day (Primicerio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Task UAV Model Indices Crop Flight Altitude (m) Sensors Task Period Reference Type Model Crop Monitoring Hexacopter Normalized Green-Red Difference Index (NGRDI) Pea, Oat 30 RGB Camera Panasonic Lumix DMC-GF1 April- August (Jannoura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015) Blue Green Pigment Index 2 (BGI2), Reformed Difference Vegetation Index (RDVI) Barley 30 Hyper-Spectral Camera Firefly ultra-high definition 185 Summer (Aasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015) Octocopter NDVI, Soil Adjusted Vegetation Index (SAVI), Optimized SAVI (OSAVI) and Li Barley 50 RGB-Sensor Panasonic Lumix GXI April- July (Bending et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015) Structure-from-Motion (SfM), airborne laser scanning (ALS) Eucalyptus Pulchella 30 RGB Camera Canon55D One day (Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016 Normalized Difference Vegetation Index (NDVI), thermal temperature Sugarbeet 55 Multiple Camera Array (MCA) Camera Tetracam mini MCA One day (Bendig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012) Sunflower 122 Multi-Spectral Camera ADC Snap (Noriega & Anderson, 2016) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Limitations in Adopting UAVs Technologies in Agriculture Sector 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='1 Technical Decisions Various types of UAVs have been produced in the commercial market by many manufacturers and companies, starting from hobby-type products up to industrial model aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Since there is no specific standard about the UAV development for agricultural purposes, it is hard to find a UAV built specifically for the agricultural context (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, suppose the available commercial software packages, which support the photogrammetric data processing, are not standardized for agricultural purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In that case, the desired UAV images may not be appropriately captured by the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Therefore, it can prevent the users from taking the right actions if unexpected situations such as a collision with another flying object occur (Abdullahi, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Another major problem associated with technical decisions is the battery usage and flight time limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The lithium-ion batteries currently used in UAVs have an advantage over conventional batteries, especially in their larger capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' However, the larger capacity affects the weight of the batteries that become heavier in return (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Unfortunately, this issue is challenging to be solved within this day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Another problem related to battery usages is battery management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Even though it is known that the batteries of UAVs need to have constant maintenance, most UAV operators often forget and do not carefully pay attention to this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' As a result, it caused periodic replacement that required additional cost (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Lastly, the possible time for UAVs to fly, which is around 20-30 minutes with a fresh battery, can still provide enough time for complete crop monitoring (Baha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Researchers try to develop optimized hybrid batteries as solutions in dealing with this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content="2 Cost The lack of awareness of the UAVs' cost, is one of the reasons for the slower adoption of this technology in the agriculture sector." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' For a starter system, agricultural UAVs can range from US$1,000 that might go up to US$10,000 or US$20,000, depending upon the cameras and the features (Stehr, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This cost is not quite affordable and surely will be an impending stop to adopt UAVs technology for smallholder farmers (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The interested farmers who could not afford the cost of UAVs may need to contract as a group to get UAV services to reduce the individual expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Another possible solution to minimize the cost of UAVs by purchasing inexpensive airframes and low-cost cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' However, this solution could build up a short endurance of the UAV platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, the low-cost UAVs are usually equipped with lightweight engines that might limit the reachable altitude of the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The low cost of cameras also limited the sensor payload both in dimension and weight, and reduced image quality (Abdullahi, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' In addition, the separate purchases of UAV components require highly skilled engineers or technicians to integrate and assembly, which may increase the total expenses (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Apart from the cost of vehicles equipped with cameras and software for aerial imagery processing, the farmers need to consider the expenditures for the operator's license." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The presence of this operator implies extra time and cost that need to be spent since not everyone is allowed to operate the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Nevertheless, all these costs will constantly decrease over the years (Bollas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='3 Payload Payload weight and size are critical factors for UAVs because they need to be carefully configured based on the specific application of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' When the UAV is ready to use, it needs to be configured by paying attention to payload design, mechanical and electrical accommodation even though there is no specific engineering guideline to be followed (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='4 Operation In the UAV operation, most UAV types do not have the capability of automated take-off and landing (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, the frequency of flying UAVs should be carefully selected because there are insufficient regulations about flying UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Even certain regions restrict the usage of UAVs as a security precaution (Eisenbeiss, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Another challenge is the UAVs' inability to take readings during extreme weather conditions like rain or storm (Abdullahi et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Therefore, highly skilled operators for remote control are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' However, the demand for skilled users to operate the UAVs is a problem for small and medium producers to adopt UAV technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Training issues and lack of demonstrated financial returns in the short and medium term are considered the reason for this issue (Abdullahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thus, autonomous flight according to georeferenced coordinates has then become a highly desirable component for practical use of UAVs in agriculture (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The swarm-control techniques can be applied to efficiently control multiple UAVs in performing a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Although swarm-control can provide practical techniques to lower the battery cost and operate more efficiently with shorter flight times, there is a need for user interface improvement so that people who are older or unfamiliar with UAVs can easily control the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The user interface improvement is made by considering multimodal feedback, including visual, auditory, and haptic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Therefore, an improvement that mainly focused on human-centered user interface and feedback are two ways that seem to be effective to deal with multiple UAVs (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Challenges in Implementing UAVs in Indonesia Kavianand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" (2016) have reported that agricultural development in Indonesia is critical since it has primarily contributed to Indonesia's GDP." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Roughly about 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content="4 percent of Indonesia's total GDP comes from the agriculture sector and has reduced the unemployment rate by absorbing 38." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='6 percent of the workforce (David & Ardiansyah, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Despite its considerable contribution to Indonesia's GDP, the contribution of agriculture to Indonesia's GDP has been remarkably decreasing for the last five decades due to low productivity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Some natural phenomena, such as extreme weather changes, have also influenced Indonesia's agriculture (Syuaib, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Many researchers have suggested implementing precision agriculture via UAVs to improve the productivity of the work in agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The application of UAVs offered many benefits that could grow the economic profit and provide a proper solution in solving current issues in agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' However, as much as Indonesia depends upon agriculture, the application of UAVs in the agriculture field is relatively far from adopting the latest technology into farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" Even though some developed countries have started to use UAVs in their precision agriculture and proved that this technology is essential in reducing farmers' workload, Indonesia seems to fall behind and keep using manual operating for farm activity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' One of the viable reasons for preventing the adoption of UAVs is the education level of farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The majority of farmers in Indonesia do not complete their high school education in which 38 percent of local farmers have graduated from primary school (Haq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, only 6 percent of the local farmers can complete high school or university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=" These numbers significantly describe the current state of Indonesia's farmers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The lack of education could cause a low understanding of the technology application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, this low level of understanding can lead to the anxiety of relearning integrating agriculture and technology (Suryanegara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Despite the reasons mentioned above, researchers have tested the application of UAV in several agricultural sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' For instance, the UAV has been practically implemented or tested in Indonesia’s agriculture, including the sugar cane plantation in PTPN (Perkebunan Nusantara Maospati East Java) and a paddy field near Menara Cigarette Factory, and the Teak Wood Forest in Madiun (Rokhmana, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides, the application of UAV has been significantly tested in one of the paddy fields in Parankasalak, Sukabumi, West Java, to monitor the crop by mapping the paddy field through differentiating them based on their spectral characteristic (Rokhmatuloh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' However, the research found that the implementation of UAV poses several limitations and challenges that become a prohibitive factor for broader use in Indonesia’s agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The application of UAV in PA requires a high investment cost to purchase the technology and the maintenance cost (Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, due to the limited space of agricultural land and the unstable market price for the crop yield, the implementation of UAV might pose another operational cost to the farmers (Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Vera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Even though the market has commercially offered an amateur and cheaper UAV, the product has several limitations related to stability, accuracy, and quality (Norasma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The cheaper UAV has a low ability to reach a certain altitude due to its low power engine (Norasma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This notion is reinforced by (Rokhmana, 2015) who notes that amateur UAVs generally have an error in their camera lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This case happens because both the stability and accuracy of the non-metric lens are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Besides, the UAV is relatively light, possesses only 3 kg weight, which causes them to be easily disturbed by the wind and air turbulence when the weather is windy (> 40km/h) and rainy days (Norasma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Rokhmana, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This case requires huge attention because Indonesia is located along the equatorial belt region to have periodic heavy rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Furthermore, the UAV requires data-intensive procedures and skilled personal for exploiting the acquainted imagery data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Hence, the farmers need to hire the expertise of UAV technology or do intensive training that may be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' This case requires intensive consideration because the average farmer in Indonesia is not in productive ages with low educational background (Haq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' It reported that 88% of the average farmer in Bantarkawung is on the 15-60 years and the remaining farmer is on non-productive ages (Haq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, it discovered that only 6% of the majority graduated from high school and university;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' the remaining only attended primary school, and 38% were in junior high school (Haq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' As a result, most farmers have a common understanding of technology, IoT and little comprehension of imagery data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Another viable reason for preventing people from using the UAV technology is its limited flight time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' (Tsouros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=', 2019) revealed that most commercial UAVs only have 20 min to 1 hour flight time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, it only covers a small restricted area for each flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Thus, the total cost expenses to purchase the UAV technology for PA might not be advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Conclusion The application of UAVs in current days has opened unlimited potential, especially in the agriculture sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Two main UAVs applications in agriculture sectors, such as spraying, and crop monitoring have been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The urgency of UAVs and the implementation of UAVs were necessary to be implemented in order to establish precision agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Numerous issues and problems that might occur in the future have also been highlighted to build awareness about the issues by providing various data and sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' The application of UAVs in spraying and crop monitoring are the main parts of this paper since we were thoroughly investigating the application of UAVs that includes the benefits obtained, various application forms of UAVs from several types of research, and the flow of operating the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Moreover, the limitation found in the application of UAVs was also identified to reveal the gap of UAVs implementation in the agriculture field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content=' Lastly, the challenges in implementing UAVs are also being discussed, especially in Indonesia.' metadata={'source': 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PWM precision spraying controller for unmanned aerial vehicles.” Journal of Bionic Engineering, 7(3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} +page_content='276-283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfhfi9/content/2301.00379v1.pdf'} diff --git a/VdAzT4oBgHgl3EQfYPwk/content/tmp_files/2301.01330v1.pdf.txt b/VdAzT4oBgHgl3EQfYPwk/content/tmp_files/2301.01330v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1d641b4152cb5cd4a143b0f708c7a0f6c3b7afc --- /dev/null +++ b/VdAzT4oBgHgl3EQfYPwk/content/tmp_files/2301.01330v1.pdf.txt @@ -0,0 +1,314 @@ +arXiv:2301.01330v1 [math.GR] 3 Jan 2023 +ON REPRESENTATIONS OF DIRECT PRODUCTS AND THE +BOUNDED GENERATION PROPERTY OF BRANCH GROUPS +STEFFEN KIONKE AND EDUARD SCHESLER +Abstract. We prove that the minimal representation dimension of a direct +product G of non-abelian groups G1, . . . , Gn is bounded below by n + 1 and +thereby answer a question of Ab´ert. If each Gi is moreover non-solvable, then +this lower bound can be improved to be 2n. By combining this with results of +Pyber, Segal, and Shusterman on the structure of boundedly generated groups +we show that branch groups cannot be boundedly generated. +Introduction +An infinite group G is called just-infinite if all of its proper quotients are finite. +Obvious examples of just-infinite groups are virtually simple groups. Other exam- +ples arise from irreducible lattices in higher rank semisimple Lie groups, such as +SLn(Z) for n ≥ 3, after dividing out their centers, see [12, Chapter IV]. Such groups +are in fact hereditarily just-infinite, which means that they are residually finite and +all of their finite index subgroups are just-infinite. Grigorchuk’s group [10] provided +the first example of a just-infinite group that is not virtually a finite direct power of +a simple or a hereditarily just infinite group. Grigorchuk’s group is a just-infinite +branch group, which means that its commensurability classes of subnormal sub- +groups form a lattice that is isomorphic to the lattice of open and closed subsets +of a Cantor set. By Wilson’s classification [18] just-infinite groups fall into three +classes. Every just-infinite group G is either a branch group or virtually a direct +power of a simple or a hereditarily just-infinite group. +Since its introduction by McCarthy [13] in the late 1960’s, the class of just-infinite +groups remained an active field of research. One reason might be that every finitely +generated infinite group admits a just-infinite quotient. Thus whenever there is +some finitely generated, infinite group G that admits a property P that is preserved +under homomorphic images, then there is also a finitely generated just-infinite group +with P. Following [3], we call a property P that is preserved under homomorphic +images an H-property. Well-known examples of H-properties include amenability, +property (T), bounded generation, being a torsion group, having subexponential +growth etc. In view of Wilson’s classification, it is natural to investigate which +of the three classes of just-infinite groups contain groups that satisfy a given H- +property P. For the H-property “being a torsion group” this question is settled. In +this case it is know that there are finitely generated simple groups [15], just-infinite +branch groups [10], and hereditarily just-infinite groups [9] that are torsion. On +the other hand, there are torsion-free, finitely generated, just-infinite groups that +are simple [4], branch [2], and hereditarily just-infinite (e.g. Z). +The purpose of this note is to study this question for the bounded generation +property. Recall that a group G is boundedly generated if it contains a finite subset +2010 Mathematics Subject Classification. Primary 20E08; Secondary 20E26. +Key words and phrases. bounded generation, branch group, faithful representations of direct +products. +Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - +441848266. +1 + +2 +S. KIONKE AND E. SCHESLER +{g1, . . . , gn} such that every g ∈ G can be written as g = gk1 +1 · · · gkn +n for appropriate +ki ∈ Z. Since infinite torsion groups are not boundedly generated, it follows that +each of the three classes of just-infinite groups contains a finitely generated group +that does not have the bounded generation property. On the other hand, it was +proven by Carter and Keller [7] that PSLn(Z) is boundedly generated for n ≥ 3, +which provides an interesting boundedly generated hereditarily just-infinite group. +The existence of boundedly generated, infinite, simple groups was established by +Muranov [14], whose construction seems to be the only one available at present. It +remains to study just-infinite branch groups. The question of existence of boundedly +generated just-infinite branch groups was raised by Bartholdi, Grigorchuk, and +ˇSuni´k [3, Question 12] and remained open to the best of our knowledge. +The +purpose of the paper is to show that the answer is negative for arbitrary branch +groups (even without the assumption of being just-infinite). +Theorem 1. There is no boundedly generated branch group. +As a consequence, it follows from Wilson’s classification of just-infinite groups +that every boundedly generated infinite group has a quotient that is virtually a +product of finitely many copies of a boundedly generated simple or hereditarily +just infinite group. The proof the Theorem 1 is a rather direct combination of +results of Pyber and Segal [16], Shusterman [17], and Ab´ert [1]. +Ab´ert proved that weakly branch groups are not linear over any field (for branch +groups this result is due to Grigorchuk and Delzant). More precisely, he defined +for every field k the natural number matk(n) to be the minimal r such that every +graph on n vertices can be represented in the matrix algebra Mr,r(k) where the +graph’s edges encode non-commutation. Ab´ert showed that +� +⌊n/2⌋ ≤ matk(n) ≤ +2(n−⌊log2(n)⌋+1) and asked for a linear lower bound [1, Question 4]. The following +result answers this question in the affirmative. +Theorem 2. Let k be a field and let r ≥ 1. Suppose that there are (r × r)-matrices +a1, . . . , an, b1, . . . , bn ∈ Mr,r(k) such that all pairwise commutators are trivial except +for [ai, bi] = aibi − biai for all i ∈ {1, . . . , n}. Then r ≥ n + 1. +The lower bound in Theorem 2 is sharp. Let λ ∈ k×. Consider the matrices +ai = I + E1,i+1, bi = I − λEi+1,i+1 ∈ Mn+1,n+1(k) for i = 1, . . . , n, where I is the +identity matrix and Ei,j denotes the elementary matrix whose (i, j)-entry is 1 and +all other entries are 0. Then ai, bi satisfy the assumptions of Theorem 2. If λ ̸= 1, +then ai, bi are invertible. If |k| > 2, this shows with [1, Prop. 5] that +�n +2 +� ++ 1 ≤ matk(n) ≤ n + 1. +The non-linearity of weakly branch groups follows since these groups contain +infinite products of non-abelian groups. Let µk(G) denote the minimal dimension +of a faithful, finite dimensional representation of a group G over a field k (we write +µk(G) = ∞ if G is not linear over k). Theorem 2 directly implies a lower bound +µk(G1 × . . . × Gn) ≥ n + 1 for direct products of non-abelian groups. Similarly +Theorem 2 provides lower bounds for representations of products non-commutative +(Lie) algebras. If the factors Gi are assumed to be non-solvable, the lower bound +can be improved further. +Theorem 3. Let k be a field, let G1, . . . , Gn be groups and let G = G1 × · · · × Gn +denote their direct product. +(1) If the groups G1, . . . , Gn are non-abelian, then µk(G) ≥ n + 1. +(2) If the groups G1, . . . , Gn are non-solvable, then µk(G) ≥ 2n. +Both lower bounds in Theorem 3 are sharp. Let ai = I + E1,i+1, bi = I − +2Ei+1,i+1 ∈ GLn+1(Q) be as above. Setting Gi = ⟨ai, bi⟩ we can therefore deduce + +BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED +3 +that µQ(G1 ×. . .×Gn) = n+1. Suppose that the groups Gi in Theorem 3 are non- +solvable subgroups of GL2(k) for some field k. Then each Gi can be embedded in a +2×2-diagonal block in GL2n(k), which gives us an embedding of G = G1 ×· · ·×Gn +in GL2n(k). Together with Theorem 3 this implies µk(G) = 2n. In particular, +this applies to the case where each Gi is a non-abelian free group and thereby +recovers [6, Theorem 3] in the SLn-case. +1. Branch groups are not boundedly generated +There are several characterizations of branch groups. The following one, which +is a slight reformulation of [3, Definition 1.1], does not involve a rooted tree which +makes it rather abstract. However it suits well for our purposes. A more geometric +definition can be found in [3, Definition 1.13]. +Definition. A group G is called a branch group if it admits a decreasing sequence +of subgroups (Hi)i∈N0 with H0 = G and +� +i∈N0 +Hi = 1, and a sequence of integers +(ki)i∈N0 with k0 = 1 such that for each i the following hold: +(1) Hi is a normal subgroup of finite index in G. +(2) Hi splits as a direct product Hi = H(1) +i +× . . . × H(ki) +i +, where the factors are +pairwise isomorphic. +(3) the quotient mi+1 := ki+1/ki is an integer with mi+1 ≥ 2, and the product +decomposition of Hi+1 refines the product decomposition of Hi in the sense +that each factor H(j) +i +of Hi contains the factors H(ℓ) +i+1 of Hi+1, where ℓ +satisfies (j − 1) · mi+1 + 1 ≤ ℓ ≤ j · mi+1. +(4) G acts transitively by conjugation on the set of factors H(j) +i +of Hi. +As indicated in the introduction, not every branch group is just-infinite. In fact +there is no need for finitely generated branch groups to admit a just-infinite quotient +that is a branch group. See [8, Theorem 2] for an example of finitely generated +branch group that maps homomorphically onto Z, which is of course just-infinite +and bounded generated. As a consequence, to prove that branch groups cannot be +boundedly generated, it is not sufficient to consider the just-infinite case, in which +the claim turns out to be a direct consequence of results of Ab´ert [1], Pyber and +Segal [16]. +Proof of Theorem 1. Suppose there is a branch group G that is boundedly gener- +ated. Then [16, Corollary 1.6] tells us that G admits an epimorphism π: G → Q, +where Q is an infinite linear group. However, by [1, Corollary 7] branch groups are +not linear over any field. Thus Q is a proper quotient of G. As such Q is virtu- +ally abelian by [8, Proposition 6]. Since G, being a boundedly generated group, +is finitely generated, it follows that Q has a (non-trivial) free abelian finite index +subgroup Q0. We can therefore consider the finite index subgroup G0 := π−1(Q0) +of G, which by construction maps onto Z. Let us now fix an arbitrary number +n ∈ N. From the definition of a branch group it follows that G contains a finite +index subgroup of the form Hi = H(1) +i +× . . . × H(ki) +i +, where the factors are pairwise +isomorphic and ki ≥ n. Then Hi ∩ G0 is a finite index subgroup of Hi. In this +case it can be easily seen that there are pairwise isomorphic finite index subgroups +K(j) +i +≤ H(j) +i +such that Ki := K(1) +i +× . . . × K(ki) +i +≤ Hi ∩ G0. In particular we see +that Ki has finite index in G0, which implies that it maps onto Z. Thus some, and +hence every, factor K(j) +i +maps onto Z. We can therefore deduce that the torsion-free +part of the abelianization of Ki has rank at least ki ≥ n. As a consequence, this +holds for every finite index subgroup of Ki. In particular this tells us that there +is no finite index subgroup of Ki that can be generated with less then n elements. + +4 +S. KIONKE AND E. SCHESLER +Since n ∈ N was arbitrary this contradicts a result of Shusterman [17, Theorem +1.1], which tells us that for every boundedly generated group H there is a constant +C > 0 such that every finite index subgroup of H contains a finite index subgroup +that can be generated by at most C elements. +□ +2. Lower bounds for the minimal representation dimension of directs +products +Let us now prove the results concerning the minimal representation dimensions. +Proof of Theorem 2. For the proof we combine ideas from [1] and [5]. Extending +scalars, we may assume that k is an infinite field. Recall that I ∈ Mr,r(k) denotes +the identity matrix. We claim that I, a1, . . . , an, b1, . . . , bn are linearly independent. +This follows along the lines of [1, Proof of Thm. 3]. Suppose that cI + � +j λjaj + +� +j λ′ +jbj = 0 for c, λ1, . . . , λn, λ′ +1, . . . , λ′ +n ∈ k. Taking commutators with ai (resp. +bi) shows λi = 0 (resp. λ′ +i = 0); since I ̸= 0 the last remaining coefficient c vanishes +as well. +Let V = kr and let C denote the linear span of {I, a1, . . . , an, b1, . . . , bn} in +Mr,r(k). Consider the linear map Ψ: C → V defined by Ψ(X) = Xv for some +v ∈ V . We will see that the image of Ψ has dimension at least n + 1 if v is chosen +appropriately. As the commutators zi = [ai, bi] are non-trivial, the kernel of each +zi is a proper subspace of V . However, V cannot be covered by a finite union of +proper subspaces (as k is infinite). Thus there is a vector v ∈ V such that ziv ̸= 0 +for all i ∈ {1, . . . , n}. Let α: V → k be a linear form such that α(v) ̸= 0 and +α(ziv) ̸= 0 for all i ∈ {1, 2, . . ., n} (such a linear form α exists, as the dual space +V ∗ cannot be covered by finitely many proper subspaces). Now β : C × C → k +defined by β(x, y) = α([x, y](v)) is an alternating form on C. It is not difficult to +see that β is non-degenerate on the subspace ⟨a1, . . . , an, b1, . . . , bn⟩ ⊆ C (e.g. the +matrix representation has full rank). Let us observe that kI +ker(Ψ) is an isotropic +subspace, since for x, y ∈ kI + ker(Ψ) we have [x, y](v) = xyv − yxv = 0. As v ̸= 0 +we have I ̸∈ ker(Ψ) and thus dimk ker(Ψ) + 1 ≤ n + 1. This allows us to conclude +that +r ≥ dimk(im(Ψ)) = 2n + 1 − dimk ker(Ψ) ≥ n + 1. +□ +Proof of Theorem 3. The first assertion follows immediately from Theorem 2. As- +sume now that each Gi is non-solvable. If G is not linear, there is nothing to show. +Assume that (ρ, V ) is a finite dimensional faithful representation over k. By exten- +sion of scalars, we may assume that k is algebraically closed. Let V 1, . . . , V t denote +the composition factors of V considered as G-module. +Since k is algebraically +closed, the composition factor V j is isomorphic to a tensor product +V j = V j +1 ⊗k V j +2 ⊗k · · · ⊗k V j +n +where V j +i is an irreducible Gi-representation; see e.g. [11, Prop. 2.3.23]. The compo- +sition factors of V |Gi are the irreducible representations V 1 +i , . . . , V t +i each one possi- +bly occurring several times. Suppose for a contradiction that V j +i is one-dimensional +for all j. +Then there is a basis of V such that ρ(Gi) is represented by upper +triangular matrices. This gives a contradiction, since Gi is not solvable. +For each j let Sj ⊆ {1, . . ., n} be the set of i such that dimk V j +i ≥ 2. By the +observation above, each i ≤ n belongs to at least one of the sets Sj. This implies +dimk V = +t +� +j=1 +n +� +i=1 +dimk V j +i ≥ +t +� +j=1 +2|Sj| ≥ +t +� +j=1 +2|Sj| ≥ 2n. +□ + +BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED +5 +References +1. Mikl´os Ab´ert, Representing graphs by the non-commuting relation, Publ. Math. Debrecen 69 +(2006), no. 3, 261–269. MR 2273978 +2. Laurent Bartholdi and Rostislav I. Grigorchuk, On parabolic subgroups and Hecke algebras of +some fractal groups, Serdica Math. J. 28 (2002), no. 1, 47–90. MR 1899368 +3. Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran ˇSuni´k, Branch groups, Handbook of +algebra, Vol. 3, Handb. Algebr., vol. 3, Elsevier/North-Holland, Amsterdam, 2003, pp. 989– +1112. MR 2035113 +4. Marc Burger and Shahar Mozes, Lattices in product of trees, Inst. Hautes ´Etudes Sci. Publ. +Math. (2000), no. 92, 151–194 (2001). MR 1839489 +5. Leandro Cagliero and Nadina Rojas, Faithful representations of minimal dimension of current +Heisenberg Lie algebras, Internat. J. Math. 20 (2009), no. 11, 1347–1362. MR 2584190 +6. Caterina Campagnolo and Holger Kammeyer, Products of free groups in lie groups, Journal +of Algebra 579 (2021), 237–255. +7. David Carter and Gordon Keller, Bounded elementary generation of SLn(O), Amer. J. Math. +105 (1983), no. 3, 673–687. MR 704220 +8. Thomas Delzant and Rostislav Grigorchuk, Homomorphic images of branch groups, and +Serre’s property (FA), Geometry and dynamics of groups and spaces, Progr. Math., vol. +265, Birkh¨auser, Basel, 2008, pp. 353–375. MR 2402409 +9. Mikhail Ershov and Andrei Jaikin-Zapirain, Property (T) for noncommutative universal lat- +tices, Invent. Math. 179 (2010), no. 2, 303–347. MR 2570119 +10. R. I. Grigorˇcuk, On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen. +14 (1980), no. 1, 53–54. MR 565099 +11. Emmanuel Kowalski, An introduction to the representation theory of groups, Graduate Studies +in Mathematics, vol. 155, American Mathematical Society, Providence, RI, 2014. MR 3236265 +12. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik +und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer- +Verlag, Berlin, 1991. MR 1090825 +13. Donald McCarthy, Infinite groups whose proper quotient groups are finite. I, Comm. Pure +Appl. Math. 21 (1968), 545–562. MR 237637 +14. Alexey Muranov, Diagrams with selection and method for constructing boundedly generated +and boundedly simple groups, Comm. Algebra 33 (2005), no. 4, 1217–1258. MR 2136699 +15. Alexander Yu. Ol’shanskii, Infinite groups with cyclic subgroups, Dokl. Akad. Nauk SSSR 245 +(1979), no. 4, 785–787. MR 527709 +16. L´aszl´o Pyber and Dan Segal, Finitely generated groups with polynomial index growth, J. Reine +Angew. Math. 612 (2007), 173–211. MR 2364077 +17. Mark Shusterman, Ranks of subgroups in boundedly generated groups, Bull. Lond. Math. Soc. +48 (2016), no. 3, 539–547. MR 3509913 +18. J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), +373–391. MR 274575 +FernUniversit¨at in Hagen, Fakult¨at f¨ur Mathematik und Informatik, 58084 Hagen +Email address: steffen.kionke@fernuni-hagen.de +Email address: eduard.schesler@fernuni-hagen.de + diff --git a/VdAzT4oBgHgl3EQfYPwk/content/tmp_files/load_file.txt b/VdAzT4oBgHgl3EQfYPwk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..94e3f16cd945eaf30a45c0279a0d5b83efbf2a55 --- /dev/null +++ b/VdAzT4oBgHgl3EQfYPwk/content/tmp_files/load_file.txt @@ -0,0 +1,352 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf,len=351 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='01330v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='GR] 3 Jan 2023 ON REPRESENTATIONS OF DIRECT PRODUCTS AND THE BOUNDED GENERATION PROPERTY OF BRANCH GROUPS STEFFEN KIONKE AND EDUARD SCHESLER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' We prove that the minimal representation dimension of a direct product G of non-abelian groups G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , Gn is bounded below by n + 1 and thereby answer a question of Ab´ert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' If each Gi is moreover non-solvable, then this lower bound can be improved to be 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Introduction An infinite group G is called just-infinite if all of its proper quotients are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Obvious examples of just-infinite groups are virtually simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Other exam- ples arise from irreducible lattices in higher rank semisimple Lie groups, such as SLn(Z) for n ≥ 3, after dividing out their centers, see [12, Chapter IV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Such groups are in fact hereditarily just-infinite, which means that they are residually finite and all of their finite index subgroups are just-infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Grigorchuk’s group [10] provided the first example of a just-infinite group that is not virtually a finite direct power of a simple or a hereditarily just infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Grigorchuk’s group is a just-infinite branch group, which means that its commensurability classes of subnormal sub- groups form a lattice that is isomorphic to the lattice of open and closed subsets of a Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' By Wilson’s classification [18] just-infinite groups fall into three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Every just-infinite group G is either a branch group or virtually a direct power of a simple or a hereditarily just-infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Since its introduction by McCarthy [13] in the late 1960’s, the class of just-infinite groups remained an active field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' One reason might be that every finitely generated infinite group admits a just-infinite quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Thus whenever there is some finitely generated, infinite group G that admits a property P that is preserved under homomorphic images, then there is also a finitely generated just-infinite group with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Following [3], we call a property P that is preserved under homomorphic images an H-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Well-known examples of H-properties include amenability, property (T), bounded generation, being a torsion group, having subexponential growth etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In view of Wilson’s classification, it is natural to investigate which of the three classes of just-infinite groups contain groups that satisfy a given H- property P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' For the H-property “being a torsion group” this question is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In this case it is know that there are finitely generated simple groups [15], just-infinite branch groups [10], and hereditarily just-infinite groups [9] that are torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' On the other hand, there are torsion-free, finitely generated, just-infinite groups that are simple [4], branch [2], and hereditarily just-infinite (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The purpose of this note is to study this question for the bounded generation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Recall that a group G is boundedly generated if it contains a finite subset 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Primary 20E08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Secondary 20E26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' bounded generation, branch group, faithful representations of direct products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 441848266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 1 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' KIONKE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' SCHESLER {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , gn} such that every g ∈ G can be written as g = gk1 1 · · · gkn n for appropriate ki ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Since infinite torsion groups are not boundedly generated, it follows that each of the three classes of just-infinite groups contains a finitely generated group that does not have the bounded generation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' On the other hand, it was proven by Carter and Keller [7] that PSLn(Z) is boundedly generated for n ≥ 3, which provides an interesting boundedly generated hereditarily just-infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The existence of boundedly generated, infinite, simple groups was established by Muranov [14], whose construction seems to be the only one available at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' It remains to study just-infinite branch groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The question of existence of boundedly generated just-infinite branch groups was raised by Bartholdi, Grigorchuk, and ˇSuni´k [3, Question 12] and remained open to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The purpose of the paper is to show that the answer is negative for arbitrary branch groups (even without the assumption of being just-infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' There is no boundedly generated branch group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As a consequence, it follows from Wilson’s classification of just-infinite groups that every boundedly generated infinite group has a quotient that is virtually a product of finitely many copies of a boundedly generated simple or hereditarily just infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The proof the Theorem 1 is a rather direct combination of results of Pyber and Segal [16], Shusterman [17], and Ab´ert [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Ab´ert proved that weakly branch groups are not linear over any field (for branch groups this result is due to Grigorchuk and Delzant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' More precisely, he defined for every field k the natural number matk(n) to be the minimal r such that every graph on n vertices can be represented in the matrix algebra Mr,r(k) where the graph’s edges encode non-commutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Ab´ert showed that � ⌊n/2⌋ ≤ matk(n) ≤ 2(n−⌊log2(n)⌋+1) and asked for a linear lower bound [1, Question 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The following result answers this question in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let k be a field and let r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Suppose that there are (r × r)-matrices a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , bn ∈ Mr,r(k) such that all pairwise commutators are trivial except for [ai, bi] = aibi − biai for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Then r ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The lower bound in Theorem 2 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let λ ∈ k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Consider the matrices ai = I + E1,i+1, bi = I − λEi+1,i+1 ∈ Mn+1,n+1(k) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , n, where I is the identity matrix and Ei,j denotes the elementary matrix whose (i, j)-entry is 1 and all other entries are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Then ai, bi satisfy the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' If λ ̸= 1, then ai, bi are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' If |k| > 2, this shows with [1, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 5] that �n 2 � + 1 ≤ matk(n) ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The non-linearity of weakly branch groups follows since these groups contain infinite products of non-abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let µk(G) denote the minimal dimension of a faithful, finite dimensional representation of a group G over a field k (we write µk(G) = ∞ if G is not linear over k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Theorem 2 directly implies a lower bound µk(G1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' × Gn) ≥ n + 1 for direct products of non-abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Similarly Theorem 2 provides lower bounds for representations of products non-commutative (Lie) algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' If the factors Gi are assumed to be non-solvable, the lower bound can be improved further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let k be a field, let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , Gn be groups and let G = G1 × · · · × Gn denote their direct product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' (1) If the groups G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , Gn are non-abelian, then µk(G) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' (2) If the groups G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , Gn are non-solvable, then µk(G) ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Both lower bounds in Theorem 3 are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let ai = I + E1,i+1, bi = I − 2Ei+1,i+1 ∈ GLn+1(Q) be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Setting Gi = ⟨ai, bi⟩ we can therefore deduce BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED 3 that µQ(G1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='×Gn) = n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Suppose that the groups Gi in Theorem 3 are non- solvable subgroups of GL2(k) for some field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Then each Gi can be embedded in a 2×2-diagonal block in GL2n(k), which gives us an embedding of G = G1 ×· · ·×Gn in GL2n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Together with Theorem 3 this implies µk(G) = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In particular, this applies to the case where each Gi is a non-abelian free group and thereby recovers [6, Theorem 3] in the SLn-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Branch groups are not boundedly generated There are several characterizations of branch groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The following one, which is a slight reformulation of [3, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='1], does not involve a rooted tree which makes it rather abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' However it suits well for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' A more geometric definition can be found in [3, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' A group G is called a branch group if it admits a decreasing sequence of subgroups (Hi)i∈N0 with H0 = G and � i∈N0 Hi = 1, and a sequence of integers (ki)i∈N0 with k0 = 1 such that for each i the following hold: (1) Hi is a normal subgroup of finite index in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' (2) Hi splits as a direct product Hi = H(1) i × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' × H(ki) i , where the factors are pairwise isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' (3) the quotient mi+1 := ki+1/ki is an integer with mi+1 ≥ 2, and the product decomposition of Hi+1 refines the product decomposition of Hi in the sense that each factor H(j) i of Hi contains the factors H(ℓ) i+1 of Hi+1, where ℓ satisfies (j − 1) · mi+1 + 1 ≤ ℓ ≤ j · mi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' (4) G acts transitively by conjugation on the set of factors H(j) i of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As indicated in the introduction, not every branch group is just-infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In fact there is no need for finitely generated branch groups to admit a just-infinite quotient that is a branch group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' See [8, Theorem 2] for an example of finitely generated branch group that maps homomorphically onto Z, which is of course just-infinite and bounded generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As a consequence, to prove that branch groups cannot be boundedly generated, it is not sufficient to consider the just-infinite case, in which the claim turns out to be a direct consequence of results of Ab´ert [1], Pyber and Segal [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Suppose there is a branch group G that is boundedly gener- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Then [16, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='6] tells us that G admits an epimorphism π: G → Q, where Q is an infinite linear group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' However, by [1, Corollary 7] branch groups are not linear over any field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Thus Q is a proper quotient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As such Q is virtu- ally abelian by [8, Proposition 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Since G, being a boundedly generated group, is finitely generated, it follows that Q has a (non-trivial) free abelian finite index subgroup Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' We can therefore consider the finite index subgroup G0 := π−1(Q0) of G, which by construction maps onto Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let us now fix an arbitrary number n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' From the definition of a branch group it follows that G contains a finite index subgroup of the form Hi = H(1) i × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' × H(ki) i , where the factors are pairwise isomorphic and ki ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Then Hi ∩ G0 is a finite index subgroup of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In this case it can be easily seen that there are pairwise isomorphic finite index subgroups K(j) i ≤ H(j) i such that Ki := K(1) i × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' × K(ki) i ≤ Hi ∩ G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In particular we see that Ki has finite index in G0, which implies that it maps onto Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Thus some, and hence every, factor K(j) i maps onto Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' We can therefore deduce that the torsion-free part of the abelianization of Ki has rank at least ki ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As a consequence, this holds for every finite index subgroup of Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' In particular this tells us that there is no finite index subgroup of Ki that can be generated with less then n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' KIONKE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' SCHESLER Since n ∈ N was arbitrary this contradicts a result of Shusterman [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='1], which tells us that for every boundedly generated group H there is a constant C > 0 such that every finite index subgroup of H contains a finite index subgroup that can be generated by at most C elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Lower bounds for the minimal representation dimension of directs products Let us now prove the results concerning the minimal representation dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' For the proof we combine ideas from [1] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Extending scalars, we may assume that k is an infinite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Recall that I ∈ Mr,r(k) denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' We claim that I, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , bn are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' This follows along the lines of [1, Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Suppose that cI + � j λjaj + � j λ′ jbj = 0 for c, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , λn, λ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , λ′ n ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Taking commutators with ai (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' bi) shows λi = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' λ′ i = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' since I ̸= 0 the last remaining coefficient c vanishes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let V = kr and let C denote the linear span of {I, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , bn} in Mr,r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Consider the linear map Ψ: C → V defined by Ψ(X) = Xv for some v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' We will see that the image of Ψ has dimension at least n + 1 if v is chosen appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As the commutators zi = [ai, bi] are non-trivial, the kernel of each zi is a proper subspace of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' However, V cannot be covered by a finite union of proper subspaces (as k is infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Thus there is a vector v ∈ V such that ziv ̸= 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let α: V → k be a linear form such that α(v) ̸= 0 and α(ziv) ̸= 0 for all i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=', n} (such a linear form α exists, as the dual space V ∗ cannot be covered by finitely many proper subspaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Now β : C × C → k defined by β(x, y) = α([x, y](v)) is an alternating form on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' It is not difficult to see that β is non-degenerate on the subspace ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , bn⟩ ⊆ C (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' the matrix representation has full rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let us observe that kI +ker(Ψ) is an isotropic subspace, since for x, y ∈ kI + ker(Ψ) we have [x, y](v) = xyv − yxv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As v ̸= 0 we have I ̸∈ ker(Ψ) and thus dimk ker(Ψ) + 1 ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' This allows us to conclude that r ≥ dimk(im(Ψ)) = 2n + 1 − dimk ker(Ψ) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The first assertion follows immediately from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' As- sume now that each Gi is non-solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' If G is not linear, there is nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Assume that (ρ, V ) is a finite dimensional faithful representation over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' By exten- sion of scalars, we may assume that k is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Let V 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , V t denote the composition factors of V considered as G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Since k is algebraically closed, the composition factor V j is isomorphic to a tensor product V j = V j 1 ⊗k V j 2 ⊗k · · · ⊗k V j n where V j i is an irreducible Gi-representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' [11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' The compo- sition factors of V |Gi are the irreducible representations V 1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' , V t i each one possi- bly occurring several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Suppose for a contradiction that V j i is one-dimensional for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Then there is a basis of V such that ρ(Gi) is represented by upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' This gives a contradiction, since Gi is not solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' For each j let Sj ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=', n} be the set of i such that dimk V j i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' By the observation above, each i ≤ n belongs to at least one of the sets Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' This implies dimk V = t � j=1 n � i=1 dimk V j i ≥ t � j=1 2|Sj| ≥ t � j=1 2|Sj| ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' □ BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED 5 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Mikl´os Ab´ert, Representing graphs by the non-commuting relation, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Debrecen 69 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 3, 261–269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 2273978 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Laurent Bartholdi and Rostislav I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 28 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 1, 47–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 1899368 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Laurent Bartholdi, Rostislav I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Grigorchuk, and Zoran ˇSuni´k, Branch groups, Handbook of algebra, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 3, Handb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Algebr.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 92, 151–194 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 1839489 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Leandro Cagliero and Nadina Rojas, Faithful representations of minimal dimension of current Heisenberg Lie algebras, Internat.' metadata={'source': 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353–375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 2402409 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Mikhail Ershov and Andrei Jaikin-Zapirain, Property (T) for noncommutative universal lat- tices, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 179 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 2, 303–347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 2570119 10.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 48 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 3, 539–547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 3509913 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Wilson, Groups with every proper quotient finite, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' 69 (1971), 373–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content=' MR 274575 FernUniversit¨at in Hagen, Fakult¨at f¨ur Mathematik und Informatik, 58084 Hagen Email address: steffen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='kionke@fernuni-hagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='de Email address: eduard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='schesler@fernuni-hagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'} diff --git a/W9E4T4oBgHgl3EQfNQwq/content/tmp_files/2301.04954v1.pdf.txt b/W9E4T4oBgHgl3EQfNQwq/content/tmp_files/2301.04954v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c5c278374727b1bf35f57aac3683ace1be8290a --- /dev/null +++ b/W9E4T4oBgHgl3EQfNQwq/content/tmp_files/2301.04954v1.pdf.txt @@ -0,0 +1,822 @@ +We are Going to the Space - Part 1: +Which device to deploy in a satellite? +Robert Bayer +IT University of Copenhagen +roba@itu.dk +Julian Priest +IT University of Copenhagen +jucp@itu.dk +Pınar Tözün +IT University of Copenhagen +pito@itu.dk +ABSTRACT +The shrinkage in sizes of components that make up satellites led to +wider and low cost availability of satellites. As a result, there has +been an advent of smaller organizations having the ability to deploy +satellites with a variety of data-intensive applications to run on +them. One popular application is image analysis to detect, for exam- +ple, land, ice, clouds, etc. However, the resource-constrained nature +of the devices deployed in satellites creates additional challenges +for this resource-intensive application. +In this paper, we investigate the performance of a variety of edge +devices for deep-learning-based image processing in space. Our +goal is to determine the devices that satisfy the latency and power +constraints of satellites while achieving reasonably accurate results. +Our results demonstrate that hardware accelerators (TPUs, GPUs) +are necessary to reach the latency requirements. On the other hand, +state-of-the-art edge devices with GPUs could have a high power +draw, making them unsuitable for deployment on a satellite. +1 +INTRODUCTION +In the last century, most innovation in real-world satellite appli- +cations were only available to the largest countries such as USA +and Russia. These innovations led to significant reductions in the +size of the components that make up a satellite and in the cost of +the manufacturing and deployment process of a satellite. This, in +turn, introduced a new CubeSat class of miniature satellites. Their +format is based on a 10cm cube, with the possibility of combining +multiple modules to create a larger satellite. This standardization +makes a batch deployment of satellites easier as the format affords +tight configuration. The reduction in costs that came as a result of +this new deployment method led to the advent of satellites owned +by small or private organizations. +However, the size of the CubeSat class satellites poses new com- +plex challenges such as the power and thermal constraints as well +as the physical dimensions of components. These CubeSats often +perform resource-intensive tasks, which clashes with the resource- +constrained nature of their format. +Image processing and analysis is one class of possible satellite +workloads. Satellites with this type of workload take large-scale +images, which they have to store and send back to a ground station. +The link between the satellite and a ground station is of limited +bandwidth and short-lived. The images must be highly compressed +to send all of them or filtered by quality and areas of interest. The +images already have a resolution of tens or hundreds of meters +per pixel, and lossy compression would lead to even more loss of +detail. Filtering preserves the details of the images, but is more +resource-intensive. +This paper is a step toward understanding the requirements +and limitations of deep-learning-based image filtering systems on +small satellites. More specifically, we characterize the requirements +and constraints of an image processing unit (IPU) deployed on a +satellite and outline possible use case scenarios of such a system, +each introducing a different degree of resource constraints and +intensity. We then further analyze the performance of multiple +edge devices on these different scenarios and their suitability for +possible deployment on one of the Danish Student CubeSat Project’s +(DISCO) [7] satellites. +These devices are based on different hardware architectures, +ranging from a microcontroller to a tensor processing unit (TPU). +While microcontrollers or more complex CPUs have extensive flight +history (have already been deployed in satellites before) and low +power footprint, they were not built with running deep learning +workloads in mind. On the other hand, GPUs have been utilized +more in the past decade in satellites, with one of the leading work- +loads being machine learning, based on the success of GPUs in +terrestrial use cases of machine learning. The latest of these archi- +tectures, TPU, has not been extensively researched for deployment +in satellites, even though TPUs were designed to perform fast neural +network inference, promising low latency with high efficiency. +The contributions of this study are as follows: +• We illustrate the differences between the terrestrial and space +edge IoT. The edge devices deployed in space have to perform +very resource-intensive workloads with high reliability and do +so in a highly resource-constrained environment. +• We characterize a set of requirements specific to IPU on a +CubeSat. We do this by outlining multiple use case scenarios +and showing how they affect the required latency and possible +power draw of such a system. +• We compare the performance of multiple edge devices based +on different architectures and show the need and the bene- +fit of using highly specialized hardware for machine learning +workloads on satellites. +2 +BACKGROUND +We first introduce DISCO project and survey related work that +specifically targets data processing in satellites. +2.1 +DISCO +The Danish Student Cubesat Program (DISCO) [7] offers Danish +university students the opportunity to design and operate a small +satellite and to gain space flight experience. DISCO is a collaboration +between four universities in Denmark, which will initially launch +three student Cubesats into Low Earth Orbit with the first launch +scheduled for 2023. +DISCO2 has been designed by students to carry an Earth imag- +ing payload, infrastructure to capture images in space, into a solar +synchronous polar orbit. The instrument is a collaboration with +arXiv:2301.04954v1 [cs.LG] 12 Jan 2023 + +Bayer, et al. +Image process- +ing Unit +Nominal +Power +Peak Power +Design Margin +Nominal Cycle +Peak Cycle +Power +Mass budget +Dimensions +Constraints +2.00 W +5.00 W +5% +20% +20% +1.48 W +0.15 Kg +10x70x80 mm +Table 1: List of constraints of the IPU on board of the DISCO2 satellite. +the Arctic Research Centre at Aarhus University and will support a +range of field research in Greenland. Initially focused on a long term +marine systems and cryosphere monitoring projects, the satellite +will supplement ground based observations with remote sensing +data, providing both good polar coverage and on demand availabil- +ity for the projects. +The payload will include one or more high resolution cameras +as well as a dedicated IPU, which will be capable of simple machine +learning applications for image classification. Students will be able +to carry out conventional on-satellite image processing in addi- +tion to running machine learning applications for classification, +discrimination, and feature identification. +2.2 +Related Work +The challenge of low bandwidth on small satellites creates the need +for data post-processing to filter the data on the satellite before +sending elsewhere. Neural-network-based filtering has been stud- +ied for this purpose in recent years. Specifically, several works have +tested the suitability of small system-on-chip (SoC) devices leverag- +ing the power of GPUs to accelerate machine learning workloads +[3, 13]. Moreover, some works have also explored the use of ASICs, +such as Intel’s Movidius Myriad vision processing units (VPUs) +[8, 9]. This VPU has even seen real-life deployment [8]. +Evaluation of TPUs for deployment on satellites is limited, how- +ever. To the best of our knowledge, only one work has explored this +option [10], with no real-life deployments. DISCO project would +therefore be the first satellite to leverage TPU for onboard deep +learning applications, based on the findings of our study. +In this work, we aim to characterize the performance of the dif- +ferent flavors of off-the-shelf edge devices to check their suitability +for being deployed on a satellite in the context of DISCO project. +Data management and processing for internet-of-things and edge +computing has been important in the data management community +as well [4, 14, 16, 18]. We are complementing these works with a +very specific application focus, which is data-intensive applications +deployed on satellites. +3 +REQUIREMENTS +The requirements and constraints used in this study are based on +the DISCO2 Arctic imaging mission use case. The power and mass +constraints of the edge device to be deployed on the satellite are +based on the power and mass budgets developed in the DISCO2 en- +gineering design. The satellite design is a modular 3U Cubesat with +off-the-shelf modules for attitude control, power, communications, +and flight control. The power budget values are typical for a 3U +earth imaging Cubesat of this type. The resulting payload capacity +is 1U (10x10x10cm) and 1.3kg and this must include the cameras, +module enclosure, optics. and image processing unit (IPU). The full +list of constraints of the IPU is shown in Table 1. +The planned orbit of DISCO2 is a solar synchronous polar or- +bit at 550 km altitude. The camera sensor is based on an Alvium +1800 C-2040 and the lens focal length is the maximum that could +be considered for the mission. Assuming a 50% overlap between +images, i.e., capturing the same land area, this provides a minimum +time of 4.42s between consecutive images (𝑇𝑖), which was derived +as follows: +𝑇𝑖 = 𝐺𝑆𝐷 ∗𝑊𝑖 ∗ ∩𝑖 +𝑣 += 14.8495𝑚/𝑝𝑥 ∗ 4512𝑝𝑥 ∗ 0.5 +7585.16𝑚/𝑠 += 4.42𝑠 +(1) +, where 𝐺𝑆𝐷 (ground sample distance) is the spatial resolution of +the image, 𝑊𝑖 is the width of image in pixels, ∩𝑖 is the overlap +between images and 𝑣 is the orbital velocity of the satellite. +Taking this latency value as a baseline, we now describe three +image processing scenarios with different levels of difficulty based +on this Arctic mission’s goals. +Scenario 1: Real-time imaging. Images are taken with a pe- +riod of 4.42s, as derived above. The inference has to be performed +and completed before the next image is captured. This would allow +for the results of an inference to be used in decision making for the +next image capture. +Scenario 2: Arctic region imaging. In this scenario, images +are taken only while the satellite is over the polar regions relaxing +the latency requirements. Images are buffered and inference is +performed in the remainder of the orbit, with the IPU available for +inference before the subsequent orbit. The capture-inference cycle +completes in 5739s. +Scenario 3: Greenland imaging. The images are taken only +while over Greenland further relaxing the latency requirements. +As the orbit is solar synchronous with a period of one day. This +results in subsequent bursts but only when the swathe passes over +Greenland. Images are again buffered and the capture-inference +cycle completes in 21600s. +4 +METHODOLOGY AND SETUP +Our goal is to characterize the performance of modern low-power +hardware for deployment as the IPU of the DISCO2 satellite. This +section presents our experimental methodology and setup to achieve +this goal. +4.1 +Devices under Test +As representative hardware for this study, we picked three devices, +each based on a different hardware architecture. These choices were +made based on their physical dimensions, weight, power draw, and +high performance considering the low power draw limits. +ARM Cortex-M Mirocontroller, based on the STM32H745 +chip, has the potential of the lowest power draw, while the least +performant of the choices. As an additional advantage, this chip +has extensive flight history and, therefore, would be a safe choice +for deployment in space applications. + +We are Going to the Space - Part 1: +Which device to deploy in a satellite? +In order to use this device as an on-board IPU, we can use Tensor- +Flow Lite for Microcontrollers [5], a framework designed to allow +neural network inference with only 16 KB of memory overhead +(core runtime) and without the need for an operating system. In +addition, the framework also provides support for the use of spe- +cialized neural network kernels, such as CMSIS-NN [12] kernels +designed specifically for Cortex-M processor cores. The X-CUBE- +AI [17] framework makes the deployment of the neural networks +with TensorFlow Lite for microcontrollers even more accessible. It +provides an intuitive GUI for the libraries or code samples based +on the provided and trained TensorFlow Lite [2] model. +Even though this device has support for floating point operations, +a quantized model was used, in order to achieve better latency +and, more importantly, lower memory footprint, which is a large +constraint of this device. +To simulate the exact performance and power characteristics of +a flight computer already available for a potential deployment in +DISCO project, OBC-P3, we used only the Cortex-M7 core of the +STM32H745 and scaled its clock from 480MHz down to 300MHz. +Parameters of the OBC-P3 system can be found in the Table 2. +Processor +2x ARM Cortex-M7 @ 300 MHz +SRAM +384 KB SRAM +FRAM +32 KB +Flash +2 MB +Storage +64 GB eMMC +Dimensions +94 x 94 x 13 mm +Weight +120 g +Table 2: Specifications of the OBC-P3 system containing +the ARM Cortex-M7 MCUs. The dimensions and weight in- +cludes the enclosure with aluminium shielding. +NVIDIA Jetson Nano is a portable SoC composed of power- +efficient ARM CPU and NVIDIA GPU designed for embedded sys- +tems that require GPU-friendly computations, e.g., image process- +ing, video encoding/decoding, and machine learning tasks. It is +the only device in our experiments that supports TensorFlow Core. +Therefore, the deployment process to this device is equivalent to +that of servers or desktops. It is also the only evaluated device +supporting batch sizes larger than one. +The device can operate at a wattage between 5W - 10W. We +configured it to operate at 5W by disabling two of the four CPU +cores. While this setting leads to lower performance, it is necessary +in order to fit into the power budget outlined in Table 1. Parameters +of this device can be found in Table 3. +CPU +Quad-core ARM A57 @ 1.43 GHz +GPU +128-core Maxwell +GFLOPS +472 +RAM +4 GB 64-bit LPDDR4 25.6 GB/s +Storage +64 GB SD card +Dimensions +100 x 80 x 29 mm +Weight +141 g +Table 3: Specifications of the NVIDIA Jetson Nano. The di- +mensions and weight are based on the developer kit version +of this device. +CoralAI Dev +Board Mini +Raspberry Pi 2B +CPU +Quad-core Arm +Cortex-A35 @ 1.5 GHz +Quad-core ARM +Cortex-A7 @ 900 MHz +RAM +2 GB +1 GB +Storage +8 GB eMMC +32 GB SD card +Dimensions +64 x 48 x 14.6 mm +85.6 x 56.5 x 17 mm +Weight +25.5 g +45 g +CoralAI TPU +TOPS +Interface +Dimensions +Weight +4 +USB2 +65 x 30 mm +4.3 g +Table 4: Specifications of the CoralAI Dev Board Mini and +Raspberry Pi 2B, as well as the CoralAI TPU chip. Note +that the CoralAI Dev Board Mini’s physical dimensions and +weight include the on-board TPU. +CoralAI TPU is an ASIC, AI accelerator developed by Google +to accelerate machine learning workloads at the edge. TPU is not +a standalone device and must be deployed together with a Linux, +Windows, or macOS system. Even though this device is highly +specialized, model deployment is performed by compiling a Ten- +sorFlow Lite model containing a subset of supported layers [1]. +This model must be quantized to an 8-bit integer, as this is only +supported data type by this device. This compiled model can then +be used with the TPU’s Python or C++ library. In our experiments, +we rely on the C++ library. +We evaluated this device in two formats: +(1) CoralAI Dev Board Mini, which couples this chip and an SoC +on a single board. +(2) CoralAI USB accelerator connected to a Raspberry Pi 2B. +Even though (1) has the TPU chip on the same board as the SoC, +the two communicate through the USB2 bus, as in the case of (2). +In addition, both of the devices run Linux. The parameters of the +different devices can be found in Table 4. +4.2 +Metrics +The metrics to evaluate the suitability of the devices as the on- +satellite IPUs are based on the requirements Section 3 outlined. +Latency is reported in seconds per inference of a sample, where +a sample is a whole image of size 4512 x 4512 pixels or 400 tiles of +size 224 x 224 pixels. We measure the latency of the neural network +inference once the data is already in the device’s memory. +Nominal power draw is the average power draw in mW mul- +tiplied by the duty cycle, which is the ratio between achieved and +required latency for each scenario (outlined in Section 3). We use +an external appliance for CoralAI TPU and the ARM Cortex-M7 +and tegrastats for Jetson Nano to measure this metric. +Peak power draw is the maximum power draw over the pe- +riod of inference in mW. It is measured using the same tools as in +nominal power draw. +Power consumption is reported per inference of a sample. This +metric shows the power efficiency of the device and will guide +the choice of a more suitable device if multiple devices fulfill the +requirements of a particular scenario. + +Bayer, et al. +Scaling factor +0.25 +0.5 +1.0 +Device +Latency (s) +Power consump- +tion (mWh) +Latency (s) +Power consump- +tion (mWh) +Latency (s) +Power consump- +tion (mWh) +ARM Cortex-M7 +118.80 +12.10 +N/A +N/A +N/A +N/A +CoralAI Dev Board Mini +3.778 +1.76 +3.991 +1.90 +4.533 +2.40 +Raspberry Pi + CoralAI TPU +3.106 +1.69 +3.331 +1.93 +3.825 +2.40 +NVIDIA Jetson Nano +13.900 +10.54 +14.109 +12.74 +18.581 +22.10 +NVIDIA Jetson Nano, batch size 64 +3.529 +3.71 +5.382 +5.94 +10.171 +12.89 +Table 5: Latency and power consumption results with different scaling factors of the model as measured on corresponding +devices. Batch size of 1 is used if not stated otherwise. +Accuracy of the deployed model is also measured to show the +effect quantization has on the model’s predictive performance. +4.3 +Workload +To simulate the imaging scenarios of interest (Section 3), we use an +image classification workload consisting of a 5-class classification +problem. For this workload, an off-the-shelf MobileNetV1 model +[11] pre-trained on the ImageNet dataset [6] was chosen for its +strong predictive power and availability in multiple scaling factors, +affecting the number of hidden layers in the model. The availability +of multiple scaling factors is an important factor as the memory of +some of the devices is highly limited and can therefore fit only the +smallest of the variants. +We further fine-tuned the model before deployment on the de- +vices using the Flowers dataset [15], containing 3670 color images +of size 224 x 224 pixels belonging to 5 different classes. While the +weights in the model do not affect the inference latency or power +required to perform the inference, the model was fine-tuned using +this dataset because it mimics one of the possible use cases closely +(in number of classes as well as size of the images) and allows us to +quantify the effects of the size and precision of the model. +Since the models running on the Cortex-M7 and CoralAI TPU +were quantized, rescaling of images was not needed before infer- +ence. For Jetson Nano, the pixel values must be rescaled to the range +of [0, 1), or, in other words, divided by 255. An additional layer was +prepended to the model for this process in order to take advantage +of the GPU parallelism on Jetson Nano. +As the size of the images would stress the memory available on +the evaluated devices, we employed a tiling method for inference. +Each image was divided into 400 patches of size 224 x 224.1 This +method results in a positive side effect, where each patch can be +filtered separately acting as a coarse-grained image segmentation. +This way only the tiles of interest are sent back to a ground station +saving us bandwidth. +The results are reported as an average of 10 inferences on the full +4512 x 4512 pixel image. The inference on Cortex-M7 and CoralAI +TPU were performed with batch size of 1 and on Jetson Nano with +the batch size of 2𝑥, where x is from range of 0-6. All of the devices +were tested with the scaling factors 0.25, 0.5, and 1.0 of the model, +with the exception of the ARM Cortex-M7 microcontroller, which +could not fit the larger models in memory. +1The image is not evenly divisible; therefore, the borders are disregarded. +5 +RESULTS +The results are split into three parts: (1) analysis of latency and +power draw results with respect to the three scenarios outlined in +Section 3, (2) impact of quantization on the accuracy of models with +various scaling factors, (3) impact of batch size on the performance +of NVIDIA Jetson Nano. +5.1 +Scenarios +Table 5 shows the latency achieved and power consumption of the +devices using the various scaling factors of the model. Furthermore, +Tables 6 and 7 show the nominal and peak power draw of different +configurations, respectively. The Sections 5.1.1-5.1.3 discuss the +suitability of each device for the different scenarios based on the +results on these tables. The reported latency, power consumption, +and the peak power draw are independent of the scenarios. Only +the nominal power draw depends on the use case scenarios, due to +the duty cycle (see Section 4.2). +5.1.1 +Scenario 1: Real-time imaging. Real-time imaging is the most +constrained scenario. Therefore, a high degree of specialization is +necessary to fulfill the requirements. +Latency. The pipeline has to achieve a 4.42s latency, which is +the spacing between images with 50% overlap. There are no passive +periods. Therefore the latency of the pipeline has to be lower than +that of the imaging for the pipeline not to get backed up. There are +multiple configurations that achieved the required latency (Table +5). These are all the configurations utilizing a TPU and the smallest +model running on Jetson Nano with batch size of 64. +Power draw. Out of the configurations that satisfy the latency +requirements for this scenario, only of the TPU configurations fit +into the nominal power budget of the satellite (Table 6). Even though +the Jetson Nano with batch size 64 achieves a latency comparable +to the TPUs with a scaling factor of 0.25, the power consumption +per inference is more than twice as high. +Furthermore, the Raspberry Pi with external TPU module does +not fit into the power budget when using the largest model, due +to exceeding the 5W requirements for peak power draw (Table 7). +NVIDIA Jetson Nano using the largest model with batch size of 64 +also exceeds this peak power requirement. Since the peak power +draw does not depend on neither the use case scenario nor the duty +cycle, this conclusion about peak power draw holds for the rest of +the scenarios as well. + +We are Going to the Space - Part 1: +Which device to deploy in a satellite? +Scaling factor +0.25 +0.5 +1.0 +Device +Power draw (mW) +Power draw (mW) +Power draw (mW) +ARM Cortex-M7 +Scenario 1 +- +- +- +Scenario 2 +- +- +- +Scenario 3 +161 +- +- +CoralAI Dev Board Mini +Scenario 1 +1433 +1546 +- +Scenario 2 +89 +95 +120 +Scenario 3 +23 +25 +32 +Raspberry Pi + CoralAI TPU +Scenario 1 +1376 +1571 +1954 +Scenario 2 +85 +97 +120 +Scenario 3 +23 +26 +32 +NVIDIA Jetson Nano +Scenario 1 +- +- +- +Scenario 2 +529 +639 +1109 +Scenario 3 +141 +170 +295 +NVIDIA Jetson Nano, batch size 64 +Scenario 1 +3021 +- +- +Scenario 2 +186 +298 +646 +Scenario 3 +49 +79 +172 +Table 6: Nominal power draw of devices with models of different sizes. The combinations of devices and model sizes, which do +not fulfil the latency requirements, are omitted as their active duty cycle is greater than 100%. The power draw is highlighted +in bold when exceeding the power budget. +Scaling factor +0.25 +0.5 +1.0 +Device +Power +draw (mW) +Power +draw (mW) +Power +draw (mW) +Arm Cortex-M7 +389 +N/A +N/A +CoralAI Dev Board Mini +2770 +2930 +4790 +Raspberry Pi + CoralAI +TPU +2595 +3405 +5050 +NVIDIA Jetson Nano +2871 +3546 +4717 +NVIDIA +Jetson +Nano, +batch size 64 +4523 +4684 +5064 +Table 7: Peak power draw of the devices during inference on +full-size image. Batch size of 1 is used if not stated otherwise. +The peak power is in bold when exceeding the power budget. +5.1.2 +Scenario 2: Arctic region imaging. By relaxing the constraints +and performing the workload equivalent to taking images of only +the areas above the arctic polar circle, we see more configurations +passing the requirements necessary to perform such a workload. +Latency. Since there would be passive imaging periods perform- +ing the workload for this scenario, the pipeline does not need to be +lower than the latency of imaging. It can instead buffer the images +and perform the inference at higher latency, given that the pipeline +can finish inference of one burst before the next burst of images +comes, which corresponds to a total latency of 5739 seconds or +71.74 seconds per image. All the configurations except the ones +with ARM Cortex-M7 achieve this latency (Table 5). +Power draw. Because of the large margin between the required +and achieved latencies for all configurations passing the latency +requirements, the active duty cycle is relatively low. Therefore, the +corresponding nominal power draw is well below the required one +in all cases (Table 6). +5.1.3 +Scenario 3: Greenland imaging. The least constrained sce- +nario corresponds to taking images of only Greenland. The low +constraints mean that all of the devices pass the requirements under +certain model configurations. +Latency. The maximum total latency for a burst of images is +21600 seconds, corresponding to 270 seconds per single image. This +requirement is easily passed by all of other configurations. +Power draw. Due to the low active duty cycle, each configura- +tion passes the nominal power draw requirement (Table 6). There is, +however, a large gap between the nominal power draw of TPU and +the rest of the devices. TPU performs the inference much more effi- +ciently, and its power draw scales more favourably with increasing +scaling factors in comparison to the other devices. +Scaling factor +Precision +0.25 +0.5 +1.0 +32 bit float +86.65% +90.46% +91.42% +8 bit integer +83.24% +90.32% +91.01% +Table 8: Accuracy of the full-precision and quantized Mo- +bilenetV1 as measured on the Flowers dataset. +5.2 +Effect of quantization and scaling factor +Table 8 shows the accuracy of the model with the various scaling +factors before and after quantization to an 8-bit integer. Models +with scaling factors 0.5 and 1.0 do not show a significant change in +predictive performance. We can, however, see a 3% drop in accuracy + +Bayer, et al. +Figure 1: Latency and power consumption at varying model +scaling factors and batch sizes on the NVIDIA Jetson Nano. +in the model with the smallest scaling factor. However, this differ- +ence is acceptable in our case and is outweighed by the benefits of +lower memory footprint and latency, and higher overall efficiency. +5.3 +Batch size impact on NVIDIA Jetson Nano +NVIDIA Jetson Nano is the only device in our evaluation that +allows doing inference with a batch size higher than 1. Figure 1 +shows the impact of increasing the batch size on latency and power +consumption of the inference. The most significant difference is +between batch sizes 1 and 8. The impact of increasing batch size then +tapers off. We can see that maximizing the batch size can lead to +45.3-74.6% lower latency and 41.7-64.8% lower power consumption. +6 +DISCUSSION +TPU shows the most promising results as it is the only device to +fulfill the requirements for real-time imaging, which is the most +constrained scenario. This device is highly specialized for this pur- +pose, utilizing the systolic array architecture. This architecture is +purpose-built to perform fast multiply-accumulate operations in a +highly parallel fashion, which allows performing matrix multiplica- +tion without the need to load/store intermediate values. +On the other end of the spectrum was the ARM Cortex-M7 micro- +controller. Due to its lack of parallelism, this device could only fulfill +the requirements of the least constrained scenario. Furthermore, the +microcontroller has a very low amount of memory available even +after heavy optimizations, such as quantization of the model and a +stripped-down version of the TensorFlow Core. The low amount of +memory means that the device cannot hold the full-size image in +memory. Therefore, in-camera tiling is used, and the results of the +operations had to be buffered to storage before the device could +perform inference on the saved tiles. +NVIDIA Jetson Nano can fulfill the requirements of the real- +time imaging scenario. However, its nominal power draw exceeds +the power budget and therefore buffering, similar to the case of +microcontroller, is used to be able to work at latencies higher than +those of the imaging. Even though the Jetson Nano could match the +latency of the devices utilizing a TPU, when doing inference on large +batches of image tiles using the smallest model, it only could do so +at the cost of a higher power draw. Even with a batch size of 64, the +Jetson Nano performs inference with power consumption 112-437% +higher compared to the devices with TPU. The GPU architecture +can perform massively parallel operations, but can only perform +matrix multiplication by loading/storing intermediate results in +shared memory, which creates inefficiency. +7 +CONCLUSION +In this paper, we characterized the performance of three devices +that are possible candidates to be deployed on a satellite focusing +on different image analysis scenarios on the satellite. Our results +demonstrated that the low latency requirements combined with the +limited budget for power draw, size, and mass necessitate highly +specialized hardware architectures in this domain. The only device +that fulfilled these requirements for all scenarios was CoralAI TPU. +While NVIDIA Jetson Nano could match its performance thanks to +its GPU, it could only do so at the cost of significantly higher power +draw, which ultimately led to exceeding the power budget. ARM +Cortex-M7, in contrast, could only fulfill the requirements of the +least constrained scenario due to the low degree of parallelism and +limited memory it has. Even though it provided the lowest peak +power draw, the high latency of inference using this device led to +nominal power draw higher than any of the other devices. +REFERENCES +[1] 2022. Edge TPU Compiler. https://coral.ai/docs/edgetpu/compiler/ +[2] 2022. TensorFlow Lite. https://www.tensorflow.org/lite/guide +[3] Adam D. Brown. 2018. Investigation of Deep Neural Network Image Processing +for CubeSat Size Satellites. MSc Thesis, Morehead State University (2018). +[4] Xenofon Chatziliadis, Eleni Tzirita Zacharatou, Steffen Zeuch, and Volker Markl. +2021. Monitoring of Stream Processing Engines Beyond the Cloud: An Overview. +OJIOT 7, 1 (2021), 71–82. +[5] Robert David, Jared Duke, Advait Jain, Vijay Janapa Reddi, Nat Jeffries, Jian Li, +Nick Kreeger, Ian Nappier, Meghna Natraj, Tiezhen Wang, Pete Warden, and +Rocky Rhodes. 2021. TensorFlow Lite Micro: Embedded Machine Learning for +TinyML Systems. In MLSys. 800–811. +[6] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009. Imagenet: +A large-scale hierarchical image database. In CVPR. 248–255. +[7] DISCO 2022. Danish student cubesat program. https://discosat.dk/ +[8] M. Esposito, S. S. Conticello, M. Pastena, and B. Carnicero Domínguez. 2019. +In-orbit demonstration of artificial intelligence applied to hyperspectral and +thermal sensing from space. In CubeSats and SmallSats for Remote Sensing III, +Vol. 11131. 111310C. +[9] Gianluca Giuffrida, Lorenzo Diana, Francesco Gioia, Gionata Benelli, Gabriele +Meoni, Massimiliano Donati, and Luca Fanucci. 2020. CloudScout: A Deep Neural +Network for On-Board Cloud Detection on Hyperspectral Images. Remote Sensing +12 (07 2020), 2205. +[10] Justin Goodwill, Gary Crum, James Mackinnon, Cody Brewer, Michael Monaghan, +Travis Wise, and Christopher Wilson. 2021. NASA SpaceCube Edge TPU SmallSat +Card for Autonomous Operations and Onboard Science-Data Analysis. In SSC. +[11] Andrew G. Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun +Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. 2017. MobileNets: +Efficient Convolutional Neural Networks for Mobile Vision Applications. CoRR +abs/1704.04861 (2017). +[12] Liangzhen Lai, Naveen Suda, and Vikas Chandra. 2018. CMSIS-NN: Efficient +Neural Network Kernels for Arm Cortex-M CPUs. CoRR abs/1801.06601 (2018). +[13] Martina Lofqvist and José Cano. 2020. Accelerating Deep Learning Applications +in Space. (2020). +[14] Dan O’Keeffe, Theodoros Salonidis, and Peter Pietzuch. 2018. Frontier: Resilient +Edge Processing for the Internet of Things. PVLDB 11, 10 (2018), 1178–1191. +[15] The TensorFlow Team. 2019. Flowers. http://download.tensorflow.org/example_ +images/flower_photos.tgz +[16] Benjamin Warnke, Johann Mantler, Sven Groppe, Yuri Cotrado Sehgelmeble, and +Stefan Fischer. 2022. A SPARQL Benchmark for Distributed Databases in IoT +Environments. In BiDEDE. 1–6. +[17] X-CUBE-AI 2022. AI expansion pack for STM32CubeMX. https://www.st.com/en/ +embedded-software/x-cube-ai.html +[18] Steffen Zeuch, Ankit Chaudhary, Bonaventura Del Monte, Haralampos Gavri- +ilidis, Dimitrios Giouroukis, Philipp M. Grulich, Sebastian Bress, Jonas Traub, +and Volker Markl. 2020. The NebulaStream Platform for Data and Application +Management in the Internet of Things. In CIDR. 1–11. + +SF=25 +SF=50 +SF=100 +25 +20 +Power consumption +20 +15 +Latency (s) +(mWh) +15 +10 +X +10 +X +X +5 +X +5 +0 +0 +2 + 4 8 16 32 64 +2 +4816 32 64 +Batch size +Batch size \ No newline at end of file diff --git a/W9E4T4oBgHgl3EQfNQwq/content/tmp_files/load_file.txt b/W9E4T4oBgHgl3EQfNQwq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5495c527810e3d2f9f47d0d16f0bf27f1ee11ac3 --- /dev/null +++ b/W9E4T4oBgHgl3EQfNQwq/content/tmp_files/load_file.txt @@ -0,0 +1,422 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf,len=421 +page_content='We are Going to the Space - Part 1: Which device to deploy in a satellite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Robert Bayer IT University of Copenhagen roba@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='dk Julian Priest IT University of Copenhagen jucp@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='dk Pınar Tözün IT University of Copenhagen pito@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='dk ABSTRACT The shrinkage in sizes of components that make up satellites led to wider and low cost availability of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' As a result, there has been an advent of smaller organizations having the ability to deploy satellites with a variety of data-intensive applications to run on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' One popular application is image analysis to detect, for exam- ple, land, ice, clouds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' However, the resource-constrained nature of the devices deployed in satellites creates additional challenges for this resource-intensive application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In this paper, we investigate the performance of a variety of edge devices for deep-learning-based image processing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Our goal is to determine the devices that satisfy the latency and power constraints of satellites while achieving reasonably accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Our results demonstrate that hardware accelerators (TPUs, GPUs) are necessary to reach the latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' On the other hand, state-of-the-art edge devices with GPUs could have a high power draw, making them unsuitable for deployment on a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 1 INTRODUCTION In the last century, most innovation in real-world satellite appli- cations were only available to the largest countries such as USA and Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' These innovations led to significant reductions in the size of the components that make up a satellite and in the cost of the manufacturing and deployment process of a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This, in turn, introduced a new CubeSat class of miniature satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Their format is based on a 10cm cube, with the possibility of combining multiple modules to create a larger satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This standardization makes a batch deployment of satellites easier as the format affords tight configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The reduction in costs that came as a result of this new deployment method led to the advent of satellites owned by small or private organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' However, the size of the CubeSat class satellites poses new com- plex challenges such as the power and thermal constraints as well as the physical dimensions of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' These CubeSats often perform resource-intensive tasks, which clashes with the resource- constrained nature of their format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Image processing and analysis is one class of possible satellite workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Satellites with this type of workload take large-scale images, which they have to store and send back to a ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The link between the satellite and a ground station is of limited bandwidth and short-lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The images must be highly compressed to send all of them or filtered by quality and areas of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The images already have a resolution of tens or hundreds of meters per pixel, and lossy compression would lead to even more loss of detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Filtering preserves the details of the images, but is more resource-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This paper is a step toward understanding the requirements and limitations of deep-learning-based image filtering systems on small satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' More specifically, we characterize the requirements and constraints of an image processing unit (IPU) deployed on a satellite and outline possible use case scenarios of such a system, each introducing a different degree of resource constraints and intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We then further analyze the performance of multiple edge devices on these different scenarios and their suitability for possible deployment on one of the Danish Student CubeSat Project’s (DISCO) [7] satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' These devices are based on different hardware architectures, ranging from a microcontroller to a tensor processing unit (TPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' While microcontrollers or more complex CPUs have extensive flight history (have already been deployed in satellites before) and low power footprint, they were not built with running deep learning workloads in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' On the other hand, GPUs have been utilized more in the past decade in satellites, with one of the leading work- loads being machine learning, based on the success of GPUs in terrestrial use cases of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The latest of these archi- tectures, TPU, has not been extensively researched for deployment in satellites, even though TPUs were designed to perform fast neural network inference, promising low latency with high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The contributions of this study are as follows: We illustrate the differences between the terrestrial and space edge IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The edge devices deployed in space have to perform very resource-intensive workloads with high reliability and do so in a highly resource-constrained environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We characterize a set of requirements specific to IPU on a CubeSat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We do this by outlining multiple use case scenarios and showing how they affect the required latency and possible power draw of such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We compare the performance of multiple edge devices based on different architectures and show the need and the bene- fit of using highly specialized hardware for machine learning workloads on satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 2 BACKGROUND We first introduce DISCO project and survey related work that specifically targets data processing in satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1 DISCO The Danish Student Cubesat Program (DISCO) [7] offers Danish university students the opportunity to design and operate a small satellite and to gain space flight experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' DISCO is a collaboration between four universities in Denmark, which will initially launch three student Cubesats into Low Earth Orbit with the first launch scheduled for 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' DISCO2 has been designed by students to carry an Earth imag- ing payload, infrastructure to capture images in space, into a solar synchronous polar orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The instrument is a collaboration with arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='04954v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='LG] 12 Jan 2023 Bayer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Image process- ing Unit Nominal Power Peak Power Design Margin Nominal Cycle Peak Cycle Power Mass budget Dimensions Constraints 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='00 W 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='00 W 5% 20% 20% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='48 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='15 Kg 10x70x80 mm Table 1: List of constraints of the IPU on board of the DISCO2 satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' the Arctic Research Centre at Aarhus University and will support a range of field research in Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Initially focused on a long term marine systems and cryosphere monitoring projects, the satellite will supplement ground based observations with remote sensing data, providing both good polar coverage and on demand availabil- ity for the projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The payload will include one or more high resolution cameras as well as a dedicated IPU, which will be capable of simple machine learning applications for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Students will be able to carry out conventional on-satellite image processing in addi- tion to running machine learning applications for classification, discrimination, and feature identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='2 Related Work The challenge of low bandwidth on small satellites creates the need for data post-processing to filter the data on the satellite before sending elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Neural-network-based filtering has been stud- ied for this purpose in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Specifically, several works have tested the suitability of small system-on-chip (SoC) devices leverag- ing the power of GPUs to accelerate machine learning workloads [3, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Moreover, some works have also explored the use of ASICs, such as Intel’s Movidius Myriad vision processing units (VPUs) [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This VPU has even seen real-life deployment [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Evaluation of TPUs for deployment on satellites is limited, how- ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' To the best of our knowledge, only one work has explored this option [10], with no real-life deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' DISCO project would therefore be the first satellite to leverage TPU for onboard deep learning applications, based on the findings of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In this work, we aim to characterize the performance of the dif- ferent flavors of off-the-shelf edge devices to check their suitability for being deployed on a satellite in the context of DISCO project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Data management and processing for internet-of-things and edge computing has been important in the data management community as well [4, 14, 16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We are complementing these works with a very specific application focus, which is data-intensive applications deployed on satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 3 REQUIREMENTS The requirements and constraints used in this study are based on the DISCO2 Arctic imaging mission use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The power and mass constraints of the edge device to be deployed on the satellite are based on the power and mass budgets developed in the DISCO2 en- gineering design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The satellite design is a modular 3U Cubesat with off-the-shelf modules for attitude control, power, communications, and flight control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The power budget values are typical for a 3U earth imaging Cubesat of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The resulting payload capacity is 1U (10x10x10cm) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3kg and this must include the cameras, module enclosure, optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' and image processing unit (IPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The full list of constraints of the IPU is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The planned orbit of DISCO2 is a solar synchronous polar or- bit at 550 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The camera sensor is based on an Alvium 1800 C-2040 and the lens focal length is the maximum that could be considered for the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Assuming a 50% overlap between images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=', capturing the same land area, this provides a minimum time of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='42s between consecutive images (𝑇𝑖), which was derived as follows: 𝑇𝑖 = 𝐺𝑆𝐷 ∗𝑊𝑖 ∗ ∩𝑖 𝑣 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='8495𝑚/𝑝𝑥 ∗ 4512𝑝𝑥 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 7585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='16𝑚/𝑠 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='42𝑠 (1) , where 𝐺𝑆𝐷 (ground sample distance) is the spatial resolution of the image, 𝑊𝑖 is the width of image in pixels, ∩𝑖 is the overlap between images and 𝑣 is the orbital velocity of the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Taking this latency value as a baseline, we now describe three image processing scenarios with different levels of difficulty based on this Arctic mission’s goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scenario 1: Real-time imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Images are taken with a pe- riod of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='42s, as derived above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The inference has to be performed and completed before the next image is captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This would allow for the results of an inference to be used in decision making for the next image capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scenario 2: Arctic region imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In this scenario, images are taken only while the satellite is over the polar regions relaxing the latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Images are buffered and inference is performed in the remainder of the orbit, with the IPU available for inference before the subsequent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The capture-inference cycle completes in 5739s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scenario 3: Greenland imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The images are taken only while over Greenland further relaxing the latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' As the orbit is solar synchronous with a period of one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This results in subsequent bursts but only when the swathe passes over Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Images are again buffered and the capture-inference cycle completes in 21600s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 4 METHODOLOGY AND SETUP Our goal is to characterize the performance of modern low-power hardware for deployment as the IPU of the DISCO2 satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This section presents our experimental methodology and setup to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1 Devices under Test As representative hardware for this study, we picked three devices, each based on a different hardware architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' These choices were made based on their physical dimensions, weight, power draw, and high performance considering the low power draw limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' ARM Cortex-M Mirocontroller, based on the STM32H745 chip, has the potential of the lowest power draw, while the least performant of the choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' As an additional advantage, this chip has extensive flight history and, therefore, would be a safe choice for deployment in space applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We are Going to the Space - Part 1: Which device to deploy in a satellite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In order to use this device as an on-board IPU, we can use Tensor- Flow Lite for Microcontrollers [5], a framework designed to allow neural network inference with only 16 KB of memory overhead (core runtime) and without the need for an operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In addition, the framework also provides support for the use of spe- cialized neural network kernels, such as CMSIS-NN [12] kernels designed specifically for Cortex-M processor cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The X-CUBE- AI [17] framework makes the deployment of the neural networks with TensorFlow Lite for microcontrollers even more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' It provides an intuitive GUI for the libraries or code samples based on the provided and trained TensorFlow Lite [2] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even though this device has support for floating point operations, a quantized model was used, in order to achieve better latency and, more importantly, lower memory footprint, which is a large constraint of this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' To simulate the exact performance and power characteristics of a flight computer already available for a potential deployment in DISCO project, OBC-P3, we used only the Cortex-M7 core of the STM32H745 and scaled its clock from 480MHz down to 300MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Parameters of the OBC-P3 system can be found in the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Processor 2x ARM Cortex-M7 @ 300 MHz SRAM 384 KB SRAM FRAM 32 KB Flash 2 MB Storage 64 GB eMMC Dimensions 94 x 94 x 13 mm Weight 120 g Table 2: Specifications of the OBC-P3 system containing the ARM Cortex-M7 MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The dimensions and weight in- cludes the enclosure with aluminium shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' NVIDIA Jetson Nano is a portable SoC composed of power- efficient ARM CPU and NVIDIA GPU designed for embedded sys- tems that require GPU-friendly computations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=', image process- ing, video encoding/decoding, and machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' It is the only device in our experiments that supports TensorFlow Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Therefore, the deployment process to this device is equivalent to that of servers or desktops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' It is also the only evaluated device supporting batch sizes larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The device can operate at a wattage between 5W - 10W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We configured it to operate at 5W by disabling two of the four CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' While this setting leads to lower performance, it is necessary in order to fit into the power budget outlined in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Parameters of this device can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' CPU Quad-core ARM A57 @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='43 GHz GPU 128-core Maxwell GFLOPS 472 RAM 4 GB 64-bit LPDDR4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='6 GB/s Storage 64 GB SD card Dimensions 100 x 80 x 29 mm Weight 141 g Table 3: Specifications of the NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The di- mensions and weight are based on the developer kit version of this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' CoralAI Dev Board Mini Raspberry Pi 2B CPU Quad-core Arm Cortex-A35 @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 GHz Quad-core ARM Cortex-A7 @ 900 MHz RAM 2 GB 1 GB Storage 8 GB eMMC 32 GB SD card Dimensions 64 x 48 x 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='6 mm 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='6 x 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 x 17 mm Weight 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 g 45 g CoralAI TPU TOPS Interface Dimensions Weight 4 USB2 65 x 30 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3 g Table 4: Specifications of the CoralAI Dev Board Mini and Raspberry Pi 2B, as well as the CoralAI TPU chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Note that the CoralAI Dev Board Mini’s physical dimensions and weight include the on-board TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' CoralAI TPU is an ASIC, AI accelerator developed by Google to accelerate machine learning workloads at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' TPU is not a standalone device and must be deployed together with a Linux, Windows, or macOS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even though this device is highly specialized, model deployment is performed by compiling a Ten- sorFlow Lite model containing a subset of supported layers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This model must be quantized to an 8-bit integer, as this is only supported data type by this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This compiled model can then be used with the TPU’s Python or C++ library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In our experiments, we rely on the C++ library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We evaluated this device in two formats: (1) CoralAI Dev Board Mini, which couples this chip and an SoC on a single board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' (2) CoralAI USB accelerator connected to a Raspberry Pi 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even though (1) has the TPU chip on the same board as the SoC, the two communicate through the USB2 bus, as in the case of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' In addition, both of the devices run Linux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The parameters of the different devices can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='2 Metrics The metrics to evaluate the suitability of the devices as the on- satellite IPUs are based on the requirements Section 3 outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Latency is reported in seconds per inference of a sample, where a sample is a whole image of size 4512 x 4512 pixels or 400 tiles of size 224 x 224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We measure the latency of the neural network inference once the data is already in the device’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Nominal power draw is the average power draw in mW mul- tiplied by the duty cycle, which is the ratio between achieved and required latency for each scenario (outlined in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We use an external appliance for CoralAI TPU and the ARM Cortex-M7 and tegrastats for Jetson Nano to measure this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Peak power draw is the maximum power draw over the pe- riod of inference in mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' It is measured using the same tools as in nominal power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Power consumption is reported per inference of a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This metric shows the power efficiency of the device and will guide the choice of a more suitable device if multiple devices fulfill the requirements of a particular scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Bayer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scaling factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='0 Device Latency (s) Power consump- tion (mWh) Latency (s) Power consump- tion (mWh) Latency (s) Power consump- tion (mWh) ARM Cortex-M7 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='80 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='10 N/A N/A N/A N/A CoralAI Dev Board Mini 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='778 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='991 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='90 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='533 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='40 Raspberry Pi + CoralAI TPU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='331 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='825 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='40 NVIDIA Jetson Nano 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='900 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='54 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='109 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='74 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='581 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='10 NVIDIA Jetson Nano, batch size 64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='529 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='382 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='94 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='171 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='89 Table 5: Latency and power consumption results with different scaling factors of the model as measured on corresponding devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Batch size of 1 is used if not stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Accuracy of the deployed model is also measured to show the effect quantization has on the model’s predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3 Workload To simulate the imaging scenarios of interest (Section 3), we use an image classification workload consisting of a 5-class classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' For this workload, an off-the-shelf MobileNetV1 model [11] pre-trained on the ImageNet dataset [6] was chosen for its strong predictive power and availability in multiple scaling factors, affecting the number of hidden layers in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The availability of multiple scaling factors is an important factor as the memory of some of the devices is highly limited and can therefore fit only the smallest of the variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We further fine-tuned the model before deployment on the de- vices using the Flowers dataset [15], containing 3670 color images of size 224 x 224 pixels belonging to 5 different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' While the weights in the model do not affect the inference latency or power required to perform the inference, the model was fine-tuned using this dataset because it mimics one of the possible use cases closely (in number of classes as well as size of the images) and allows us to quantify the effects of the size and precision of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Since the models running on the Cortex-M7 and CoralAI TPU were quantized, rescaling of images was not needed before infer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' For Jetson Nano, the pixel values must be rescaled to the range of [0, 1), or, in other words, divided by 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' An additional layer was prepended to the model for this process in order to take advantage of the GPU parallelism on Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' As the size of the images would stress the memory available on the evaluated devices, we employed a tiling method for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Each image was divided into 400 patches of size 224 x 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1 This method results in a positive side effect, where each patch can be filtered separately acting as a coarse-grained image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This way only the tiles of interest are sent back to a ground station saving us bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The results are reported as an average of 10 inferences on the full 4512 x 4512 pixel image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The inference on Cortex-M7 and CoralAI TPU were performed with batch size of 1 and on Jetson Nano with the batch size of 2𝑥, where x is from range of 0-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' All of the devices were tested with the scaling factors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='0 of the model, with the exception of the ARM Cortex-M7 microcontroller, which could not fit the larger models in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 1The image is not evenly divisible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' therefore, the borders are disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5 RESULTS The results are split into three parts: (1) analysis of latency and power draw results with respect to the three scenarios outlined in Section 3, (2) impact of quantization on the accuracy of models with various scaling factors, (3) impact of batch size on the performance of NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1 Scenarios Table 5 shows the latency achieved and power consumption of the devices using the various scaling factors of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Furthermore, Tables 6 and 7 show the nominal and peak power draw of different configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3 discuss the suitability of each device for the different scenarios based on the results on these tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The reported latency, power consumption, and the peak power draw are independent of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Only the nominal power draw depends on the use case scenarios, due to the duty cycle (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1 Scenario 1: Real-time imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Real-time imaging is the most constrained scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Therefore, a high degree of specialization is necessary to fulfill the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The pipeline has to achieve a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='42s latency, which is the spacing between images with 50% overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' There are no passive periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Therefore the latency of the pipeline has to be lower than that of the imaging for the pipeline not to get backed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' There are multiple configurations that achieved the required latency (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' These are all the configurations utilizing a TPU and the smallest model running on Jetson Nano with batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Out of the configurations that satisfy the latency requirements for this scenario, only of the TPU configurations fit into the nominal power budget of the satellite (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even though the Jetson Nano with batch size 64 achieves a latency comparable to the TPUs with a scaling factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='25, the power consumption per inference is more than twice as high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Furthermore, the Raspberry Pi with external TPU module does not fit into the power budget when using the largest model, due to exceeding the 5W requirements for peak power draw (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' NVIDIA Jetson Nano using the largest model with batch size of 64 also exceeds this peak power requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Since the peak power draw does not depend on neither the use case scenario nor the duty cycle, this conclusion about peak power draw holds for the rest of the scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We are Going to the Space - Part 1: Which device to deploy in a satellite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scaling factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='0 Device Power draw (mW) Power draw (mW) Power draw (mW) ARM Cortex-M7 Scenario 1 Scenario 2 Scenario 3 161 CoralAI Dev Board Mini Scenario 1 1433 1546 Scenario 2 89 95 120 Scenario 3 23 25 32 Raspberry Pi + CoralAI TPU Scenario 1 1376 1571 1954 Scenario 2 85 97 120 Scenario 3 23 26 32 NVIDIA Jetson Nano Scenario 1 Scenario 2 529 639 1109 Scenario 3 141 170 295 NVIDIA Jetson Nano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' batch size 64 Scenario 1 3021 Scenario 2 186 298 646 Scenario 3 49 79 172 Table 6: Nominal power draw of devices with models of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The combinations of devices and model sizes, which do not fulfil the latency requirements, are omitted as their active duty cycle is greater than 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The power draw is highlighted in bold when exceeding the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scaling factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='0 Device Power draw (mW) Power draw (mW) Power draw (mW) Arm Cortex-M7 389 N/A N/A CoralAI Dev Board Mini 2770 2930 4790 Raspberry Pi + CoralAI TPU 2595 3405 5050 NVIDIA Jetson Nano 2871 3546 4717 NVIDIA Jetson Nano, batch size 64 4523 4684 5064 Table 7: Peak power draw of the devices during inference on full-size image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Batch size of 1 is used if not stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The peak power is in bold when exceeding the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='2 Scenario 2: Arctic region imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' By relaxing the constraints and performing the workload equivalent to taking images of only the areas above the arctic polar circle, we see more configurations passing the requirements necessary to perform such a workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Since there would be passive imaging periods perform- ing the workload for this scenario, the pipeline does not need to be lower than the latency of imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' It can instead buffer the images and perform the inference at higher latency, given that the pipeline can finish inference of one burst before the next burst of images comes, which corresponds to a total latency of 5739 seconds or 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='74 seconds per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' All the configurations except the ones with ARM Cortex-M7 achieve this latency (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Because of the large margin between the required and achieved latencies for all configurations passing the latency requirements, the active duty cycle is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Therefore, the corresponding nominal power draw is well below the required one in all cases (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3 Scenario 3: Greenland imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The least constrained sce- nario corresponds to taking images of only Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The low constraints mean that all of the devices pass the requirements under certain model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The maximum total latency for a burst of images is 21600 seconds, corresponding to 270 seconds per single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This requirement is easily passed by all of other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Due to the low active duty cycle, each configura- tion passes the nominal power draw requirement (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' There is, however, a large gap between the nominal power draw of TPU and the rest of the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' TPU performs the inference much more effi- ciently, and its power draw scales more favourably with increasing scaling factors in comparison to the other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Scaling factor Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='0 32 bit float 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='65% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='46% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='42% 8 bit integer 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='24% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='32% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='01% Table 8: Accuracy of the full-precision and quantized Mo- bilenetV1 as measured on the Flowers dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='2 Effect of quantization and scaling factor Table 8 shows the accuracy of the model with the various scaling factors before and after quantization to an 8-bit integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Models with scaling factors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='0 do not show a significant change in predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We can, however, see a 3% drop in accuracy Bayer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Figure 1: Latency and power consumption at varying model scaling factors and batch sizes on the NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' in the model with the smallest scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' However, this differ- ence is acceptable in our case and is outweighed by the benefits of lower memory footprint and latency, and higher overall efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3 Batch size impact on NVIDIA Jetson Nano NVIDIA Jetson Nano is the only device in our evaluation that allows doing inference with a batch size higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Figure 1 shows the impact of increasing the batch size on latency and power consumption of the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The most significant difference is between batch sizes 1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The impact of increasing batch size then tapers off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' We can see that maximizing the batch size can lead to 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='3-74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='6% lower latency and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='7-64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content='8% lower power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 6 DISCUSSION TPU shows the most promising results as it is the only device to fulfill the requirements for real-time imaging, which is the most constrained scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This device is highly specialized for this pur- pose, utilizing the systolic array architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' This architecture is purpose-built to perform fast multiply-accumulate operations in a highly parallel fashion, which allows performing matrix multiplica- tion without the need to load/store intermediate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' On the other end of the spectrum was the ARM Cortex-M7 micro- controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Due to its lack of parallelism, this device could only fulfill the requirements of the least constrained scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Furthermore, the microcontroller has a very low amount of memory available even after heavy optimizations, such as quantization of the model and a stripped-down version of the TensorFlow Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The low amount of memory means that the device cannot hold the full-size image in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Therefore, in-camera tiling is used, and the results of the operations had to be buffered to storage before the device could perform inference on the saved tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' NVIDIA Jetson Nano can fulfill the requirements of the real- time imaging scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' However, its nominal power draw exceeds the power budget and therefore buffering, similar to the case of microcontroller, is used to be able to work at latencies higher than those of the imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even though the Jetson Nano could match the latency of the devices utilizing a TPU, when doing inference on large batches of image tiles using the smallest model, it only could do so at the cost of a higher power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even with a batch size of 64, the Jetson Nano performs inference with power consumption 112-437% higher compared to the devices with TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The GPU architecture can perform massively parallel operations, but can only perform matrix multiplication by loading/storing intermediate results in shared memory, which creates inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 7 CONCLUSION In this paper, we characterized the performance of three devices that are possible candidates to be deployed on a satellite focusing on different image analysis scenarios on the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Our results demonstrated that the low latency requirements combined with the limited budget for power draw, size, and mass necessitate highly specialized hardware architectures in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' The only device that fulfilled these requirements for all scenarios was CoralAI TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' While NVIDIA Jetson Nano could match its performance thanks to its GPU, it could only do so at the cost of significantly higher power draw, which ultimately led to exceeding the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' ARM Cortex-M7, in contrast, could only fulfill the requirements of the least constrained scenario due to the low degree of parallelism and limited memory it has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Even though it provided the lowest peak power draw, the high latency of inference using this device led to nominal power draw higher than any of the other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' REFERENCES [1] 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' Edge TPU Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' https://coral.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} +page_content=' SF=25 SF=50 SF=100 25 20 Power consumption 20 15 Latency (s) (mWh) 15 10 X 10 X X 5 X 5 0 0 2 4 8 16 32 64 2 4816 32 64 Batch size Batch size' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'} diff --git a/XtE5T4oBgHgl3EQfdA-T/content/tmp_files/2301.05608v1.pdf.txt b/XtE5T4oBgHgl3EQfdA-T/content/tmp_files/2301.05608v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4699179382dea6c2a284a6599976c4f2574f938c --- /dev/null +++ b/XtE5T4oBgHgl3EQfdA-T/content/tmp_files/2301.05608v1.pdf.txt @@ -0,0 +1,1650 @@ +Investigating the Combination of Planning-Based and +Data-Driven Methods for Goal Recognition +Nils Wilken +nils.wilken@uni-mannheim.de +Institute for Enterprise Systems, University of Mannheim +69118 Mannheim, Germany +Lea Cohausz +lea.cohausz@uni-mannheim.de +Data and Web Science Group, University of Mannheim +69118 Mannheim, Germany +Johannes Schaum +jschaum@mail.uni-mannheim.de +Institute for Enterprise Systems, University of Mannheim +69118 Mannheim, Germany +Stefan L¨udtke +stefan.l¨udtke@uni-mannheim.de +Institute for Enterprise Systems, University of Mannheim +69118 Mannheim, Germany +Heiner Stuckenschmidt +heiner.stuckenschmidt@uni-mannheim.de +Data and Web Science Group, University of Mannheim +69118 Mannheim, Germany +Abstract +An important feature of pervasive, intelligent assistance systems is the ability to dy- +namically adapt to the current needs of their users. Hence, it is critical for such systems +to be able to recognize those goals and needs based on observations of the user’s actions +and state of the environment. +In this work, we investigate the application of two state-of-the-art, planning-based plan +recognition approaches in a real-world setting. So far, these approaches were only evaluated +in artificial settings in combination with agents that act perfectly rational. We show that +such approaches have difficulties when used to recognize the goals of human subjects, +because human behaviour is typically not perfectly rational. To overcome this issue, we +propose an extension to the existing approaches through a classification-based method +trained on observed behaviour data. We empirically show that the proposed extension +not only outperforms the purely planning-based- and purely data-driven goal recognition +methods but is also able to recognize the correct goal more reliably, especially when only +a small number of observations were seen. This substantially improves the usefulness of +hybrid goal recognition approaches for intelligent assistance systems, as recognizing a goal +early opens much more possibilities for supportive reactions of the system. +1. Introduction +The ultimate goal of smart assistance technologies is to dynamically adapt the infrastructure +of a building to best meet the needs of their users by observing their behaviour and deducing +their current needs. +Identifying users’ goals and intentions based on their current and +past activities is an important task in this context. +While there is some work on goal +recognition in the context of smart assistance systems (Yordanova, Whitehouse, Paiement, +1 +arXiv:2301.05608v1 [cs.AI] 13 Jan 2023 + +Mirmehdi, Kirste, & Craddock, 2017; Yordanova, L¨udtke, Whitehouse, Kr¨uger, Paiement, +Mirmehdi, Craddock, & Kirste, 2019; Kr¨uger, Nyolt, Yordanova, Hein, & Kirste, 2014), so +far research has mostly focused on recognizing users’ current activities (Helaoui, Riboni, +& Stuckenschmidt, 2013; Rashidi, Cook, Holder, & Schmitter-Edgecombe, 2010; Hoque & +Stankovic, 2012; Yao, Nie, Sheng, Gu, Li, & Wang, 2016; Sztyler & Stuckenschmidt, 2017). +In this paper, we address the problem of identifying user goals as a basis for automatic +support. For this purpose, we look at the related problem of plan recognition, which is a +long-standing topic in the Artificial Intelligence community (Kautz & Allen, 1986; Charniak +& Goldman, 1993). We believe that plan recognition methods are particularly suited for +this task as they do not only identify the goal a user intends to achieve, but also aim to +recognize the most probable plan (i.e., ordered sequence of actions) for achieving this goal. +Knowing such a plan provides us with a better basis for supporting the user. +In this paper, we investigate the application of two state-of-the-art plan recognition ap- +proaches that are based on the principle of Plan Recognition As Planning (PRAP) (Ram´ırez +& Geffner, 2010). More explicitly, the contributions of this paper are: +• In Section 4, we reveal and analyze some major shortcomings of PRAP approaches +when applied to real-world scenarios. The main consequence of these shortcomings is +that some goals can only be identified relatively closely before they are reached, which +significantly reduces their potential benefits to an intelligent assistance system. +• As a possible solution to this problem, we propose a hybrid plan recognition method +in Section 5. The proposed method combines the principle of PRAP with a data- +driven probabilistic model that captures statistical relations between certain states of +the environment and goals that can be learned from past observations. +• Finally, we empirically evaluate the proposed hybrid method in sections 6 and 7 and +compare its performance to the performances of purely planning-based and purely +data-driven approaches. The evaluation shows that both approaches can be applied +to identify the goals of a user in real-world scenarios. Further, the results show that +using a hybrid goal recognition method leads to a much earlier identification of the +correct goal while only requiring very small amounts of training data. +2. Problem Definition +Probabilistic goal recognition is the problem of inferring a probability distribution over a set +of intended goals of an observed agent, given a possibly incomplete sequence of observed +actions and a domain model that describes the domain in which the observed agent acts. +More formally, the aim of goal recognition approaches is to find a posterior probability +distribution P(G|O) over all goals g ∈ G, given a sequence of observed actions o. +This work considers the smart home domain as an example environment for goal recog- +nition. Figure 1 shows the layout of a smart flat and partial action sequences that sketch a +simple use case. This use case will be employed to analyze the shortcomings of the investi- +gated planning-based goal recognition approaches. It is important to note that this use case +is completely synthetic and does not correspond to a real-world experimental setup. The +flat has four rooms and a hallway that connects all rooms. In each room, different devices +2 + +h1 +h2 +h3 +h4 +Be1 +Be2 +be3 +be4 +ba1 +ba2 +ba3 +ba4 +k1 +k2 +k3 +k4 +l1 +l2 +l3 +l4 +Livingroom +Bedroom +Bathroom +Kitchen +Shower +Toilet +Basin +Fridge +Oven +Bed +Closet +A +Washing +Machine +Laundry +Basket +T +T +T +T +A +A +A +A +A +Window +Window +Window +Window +Window +20°C +good +18°C +very good +22°C +good +19°C +bad +Outdoor +TA +5°C +very good +Sensing functionality +Acting functionality +Sensing/Acting +functionality +Agent +Heater +Heater +Heater +Heater +Possible Goals: +• +A (Prepare Meal) +• +B (Watch TV) +• +C (Use toilet) +• +D (Use shower) +(a) +h1 +h2 +h3 +h4 +Be1 +Be2 +be3 +be4 +ba1 +ba2 +ba3 +ba4 +k1 +k2 +k3 +k4 +l1 +l2 +l3 +l4 +Livingroom +Bedroom +Bathroom +Kitchen +Shower +Toilet +Basin +Fridge +Oven +Bed +Closet +A +Washing +Machine +Laundry +Basket +T +T +T +T +A +A +A +A +A +Window +Window +Window +Window +Window +20°C +good +18°C +very good +22°C +good +19°C +bad +Outdoor +TA +5°C +very good +Sensing functionality +Acting functionality +Sensing/Acting +functionality +Agent +Heater +Heater +Heater +Heater +Possible Goals: +• +A (Prepare Meal) +• +B (Watch TV) +• +C (Use toilet) +• +D (Use shower) +(b) +Figure 1: Illustration of an exemplary smart flat and a simple example use case (i.e., “Beer +Use Case”). +and furnishings are located. Some of these objects can possibly function as sensor, actuator, +or a mixture of both, which is indicated by the green and orange dots. Furthermore, it is +assumed that the current location of the agent can be sensed at all times and that the agent +can navigate the cells in the flat by moving in all possible directions, including diagonal +moves. +Example 1 (Beer Use Case) Figures 1a and 1b roughly sketch parts of a small use case +in this smart flat, which we will refer to as “Beer Use Case” (BUC) from here on. In +the BUC, a single agent is initially located in the cell “l3” in the livingroom, moves to the +fridge, takes out a beer, and moves back to the couch in the livingroom (Fig. 1a). When +the agent arrives at the couch in the livingroom, she sits down on the couch, opens the beer, +and drinks it while watching TV. After a while, the agent decides to get another beer from +the fridge. When the second beer is empty, the agent gets up from the couch, moves back to +the kitchen, and subsequently via the hallway to the bathroom to use the toilet (Fig. 1b). +3. Background +In this work, we investigate the application of two state-of-the-art approaches to plan recog- +nition to a real-world goal recognition scenario. In contrast to probabilistic goal recognition, +probabilistic plan recognition not only describes the problem of inferring a probability dis- +tribution over a set of goals, but also the probability distribution over a set of possible plans +that an agent might follow to reach it’s intended goal. From a solution to a plan recognition +problem, the solution of the corresponding goal recognition problem can be derived by only +considering the goals of the recognized plans. Plan recognition is a long standing research +area in the Artificial Intelligence community. Recent plan recognition systems mostly rely +on the Plan Recognition As Planning (PRAP) (Ram´ırez & Geffner, 2009) principle and +hence, utilize symbolic planning systems to solve plan- and goal recognition problems. +3.1 Symbolic Planning +Symbolic planning is based on a symbolic model of the planning domain that defines possible +actions, their preconditions and effects on the environment. Given a current state and goals +3 + +in terms of partial state descriptions, planning methods aim to construct an optimal plan for +reaching the goals from the current state consisting of a (possibly partial) order of actions to +be executed. We adopt the formalization of a planning problem from (Ram´ırez & Geffner, +2010). +Definition 1 (Planning Problem) A Planning Problem is a tuple P = ⟨F, s0, A, G⟩ where +F is a set of fluents (boolean statements about properties of the modeled environment), +s0 ⊆ F and G ⊆ F are the initial state and the goal description and A is a set of actions +with preconditions Pre(a) ⊆ F and lists of fluents Add(a) ⊆ F and Del(a) ⊆ F that de- +scribe the effects of an action a in terms of fluents that are added and deleted from the +current state. Actions have a non-negative cost c(a). A state is described by the subset of +fluents which are currently considered to be true. A goal state is a state s with s ⊇ G. An +action a is applicable in a state s if and only if Pre(a) ⊆ s. Applying an action a in a state +s leads to a new state s′ = (s ∪ Add(a) \ Del(a)). A solution for a planning problem (i.e., a +plan) is a sequence of applicable actions π = a1, · · · an that transforms the initial state into +a goal state. The cost of the plan is defined as c(π) = � +i +c(ai). A plan is optimal if the cost +of the plan is minimal. +This basic model has been extended in different directions. In this paper, we make use +of two extensions. One allows us to specify goals of form ¬f that claim that a certain +fluent f is absent in the goal state. The other enables the use of a conditional effect of +form p → q, where p and q are single fluents. This means that when an action x has such +a conditional effect, fluent q only becomes true after the execution of x when p was true +before the execution (Ram´ırez & Geffner, 2010). +3.2 Plan Recognition As Planning: State-of-the-Art +As already mentioned, many recent plan- and goal recognition approaches rely onto the +principle of Plan Recognition as Planning (PRAP), which was first introduced by Ram´ırez +and Geffner (Ram´ırez & Geffner, 2009). All approaches that follow this principle have in +common that they utilize concepts from the area of classical planning to compute probability +distributions over a set of possible plans or goals, respectively. +Definition 2 (Probabilistic Plan Recognition Problem) A probabilistic plan recog- +nition problem is a tuple T = ⟨D, G, O, P(G)⟩ where D = ⟨F, s0, A, ∅⟩ is a planning domain, +G is a set of possible goals g ⊆ F, o = o1, · · · om, where oi ∈ A is a sequence of actions +that have been observed and P(G) is the prior probability distribution over G. A solution +to the probabilistic plan recognition problem is the conditional probability of the goals given +the observation sequence o (i.e., P(G = g|O = o)∀g ∈ G). +Estimating Goal Probabilities +Both plan recognition methods that are used in this +work are based on the idea of using Bayes Rule to compute the posterior probabilities of +the goals: +P(G|O) = αP(O|G)P(G) +(1) +It is assumed that the prior probabilities P(G = g) of goals g ∈ G are given in the problem +definition. Hence, the problem of probabilistic goal recognition boils down to the estimation +4 + +of P(O|G). Both investigated approaches utilize symbolic planning systems to estimate this +probability. +The idea behind this is based on the assumption that agents act perfectly rational and +hence, use strictly optimal plans (i.e, plans that minimize costs) to achieve their goals. +Furthermore, it is assumed that the probability of a goal to be the agent’s actual goal can +be estimated by relating the costs of an optimal plan that includes a given observation +sequence o and an optimal plan that does not include o, while reaching a given goal g ∈ G. +This can be done because an optimal plan that does not have to fulfill the requirement of +including o is, according to the planning domain, a perfectly rational plan from the given +initial state to a given goal g. Hence, when the costs of an optimal plan that includes o +are higher, this means that the agent is taking a detour compared to a perfectly rational +plan. More precisely, Ram´ırez and Geffner (Ram´ırez & Geffner, 2010) propose to calculate +P(o|g) as follows: +P(o|g) = α′ +exp{−β∆(g)} +1 + exp{−β∆(g)} +(2) +Where α′ is a normalization factor and ∆(g) = c(o, g)−c(o, g) is the cost difference between +an optimal plan for g that satisfies o and an optimal plan for g that does not satisfy o. The +costs c(o, g) and c(o, g) can be computed out of the box using classical planning systems. +Translating a Plan Recognition Problem into Planning Problems +The two state- +of-the-art plan recognition methods used in this paper were proposed by Ram´ırez and +Geffner (Ram´ırez & Geffner, 2010) (referred to as “RG” from here on) and Vered et al. +(Vered, Kaminka, & Biham, 2016) (referred to as “GM” (Goal Mirroring) from here on). +They mainly differ in the way they transform the original planning problem, which is +necessary to ensure that the computed plans fulfill some necessary requirements. +To compute the probabilities P(O|G), the RG approach compiles a plan recognition +problem T = ⟨P, G, o, Prob⟩ into 2|G| planning problems. For each goal g ∈ G the two +planning problems Po(g) and Po(g) have to be compiled and solved. Classical planning +systems naturally cannot handle the requirement of satisfying a given sequence of observed +actions in a computed plan. To ensure that the computed solutions fulfill this requirement, +the original planning domain D has to be slightly modified. +Definition 3 (Transformation of the Planning Domain (RG)) For a given planning +domain D = ⟨F, s0, A⟩ and a given observation sequence o, the transformed domain is de- +fined as D′ = ⟨F ′, I, A′⟩ with F ′ = F ∪ {poi|oi ∈ (oi)n +0}, where poi is a new fluent and the +actions o ∈ A′ that are in o have an additional effect poi when i = 0 and poi−1 → poi has to +hold otherwise. +For this transformation it is assumed that no action appears twice in o. When this is +the case, the corresponding actions are duplicated and renamed to ensure that the order +of observed actions is unmodified in the resulting plans. Now the costs c(o, g) and c(o, g) +can be calculated by solving the planning problems Po(g) = ⟨F ′, s′ +0, A′, g ∪ {pon}⟩ and +Po(g) = ⟨F ′, s′ +0, A′, g ∪ {¬pon}⟩. +Goal Mirroring +The main difference between RG and GM is the domain translation +procedure: While RG adapts the actions in a given planning domain, GM uses a different +initial state to generate plans that embed o each time a new observation is observed. +5 + +Definition 4 (Transformation of Planning Problem (GM)) For a given planning do- +main D = ⟨F, s0, A⟩ and a given observation sequence o, the transformed domain is defined +as D′ = ⟨F ′, s′ +0, A′⟩ with F ′ = F, where s′ +0 = s0[[o]] and s[[o]] returns as a result the +planning state that is obtained when the action sequence o is applied to a planning state s. +When this transformation is completed, analogously to the RG approach, GM calculates +the costs c(o, g) and c(o, g). However, in contrast to RG, GM assumes for the calculation +that an optimal plan from s0 to a goal g can be obtained by concatenating o with a plan +for g that starts at the adjusted initial state s′ +0. From such a plan, again the costs c(o, g) +can be determined. Furthermore, GM does not generate plans that strictly do not embed o, +but instead computes an optimal plan from s0 to each goal and uses the costs of these plans +analogously to the costs of plans that do not embed o as RG does (i.e., c(o, g)). Apart from +this, GM uses the same heuristic as RG (i.e., Equation 1) to compute goal probabilities +P(G|O) from these costs. +One major benefit of GM compared to RG is that it is expected to be much more +time efficient in the case of online probabilistic goal recognition. This becomes increasingly +important with increasing complexity of the involved planning problems. +Definition 5 (Online Probabilistic Goal Recognition) We define online probabilistic +goal recognition as a special variant of the probabilistic goal recognition problem defined +earlier. In online goal recognition, we assume that the observation sequence o is revealed +incrementally. More explicitly, we introduce the notion of time t ∈ {0, . . . , T}, where T = +|o|. For every value of t, one probabilistic goal recognition problem R(t) can be induced +as R(t) = ⟨D, G, ot, Prob⟩ where D = ⟨F, s0, A, ∅⟩ and ot = {oi|0 ≤ i ≤ t, oi ∈ o}. A +solution to the online probabilistic goal recognition problem are the conditional probabilities +Pt(G = g|ot); ∀g ∈ G, t ∈ [0, T]. +Hence, in the case of online probabilistic goal recognition, GM solves, due to the different +transformation procedure, only |G||O| + |G| planning problems instead of 2|G||O| planning +problems that RG solves. +4. Case Study: Goal Recognition in the Beer Use Case +In this section we evaluate the performance of the RG and GM goal recognition approaches +when applied to the synthetic BUC example (see Section 2). Furthermore, we demonstrate +and discuss some major limitations of them. +4.1 Experimental Setup +For the experiments, we modeled a planning domain DBUC in the Planning Domain Def- +inition Language (PDDL) (McDermott, Ghallab, Howe, Knoblock, Ram, Veloso, Weld, & +Wilkins, 1998). The goal set of the corresponding plan recognition problems (see Definition +2) is defined as GBUC = {gprepare meal, gwatch TV , guse shower, guse toilet}. Further, following +the approach of Ram´ırez and Geffner (Ram´ırez & Geffner, 2010), we assume uniform prior +probabilities PBUC(G) for all goals in GBUC. +Based on this experimental setup, we conducted two experiments E1 and E2 with +both recognition approaches, which, however, differ in the observation sequences that are +6 + +Table 1: Evaluation results for the RG and GM goal recognition approaches when applied +to E1 and E2 with the LAMA planner in anytime mode. The results for both approaches +are identical for both, E1 and E2. Each row describes the probabilities P(G|O) for all +goals G ∈ GBUC for different lengths of O (|O|). +g1 = gprepare meal, g2 = gwatch TV , +g3 = guse shower, g4 = guse toilet. +(a) Results for E1 +P(G|O) +|O| +g1 +g2 +g3 +g4 +28 + 0 +0.25 +0.25 +0.25 +0.25 +28 + 1 +0.319 +0.043 +0.319 +0.319 +28 + 2 +0.331 +0.006 +0.331 +0.331 +28 + 3 +0.063 +0.001 +0.468 +0.468 +28 + 4 +0.009 +0.0 +0.495 +0.495 +28 + 5 +0.002 +0.0 +0.268 +0.73 +28 + 6 +0.001 +0.0 +0.119 +0.88 +(b) Results for E2 +P(G|O) +|O| +g1 +g2 +g3 +g4 +0 +0.25 +0.25 +0.25 +0.25 +1 +0.316 +0.052 +0.316 +0.316 +2 +0.329 +0.012 +0.329 +0.329 +3 +0.106 +0.002 +0.446 +0.446 +4 +0.018 +0.0 +0.491 +0.491 +5 +0.003 +0.0 +0.349 +0.648 +6 +0.001 +0.0 +0.192 +0.806 +used to compile the involved planning problems. For experiment E1, the actions in the +utilized observation sequence represent the entire BUC (see Example 1). For experiment +E2, to evaluate how much the goal probability estimates depend on information gained +from the observations of the agent getting and drinking beer, only the last six actions of the +observation sequence used in E1 are used (i.e., in E2 the observations of the agent getting +and drinking beer are not included in the observation sequences). The remaining setups are +similar for both experiments. To solve the planning problems compiled from these setups, +we used the LAMA (Richter, Helmert, & Westphal, 2008) planner, which is a satisficing +planner that can be used either in greedy or anytime mode. When used in greedy mode, the +planner stops immediately when a plan is found, whereas in anytime mode LAMA returns +the best plan found in a given time window. Here, we used LAMA in anytime mode. +4.2 Results and Limitations +The results of the experiments are shown in Table 1. Each row reports the probabilities +P(G|O) for all g ∈ GBUC for different lengths of O. As the RG and GM approaches are +based on the same underlying principle and the BUC domain is small enough to be handled +efficiently by both approaches, the results for the experiments E1 and E2 are identical for +both of them. Hence, we only report one result table for each experiment. The results of +the experiments reveal two major limitations of the two planning-based approaches. +1. In both experiments, the correct goal g4 is only recognized at |O| = 28 + 5 and +|O| = 5 respectively, i.e., shortly before the goal is actually reached. This timestep +corresponds to the observation where the agent has moved to the location ba3, which +is also where the toilet is located. This circumstance significantly reduces the possible +usefulness for an intelligent assistance system, as the goal is recognized too late to +provide support through adaptations of the environment. +7 + +2. The additional information that is included in the observation sequence used for E1, +has no impact on the estimated probabilities P(G|O). Instead, the estimates are only +based on information that are gained from the up to last six actions that are observed +afterwards. Intuitively, however, the additional information should have an impact +onto the estimate, because there exists a causal relation between drinking beer and the +probability that this agent pursues the goal to use a toilet afterwards. Thus, it should +be possible to recognize the correct goal much earlier in the observation sequence. +The main reason for these limitations of the two planning-based approaches is that the +additional actions that are contained in the observation sequence used for E1 are not strictly +necessary to fulfill the preconditions of any action that is required to reach one of the +possible goals in an optimal way, i.e., drinking beer is not a necessary requirement to visit +the bathroom. As a consequence, these observations have the same impact on the estimate +of P(G|O) for all possible goals, although they might contain valuable information about +the true probability of P(G|O). +5. A Hybrid Goal Recognition Approach +As shown in the previous section, a significant shortcoming of PRAP based approaches is +that causal relations between goals and observations in the environment cannot be exploited +in general. This is due to the fact that the costs of plans are used as the only criterion to es- +timate the probabilities P(O|G). To solve this problem, we propose to combine these PRAP +approaches with a data-driven probabilistic causal model of agent goals and observations. +5.1 A Statistical Causal Model of Observations +We propose to model the relationship between goals and observations via a probability +distribution P(F1, . . . , FN|G), where F1, . . . , FN are fluents of the planning state st. This +distribution models how goals (e.g., making a sandwich) affect the probability of specific +fluents in the planning states (e.g., whether the agent holds a cucumber). In contrast to +the PRAP models, the idea here is to learn the parameters of P(F1, . . . , Fn|g) from training +data. This way, the probabilistic model can capture the relations between fluents and goals +that cannot be captured by planning-based approaches: Specifically, the planning-based +models cannot capture statistical relations between fluents and goals that are not necessary +for an optimal plan. For example, in the BUC planning domain, drinking beer is not a +necessary precondition for visiting the bathroom. Still, drinking beer makes an eventual +bathroom visit more likely. +In general, we could use any probabilistic model, like Bayesian Networks, deep generative +models etc., to represent P(F1, . . . , FN|G). Here, we focus on a Naive Bayes model (NBM), +i.e., assuming P(F1, . . . , Fn|g) = �n +i=1 P(Fi|g). The model is visualized in Figure 2. On +the one hand, the strong independence assumptions between all fluents do not necessarily +hold in practice. On the other hand, a Naive Bayes model has few parameters (linear in +the number of variables). Thus, training is possible even when training data is scarce – as +is often the case for activity sequences of real human subjects. +Note that we made another simplifying assumption here: The distribution P(F1, . . . , FN|G) +only models the dependency of the current planning state on the goal g, but not the de- +8 + +pendency of the action sequence o on g. More specifically, different observation sequences +that result in the same planning state could be associated with different goals, but we +deliberately neglected this information to make the learning problem tractable. +F_N +… +g +F_1 +Figure 2: Bayesian network representation of the Naive Bayes Model. +5.2 A Hybrid Goal Recognition Method +To overcome the shortcomings of the PRAP approaches discussed in Section 4.2, we propose +to combine it with the proposed probabilistic learned model of P(o|g). +For the combination of the goal probabilities, we use model stacking, which is a common +ensemble learning method. In model stacking, a so-called meta-model is used to generate +a combined estimate from the estimates of heterogeneous base models. In this work, we +studied two different, manually designed, meta-models to combine the estimate of one of +the PRAP approaches and the estimate of the NBM. +It is important to note that there are some major differences between the ways in which +the two utilized base models estimate the goal probabilities P(G|O). One of these differences +is that the PRAP models are based purely on manually specified knowledge (in the form +of the planning domain), whereas the NBM only relies on this manually defined domain +knowledge to establish its structure but learns its parameters from training data. Another +major difference between the models is the set of features which are used to estimate the +probability P(O|G) and hence, also the probability P(G|O). Although the predictions of +both models are based on an observed action sequence and an observed initial state of the +environment, they estimate the probability P(O|G) based on different features that can +be derived from these observations. The planning-based methods use two entire plans to +estimate the probability P(o|g). In contrast, the NBM model uses only the planning state +that results from applying the sequence o to the initial state to estimate P(o|g). +Weighted Sum of Predictions +As a first meta-model, we use a weighted sum (WS) of +the two base-model estimates from one of the planning-based approaches and the NBM. +More formally, this meta-model is defined as follows: +P(g|o) = wsPs(g|o) + wdPd(g|o) +(3) +Ps is the goal probability estimate of one of the symbolic PRAP approaches, ws is the +weight for the symbolic approach, Pd is the goal probability estimate of the data-driven +NBM, and wd is the weight of the data-driven NBM. +Tiebreaking of the PRAP approaches +The second meta-model, which we refer to +as tiebreaking (TB), only considers the estimate of the NBM when more than one goal is +9 + +Table 2: Evaluation results for E1 for both meta-models. Each row describes the proba- +bilities P(O|G) for all goals G ∈ GBUC for different lengths of O (|O|). g1 = gprepare meal, +g2 = gwatch TV , g3 = guse shower, g4 = guse toilet. +(a) Results for the tiebreaking (TB) meta-model. +P(G|O) +|O| +g1 +g2 +g3 +g4 +28 + 0 +0.131 +0.61 +0.13 +0.13 +28 + 1 +0.287 +0.176 +0.214 +0.323 +28 + 2 +0.293 +0.158 +0.22 +0.33 +28 + 3 +0.146 +0.14 +0.283 +0.431 +28 + 4 +0.005 +0.16 +0.361 +0.474 +28 + 5 +0.002 +0.0 +0.268 +0.73 +28 + 6 +0.0 +0.0 +0.12 +0.88 +(b) Results for the weighted sum (WS) meta- +model. +P(G|O) +|O| +g1 +g2 +g3 +g4 +28 + 0 +0.131 +0.61 +0.13 +0.13 +28 + 1 +0.287 +0.176 +0.214 +0.323 +28 + 2 +0.293 +0.158 +0.22 +0.329 +28 + 3 +0.146 +0.14 +0.283 +0.431 +28 + 4 +0.005 +0.16 +0.361 +0.474 +28 + 5 +0.037 +0.088 +0.165 +0.71 +28 + 6 +0.037 +0.088 +0.091 +0.785 +assigned with the highest likelihood by the PRAP approach. The intuition behind this +meta-model is based on the observations from the results of experiments E1 and E2: Here, +the PRAP approaches never ranked wrong goals as most probable, but not only ranked +the true goal as most probable. +When this is the case, we combine the two estimated +probabilities of the NBM and the planning-based approaches again by taking the weighted +sum of them. More formally, the meta-model is defined as follows: +P(g|o) = +� +Ps(g|o), +if | arg maxg∈G Ps(g|o)| = 1 +wsPs(g|o) + wdPd(g|o), +if | arg maxg∈G Ps(g|o)| > 1, +(4) +Hybrid Goal Recognition Performance for E1 +To evaluate whether the proposed +hybrid method is able to leverage on the additional information contained in the observation +sequence used for E1, we applied the hybrid approach to E1. +For the experiments, we +assumed both weights ws and wd to be 0.5 and, due to the lack of training data for the +BUC use case, modeled the parameters of the data-driven model manually. The results +of the experiments are summarized in Table 2. The results show that goal g4 is ranked +as most probable once one additional observation is made for both meta-models. Before +this point, g2 is considered to be the most probable goal. Hence, the results show that a +hybrid goal recognition model is able to leverage on the information that is contained in +the observations that the agent drank beer. Consequently, it can recognize the true goal +much earlier than the PRAP approaches for this example. Also interesting to note is that +the results are identical for both meta-models until the fifth observation is observed. This +makes sense as the results in Table 1 show that the PRAP approaches are undecided for +the first four observation steps. Hence, both meta-models use the same weighted sum to +combine goal probability estimates. For the fifth and sixth observation steps, both meta- +models estimate the highest probability for the correct goal (i.e., g4). However, the TB +meta-model estimates slightly higher probabilities for this goal and hence, is slightly more +confident in g4 being the actual goal of the agent. +10 + +6. Evaluation Setup +This section describes the experimental setup of the empirical evaluation. The empirical +evaluation aims to achieve the following goals: +1. Evaluate the performance of the planning-based methods, the NBM, and other well- +known data-driven techniques, when applied to goal recognition problems in a real- +world scenario, to determine which methods are best suited to be used in a hybrid +goal recognition method. +2. Show that a hybrid probabilistic goal recognition method is able to achieve superior +performance, compared to purely planning-based and purely data-driven methods. +3. Investigate how the performance of the hybrid method is affected by increasing the +number of possible goals. +We used real-world and artificial datasets for empirically evaluating the goal recognition +methods. In all experiments, the online goal recognition problem was considered (as defined +by Definition 5). +6.1 Datasets +This section presents the real-world and artificial datasets that were used for evaluation. +Real-World Kitchen Dataset +As a real-world data set, we used the CMU-MMAC +Kitchen Dataset (De la Torre, Hodgins, Montano, Valcarcel, Forcada, & Macey, ) to which +we will refer to as “CMU” from here on. It contains data from different sources (e.g., video, +motion capture, etc.) that were recorded by observing different people while cooking one +out of five recipes. We will consider the different dishes the observed participants might +cook as possible goals. We first transformed the existing “raw” data into a suitable format +for our purpose. As a starting point of this transformation, we used the results of a semantic +annotation project (Yordanova, Kr¨uger, & Kirste, 2018). In this project, planning domains +in PDDL format and annotated observation sequences were created for three of the five +recipes (i.e., brownies, eggs, and sandwich). In addition, we created annotations for the +remaining two recipes (i.e., pizza and salad). Table 3 displays some summarizing statistics +of the observation sequence lengths per goal. Note that the CMU dataset only includes the +first five goals. Interesting to note is that the average- and median observation sequence +lengths substantially differ between the recipes. In addition, the standard deviations of the +sequence lengths are relatively high. This indicates that the different observed persons used +significantly different paths to reach one of the goals. +One limitation of the CMU Dataset is that although it is based on sensor recordings +of real human participants that were recorded while they were cooking different recipes, +the general setup that was used during the sensor recordings is still rather artificial and, +therefore, does not necessarily reflect all aspects of natural behaviour in a cooking scenario. +Nevertheless, we still think that it is able to provide a solid basis to judge whether the +investigated goal recognition approaches are able to handle recognition scenarios of real- +world complexity, which is the aim of this work. +11 + +Table 3: Statistics of the observation sequence lengths per recipe in the CMU and artifi- +cially extended CMU datasets (g1=brownies, g2=eggs, g3=sandwich, g4=salad, g5=pizza, +g6=bread, g7=briocheBraid, g8=cheeseburger, g9=spaghetti, g10=spinachFetaPastry). The +original CMU dataset only contains the first five goals. +g1 +g2 +g3 +g4 +g5 +g6 +g7 +g8 +g9 +g10 +Average +111.17 +87.08 +57.85 +110.56 +90.40 +69.73 +50.27 +122.96 +67.63 +108.12 +Median +108.0 +88.0 +58.0 +108.5 +89.5 +70.0 +50.0 +114.0 +65.5 +96.5 +Std. Dev. +15.43 +17.25 +9.02 +17.94 +15.35 +7.57 +7.61 +27.29 +9.43 +39.01 +Extending the CMU Dataset with Artificially Generated Data +To evaluate the +scalability of the proposed hybrid approach to a higher number of goals, we extended the +data from the CMU dataset with five artificially generated goals. +In the remainder of +this work, we will refer to this extended dataset as “ACMU”. The added goals (bread, +brioche braid, cheeseburger, spaghetti, and spinach feta pastry) were manually defined +in the planning domain. +This domain, which was used in both experiments (just with +different sets of possible goals), contains 3411 different actions and 1627 different fluents. +To generate artificial observation sequences for these goals, which is required for training +the data-driven methods, we developed a procedure to sample such observation sequences +for a given planning problem. The intuition underlying our proposed hybrid approach is the +assumption that human behavior includes carrying out actions that are not strictly necessary +in order to reach the current goal (i.e., which are not rational according to the planning +domain). Thus, the sampling procedure should reflect this intuition. Consequently, it is +not sufficient to use optimal plans as generated by existing planning systems: These plans +generally only contain actions that are strictly necessary to reach a given goal. Additionally, +a planning system will always generate very similar plans when it is presented with the same +planning problem. As a solution, we propose a sampling algorithm that generates artificial +observations sequentially. At each step, the algorithm either returns the action used in an +optimal plan, or a randomly drawn action (where the probability of drawing a certain action +depends on the goal). More details on the sampling algorithm can be found in Appendix +A. +Table 3 presents some summarizing statistics for the artificially generated observation +sequences for g6 - g10. The average- and median lengths of the artificially generated obser- +vation sequences are comparable to the sequences that are based on real observed sensor +data. In addition, there is also a comparable amount of variation among these sequences, +which is important to reflect the properties of real-world observations. +Artificial Dataset +To further investigate the scalability of the proposed hybrid approach +and make the results better comparable to existing work, we used a well-known artificial +planning domain, which was commonly used as a benchmark in the existing literature +(Ram´ırez & Geffner, 2010),(Pereira, Oren, & Meneguzzi, 2020). +This domain models a +logistics problem, where different objects have to be delivered to several destinations. In +contrast to the CMU domain, this domain is much smaller and contains only 356 actions +and 84 fluents. We will refer to the resulting dataset as “LOG” from here on. As this +domain is purely synthetic, no real observation sequences existed. +Hence, we used the +12 + +same sampling procedure that was used to extend the CMU dataset to generate artificial +observation sequences for this domain. Table 4 displays some summarizing statistics for +the resulting observation sequences. An important difference to the CMU datasets is that +the average observations sequence length is much smaller in the logistics dataset, which +is mainly caused by the synthetic nature of this domain. Nevertheless, although this is +the case, there is still a recognizable amount of variance among the sampled observation +sequences. +Table 4: Statistics of the observation sequence lengths per recipe in logistics dataset. +g1 +g2 +g3 +g4 +g5 +g6 +g7 +g8 +g9 +g10 +Average +21.33 +22.47 +23.23 +23.43 +21.40 +22.80 +23.67 +22.27 +24.27 +22.37 +Median +21.0 +21.0 +23.0 +22.0 +21.0 +22.0 +23.0 +22.0 +23.5 +22.0 +Std. Dev. +2.31 +3.51 +2.64 +3.48 +2.93 +2.51 +2.27 +2.80 +2.32 +4.52 +6.2 Goal Recognition Methods +In the empirical experiments, we applied different symbolic, data-driven, and hybrid goal +recognition methods. +Symbolic Goal Recognition Methods +We used two state-of-the-art planning-based +methods: GM and RG, which were presented in Section 4. To solve the planning problems +that are generated by the two PRAP approaches, we used the MetricFF (Hoffmann, 2003) +planner. MetricFF is a satisficing planner that supports metric fluents.Following Ram´ırez +and Geffner (Ram´ırez & Geffner, 2010) and Vered et al. (Vered et al., 2016), we assumed +equal cost for all actions and used a value of β = 1. As the timeout for the MetricFF +planner, we used 340 seconds. After this timeout, the planner will be forced to stop and +the associated planning problem is considered not solvable. +Data-Driven Goal Recognition Methods +We evaluated four different data-driven +methods: The NBM presented earlier in Section 5 and three additional data-driven ap- +proaches, which were selected following a recent study by Borrajo et al. (Borrajo, Gopalakr- +ishnan, & Potluru, 2020). Specifically, we used K-Nearest-Neighbors (KNN) (Russell, 2016, +pp. 738-740), XGBoost (Chen & Guestrin, 2016), and a Long-Short-Term Memory (LSTM) +network (Hochreiter & Schmidhuber, 1997). +For experiments with the KNN and XGBoost approaches, we evaluated two different +data encodings. First, we used a binary encoding of the planning states, consisting of the +state of all planning fluents. Second, following the work of Borrajo et al. (Borrajo et al., +2020), we used a vector encoding of the observed action sequence. +We performed a grid search to determine the hyper-parameters for the KNN and XG- +Boost methods. For the planning fluent-based encoding, we used a leaf size of 30 and k = 8 +for KNN and a learning rate of 0.24, minimum child weight of 5, α = 0.42, λ = 1.15, maxi- +mum depth of 5, and γ = 0.99 for XGBoost. In contrast, for the action vector encoding, we +used a leaf size of 40 and k = 3 for KNN and a learning rate of 0.31, minimum child weight +of 2, α = 3.26, λ = 1.03, maximum depth of 3, and γ = 0.96 for XGBoost. +13 + +For the LSTM, we also evaluated two different data encodings: A one hot representation +of both the observed action sequence and the observed state sequence. +Here, we used +different vector encodings than for the other data-driven methods because LSTM models, +in contrast to the other considered methods, are explicitly designed to handle temporal data +sequences which are better captured by one hot data encodings (Borrajo et al., 2020). For +both setups, we used the ADAM optimizer with a learning rate of 0.01, a batch size of 32, +and 100 epochs. +Hybrid Goal Recognition Methods +We evaluated the two different meta-models de- +scribed in Subsection 5.2. For both meta-models, we computed the weight for the NBM +as wNBM(n, t) = +a +1+e−b(t−((cn)+d)) , where n is the number of training examples used to train +the NBM, t is the number of observations used for goal recognition, and a, b, c, and d +are fitting parameters. As for the data-driven approaches, we performed a grid search to +determine the best performing values for a, b, c, and d. Accordingly, we set the parameters +to a = 0.5, b = −0.15, c = 4, d = 2.5 (CMU), and a = 0.5, b = −0.15, c = 5, d = 1 (ACMU) +respectively for the CMU datasets. For the LOG dataset, we used the following parameter +values: a = 0.2, b = −0.15, c = 0, and d = 0. The weight for the PRAP approaches is +calculated as wPRAP = 1 − wNBM(n) for all datasets. +6.3 Experimental Design +To assess the goal recognition performance of the different methods, we used the mean goal +recognition accuracy. To calculate the accuracies, in contrast to most previous works, we +did not consider a goal to be recognized correctly if it is part of a set of goals that were +assigned with the highest likelihood. Instead, we only considered a goal to be recognized +correctly if it is the only goal that was assigned with the highest probability. We decided for +this evaluation method as it, in our opinion, better reflects the usefulness of the prediction +for practical application in an assistance system. If such a system is provided with more +than one most probable goal, it has to randomly decide for one goal. +Furthermore, as +we consider online goal recognition problems in this evaluation, we calculated the mean +accuracy for different fractions of total observations that were used for goal recognition. +Here we used relative numbers because the lengths of the involved observation sequences +substantially differ. Hence, the mean accuracy Acc for a relative number of observations +λ ∈ [0, 1] is calculated as follows: +Acc(λ, D) = +� +R∈D [R(⌊TRλ⌋) = ˜ +gR] +|D| +(5) +Here, D is a set of online goal recognition problems R, ˜ +gR denotes the correct goal of goal +recognition problem R, TR is the maximum value of t for online goal recognition problem +R (i.e., length of observation sequence that is associated with R), and [R(t) = ˜ +gR] equals 1 +if the correct goal is recognized for R(t) and 0 otherwise. +To evaluate the performance of the symbolic goal recognition methods, we calculated +Acc(λ, D) for different values of λ and for different domains D. To investigate the per- +formance of the data-driven approaches in relation to the number of available training +examples (i.e., number n of complete observation sequences), we used a slightly adjusted +cross-validation procedure: For a given value of n, we split a set of online goal recognition +14 + +0 +1 +2 +3 +4 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +Θ(λ, CMU) +GM +RG +Figure 3: Mean accuracy of the planning-based methods (RG and GM) on the CMU Dataset +without artifical samples. +problems D into k partitions, where k = |D|/n. Then, k models were trained, but in con- +trast to the typical cross-validation procedure, only one of the partitions was used as the +training set and the remaining partitions were used for validation. In cases where D cannot +be splitted into k partitions of equal size n, we randomly sampled sequences from the other +partitions to complete the partitions with a size smaller than n. To assess the performance +of a data-driven method, we calculated Acc(λ, D) for all k models and, subsequently, took +the average over these accuracies. For the evaluation of the hybrid methods, we calculated +the combined estimates for all results obtained from the cross-validation procedure for the +data-driven approaches and then also took the average of the accuracies of all k models. +7. Experimental Results +In the following, we present and discuss the results of the experiments corresponding to +each of the three evaluation goals defined in Section 6. +7.1 Symbolic and Data-Driven Goal Recognition +Symbolic Goal Recognition Results +We start by comparing the two planning-based +goal recognition methods on the CMU dataset. Figure 3 shows the average goal recognition +accuracy of the RG and GM approaches on this dataset. +It can be seen that the GM +approach outperformed the RG approach consistently, except for the case when only very +small fractions of the observation sequences are used. Furthermore, the accuracy of the RG +approach decreases when more than 20% of the observations are used. The main reason +for this behavior is the fact that the involved planning problems became too complex to +be solved optimally, or were not solvable at all in the given time limit. This fact is also +the reason for the large difference between GM and RG which could not be observed for +the (much simpler) BUC before: The higher complexity of the planning problem made +the solutions that are found within the given time limit less optimal. Hence, the results +show that the compilation process of the RG approach had a much higher impact onto +the optimality of the solutions than the transformation procedure of the GM approach. +Through a detailed analysis of the generated plans, we found that this is mainly caused by +the fact that, in contrast to the GM approach, the RG approach changes the structure of +the actions space of a planning problem in a way that most planning heuristics are not able +to deal with. +15 + +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, CMU) +n=1 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=3 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, CMU) +n=5 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=7 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +Acc(λ, CMU) +n=9 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +n=11 +NBM +XGBoost (actions) +XGBoost (states) +KNN (actions) +KNN (states) +Figure 4: Mean accuracy of the data-driven methods (NBM, KNN and XGBoost) on the +CMU Dataset without additional samples for different sizes of the training set n. +As the GM approach provided a much better overall performance, in all following ex- +periments, we only considered the GM approach. +Data-Driven Goal Recognition Results +Next, we compare the different data-driven +goal recognition methods. +Figure 4 shows the average, cross-validated goal recognition +accuracies of the NBM, KNN, and XGBoost for the CMU dataset. As the LSTM approach +did not achieve accuracy values above 25% for any training set size, we did not include the +results in Figure 4. For KNN and XGBoost, we compare performances of the fluent-based +and action-based data encodings, as introduced in Section 6.2. +The results show that all approaches performed much better, especially early in the +observation sequence, when the planning state-based data encoding was used. This shows +that, in case of the CMU domain, the symbolic planning states encode more useful informa- +tion regarding the actual goal of an observed agent than the sequence of observed actions. +Furthermore, the accuracies of all three methods did not depend strongly on the amount +of available training data. Interesting to note is that even though the NBM is the model +with the lowest computational complexity, it was still not outperformed by the (slightly) +more complex KNN and XGBoost models. Hence, overall, the NBM is the most favor- +able data-driven model for this scenario, especially in mobile computing scenarios, where +computational efficiency is of high relevance. +16 + +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, CMU) +n=1 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=3 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, CMU) +n=5 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=7 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +Acc(λ, CMU) +n=9 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +n=11 +GM +NBM +WS +TB +Figure 5: Mean accuracy of the Goal Mirroring (GM) and Naive Bayes Model (NBM) +approaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on +the CMU Dataset without artifical samples for different sizes of the training set n. +7.2 Hybrid Goal Recognition +In this section, we assess the performance of the hybrid goal recognition models (i.e., +Weighted Sum (WS) and Tiebreaking (TB)) in comparison to the purely data-driven NBM +approach and the purely planning-based GM method. Figure 5 shows the average, cross- +validated goal recognition accuracies of these approaches for the CMU dataset. +The results show that both hybrid approaches were at least as good as the GM and +NBM approaches for small training set sizes. For larger training set sizes (i.e., n ≥ 3), the +TB approach was increasingly outperformed by the NBM early in the recognition process +(i.e., when only a small fraction of the observations were seen). The reason for this is that +the TB approach relies strongly on the predictions of the GM approach, which also became +increasingly outperformed by the NBM early in the observation sequence with increasing +training set sizes. In contrast, the WS approach was not outperformed by the NBM, but +reached at least similar performance as the NBM also when only a small fraction of the +observations were seen. The WS approach was even able to substantially outperform both +the NBM and the GM approaches early in the observation sequences when, depending on +the training set size, between 3% and 30% of the observations were used. This effect was +most prominent when training set sizes between n = 3 and n = 7 were used. +The results show that for n ≥ 3, the planning-based and data-driven methods com- +plemented each other well regarding recognition performance. While the NBM approach +17 + +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, ACMU) +n=1 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=3 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, ACMU) +n=5 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=7 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +Acc(λ, ACMU) +n=9 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +n=11 +GM +NBM +WS +TB +Figure 6: Mean accuracy of the Goal Mirroring (GM), Naive Bayes Model (NBM) ap- +proaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the +artificially extended CMU Dataset for different sizes of the training set n. +achieved the best performances early in the observation sequences (i.e., less than 10% - 20% +of the observations), the GM approach outperformed the NBM later in the observation +sequences (i.e., more than 10% - 20% of the observations). The hybrid WS approach was +able to leverage on the strengths of the two individual approaches, constantly performing +as good or better as each of them. +7.3 Scalability of Hybrid Goal Recognition +Evaluating Scalability on Extended Real-World Dataset +Next, we investigate the +scalability of the methods, by assessing goal recognition performance when the number of +goals is increased. Figure 6 shows the cross-valiated mean accuracy of the GM, NBM, TB, +and WS approaches for the ACMU dataset (i.e., with sampled observation sequences). +Due to the doubled number of goals, the GM and the NBM approaches achieve a sig- +nificantly lower recognition accuracy compared to the results for the CMU dataset that has +not been extended with artificial data. Nevertheless, it can be observed that the recognition +performance of the GM approach converges towards the GM performance on the not artifi- +cially extended CMU dataset with an increasing fraction of observations that were used for +recognition. The results also show that the NBM approach, even when only a small number +of training examples were used (i.e., n = 3), is able to achieve a better goal recognition +performance than the GM when less than 5% of the observation sequences were seen. In +18 + +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, LOG) +n=1 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=3 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +Acc(λ, LOG) +n=5 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +n=7 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +Acc(λ, LOG) +n=9 +0 1 2 3 4 5 101520253035404550556065707580859095 +0.2 +0.4 +0.6 +0.8 +1 +λ (%) +n=11 +GM +NBM +WS +TB +Figure 7: Mean accuracy of the Goal Mirroring (GM), Naive Bayes Model (NBM) ap- +proaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the +logistics domain for different sizes of the training set n. +addition, the results show that the WS approach again performs similarly well or better +than the two individual approaches. +The differences between the achieved recognition performances of the GM and NBM +approaches are even larger early and late in the observation sequences than for the stan- +dard CMU dataset. This indicates that increasing the number of possible goals makes the +weaknesses of the individual approaches even more prominent and hence, using a hybrid +approach that is able to compensate for them is even more favorable. Note that this obser- +vation only holds when a limited number of training data is available as the performance +of the NBM naturally will increase when more training data is available. +Nevertheless, +it is very common in practice that annotated training examples are scarce as manually +annotating observation sequences is costly and error-prone. +In summary, the results show that the hybrid recognition approach still achieves good +goal recognition performance when the number of possible goals increases. Moreover, the +results indicate that using a hybrid approach is even more beneficial when the number of +goals increases, compared to purely data-driven or purely planning-based methods. +Evaluating Scalability on an Artificial Dataset +Finally, we further investigate the +scalability of the methods by applying them to a benchmark plan recognition domain (which +has simpler plans, but more possible goals than the real-world CMU domain). Figure 7 +shows the mean goal recognition accuracy of the GM, NBM, TB, and WS approaches for +19 + +the logistics planning domain. As for the CMU domain, the NBM performed better than +the GM approach early in the observation sequences (i.e., when less than, depending on +n, 5% - 20% of the observations were seen) and the GM performed better later in the +observation sequences. However, for the logistics domain, the NBM only achieved slightly +better performance than the GM approach. +Interestingly, in contrast to the experiments with the CMU Dataset, the Tiebreaking +(TB) approach also constantly performed as good or better than the two individual ap- +proaches (in addition to WS, as for the CMU dataset). The main reason for this behavior +is the fact that the assumptions underlying the TB approach hold more firmly for the LOG +domain: TB assumes that the planning-based approach (GM, in this case) never predicts +a wrong goal to be most probable, but only predicts multiple, equally likely goals (one of +which is correct). This assumption only holds if the involved plans are optimal. The logis- +tics domain, however, has substantially lower complexity than the CMU domain, such that +the MetricFF planner was able to find more optimal plans in the given time limit. Thus, +the assumptions of TB hold and TB could achieve better results than for the CMU and +ACMU datasets. In summary, the results show that our hybrid goal recognition approach +is also beneficial in artificial planning domains where the number of goals is substantially +higher than in the investigated real-world domain. +8. Related Work +Existing approaches to goal- and plan recognition can be divided into model-based and +model-free approaches. Model-based approaches typically reason over handcrafted symbolic +domain models to solve the recognition task. In contrast, model-free approaches treat the +recognition problem as a classification problem and learn to predict the current user goal +from data and, thus, are data-driven. +Early model-based approaches to plan recognition relied on complete plan libraries that +encode possible user behavior to recognize the current plan from observed user actions +(Kautz & Allen, 1986; Charniak & Goldman, 1993). However, these approaches require a +large manual modeling effort, which is infeasible in large domains. To overcome this issue, a +new class of approaches to plan recognition that no longer required complete plan libraries, +but only a domain model that defines possible states and actions, was proposed. +The +PRAP approaches considered in this work (Ram´ırez & Geffner, 2010; Pereira et al., 2020), +(Vered et al., 2016) belong to this class. Another example approach that relies on the use +of classical planning systems is the approach by Sohrabi et al. (Sohrabi, Riabov, & Udrea, +2016). They propose to use a top-k planner to generate the top-k plans for all possible +goals in order to obtain which goal a user currently intents to achieve. Nevertheless, most +of these approaches have, so far, only been evaluated on relatively small, artificial domains, +and hence, it is not clear whether they are also applicable to real-world scenarios. +We +have shown that these PRAP approaches indeed show good performance in a real-world +setting, but have problems in capturing relations between observations and user goals that +cannot be properly modeled manually. Some other recent approaches to goal recognition in +smart environments also belong to this class of approaches (Yordanova et al., 2017, 2019). +Consequently, they have the same problems as the approaches considered in this work. +20 + +In contrast, model-free approaches learn to predict the most probable user goal directly +from data. Hence, they have the potential to learn the relations between actions and user +goals that are not properly captured by model-based approaches. In (Albrecht, Zukerman, +Nicholson, & Bud, 1997), the authors propose to use a BN model to predict the current quest +of an observed player of a computer game. Recently, approaches that applied deep learning +methods to goal recognition problems have been proposed (Min, Mott, Rowe, Liu, & Lester, +2016; Amado, Aires, Pereira, Magnaguagno, Granada, & Meneguzzi, 2018). For example, +Min et al. (Min et al., 2016) applied a LSTM for player goal recognition in digital games. +However, model-free approaches usually require large amounts of training data to produce +reasonable results. In the case of deep learning models, several thousands of annotated +training examples are required to train the model adequately. Such amounts of training +data are usually not easily available for real-world scenarios. Regarding this aspect, model- +based approaches have a clear advantage because they can rely on handcrafted domain +knowledge. Thus, to benefit from both paradigms’ strengths, we propose a hybrid approach +that combines a model-based and a model-free method. +9. Conclusion and Future Work +In this work, we investigated whether existing plan recognition as planning (PRAP) ap- +proaches can be applied to solve the online goal recognition problem in a real-world kitchen +scenario. More explicitly, we conducted several empirical goal recognition experiments on +the basis of the well-known CMU Kitchen Dataset, which contains observation sequences +for five possible goals of up to 36 different subjects. We found that such PRAP approaches +can indeed be used to solve the online goal recognition problems in real-world scenarios. +Nevertheless, we also revealed and analyzed some major limitations of PRAP approaches +when applied to such scenarios. As a possible solution, we proposed a hybrid goal recog- +nition method, which combines a symbolic PRAP approach and a data-driven model. We +showed that the hybrid approach is able to recognize an agent’s true goal more reliably than +the PRAP approaches, especially early in an observation sequence (i.e., when only a small +fraction of the observations were seen). To investigate the scalability of the proposed hybrid +approach in terms of the number of possible goals, we conducted an experiment based on an +artificially extended version of the CMU Kitchen Dataset. The results of these experiments +indicate that the advantages of using a hybrid approach are becoming even more prominent +with an increasing number of possible goals. +In summary, we showed that using a hybrid goal recognition method provides a valuable +improvement compared to state-of-the-art purely symbolic and data-driven goal recognition +methods. It was found that our proposed hybrid method is able to outperform purely sym- +bolic and data-driven methods and recognize the correct goal more reliably based on a +lower number of observations, although only a small number of training examples are used. +This result substantially improves the usefulness of goal recognition for intelligent assistance +systems, as recognizing a goal early opens much more possibilities for supportive reactions +of the system. Furthermore, it is usually very expansive to obtain annotated training ex- +amples for real-world application scenarios. Hence, being able to provide valuable results +based on limited numbers of training examples is an important requirement for potential +goal recognition methods that should be applied to real-world application scenarios. Nev- +21 + +ertheless, we still see some potential for improvements of the extended approach in future +work. One direction is to optimize the procedure that is used to combine the results of the +two individual approaches. Another direction that we plan to investigate in future work is +to use more complex tractable probabilistic models, like Sum-Product Networks (Poon & +Domingos, 2011). +10. Acknowledgements +The data used in this paper was obtained from kitchen.cs.cmu.edu and the data collection +was funded in part by the National Science Foundation [grant number EEEC-0540865]. This +work was supported by the German Federal Ministry of Education and Research (BMBF) +[grant number 01lS18079C]. +Appendix A. Sampling Artificial Observation Sequences +Algorithm 1 summarizes the sampling procedure that is used in this work to sample artificial +observation sequences. Here, pplanAction is a parameter that specifies the goal-directedness, +i.e., the probability that the next action is taken from a precomputed optimal plan. When +this is not the case, the next action is sampled out of the set of actions that are currently +applicable in the current planning state. These actions are randomly drawn from all actions +that are applicable in a certain state of the planning domain, following two predefined +probability models that model the probability that an interaction with a certain object O is +observed given that we want to reach a goal G (P(O|G)), and the probability that a certain +kind of action A is observed given that we want to reach goal G (P(A|G)). +The distribution P(A|G, S) that is used to sample an action at random is defined as +follows: +P(A = ai|g, s) ∝ +� +wi +if ai applicable in s +0 +otherwise +(6) +That is, only applicable actions can be selected. The weight wi of an action depends on the +corresponding “action type” AT(ai) and the set of objects OB(ai) with that each action +interacts. The underlying intuition is the observation that depending on the current goal, +the agent will choose actions of different action types with higher probabilities than others +and also interact with certain objects with higher probabilities. Based on this intuition, we +use randomly initialized weight score distributions W(AT|G) and W(OB|G) to determine +the weights wi via +wi = W(AT(ai)|g) +� +x∈OB(ai) +W(x|g), +(7) +We initialize the parameters of P(A|G) and P(O|G) randomly and use the MetricFF +planner to compute the initial plan. +Each time an action from the sampling model is +selected, the optimal plan from the resulting state is recomputed. +22 + +Algorithm 1 Sample artificial observation sequence for goal g. +sampledPlan ← () +cState ← initialPlanningState +optPlan ← computeOptimalPlan(cState, g) +i = 0 +while goal not reached do +r ← random(0, 1) +if r < pplanAction then +sAction ← optPlan.getAction(i) +cState ← cState.apply(sAction) +i ← i + 1 +else +sAction ← sampleActionFromApplicableActions(cState) +cState ← cState.apply(sAction) +optPlan ← computeOptimalPlan(cState, g) +i = 0 +end if +sampledPlan ← concat(sampledPlan, sAction) +end while +References +Albrecht, D. W., Zukerman, I., Nicholson, A. E., & Bud, A. 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Accessed: +2021-11-15. +Yordanova, K., L¨udtke, S., Whitehouse, S., Kr¨uger, F., Paiement, A., Mirmehdi, M., Crad- +dock, I., & Kirste, T. (2019). Analysing cooking behaviour in home settings: Towards +health monitoring. Sensors, 19(3), 646. +Yordanova, K., Whitehouse, S., Paiement, A., Mirmehdi, M., Kirste, T., & Craddock, I. +(2017). What’s cooking and why? behaviour recognition during unscripted cooking +tasks for health monitoring. In 2017 IEEE International Conference on Pervasive +Computing and Communications Workshops (PerCom Workshops), pp. 18–21. +25 + diff --git a/XtE5T4oBgHgl3EQfdA-T/content/tmp_files/load_file.txt b/XtE5T4oBgHgl3EQfdA-T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e2abda94e87b030996f436e45ccbb93aa04fb17 --- /dev/null +++ b/XtE5T4oBgHgl3EQfdA-T/content/tmp_files/load_file.txt @@ -0,0 +1,1221 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf,len=1220 +page_content='Investigating the Combination of Planning-Based and Data-Driven Methods for Goal Recognition Nils Wilken nils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='wilken@uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='de Institute for Enterprise Systems, University of Mannheim 69118 Mannheim, Germany Lea Cohausz lea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='cohausz@uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='de Data and Web Science Group, University of Mannheim 69118 Mannheim, Germany Johannes Schaum jschaum@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='de Institute for Enterprise Systems, University of Mannheim 69118 Mannheim, Germany Stefan L¨udtke stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='l¨udtke@uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='de Institute for Enterprise Systems, University of Mannheim 69118 Mannheim, Germany Heiner Stuckenschmidt heiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='stuckenschmidt@uni-mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='de Data and Web Science Group, University of Mannheim 69118 Mannheim, Germany Abstract An important feature of pervasive, intelligent assistance systems is the ability to dy- namically adapt to the current needs of their users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, it is critical for such systems to be able to recognize those goals and needs based on observations of the user’s actions and state of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In this work, we investigate the application of two state-of-the-art, planning-based plan recognition approaches in a real-world setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' So far, these approaches were only evaluated in artificial settings in combination with agents that act perfectly rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We show that such approaches have difficulties when used to recognize the goals of human subjects, because human behaviour is typically not perfectly rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To overcome this issue, we propose an extension to the existing approaches through a classification-based method trained on observed behaviour data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We empirically show that the proposed extension not only outperforms the purely planning-based- and purely data-driven goal recognition methods but is also able to recognize the correct goal more reliably, especially when only a small number of observations were seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This substantially improves the usefulness of hybrid goal recognition approaches for intelligent assistance systems, as recognizing a goal early opens much more possibilities for supportive reactions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Introduction The ultimate goal of smart assistance technologies is to dynamically adapt the infrastructure of a building to best meet the needs of their users by observing their behaviour and deducing their current needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Identifying users’ goals and intentions based on their current and past activities is an important task in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' While there is some work on goal recognition in the context of smart assistance systems (Yordanova, Whitehouse, Paiement, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='05608v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='AI] 13 Jan 2023 Mirmehdi, Kirste, & Craddock, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Yordanova, L¨udtke, Whitehouse, Kr¨uger, Paiement, Mirmehdi, Craddock, & Kirste, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Kr¨uger, Nyolt, Yordanova, Hein, & Kirste, 2014), so far research has mostly focused on recognizing users’ current activities (Helaoui, Riboni, & Stuckenschmidt, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Rashidi, Cook, Holder, & Schmitter-Edgecombe, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hoque & Stankovic, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Yao, Nie, Sheng, Gu, Li, & Wang, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Sztyler & Stuckenschmidt, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In this paper, we address the problem of identifying user goals as a basis for automatic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For this purpose, we look at the related problem of plan recognition, which is a long-standing topic in the Artificial Intelligence community (Kautz & Allen, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Charniak & Goldman, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We believe that plan recognition methods are particularly suited for this task as they do not only identify the goal a user intends to achieve, but also aim to recognize the most probable plan (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', ordered sequence of actions) for achieving this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Knowing such a plan provides us with a better basis for supporting the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In this paper, we investigate the application of two state-of-the-art plan recognition ap- proaches that are based on the principle of Plan Recognition As Planning (PRAP) (Ram´ırez & Geffner, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More explicitly, the contributions of this paper are: In Section 4, we reveal and analyze some major shortcomings of PRAP approaches when applied to real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The main consequence of these shortcomings is that some goals can only be identified relatively closely before they are reached, which significantly reduces their potential benefits to an intelligent assistance system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As a possible solution to this problem, we propose a hybrid plan recognition method in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The proposed method combines the principle of PRAP with a data- driven probabilistic model that captures statistical relations between certain states of the environment and goals that can be learned from past observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Finally, we empirically evaluate the proposed hybrid method in sections 6 and 7 and compare its performance to the performances of purely planning-based and purely data-driven approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The evaluation shows that both approaches can be applied to identify the goals of a user in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Further, the results show that using a hybrid goal recognition method leads to a much earlier identification of the correct goal while only requiring very small amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Problem Definition Probabilistic goal recognition is the problem of inferring a probability distribution over a set of intended goals of an observed agent, given a possibly incomplete sequence of observed actions and a domain model that describes the domain in which the observed agent acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More formally, the aim of goal recognition approaches is to find a posterior probability distribution P(G|O) over all goals g ∈ G, given a sequence of observed actions o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This work considers the smart home domain as an example environment for goal recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Figure 1 shows the layout of a smart flat and partial action sequences that sketch a simple use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This use case will be employed to analyze the shortcomings of the investi- gated planning-based goal recognition approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' It is important to note that this use case is completely synthetic and does not correspond to a real-world experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The flat has four rooms and a hallway that connects all rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In each room,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' different devices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='h1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='h3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='h4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='19°C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='bad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Outdoor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='TA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5°C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='very good ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Sensing functionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Acting functionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Sensing/Acting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='functionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Agent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Heater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Heater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Heater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Heater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Possible Goals: A (Prepare Meal) B (Watch TV) C (Use toilet) D (Use shower) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Figure 1: Illustration of an exemplary smart flat and a simple example use case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', “Beer Use Case”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' and furnishings are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Some of these objects can possibly function as sensor, actuator, or a mixture of both, which is indicated by the green and orange dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, it is assumed that the current location of the agent can be sensed at all times and that the agent can navigate the cells in the flat by moving in all possible directions, including diagonal moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Example 1 (Beer Use Case) Figures 1a and 1b roughly sketch parts of a small use case in this smart flat, which we will refer to as “Beer Use Case” (BUC) from here on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In the BUC, a single agent is initially located in the cell “l3” in the livingroom, moves to the fridge, takes out a beer, and moves back to the couch in the livingroom (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When the agent arrives at the couch in the livingroom, she sits down on the couch, opens the beer, and drinks it while watching TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' After a while, the agent decides to get another beer from the fridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When the second beer is empty, the agent gets up from the couch, moves back to the kitchen, and subsequently via the hallway to the bathroom to use the toilet (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Background In this work, we investigate the application of two state-of-the-art approaches to plan recog- nition to a real-world goal recognition scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast to probabilistic goal recognition, probabilistic plan recognition not only describes the problem of inferring a probability dis- tribution over a set of goals, but also the probability distribution over a set of possible plans that an agent might follow to reach it’s intended goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' From a solution to a plan recognition problem, the solution of the corresponding goal recognition problem can be derived by only considering the goals of the recognized plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Plan recognition is a long standing research area in the Artificial Intelligence community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Recent plan recognition systems mostly rely on the Plan Recognition As Planning (PRAP) (Ram´ırez & Geffner, 2009) principle and hence, utilize symbolic planning systems to solve plan- and goal recognition problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='1 Symbolic Planning Symbolic planning is based on a symbolic model of the planning domain that defines possible actions, their preconditions and effects on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Given a current state and goals 3 in terms of partial state descriptions, planning methods aim to construct an optimal plan for reaching the goals from the current state consisting of a (possibly partial) order of actions to be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We adopt the formalization of a planning problem from (Ram´ırez & Geffner, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Definition 1 (Planning Problem) A Planning Problem is a tuple P = ⟨F, s0, A, G⟩ where F is a set of fluents (boolean statements about properties of the modeled environment), s0 ⊆ F and G ⊆ F are the initial state and the goal description and A is a set of actions with preconditions Pre(a) ⊆ F and lists of fluents Add(a) ⊆ F and Del(a) ⊆ F that de- scribe the effects of an action a in terms of fluents that are added and deleted from the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Actions have a non-negative cost c(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A state is described by the subset of fluents which are currently considered to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A goal state is a state s with s ⊇ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' An action a is applicable in a state s if and only if Pre(a) ⊆ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Applying an action a in a state s leads to a new state s′ = (s ∪ Add(a) \\ Del(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A solution for a planning problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', a plan) is a sequence of applicable actions π = a1, · · · an that transforms the initial state into a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The cost of the plan is defined as c(π) = � i c(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A plan is optimal if the cost of the plan is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This basic model has been extended in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In this paper, we make use of two extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' One allows us to specify goals of form ¬f that claim that a certain fluent f is absent in the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The other enables the use of a conditional effect of form p → q, where p and q are single fluents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This means that when an action x has such a conditional effect, fluent q only becomes true after the execution of x when p was true before the execution (Ram´ırez & Geffner, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 Plan Recognition As Planning: State-of-the-Art As already mentioned, many recent plan- and goal recognition approaches rely onto the principle of Plan Recognition as Planning (PRAP), which was first introduced by Ram´ırez and Geffner (Ram´ırez & Geffner, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' All approaches that follow this principle have in common that they utilize concepts from the area of classical planning to compute probability distributions over a set of possible plans or goals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Definition 2 (Probabilistic Plan Recognition Problem) A probabilistic plan recog- nition problem is a tuple T = ⟨D, G, O, P(G)⟩ where D = ⟨F, s0, A, ∅⟩ is a planning domain, G is a set of possible goals g ⊆ F, o = o1, · · · om, where oi ∈ A is a sequence of actions that have been observed and P(G) is the prior probability distribution over G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A solution to the probabilistic plan recognition problem is the conditional probability of the goals given the observation sequence o (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', P(G = g|O = o)∀g ∈ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Estimating Goal Probabilities Both plan recognition methods that are used in this work are based on the idea of using Bayes Rule to compute the posterior probabilities of the goals: P(G|O) = αP(O|G)P(G) (1) It is assumed that the prior probabilities P(G = g) of goals g ∈ G are given in the problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, the problem of probabilistic goal recognition boils down to the estimation 4 of P(O|G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Both investigated approaches utilize symbolic planning systems to estimate this probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The idea behind this is based on the assumption that agents act perfectly rational and hence, use strictly optimal plans (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e, plans that minimize costs) to achieve their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, it is assumed that the probability of a goal to be the agent’s actual goal can be estimated by relating the costs of an optimal plan that includes a given observation sequence o and an optimal plan that does not include o, while reaching a given goal g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This can be done because an optimal plan that does not have to fulfill the requirement of including o is, according to the planning domain, a perfectly rational plan from the given initial state to a given goal g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, when the costs of an optimal plan that includes o are higher, this means that the agent is taking a detour compared to a perfectly rational plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More precisely, Ram´ırez and Geffner (Ram´ırez & Geffner, 2010) propose to calculate P(o|g) as follows: P(o|g) = α′ exp{−β∆(g)} 1 + exp{−β∆(g)} (2) Where α′ is a normalization factor and ∆(g) = c(o, g)−c(o, g) is the cost difference between an optimal plan for g that satisfies o and an optimal plan for g that does not satisfy o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The costs c(o, g) and c(o, g) can be computed out of the box using classical planning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Translating a Plan Recognition Problem into Planning Problems The two state- of-the-art plan recognition methods used in this paper were proposed by Ram´ırez and Geffner (Ram´ırez & Geffner, 2010) (referred to as “RG” from here on) and Vered et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (Vered, Kaminka, & Biham, 2016) (referred to as “GM” (Goal Mirroring) from here on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' They mainly differ in the way they transform the original planning problem, which is necessary to ensure that the computed plans fulfill some necessary requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To compute the probabilities P(O|G), the RG approach compiles a plan recognition problem T = ⟨P, G, o, Prob⟩ into 2|G| planning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For each goal g ∈ G the two planning problems Po(g) and Po(g) have to be compiled and solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Classical planning systems naturally cannot handle the requirement of satisfying a given sequence of observed actions in a computed plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To ensure that the computed solutions fulfill this requirement, the original planning domain D has to be slightly modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Definition 3 (Transformation of the Planning Domain (RG)) For a given planning domain D = ⟨F, s0, A⟩ and a given observation sequence o, the transformed domain is de- fined as D′ = ⟨F ′, I, A′⟩ with F ′ = F ∪ {poi|oi ∈ (oi)n 0}, where poi is a new fluent and the actions o ∈ A′ that are in o have an additional effect poi when i = 0 and poi−1 → poi has to hold otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For this transformation it is assumed that no action appears twice in o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When this is the case, the corresponding actions are duplicated and renamed to ensure that the order of observed actions is unmodified in the resulting plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Now the costs c(o, g) and c(o, g) can be calculated by solving the planning problems Po(g) = ⟨F ′, s′ 0, A′, g ∪ {pon}⟩ and Po(g) = ⟨F ′, s′ 0, A′, g ∪ {¬pon}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Goal Mirroring The main difference between RG and GM is the domain translation procedure: While RG adapts the actions in a given planning domain, GM uses a different initial state to generate plans that embed o each time a new observation is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 5 Definition 4 (Transformation of Planning Problem (GM)) For a given planning do- main D = ⟨F, s0, A⟩ and a given observation sequence o, the transformed domain is defined as D′ = ⟨F ′, s′ 0, A′⟩ with F ′ = F, where s′ 0 = s0[[o]] and s[[o]] returns as a result the planning state that is obtained when the action sequence o is applied to a planning state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When this transformation is completed, analogously to the RG approach, GM calculates the costs c(o, g) and c(o, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' However, in contrast to RG, GM assumes for the calculation that an optimal plan from s0 to a goal g can be obtained by concatenating o with a plan for g that starts at the adjusted initial state s′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' From such a plan, again the costs c(o, g) can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, GM does not generate plans that strictly do not embed o, but instead computes an optimal plan from s0 to each goal and uses the costs of these plans analogously to the costs of plans that do not embed o as RG does (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', c(o, g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Apart from this, GM uses the same heuristic as RG (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', Equation 1) to compute goal probabilities P(G|O) from these costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' One major benefit of GM compared to RG is that it is expected to be much more time efficient in the case of online probabilistic goal recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This becomes increasingly important with increasing complexity of the involved planning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Definition 5 (Online Probabilistic Goal Recognition) We define online probabilistic goal recognition as a special variant of the probabilistic goal recognition problem defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In online goal recognition, we assume that the observation sequence o is revealed incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More explicitly, we introduce the notion of time t ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , T}, where T = |o|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For every value of t, one probabilistic goal recognition problem R(t) can be induced as R(t) = ⟨D, G, ot, Prob⟩ where D = ⟨F, s0, A, ∅⟩ and ot = {oi|0 ≤ i ≤ t, oi ∈ o}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A solution to the online probabilistic goal recognition problem are the conditional probabilities Pt(G = g|ot);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' ∀g ∈ G, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, in the case of online probabilistic goal recognition, GM solves, due to the different transformation procedure, only |G||O| + |G| planning problems instead of 2|G||O| planning problems that RG solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Case Study: Goal Recognition in the Beer Use Case In this section we evaluate the performance of the RG and GM goal recognition approaches when applied to the synthetic BUC example (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, we demonstrate and discuss some major limitations of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='1 Experimental Setup For the experiments, we modeled a planning domain DBUC in the Planning Domain Def- inition Language (PDDL) (McDermott, Ghallab, Howe, Knoblock, Ram, Veloso, Weld, & Wilkins, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The goal set of the corresponding plan recognition problems (see Definition 2) is defined as GBUC = {gprepare meal, gwatch TV , guse shower, guse toilet}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Further, following the approach of Ram´ırez and Geffner (Ram´ırez & Geffner, 2010), we assume uniform prior probabilities PBUC(G) for all goals in GBUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Based on this experimental setup, we conducted two experiments E1 and E2 with both recognition approaches, which, however, differ in the observation sequences that are 6 Table 1: Evaluation results for the RG and GM goal recognition approaches when applied to E1 and E2 with the LAMA planner in anytime mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results for both approaches are identical for both, E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Each row describes the probabilities P(G|O) for all goals G ∈ GBUC for different lengths of O (|O|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' g1 = gprepare meal, g2 = gwatch TV , g3 = guse shower, g4 = guse toilet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (a) Results for E1 P(G|O) |O| g1 g2 g3 g4 28 + 0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='648 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='806 used to compile the involved planning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For experiment E1, the actions in the utilized observation sequence represent the entire BUC (see Example 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For experiment E2, to evaluate how much the goal probability estimates depend on information gained from the observations of the agent getting and drinking beer, only the last six actions of the observation sequence used in E1 are used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', in E2 the observations of the agent getting and drinking beer are not included in the observation sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The remaining setups are similar for both experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To solve the planning problems compiled from these setups, we used the LAMA (Richter, Helmert, & Westphal, 2008) planner, which is a satisficing planner that can be used either in greedy or anytime mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When used in greedy mode, the planner stops immediately when a plan is found, whereas in anytime mode LAMA returns the best plan found in a given time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Here, we used LAMA in anytime mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 Results and Limitations The results of the experiments are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Each row reports the probabilities P(G|O) for all g ∈ GBUC for different lengths of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As the RG and GM approaches are based on the same underlying principle and the BUC domain is small enough to be handled efficiently by both approaches, the results for the experiments E1 and E2 are identical for both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, we only report one result table for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results of the experiments reveal two major limitations of the two planning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In both experiments, the correct goal g4 is only recognized at |O| = 28 + 5 and |O| = 5 respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', shortly before the goal is actually reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This timestep corresponds to the observation where the agent has moved to the location ba3, which is also where the toilet is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This circumstance significantly reduces the possible usefulness for an intelligent assistance system, as the goal is recognized too late to provide support through adaptations of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The additional information that is included in the observation sequence used for E1, has no impact on the estimated probabilities P(G|O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Instead, the estimates are only based on information that are gained from the up to last six actions that are observed afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Intuitively, however, the additional information should have an impact onto the estimate, because there exists a causal relation between drinking beer and the probability that this agent pursues the goal to use a toilet afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Thus, it should be possible to recognize the correct goal much earlier in the observation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The main reason for these limitations of the two planning-based approaches is that the additional actions that are contained in the observation sequence used for E1 are not strictly necessary to fulfill the preconditions of any action that is required to reach one of the possible goals in an optimal way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', drinking beer is not a necessary requirement to visit the bathroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As a consequence, these observations have the same impact on the estimate of P(G|O) for all possible goals, although they might contain valuable information about the true probability of P(G|O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' A Hybrid Goal Recognition Approach As shown in the previous section, a significant shortcoming of PRAP based approaches is that causal relations between goals and observations in the environment cannot be exploited in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This is due to the fact that the costs of plans are used as the only criterion to es- timate the probabilities P(O|G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To solve this problem, we propose to combine these PRAP approaches with a data-driven probabilistic causal model of agent goals and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='1 A Statistical Causal Model of Observations We propose to model the relationship between goals and observations via a probability distribution P(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , FN|G), where F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , FN are fluents of the planning state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This distribution models how goals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', making a sandwich) affect the probability of specific fluents in the planning states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', whether the agent holds a cucumber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast to the PRAP models, the idea here is to learn the parameters of P(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , Fn|g) from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This way, the probabilistic model can capture the relations between fluents and goals that cannot be captured by planning-based approaches: Specifically, the planning-based models cannot capture statistical relations between fluents and goals that are not necessary for an optimal plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For example, in the BUC planning domain, drinking beer is not a necessary precondition for visiting the bathroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Still, drinking beer makes an eventual bathroom visit more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In general, we could use any probabilistic model, like Bayesian Networks, deep generative models etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', to represent P(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , FN|G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Here, we focus on a Naive Bayes model (NBM), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', assuming P(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , Fn|g) = �n i=1 P(Fi|g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The model is visualized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' On the one hand, the strong independence assumptions between all fluents do not necessarily hold in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' On the other hand, a Naive Bayes model has few parameters (linear in the number of variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Thus, training is possible even when training data is scarce – as is often the case for activity sequences of real human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Note that we made another simplifying assumption here: The distribution P(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' , FN|G) only models the dependency of the current planning state on the goal g, but not the de- 8 pendency of the action sequence o on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More specifically, different observation sequences that result in the same planning state could be associated with different goals, but we deliberately neglected this information to make the learning problem tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' F_N … g F_1 Figure 2: Bayesian network representation of the Naive Bayes Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 A Hybrid Goal Recognition Method To overcome the shortcomings of the PRAP approaches discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2, we propose to combine it with the proposed probabilistic learned model of P(o|g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For the combination of the goal probabilities, we use model stacking, which is a common ensemble learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In model stacking, a so-called meta-model is used to generate a combined estimate from the estimates of heterogeneous base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In this work, we studied two different, manually designed, meta-models to combine the estimate of one of the PRAP approaches and the estimate of the NBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' It is important to note that there are some major differences between the ways in which the two utilized base models estimate the goal probabilities P(G|O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' One of these differences is that the PRAP models are based purely on manually specified knowledge (in the form of the planning domain), whereas the NBM only relies on this manually defined domain knowledge to establish its structure but learns its parameters from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Another major difference between the models is the set of features which are used to estimate the probability P(O|G) and hence, also the probability P(G|O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Although the predictions of both models are based on an observed action sequence and an observed initial state of the environment, they estimate the probability P(O|G) based on different features that can be derived from these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The planning-based methods use two entire plans to estimate the probability P(o|g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast, the NBM model uses only the planning state that results from applying the sequence o to the initial state to estimate P(o|g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Weighted Sum of Predictions As a first meta-model, we use a weighted sum (WS) of the two base-model estimates from one of the planning-based approaches and the NBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More formally, this meta-model is defined as follows: P(g|o) = wsPs(g|o) + wdPd(g|o) (3) Ps is the goal probability estimate of one of the symbolic PRAP approaches, ws is the weight for the symbolic approach, Pd is the goal probability estimate of the data-driven NBM, and wd is the weight of the data-driven NBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Tiebreaking of the PRAP approaches The second meta-model, which we refer to as tiebreaking (TB), only considers the estimate of the NBM when more than one goal is 9 Table 2: Evaluation results for E1 for both meta-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Each row describes the proba- bilities P(O|G) for all goals G ∈ GBUC for different lengths of O (|O|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' g1 = gprepare meal, g2 = gwatch TV , g3 = guse shower, g4 = guse toilet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (a) Results for the tiebreaking (TB) meta-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' P(G|O) |O| g1 g2 g3 g4 28 + 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='13 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='474 28 + 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='73 28 + 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='88 (b) Results for the weighted sum (WS) meta- model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' P(G|O) |O| g1 g2 g3 g4 28 + 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='131 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='431 28 + 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='474 28 + 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='71 28 + 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='785 assigned with the highest likelihood by the PRAP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The intuition behind this meta-model is based on the observations from the results of experiments E1 and E2: Here, the PRAP approaches never ranked wrong goals as most probable, but not only ranked the true goal as most probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When this is the case, we combine the two estimated probabilities of the NBM and the planning-based approaches again by taking the weighted sum of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More formally, the meta-model is defined as follows: P(g|o) = � Ps(g|o), if | arg maxg∈G Ps(g|o)| = 1 wsPs(g|o) + wdPd(g|o), if | arg maxg∈G Ps(g|o)| > 1, (4) Hybrid Goal Recognition Performance for E1 To evaluate whether the proposed hybrid method is able to leverage on the additional information contained in the observation sequence used for E1, we applied the hybrid approach to E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For the experiments, we assumed both weights ws and wd to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 and, due to the lack of training data for the BUC use case, modeled the parameters of the data-driven model manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results of the experiments are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results show that goal g4 is ranked as most probable once one additional observation is made for both meta-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Before this point, g2 is considered to be the most probable goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, the results show that a hybrid goal recognition model is able to leverage on the information that is contained in the observations that the agent drank beer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Consequently, it can recognize the true goal much earlier than the PRAP approaches for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Also interesting to note is that the results are identical for both meta-models until the fifth observation is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This makes sense as the results in Table 1 show that the PRAP approaches are undecided for the first four observation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, both meta-models use the same weighted sum to combine goal probability estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For the fifth and sixth observation steps, both meta- models estimate the highest probability for the correct goal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', g4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' However, the TB meta-model estimates slightly higher probabilities for this goal and hence, is slightly more confident in g4 being the actual goal of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Evaluation Setup This section describes the experimental setup of the empirical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The empirical evaluation aims to achieve the following goals: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Evaluate the performance of the planning-based methods, the NBM, and other well- known data-driven techniques, when applied to goal recognition problems in a real- world scenario, to determine which methods are best suited to be used in a hybrid goal recognition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Show that a hybrid probabilistic goal recognition method is able to achieve superior performance, compared to purely planning-based and purely data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Investigate how the performance of the hybrid method is affected by increasing the number of possible goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We used real-world and artificial datasets for empirically evaluating the goal recognition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In all experiments, the online goal recognition problem was considered (as defined by Definition 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='1 Datasets This section presents the real-world and artificial datasets that were used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Real-World Kitchen Dataset As a real-world data set, we used the CMU-MMAC Kitchen Dataset (De la Torre, Hodgins, Montano, Valcarcel, Forcada, & Macey, ) to which we will refer to as “CMU” from here on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' It contains data from different sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', video, motion capture, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=') that were recorded by observing different people while cooking one out of five recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We will consider the different dishes the observed participants might cook as possible goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We first transformed the existing “raw” data into a suitable format for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As a starting point of this transformation, we used the results of a semantic annotation project (Yordanova, Kr¨uger, & Kirste, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In this project, planning domains in PDDL format and annotated observation sequences were created for three of the five recipes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', brownies, eggs, and sandwich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In addition, we created annotations for the remaining two recipes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', pizza and salad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Table 3 displays some summarizing statistics of the observation sequence lengths per goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Note that the CMU dataset only includes the first five goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Interesting to note is that the average- and median observation sequence lengths substantially differ between the recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In addition, the standard deviations of the sequence lengths are relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This indicates that the different observed persons used significantly different paths to reach one of the goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' One limitation of the CMU Dataset is that although it is based on sensor recordings of real human participants that were recorded while they were cooking different recipes, the general setup that was used during the sensor recordings is still rather artificial and, therefore, does not necessarily reflect all aspects of natural behaviour in a cooking scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nevertheless, we still think that it is able to provide a solid basis to judge whether the investigated goal recognition approaches are able to handle recognition scenarios of real- world complexity, which is the aim of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 11 Table 3: Statistics of the observation sequence lengths per recipe in the CMU and artifi- cially extended CMU datasets (g1=brownies, g2=eggs, g3=sandwich, g4=salad, g5=pizza, g6=bread, g7=briocheBraid, g8=cheeseburger, g9=spaghetti, g10=spinachFetaPastry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The original CMU dataset only contains the first five goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 Average 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='17 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='08 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='85 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='56 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='40 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='73 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='27 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='96 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='63 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='12 Median 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='43 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='02 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='94 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='61 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='29 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='43 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='01 Extending the CMU Dataset with Artificially Generated Data To evaluate the scalability of the proposed hybrid approach to a higher number of goals, we extended the data from the CMU dataset with five artificially generated goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In the remainder of this work, we will refer to this extended dataset as “ACMU”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The added goals (bread, brioche braid, cheeseburger, spaghetti, and spinach feta pastry) were manually defined in the planning domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This domain, which was used in both experiments (just with different sets of possible goals), contains 3411 different actions and 1627 different fluents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To generate artificial observation sequences for these goals, which is required for training the data-driven methods, we developed a procedure to sample such observation sequences for a given planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The intuition underlying our proposed hybrid approach is the assumption that human behavior includes carrying out actions that are not strictly necessary in order to reach the current goal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', which are not rational according to the planning domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Thus, the sampling procedure should reflect this intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Consequently, it is not sufficient to use optimal plans as generated by existing planning systems: These plans generally only contain actions that are strictly necessary to reach a given goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Additionally, a planning system will always generate very similar plans when it is presented with the same planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As a solution, we propose a sampling algorithm that generates artificial observations sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' At each step, the algorithm either returns the action used in an optimal plan, or a randomly drawn action (where the probability of drawing a certain action depends on the goal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More details on the sampling algorithm can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Table 3 presents some summarizing statistics for the artificially generated observation sequences for g6 - g10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The average- and median lengths of the artificially generated obser- vation sequences are comparable to the sequences that are based on real observed sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In addition, there is also a comparable amount of variation among these sequences, which is important to reflect the properties of real-world observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Artificial Dataset To further investigate the scalability of the proposed hybrid approach and make the results better comparable to existing work, we used a well-known artificial planning domain, which was commonly used as a benchmark in the existing literature (Ram´ırez & Geffner, 2010),(Pereira, Oren, & Meneguzzi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This domain models a logistics problem, where different objects have to be delivered to several destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast to the CMU domain, this domain is much smaller and contains only 356 actions and 84 fluents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We will refer to the resulting dataset as “LOG” from here on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As this domain is purely synthetic, no real observation sequences existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, we used the 12 same sampling procedure that was used to extend the CMU dataset to generate artificial observation sequences for this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Table 4 displays some summarizing statistics for the resulting observation sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' An important difference to the CMU datasets is that the average observations sequence length is much smaller in the logistics dataset, which is mainly caused by the synthetic nature of this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nevertheless, although this is the case, there is still a recognizable amount of variance among the sampled observation sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Table 4: Statistics of the observation sequence lengths per recipe in logistics dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 Average 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='33 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='47 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='23 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='43 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='40 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='80 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='67 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='27 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='27 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='37 Median 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='0 Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 Goal Recognition Methods In the empirical experiments, we applied different symbolic, data-driven, and hybrid goal recognition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Symbolic Goal Recognition Methods We used two state-of-the-art planning-based methods: GM and RG, which were presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To solve the planning problems that are generated by the two PRAP approaches, we used the MetricFF (Hoffmann, 2003) planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' MetricFF is a satisficing planner that supports metric fluents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='Following Ram´ırez and Geffner (Ram´ırez & Geffner, 2010) and Vered et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (Vered et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2016), we assumed equal cost for all actions and used a value of β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As the timeout for the MetricFF planner, we used 340 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' After this timeout, the planner will be forced to stop and the associated planning problem is considered not solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Data-Driven Goal Recognition Methods We evaluated four different data-driven methods: The NBM presented earlier in Section 5 and three additional data-driven ap- proaches, which were selected following a recent study by Borrajo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (Borrajo, Gopalakr- ishnan, & Potluru, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Specifically, we used K-Nearest-Neighbors (KNN) (Russell, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 738-740), XGBoost (Chen & Guestrin, 2016), and a Long-Short-Term Memory (LSTM) network (Hochreiter & Schmidhuber, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For experiments with the KNN and XGBoost approaches, we evaluated two different data encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' First, we used a binary encoding of the planning states, consisting of the state of all planning fluents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Second, following the work of Borrajo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (Borrajo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2020), we used a vector encoding of the observed action sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We performed a grid search to determine the hyper-parameters for the KNN and XG- Boost methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For the planning fluent-based encoding, we used a leaf size of 30 and k = 8 for KNN and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='24, minimum child weight of 5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='42, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='15, maxi- mum depth of 5, and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='99 for XGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast, for the action vector encoding, we used a leaf size of 40 and k = 3 for KNN and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='31, minimum child weight of 2, α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='26, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='03, maximum depth of 3, and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='96 for XGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 13 For the LSTM, we also evaluated two different data encodings: A one hot representation of both the observed action sequence and the observed state sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Here, we used different vector encodings than for the other data-driven methods because LSTM models, in contrast to the other considered methods, are explicitly designed to handle temporal data sequences which are better captured by one hot data encodings (Borrajo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For both setups, we used the ADAM optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='01, a batch size of 32, and 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hybrid Goal Recognition Methods We evaluated the two different meta-models de- scribed in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For both meta-models, we computed the weight for the NBM as wNBM(n, t) = a 1+e−b(t−((cn)+d)) , where n is the number of training examples used to train the NBM, t is the number of observations used for goal recognition, and a, b, c, and d are fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As for the data-driven approaches, we performed a grid search to determine the best performing values for a, b, c, and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Accordingly, we set the parameters to a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5, b = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='15, c = 4, d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5 (CMU), and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='5, b = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='15, c = 5, d = 1 (ACMU) respectively for the CMU datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For the LOG dataset, we used the following parameter values: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2, b = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='15, c = 0, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The weight for the PRAP approaches is calculated as wPRAP = 1 − wNBM(n) for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='3 Experimental Design To assess the goal recognition performance of the different methods, we used the mean goal recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To calculate the accuracies, in contrast to most previous works, we did not consider a goal to be recognized correctly if it is part of a set of goals that were assigned with the highest likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Instead, we only considered a goal to be recognized correctly if it is the only goal that was assigned with the highest probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We decided for this evaluation method as it, in our opinion, better reflects the usefulness of the prediction for practical application in an assistance system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' If such a system is provided with more than one most probable goal, it has to randomly decide for one goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, as we consider online goal recognition problems in this evaluation, we calculated the mean accuracy for different fractions of total observations that were used for goal recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Here we used relative numbers because the lengths of the involved observation sequences substantially differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, the mean accuracy Acc for a relative number of observations λ ∈ [0, 1] is calculated as follows: Acc(λ, D) = � R∈D [R(⌊TRλ⌋) = ˜ gR] |D| (5) Here, D is a set of online goal recognition problems R, ˜ gR denotes the correct goal of goal recognition problem R, TR is the maximum value of t for online goal recognition problem R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', length of observation sequence that is associated with R), and [R(t) = ˜ gR] equals 1 if the correct goal is recognized for R(t) and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To evaluate the performance of the symbolic goal recognition methods, we calculated Acc(λ, D) for different values of λ and for different domains D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To investigate the per- formance of the data-driven approaches in relation to the number of available training examples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', number n of complete observation sequences), we used a slightly adjusted cross-validation procedure: For a given value of n, we split a set of online goal recognition 14 0 1 2 3 4 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) Θ(λ, CMU) GM RG Figure 3: Mean accuracy of the planning-based methods (RG and GM) on the CMU Dataset without artifical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' problems D into k partitions, where k = |D|/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Then, k models were trained, but in con- trast to the typical cross-validation procedure, only one of the partitions was used as the training set and the remaining partitions were used for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In cases where D cannot be splitted into k partitions of equal size n, we randomly sampled sequences from the other partitions to complete the partitions with a size smaller than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To assess the performance of a data-driven method, we calculated Acc(λ, D) for all k models and, subsequently, took the average over these accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For the evaluation of the hybrid methods, we calculated the combined estimates for all results obtained from the cross-validation procedure for the data-driven approaches and then also took the average of the accuracies of all k models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Experimental Results In the following, we present and discuss the results of the experiments corresponding to each of the three evaluation goals defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='1 Symbolic and Data-Driven Goal Recognition Symbolic Goal Recognition Results We start by comparing the two planning-based goal recognition methods on the CMU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Figure 3 shows the average goal recognition accuracy of the RG and GM approaches on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' It can be seen that the GM approach outperformed the RG approach consistently, except for the case when only very small fractions of the observation sequences are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, the accuracy of the RG approach decreases when more than 20% of the observations are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The main reason for this behavior is the fact that the involved planning problems became too complex to be solved optimally, or were not solvable at all in the given time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This fact is also the reason for the large difference between GM and RG which could not be observed for the (much simpler) BUC before: The higher complexity of the planning problem made the solutions that are found within the given time limit less optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, the results show that the compilation process of the RG approach had a much higher impact onto the optimality of the solutions than the transformation procedure of the GM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Through a detailed analysis of the generated plans, we found that this is mainly caused by the fact that, in contrast to the GM approach, the RG approach changes the structure of the actions space of a planning problem in a way that most planning heuristics are not able to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 15 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, CMU) n=1 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=3 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, CMU) n=5 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=7 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) Acc(λ, CMU) n=9 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) n=11 NBM XGBoost (actions) XGBoost (states) KNN (actions) KNN (states) Figure 4: Mean accuracy of the data-driven methods (NBM, KNN and XGBoost) on the CMU Dataset without additional samples for different sizes of the training set n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As the GM approach provided a much better overall performance, in all following ex- periments, we only considered the GM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Data-Driven Goal Recognition Results Next, we compare the different data-driven goal recognition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Figure 4 shows the average, cross-validated goal recognition accuracies of the NBM, KNN, and XGBoost for the CMU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As the LSTM approach did not achieve accuracy values above 25% for any training set size, we did not include the results in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For KNN and XGBoost, we compare performances of the fluent-based and action-based data encodings, as introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results show that all approaches performed much better, especially early in the observation sequence, when the planning state-based data encoding was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This shows that, in case of the CMU domain, the symbolic planning states encode more useful informa- tion regarding the actual goal of an observed agent than the sequence of observed actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, the accuracies of all three methods did not depend strongly on the amount of available training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Interesting to note is that even though the NBM is the model with the lowest computational complexity, it was still not outperformed by the (slightly) more complex KNN and XGBoost models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, overall, the NBM is the most favor- able data-driven model for this scenario, especially in mobile computing scenarios, where computational efficiency is of high relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 16 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, CMU) n=1 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=3 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, CMU) n=5 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=7 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) Acc(λ, CMU) n=9 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) n=11 GM NBM WS TB Figure 5: Mean accuracy of the Goal Mirroring (GM) and Naive Bayes Model (NBM) approaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the CMU Dataset without artifical samples for different sizes of the training set n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 Hybrid Goal Recognition In this section, we assess the performance of the hybrid goal recognition models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', Weighted Sum (WS) and Tiebreaking (TB)) in comparison to the purely data-driven NBM approach and the purely planning-based GM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Figure 5 shows the average, cross- validated goal recognition accuracies of these approaches for the CMU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results show that both hybrid approaches were at least as good as the GM and NBM approaches for small training set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For larger training set sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', n ≥ 3), the TB approach was increasingly outperformed by the NBM early in the recognition process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', when only a small fraction of the observations were seen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The reason for this is that the TB approach relies strongly on the predictions of the GM approach, which also became increasingly outperformed by the NBM early in the observation sequence with increasing training set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast, the WS approach was not outperformed by the NBM, but reached at least similar performance as the NBM also when only a small fraction of the observations were seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The WS approach was even able to substantially outperform both the NBM and the GM approaches early in the observation sequences when, depending on the training set size, between 3% and 30% of the observations were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This effect was most prominent when training set sizes between n = 3 and n = 7 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results show that for n ≥ 3, the planning-based and data-driven methods com- plemented each other well regarding recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' While the NBM approach 17 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, ACMU) n=1 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=3 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, ACMU) n=5 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=7 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) Acc(λ, ACMU) n=9 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) n=11 GM NBM WS TB Figure 6: Mean accuracy of the Goal Mirroring (GM), Naive Bayes Model (NBM) ap- proaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the artificially extended CMU Dataset for different sizes of the training set n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' achieved the best performances early in the observation sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', less than 10% - 20% of the observations), the GM approach outperformed the NBM later in the observation sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', more than 10% - 20% of the observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The hybrid WS approach was able to leverage on the strengths of the two individual approaches, constantly performing as good or better as each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='3 Scalability of Hybrid Goal Recognition Evaluating Scalability on Extended Real-World Dataset Next, we investigate the scalability of the methods, by assessing goal recognition performance when the number of goals is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Figure 6 shows the cross-valiated mean accuracy of the GM, NBM, TB, and WS approaches for the ACMU dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', with sampled observation sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Due to the doubled number of goals, the GM and the NBM approaches achieve a sig- nificantly lower recognition accuracy compared to the results for the CMU dataset that has not been extended with artificial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nevertheless, it can be observed that the recognition performance of the GM approach converges towards the GM performance on the not artifi- cially extended CMU dataset with an increasing fraction of observations that were used for recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results also show that the NBM approach, even when only a small number of training examples were used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', n = 3), is able to achieve a better goal recognition performance than the GM when less than 5% of the observation sequences were seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In 18 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, LOG) n=1 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=3 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 Acc(λ, LOG) n=5 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 n=7 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) Acc(λ, LOG) n=9 0 1 2 3 4 5 101520253035404550556065707580859095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='8 1 λ (%) n=11 GM NBM WS TB Figure 7: Mean accuracy of the Goal Mirroring (GM), Naive Bayes Model (NBM) ap- proaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the logistics domain for different sizes of the training set n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' addition, the results show that the WS approach again performs similarly well or better than the two individual approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The differences between the achieved recognition performances of the GM and NBM approaches are even larger early and late in the observation sequences than for the stan- dard CMU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This indicates that increasing the number of possible goals makes the weaknesses of the individual approaches even more prominent and hence, using a hybrid approach that is able to compensate for them is even more favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Note that this obser- vation only holds when a limited number of training data is available as the performance of the NBM naturally will increase when more training data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nevertheless, it is very common in practice that annotated training examples are scarce as manually annotating observation sequences is costly and error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In summary, the results show that the hybrid recognition approach still achieves good goal recognition performance when the number of possible goals increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Moreover, the results indicate that using a hybrid approach is even more beneficial when the number of goals increases, compared to purely data-driven or purely planning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Evaluating Scalability on an Artificial Dataset Finally, we further investigate the scalability of the methods by applying them to a benchmark plan recognition domain (which has simpler plans, but more possible goals than the real-world CMU domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Figure 7 shows the mean goal recognition accuracy of the GM, NBM, TB, and WS approaches for 19 the logistics planning domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As for the CMU domain, the NBM performed better than the GM approach early in the observation sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', when less than, depending on n, 5% - 20% of the observations were seen) and the GM performed better later in the observation sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' However, for the logistics domain, the NBM only achieved slightly better performance than the GM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Interestingly, in contrast to the experiments with the CMU Dataset, the Tiebreaking (TB) approach also constantly performed as good or better than the two individual ap- proaches (in addition to WS, as for the CMU dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The main reason for this behavior is the fact that the assumptions underlying the TB approach hold more firmly for the LOG domain: TB assumes that the planning-based approach (GM, in this case) never predicts a wrong goal to be most probable, but only predicts multiple, equally likely goals (one of which is correct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This assumption only holds if the involved plans are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The logis- tics domain, however, has substantially lower complexity than the CMU domain, such that the MetricFF planner was able to find more optimal plans in the given time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Thus, the assumptions of TB hold and TB could achieve better results than for the CMU and ACMU datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In summary, the results show that our hybrid goal recognition approach is also beneficial in artificial planning domains where the number of goals is substantially higher than in the investigated real-world domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Related Work Existing approaches to goal- and plan recognition can be divided into model-based and model-free approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Model-based approaches typically reason over handcrafted symbolic domain models to solve the recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In contrast, model-free approaches treat the recognition problem as a classification problem and learn to predict the current user goal from data and, thus, are data-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Early model-based approaches to plan recognition relied on complete plan libraries that encode possible user behavior to recognize the current plan from observed user actions (Kautz & Allen, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Charniak & Goldman, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' However, these approaches require a large manual modeling effort, which is infeasible in large domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To overcome this issue, a new class of approaches to plan recognition that no longer required complete plan libraries, but only a domain model that defines possible states and actions, was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The PRAP approaches considered in this work (Ram´ırez & Geffner, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2020), (Vered et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2016) belong to this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Another example approach that relies on the use of classical planning systems is the approach by Sohrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (Sohrabi, Riabov, & Udrea, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' They propose to use a top-k planner to generate the top-k plans for all possible goals in order to obtain which goal a user currently intents to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nevertheless, most of these approaches have, so far, only been evaluated on relatively small, artificial domains, and hence, it is not clear whether they are also applicable to real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We have shown that these PRAP approaches indeed show good performance in a real-world setting, but have problems in capturing relations between observations and user goals that cannot be properly modeled manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Some other recent approaches to goal recognition in smart environments also belong to this class of approaches (Yordanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2017, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Consequently, they have the same problems as the approaches considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 20 In contrast, model-free approaches learn to predict the most probable user goal directly from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, they have the potential to learn the relations between actions and user goals that are not properly captured by model-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In (Albrecht, Zukerman, Nicholson, & Bud, 1997), the authors propose to use a BN model to predict the current quest of an observed player of a computer game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Recently, approaches that applied deep learning methods to goal recognition problems have been proposed (Min, Mott, Rowe, Liu, & Lester, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Amado, Aires, Pereira, Magnaguagno, Granada, & Meneguzzi, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' For example, Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', 2016) applied a LSTM for player goal recognition in digital games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' However, model-free approaches usually require large amounts of training data to produce reasonable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In the case of deep learning models, several thousands of annotated training examples are required to train the model adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Such amounts of training data are usually not easily available for real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Regarding this aspect, model- based approaches have a clear advantage because they can rely on handcrafted domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Thus, to benefit from both paradigms’ strengths, we propose a hybrid approach that combines a model-based and a model-free method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Conclusion and Future Work In this work, we investigated whether existing plan recognition as planning (PRAP) ap- proaches can be applied to solve the online goal recognition problem in a real-world kitchen scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' More explicitly, we conducted several empirical goal recognition experiments on the basis of the well-known CMU Kitchen Dataset, which contains observation sequences for five possible goals of up to 36 different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We found that such PRAP approaches can indeed be used to solve the online goal recognition problems in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nevertheless, we also revealed and analyzed some major limitations of PRAP approaches when applied to such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' As a possible solution, we proposed a hybrid goal recog- nition method, which combines a symbolic PRAP approach and a data-driven model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' We showed that the hybrid approach is able to recognize an agent’s true goal more reliably than the PRAP approaches, especially early in an observation sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', when only a small fraction of the observations were seen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' To investigate the scalability of the proposed hybrid approach in terms of the number of possible goals, we conducted an experiment based on an artificially extended version of the CMU Kitchen Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The results of these experiments indicate that the advantages of using a hybrid approach are becoming even more prominent with an increasing number of possible goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In summary, we showed that using a hybrid goal recognition method provides a valuable improvement compared to state-of-the-art purely symbolic and data-driven goal recognition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' It was found that our proposed hybrid method is able to outperform purely sym- bolic and data-driven methods and recognize the correct goal more reliably based on a lower number of observations, although only a small number of training examples are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This result substantially improves the usefulness of goal recognition for intelligent assistance systems, as recognizing a goal early opens much more possibilities for supportive reactions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Furthermore, it is usually very expansive to obtain annotated training ex- amples for real-world application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Hence, being able to provide valuable results based on limited numbers of training examples is an important requirement for potential goal recognition methods that should be applied to real-world application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Nev- 21 ertheless, we still see some potential for improvements of the extended approach in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' One direction is to optimize the procedure that is used to combine the results of the two individual approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Another direction that we plan to investigate in future work is to use more complex tractable probabilistic models, like Sum-Product Networks (Poon & Domingos, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Acknowledgements The data used in this paper was obtained from kitchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='edu and the data collection was funded in part by the National Science Foundation [grant number EEEC-0540865].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' This work was supported by the German Federal Ministry of Education and Research (BMBF) [grant number 01lS18079C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Sampling Artificial Observation Sequences Algorithm 1 summarizes the sampling procedure that is used in this work to sample artificial observation sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Here, pplanAction is a parameter that specifies the goal-directedness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', the probability that the next action is taken from a precomputed optimal plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' When this is not the case, the next action is sampled out of the set of actions that are currently applicable in the current planning state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' These actions are randomly drawn from all actions that are applicable in a certain state of the planning domain, following two predefined probability models that model the probability that an interaction with a certain object O is observed given that we want to reach a goal G (P(O|G)), and the probability that a certain kind of action A is observed given that we want to reach goal G (P(A|G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The distribution P(A|G, S) that is used to sample an action at random is defined as follows: P(A = ai|g, s) ∝ � wi if ai applicable in s 0 otherwise (6) That is, only applicable actions can be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The weight wi of an action depends on the corresponding “action type” AT(ai) and the set of objects OB(ai) with that each action interacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' The underlying intuition is the observation that depending on the current goal, the agent will choose actions of different action types with higher probabilities than others and also interact with certain objects with higher probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Based on this intuition, we use randomly initialized weight score distributions W(AT|G) and W(OB|G) to determine the weights wi via wi = W(AT(ai)|g) � x∈OB(ai) W(x|g), (7) We initialize the parameters of P(A|G) and P(O|G) randomly and use the MetricFF planner to compute the initial plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Each time an action from the sampling model is selected, the optimal plan from the resulting state is recomputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 22 Algorithm 1 Sample artificial observation sequence for goal g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' sampledPlan ← () cState ← initialPlanningState optPlan ← computeOptimalPlan(cState, g) i = 0 while goal not reached do r ← random(0, 1) if r < pplanAction then sAction ← optPlan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='getAction(i) cState ← cState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='apply(sAction) i ← i + 1 else sAction ← sampleActionFromApplicableActions(cState) cState ← cState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content='apply(sAction) optPlan ← computeOptimalPlan(cState, g) i = 0 end if sampledPlan ← concat(sampledPlan, sAction) end while References Albrecht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', Zukerman, I.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Cog- nitive Systems Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Annual Conference on Advances in Cognitive Systems 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Conference date: 23-06-2016 Through 26-06-2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', Nie, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=', Sheng, Q.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' behaviour recognition during unscripted cooking tasks for health monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' In 2017 IEEE International Conference on Pervasive Computing and Communications Workshops (PerCom Workshops), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 18–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE5T4oBgHgl3EQfdA-T/content/2301.05608v1.pdf'} diff --git a/ZtFPT4oBgHgl3EQfuzVj/content/tmp_files/2301.13157v1.pdf.txt b/ZtFPT4oBgHgl3EQfuzVj/content/tmp_files/2301.13157v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..94645a7674daf946f37eaae58653438061dd272e --- /dev/null +++ b/ZtFPT4oBgHgl3EQfuzVj/content/tmp_files/2301.13157v1.pdf.txt @@ -0,0 +1,4126 @@ +arXiv:2301.13157v1 [math.AG] 30 Jan 2023 +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +HONGJIE YU +Abstract +Let X be a smooth, projective, and geometrically connected curve defined over a finite field Fq and S ⊆ X +a subset of closed points. Let X and S be their base changes to an algebraic closure of Fq. We study the +number of ℓ-adic local systems in rank 2 over X − S with prescribed tame local monodromies fixed by k- +fold iterated action of Frobenius endomorphism for every k ⩾ 1. We confirm some conjectures of Deligne +predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, +in all cases, our counting results are expressed in terms of the numbers of some Higgs bundles. +CONTENTS +1. +Introduction +1 +2. +Notations +7 +3. +Global and local Langlands correspondence for GL2 +8 +4. +Spectral side of the trace formula +11 +5. +Geometric side of the trace formula and Hitchin moduli spaces +29 +6. +Proof of the main theorems +42 +7. +Case that g = 0 +50 +References +53 +1. INTRODUCTION +Let X be a smooth, projective, and geometrically connected curve defined over a finite field Fq +of genus g. In the two pages article [Dr81] of Drinfeld, he counts the number of two-dimensional +geometrically irreducible ℓ-adic (in Qℓ-coefficients with ℓ ∤ q) representations of π1(X ⊗ Fq) which +can be extended to a representation of π1(X) (here we ignore the base point in the notation). These +numbers behave as if they were expressed by a Lefschetz fixed-point formula on an algebraic +variety over the finite field. Moreover, they’re independent of ℓ. +It is equivalent to consider ℓ-adic local systems (smooth Qℓ-sheaves). Although the Langlands +correspondence established by Drinfeld and Lafforgue shows the motivic nature of ℓ-adic local +systems counted by Drinfeld, their definition depends very much on ℓ. We don’t know how +to construct a moduli space of ℓ-adic local systems in a reasonable sense that can explain these +counting results. +Deligne has made some conjectures ([De15]) on counting ℓ-adic local systems over curves over +finite fields with prescribed local monodromies, i.e., with prescribed ramification types, to ex- +tend and understand Drinfeld’s result. We mention that Kontsevich [Ko09] has also made some +proposals toward understanding Drinfeld’s result. Some progress has been made since Deligne +raised his conjectures. Indeed, when the ramifications are split semisimple and in general position +(which ensures that an ℓ-adic local system is automatically irreducible), Arinkin has verified that +in these cases, similar results hold ([De15]). When the ramifications are unipotent with one Jor- +dan block, and there are at least two such ramifications, Deligne’s conjecture has been verified by +Deligne-Flicker [DF13]. The case in rank 2 with one unipotent ramification is verified by Flicker +[Fl15]. We have generalized Drinfeld’s result to a higher rank in [Yu18], and Arinkin’s result to +allow semisimple regular in general position but possibly non-split ramifications in [Yu21b]. +1 + +2 +HONGJIE YU +This article aims to verify some of Deligne’s predictions on counting of ℓ-adic local systems +in rank 2 for all possible tame ramifications. We show that the number is always related to the +number of Higgs bundles. The results show an interesting analogy with Simpson’s non-abelian +Hodge theory, especially when g = 0 and the ramifications are in general position and the par- +abolic weights of the parabolic Higgs bundles are also in general position (these two conditions +correspond in Simpson’s theory). We will discuss it in more detail at the end of the Introduction. +1.1. Main results. +1.1.1. +Let’s recall Deligne’s conjectures which will be treated in this article. We follow Deligne’s +presentation in [De15], but we restrict to the rank 2 cases. +Let X be a smooth, projective, and geometrically connected curve defined over a finite field Fq. +Let S ⊆ X be a subset of closed points. We fix an algebraic closure Fq of Fq. Let X := X ⊗ Fq +and S := S ⊗ Fq. For each point x ∈ S, let X∗ +x = Xx − {x} be a punctured disc in x (Xx is defined +as either the Henselization or the completion of X in x). We fix a rank 2 ℓ-adic local system (Qℓ- +smooth sheaf) Rx over X∗ +x. Let E2(R) be the set isomorphism classes of irreducible rank 2 ℓ-adic +local systems over X − S. Let Frob be the Frobenius endomorphism of X, i.e., the base change to +Fq of the morphism induced by the map a �→ aq on X. If +(1) +Frob∗(RFrob(x)) ∼= Rx +for every x ∈ S, then the pullback of Frob permutes E2(R). Let E2(R)Frob∗k be the set of fixed +elements of k-iterated action of Frob∗. +Deligne conjectured that if all Rx are tamely ramified, then there are q-Weil integers α and +integers mα such that +|E2(R)Frob∗k| = ∑ +α +mααk, +∀k ⩾ 1, +where |E2(R)Frob∗k| is the cardinality of subset of the fixed points by k-fold iterated action of Frob∗. +To formalize this property, let’s introduce some integral valued functions on N∗. We say that a +function k �→ h(k) form N∗ to Z is of Lefschetz type if there are q-Weil integers α and integers +mα ∈ Z such that +h(k) = ∑ +α +mααk. +Therefore, the conjecture is to prove that +k �→ |E2(R)Frob∗k| +is of Lefschetz type. A typical example is a function k �→ |V(Fqk)| for a variety V defined over Fq. +In particular, given a permutation σ on a finite set P, the function k �→ |Pσk| is a periodic function +of Lefschetz type. Note that not all integral valued periodic functions are Lefschetz type as the +integrality of mα is essential. +1.1.2. +The tame ´etale fundamental group of X∗ +x is topologically generated by one element. There- +fore, an isomorphism class of tame local system of rank 2 over X∗ +x corresponds bijectively to con- +jugacy classes in GL2(Qℓ). +The set S ⊆ X(Fq) is fixed by Frob and its orbits correspond bijectively to S. Following the +types of prescribed local monodromies, we can define a partition on S hence S into a disjoint +union of subsets: +S = Ss ∪ Su ∪ Scr, +where Rx has different eigenvalues for x ∈ Scr, Rx induces a scalar matrix in GL2(Qℓ) for x ∈ Ss +and Rx induces a quasi-unipotent conjugacy class with non-trivial Jordan block for x ∈ Su. As +each of these sets is stable under Frob, we have a partition +S = Ss ∪ Su ∪ Scr. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +3 +Let x1 ∈ Scr. Suppose x1 +Frob +−−→ x2 +Frob +−−→ · · · +Frob +−−→ xd+1 = x1 be the orbit containing x1 of the +Frobenius action (xi ̸= x1 for any 1 < i ⩽ d). There are two non-isomorphic rank 1 ℓ-adic local +systems L1 and L2 over X∗ +x1 such that +Rx1 ∼= L1 ⊕ L2. +The condition (1) implies that +Frob∗dL1 ∼= Li, +for i = 1 or i = 2. This allows us to further subdivide Scr so that +Scr = Sc ∪ Sr, +where Sr is the set of points such that i = 1 and Sc consists of those points such that i = 2. Again, +we deduce a partition +Scr = Sc ∪ Sr. +1.1.3. +Now we need to introduce some functions of Lefschetz type that are used to express the +final results. +Let R be a collection of tame local monodromies as above so that the condition (1) is satisfied. +Its eigenvalues for each x ∈ S define a couple of numbers (εx(1), εx(2)) ∈ Q× +ℓ which could be the +same. Let S = {x1, · · · , xr}. We define a set PR by +(2) +PR := {(εx1(i1), . . . , εxr(ir)) | +r +∏ +j=1 +εx(ij) = 1; ij ∈ {1, 2}, j = 1, 2, . . . , r}. +Let Frob∗ be a permutation on PR defined so that for any (εx)x∈S ∈ (Q× +ℓ )S, we have +Frob∗((εx)x∈S) = (ε′ +x)x∈S, +with +ε′ +x = εq +Frob(x), +∀x ∈ S. +The relation (1) tells us that it is a well-defined permutation, since εFrob(x)(1)q equals either εx(1) +or εx(2). We define a function cR : N∗ −→ Z by +cR(k) := |PFrob∗k +R +|, +the number of the fixed points of Frob∗k on PR. It is of Lefschetz type. Let σ be an involution on +PR that sends (εx(ix))x∈S to (εx(3 − ix))x∈S. Define +bR(k) := |Pσ=Frob∗k +R +| +as the cardinality of the fixed point set of the action of σ ◦ Frob∗k. We prove in Proposition 6.1 that +it is also of Lefschetz type when Scr ̸= ∅. +Now we introduce some functions of Lefschetz type coming from points count of Hitchin bun- +dles. Suppose that k ∈ N∗, and V ⊆ Su ⊗ Fqk. Let V = V ⊗Fqk Fq and +D = KX + +∑ +x∈V∪Scr +x +be a divisor over X where KX is a canonical divisor on X, i.e., a divisor associated to the canonical +line bundle Ω1 +X/Fq. A parabolic Hitchin bundle of rank 2 and degree 1 with parabolic structures +in V for the divisor D is a triple (E, ϕ, (Lx)x∈V) consisting of a vector bundle of rank 2 and degree +1 over X, a bundle morphism +ϕ : E → E ⊗ OX(D), + +4 +HONGJIE YU +and a family of one dimensional Fq-subspace Lx of Ex (x ∈ V), the fiber of E in x, such that +ϕx(Lx) = 0 and Im(ϕx) ⊆ Lx, for any x ∈ V. We say that (E, ϕ, (Lx)x∈V) is semistable if for any +subline bundle L of E satisfying ϕ(L) ⊆ L ⊗ OX(D), we have +deg(L) ⩽ deg E +2 +. +We denote M1 +2,V(D) the moduli space of these parabolic Hitchin bundles. It is a variety defined +over Fq (more details are given in Section 5). We show in Section 5.2 that it admits canonical +Fqk-structure (i.e. comeing from the base change of a variety defined over Fqk) whose Fqk-points +classify semistable parabolic Hitchin bundles over X ⊗ Fqk, which we denote by M1 +2,V(D). +For each v ∈ Scr, we choose ov ∈ κv[t] a unitary polynomial of degree 2 with coefficients in κv +(the residue field of the point v), so that ov is irreducible for v ∈ Sc and ov has distinct roots in +κv if v ∈ Sr. It defines a polynomial in κx[t], for every closed point x of X lying over v via the +isomorphism: +κv[t] ⊗Fq Fq ∼= ∏ +x�→v +Fq[t]. +We define M1 +2,V(o) as the closed sub-variety of M1 +2,V(D) over Fqk consisting of those parabolic +Hitchin bundles so that the characteristic polynomial of ϕx at x ∈ Scr is given by ox. We suppose +that the sum of roots of ox (x ∈ V) is zero. We refer to 5.3 for a more precise and detailed definition. +We define for each k ⩾ 1, +HiggR(k) = +∑ +V⊆Su⊗Fqk +(−1)|Su⊗Fqk−V|2|V|q−k(4g−3+|V|+|Scr|)|M1 +2,V(o)(Fqk)|. +We show in Theorem 6.2 that this is a function of Lefschetz type in k. +Let Pic0 +X be the Jacobian variety of X. We also define for every k ⩾ 1 +Pic(k) := |Pic0 +X(Fqk)|, +and +Pic(2)(k) := |S2Pic0 +X(Fqk)|, +where S2Pic0 +X := (Pic0 +X)2/S2 is the symmetric square of Pic0 +X. They’re surely also functions of +Lefschetz type in k ∈ N∗. +1.1.4. +The following theorem proves Deligne’s conjectures [De15, 2.15 (i)(iii)] when n = 2 and +ramifications are tame. +Theorem 1.1. Suppose that (1) is satisfied, so that Frob∗ acts on E2(R). Suppose that +(3) +∏ +x∈S +εx(1)εx(2) = 1, +otherwise E2(R) is empty. The function +k �→ |E2(R)Frob∗k| +is of Lefschetz type. +More precisely, we have the following explicit identities that express |E2(R)Frob∗k| following different +cases. +(1) Scr = Su = ∅. Then |E2(R)Frob∗k| equals +HiggR(k) − cR(k) +� +Pic(k)2(g − 1) + Pic(k) +� +. +(2) Scr = ∅, Su ̸= ∅. Then |E2(R)Frob∗k| equals +HiggR(k) − cR(k) +� +βSu(k)(−1)|Su|+1Pic(2)(k) + γSu(k)Pic(k) + ωSuPic(k)2 +� +. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +5 +(3) Scr ̸= ∅, Su = ∅. If |Sc| is even, then |E2(R)Frob∗k| equals +HiggR(k) − cR(k)(2g − 2 + |Scr|) +2 +Pic(k)2. +(4) Scr ̸= ∅, Su = ∅. If |Sc| is odd, then |E2(R)Frob∗k| equals +HiggR(k)− +� +cR(k)2g − 1 + |Scr| +2 +− cR(k) + bR(k) +2 +� +Pic(k)2 − bR(k)Pic(2)(k). +(5) Scr ̸= ∅, Su ̸= ∅. If |Sc| is even, then |E2(R)Frob∗k| equals +HiggR(k)− +�cR(k)αSu(k) +2 ++ (−1)|Su| bR(k)βSu(k) +2 +� +Pic(k)2 + (−1)|Su|bR(k)βSu(k)Pic(2)(k). +(6) Scr ̸= ∅, Su ̸= ∅. If |Sc| is odd, then |E2(R)Frob∗k| equals +HiggR(k)− +�cR(k)αSu(k) +2 +− (−1)|Su| bR(k)βSu(k) +2 +� +Pic(k)2 − (−1)|Su|bR(k)βSu(k)Pic(2)(k). +In the above expressions αSu, βSu, γSu and ωSu are periodic functions of Lefschetz type (see Proposition +6.1 for their explicit expressions). When Scr is non-empty, cR/2 + bR/2 and cRαSu/2 ± bRβSu/2 are of +Lefschetz type. If |Sr| is odd, then cR/2 is of Lefschetz type and bR is constantly zero. +Remark 1.2. +(1) The necessity of (3) for E2(R) to be non-empty is explained in [De15, 2.10]. It +can be checked by passing to characteristic 0 and then passing to C, where one can use an explicit +presentation of the topological fundamental group of a punctured Riemann surface (see the proof +of Corollary 7.7 of [DF13]). Using Langlands correspondence, one can prove that E2(R) has no +Frob∗k fixed points for k ⩾ 1 if (3) does not hold as a corollary. +(2) The ramifications Rx for x ∈ Ss only affects cR(k) and bR(k) and is not involved in any other +term. +(3) In the case of general position, i.e., when the set PR = ∅, we have +|E2(R)Frob∗k| = HiggR(k), +∀k ⩾ 1. +Note that the extra terms (those different from HiggR(k)) appear only if PR ̸= ∅. This phe- +nomenon is possibly related to the singularity of the moduli space of (S-equivalent classes of) +semistable parabolic Higgs bundles in the non-coprime cases or the cases not in general position +in the terminology of this article. +The following corollary confirms Deligne’s conjectures [De15, 6.3]. +Corollary 1.3. The cardinality |E2(R)Frob∗k| is divisible by Pic(k) and +k �→ |E2(R)Frob∗k|/Pic(k) +is still a function of Lefschetz type. +Although we deal only with the tame local monodromies, the method of this article allows us +to treat some wild ramified cases as well. For example, we can allow some places to give the +so-called simple supercuspidal representation on the automorphic side. There could be a similar +result involving wild Hitchin bundles. In a private note by Zhiwei Yun, he has a simple geometric +method to deal with some cases with wild ramifications. +1.2. When g = 0. + +6 +HONGJIE YU +1.2.1. +Suppose that g = 0, i.e., the curve X is P1, and Sc = ∅. We will present an analogy with +Simpson’s non-abelian Hodge theory. It does not hold in a more general case which we hope to +understand in the spirit of conjectures [De15, 2.18, 2.21]. We are interested in the case that R is in +general position, i.e. when PR is empty. We assume that Fq ̸= F2 to ensure that such cases are +possible. +Let +R = Sr ∪ Su, +and +D = KX + ∑ +v∈R +v. +We view D also as a line bundle over X. Let ξ = (ξx)x∈R ∈ (Q2)R such that ξx,1 ⩾ ξx,2 ⩾ ξx,1 − 1 +and ξx = ξy for any x, y lying over the same closed point v ∈ R. These vectors serve as parabolic +weights (stability parameters). Let (E, ϕ, (Lx)x∈R) be a parabolic Higgs bundle (we call a parabolic +Hitchin bundle a Higgs bundle if D = KX + ∑v v and parabolic structures are imposed in R) over +X. Let L be a sub-line bundle of E, we define the parabolic degree p-deg(L) by +p-deg(L) := deg(L) + ∑ +x∈R +� +ξx,1, if Lx = Lx; +ξx,2, if Lx ̸= Lx. +We say that (E, ϕ, (Lx)x∈R) is ξ-semistable if for any sub-line bundle L of E satisfying ϕ(L) ⊆ +L ⊗ OX(D), we have +p-deg(L) ⩽ deg E + ∑x∈R(ξx,1 + ξx,2) +2 +. +Note that if +deg(E) + ∑ +x∈R +±(ξx,1 − ξx,2) /∈ 2Z, +then the equality can never be achieved. We say that such cases are in general position. +Choose ξ as above and suppose that it is in general position. The moduli space of ξ-semistable +parabolic Higgs bundles of rank 2 and of degree e which are semistable with parabolic weights +(ξx)x∈R over X has a canonical Fq-structure (see Section 5.2). We denote the moduli space by +Me,ξ +2,R = Me,ξ +2,R(D). We show in Theorem 5.8 that |Me,ξ +2,R(Fq)| is independent of the choice of the +parabolic weights as long as ξ is in general position. The space Me,ξ +2,R has a Gm-action via dilation +of the Higgs field. Let grMe,ξ +2,R := (Me,ξ +2,R)Gm, and grMe,ξ +2,R(Su) be its open subvariety consisting of +those parabolic Higgs bundles whose Higgs field does not vanish at x ∈ Su. +Theorem 1.4. Suppose that g = 0 and Sc = ∅. Suppose that (e, ξ) is in general position and that +ξx,1 = ξx,2 +for x ∈ Su. Suppose that Fq ̸= F2 and R is in general position in the sense that PR = ∅. Suppose that +∏ +x∈S +εx(1)εx(2) = 1. +We have +|grMe,ξ +2,R(Su)(Fqk)| = |E2(R)Frob∗k|. +1.3. +Given a compact Riemann surface Σ and a finite set of points R ⊆ Σ, Simpson has estab- +lished a correspondence between semistable C-local systems over Σ − R of degree 0 and semistable +quasi-parabolic Higgs bundles over Σ with parabolic structures in R of parabolic degree 0 (we re- +fer to Simpson’s original article [Si90] for more details). On the local system side, Simpson defines +residual data for each x ∈ R using the local monodromy and stability weight at the puncture x. +Similarly, Simpson defines residual data for each x ∈ R using the Higgs field and the parabolic +weight in the Higgs bundle side. His correspondence preserves the nilpotent part of the resid- +ual data and permutes the stability weights and eigenvalues of the residual datum. Theorem 1.4 + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +7 +presents an analogy with Simpson’s theory if we choose the stability weights of the local systems +to be trivial and we choose (e, ξ) in accordance with Simpson’s correspondence (cf. the diagram +in [Si90, p.720]). An interesting phenomenon is that R being in general position corresponds to +that (e, ξ) being in general position under Simpson’s correspondence. +The dominant term in Theorem 1.1 when k varies is (q4g−3+|Su|+|Scr|)k. It is half of the dimen- +sion of the moduli space of parabolic Higgs bundles of the relevant complex analogy in Simpson’s +theory. This may be related to the motivic nature of ℓ-adic local systems over a curve over Fq. +We can not expect a naive generalization of Theorem 1.4 to the cases g > 0. However, we +expect it to be a special case of (a possible modification of) Deligne’s conjecture in [De15, 2.21]. +Indeed, suppose that all ramifications are split regular semisimple (for n = 2, it is the case that +Ss = Sc = Su = ∅), in the cases of in general position, we have ([Yu21b, Th. 1.4] for any rank) +|En(R)Frob∗k| = ∑ +i +(−1)iTr(V∗k|Hi +c((Me,ξ +n,Sr)Fq, Qℓ)), +for any endomorphism V∗ which is conjugate to q− 1 +2 (n2(g−1)+|Sr|)F∗ +q . We can expect to generalize +it to the cases where ramifications are only supposed to be semisimple but remain in general +position. The more demanding question is to generalize it to allow non-trivial quasi-unipotent +ramifications or even cases not in general positions. Now we may ask if V∗ is induced from a +morphism of (Me,ξ +n,Sr)Fq. This does not seem to be the case if we consider only algebraic varieties +over Fq. We likely have to consider a lifting of the curve to characteristic 0 and consider p-adic +geometry which I’m not competent to comment on. Instead, we refer the reader to Deligne’s +course at IHES [De13]. +In this article, we only need semistable parabolic Higgs bundles with parabolic weights in gen- +eral position. It is not necessary to do so. However, with our method, it is more natural to consider +the algebraic stack version of the moduli of semistable parabolic Higgs bundles when the para- +bolic weights are not in general position, and we should expect a more complicated expression +for the point counting problem in this case. +Acknowledgement. This work continues the project that started from my thesis; I thank Pierre- +Henri Chaudouard for giving this project to me. I thank Kang Zuo for the fruitful discussions. +The work is finished during my stay at IST Austria and Weizmann Institute of Science. I thank +both institutes for providing me with excellent work conditions. Part of the work is finished with +the support of the BSF grant 2019274. +2. NOTATIONS +We gather some notations that will be used throughout the article. Other notations will be +defined where they appear. +•F, |X|, Fv, Ov, ℘v, κv, qv, A, O. Let F = Fq(X) be the global function field of the curve X. Let +|X| be the set of closed points of X, which is identified with the set of places of F. For every +v ∈ |X|, let Fv be the local field in v, Ov the ring of integers in Fv and κv the residue field of Ov. Let +℘v be the maximal ideal in Ov, and we choose a uniformizer ̟v. Suppose that κv has cardinality +qv, therefore κv ∼= Fqv. Let A be the ring of ad`eles of F and O be the sub-ring of integral ad`eles. +•G, B, N, T, B, N. If not specified otherwise, we use G for GL2. Let B be the Borel subgroup of G +consisting of upper triangular matrices and T be the torus consisting of diagonal matrices. Let N +be the unipotent radical of B, i.e., the group of upper triangular matrices with 1 on the diagonal. +Let B be the Borel subgroup that is opposite to B, i.e., consisting of lower triangular matrices, and +N be the unipotent radical of B. +•g, b, n, t. Let g, b, n, and t be respectively the Lie algebra of G, B, N, and T. +•Gv, Bv, Kv, Iv. Given a variety V defined over Fq, we will use Vv to denote V(Fv) for any +places v ∈ |X|. This notation applies in particular to Gv, Bv. We will denote G(Ov) by Kv. Let Iv +be the Iwahori subgroup consisting of matrix in Kv whose reduction modulo ℘v lies in B(κv). + +8 +HONGJIE YU +•G(A)e. For any e ∈ Z, let +G(A)e = {x ∈ G(A)| deg det x = e}. +• We fix Haar measures on G(A), N(A) so that G(O) and N(F)\N(A) (with counting mea- +sure on N(F)) have volume 1. The local Haar measures on Gv, Bv and Nv are defined so that +respectively the volumes of Kv, B(Ov) and N(Ov) are 1. +•Ev, ϕv. Given a vector bundle E over X, and a place v ∈ |X| identified as a κv-point of X, we +use Ev to denote the fiber over v which is a κv-vector scheme. Suppose ϕ : E −→ F be a bundle +morphism over X, then it induces a κv-linear map ϕv : Ev −→ Fv. +3. GLOBAL AND LOCAL LANGLANDS CORRESPONDENCE FOR GL2 +We are going to reduce the calculation of the cardinality of E2(R)Frob∗k to a question of counting +certain automorphic representations of GL2 with the help of global Langlands correspondence in +rank 2 established by Drinfeld. Note that +X = (X ⊗Fq Fqk) ⊗Fqk ⊗Fq, +and the Frobenius endomorphism of X deduced from X ⊗Fq Fqk is Frobk. Therefore, we can do +the calculation for k = 1 and apply the results to the curves X ⊗Fq Fqk over Fqk (k ⩾ 1) later. +3.1. Galois representations. It has been explained by Deligne [De15, 2.1-2.9] how to pass to the +automorphic side, and the reader is invited there for more details. This section aims to give precise +information on the ramifications of automorphic representations determined by the Frobenius ac- +tion on R. The data on the local monodromies are carried over to the automorphic side, described +by local Langlands correspondence. +We continue to use notations in the introduction. Let v ∈ S and x ∈ S that lies over v. +We fix an algebraic closure F of F. Then η := Spec(F) is a geometric point lying over the +generic point of X. Let X(x) be the Henselization of X in x and X∗ +(x) = X(x) − {x}. If we choose +an embedding of Fq(X∗ +(x)) in F, then the ´etale fundamental group π1(X∗ +x, η) (we will ignore the +choice of a base point in the notation in what follows) is canonically isomorphic to the inertial +group Ix = Gal(F|Fq(X∗ +(x))). An ℓ-adic local system over X − S (resp. X∗ +(x)) is equivalent to an +ℓ-adic representation of π1(X − S, η) (resp. Ix). Let It +x = π1(X∗ +(x), η)t be the tame fundamental +group of X∗ +(x) which is the prime-to-p quotient of π1(X∗ +(x), η). A tame ℓ-adic local system of X∗ +(x) +is equivalent to an ℓ-adic representation of It +x. +For an algebraic closed field k, let +�Zp′(1)(k) := lim µn(k). +We denote �Zp′(1) for �Zp′(1)(Fq). Let κx be the residue field of the point x, we have a canonical +isomorphism +It +x ∼= �Zp′(1)(κx). +Choose an embedding κv ֒→ Fq, we deduce from +κv ⊗Fq Fq ∼= ∏ +x�→v +Fq, +isomorphisms κx ∼= Fq. Which gives us isomorphisms +It +x ∼= �Zp′(1). +The morphism Frob : X∗ +x −→ X∗ +Frob(x), induces an isomorphism between tame fundamental +groups. It is the multiplication by q map on �Zp′(1) via the above isomorphisms. +To make the set E2(R) Frob∗-stable, the ℓ-adic local systems (Rx)x�→v have to satisfy the com- +patibility condition +(4) +Frob∗(RFrob(x)) ∼= Rx. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +9 +Let Iv and Dv be, respectively, the inertial subgroup and decomposition group of F at v. The +condition (4) implies that (Rx)x�→v come from a representation ρv of Dv. By Grothendick’s local +monodromy theorem, ρv|Iv is quasi-unipotent in the sense that it becomes unipotent on an open +subgroup. We will use Rv to denote ρv|Iv. +Let WF be the Weil group of F. Then π1(X − S, η) is a quotient of the degree 0 part of WF. +Let G2(F) be the set of isomorphism classes of ℓ-adic representation of WF. We call two ℓ-adic +representations σ1 and σ2 in G2(F) inertially equivalent: σ1 ∼ σ2 if there is a character λ : WF +deg +−−→ +Z −→ Q× +ℓ such that σ1 ∼= σ2 ⊗ λ. +Proposition 3.1. The set +E2(R)Frob∗k +is in bijection with the subset of inertially equivalent classes in G2(F ⊗ Fqk)/ ∼ consisting of σ such that +(5) +σ ⊗ λ ∼= σ =⇒ λ = 1, +σ|Iv is trivial for v /∈ S and +σ|Iv ∼= Rv +for v ∈ S. +Proof. It has been explained in [De15, Section 2], see also [Yu18, 2.1.3] for condition (5). +□ +Let Wv be the local Weil group at v. We choose an isomorphism ι : Qℓ +∼ +−→ C. Recall that +the local Langlands correspondence is a canonical bijection between the set of smooth irreducible +C-representations of Gv and the set of rank 2, Frobenius semisimple ℓ-adic (continuous) represen- +tations of Wv. +For any v ∈ S, let IrrR(Gv) be the set of irreducible representations of Gv whose associated +ℓ-adic representation of the local Weil group Wv under local Langlands correspondence extends +Rv. For a place v /∈ S, we define IrrR(Gv) to be the set of unramified representations of Gv, +i.e., those representations whose associated ℓ-adic representation of Wv under local Langlands +correspondence is trivial when restricting to Iv. +We have the following theorem that characterizes the set IrrR(Gv) purely by their representa- +tion theoretic structures. +Theorem 3.2. Let Rv be tame. We have one of the following cases. +(r) We say that Rv is (split) regular if +Rv ∼= χ1 ⊕ χ2, +is the direct sum of two distinct characters χ1, χ2 of Iv. Each χi (i = 1, 2) has exponent qv − 1 and +can be factored as Iv −→ κ× +v +χ′ +i +−→ Q× +ℓ . +In this case for any irreducible smooth representation π of Gv, we have π ∈ IrrR(Gv) if and +only if HomIv(χv, π) ∼= HomKv(ρv, π) ̸= 0, where +χv : Iv −→ B(κv) + +a +b +0 +d + +�→ι(χ′ +1(a)χ′ +2(d)) +−−−−−−−−−−−−−−→ C×, +and ρv is the induced representation of χv to Kv. Moreover, dim HomIv(χv, π) = 1 for any +π ∈ IrrR(Gv). +(c) We say that Rv is cuspidal (or anisotropic regular), if +Rv ∼= χ1 ⊕ χ2, +is the direct sum of two distinct characters of Iv such that χqv +1 += χ2 (necessarily we also have +χqv +2 = χ1), where χi can be factored as Iv −→ F× +q2v +χ′ +i +−→ Q× +ℓ . + +10 +HONGJIE YU +In this case for any irreducible smooth representation π of Gv, we have π ∈ IrrR(Gv) if and +only if dim HomKv(ρv, π) ̸= 1, where ρv is the irreducible representation of Kv inflated from the +Deligne-Lusztig induced representation +−ι(RG(κv) +U(κv)χ′ +1), +with U being any non-split maximal subtorus of G defined over κv so that we have U(κv) ∼= F× +q2v. +Moreover, for any π ∈ IrrR(Gv), we have dim HomKv(ρv, π) = 1 and π is supercuspidal. +(s) We say that Rv is scalar if +Rv ∼= χ⊕2, +where χ is a character of Iv that can be factored as Iv −→ κ× +v +χ′ +−→ Q× +ℓ . +In this case for any irreducible smooth representation π of Gv, we have π ∈ IrrR(Gv) if and +only if HomKv(θv, π) ̸= 0 where +θv : Kv +det +−→ O× +v −→ κ× +v +ι◦χ′ +−−→ C×. +Moreover, we have dim HomKv(θv, π) = 1 for any π ∈ IrrR(Gv). And the set IrrR(Gv) consists +of the one dimensional representation η of Gv which extends θv and the twist by η of irreducible +unramified representations of Gv. +(u) We say that Rv is principal quasi-unipotent if +Rv ∼= χ ⊗ ν, +is a quasi-unipotent with one principal Jordan block: χ is a character of Iv which can be factored as +Iv −→ κ× +v +χ′ +−→ Q× +ℓ and ν is a non-trivial unipotent representation of Iv. +In this case π ∈ IrrR(Gv) if and only if π = St ⊗ λ for some character λ of the form Gv +det +−→ +F× +v +λ′ +−→ C× so that θv = λ′|O× +v inflates ι ◦ χ′. Here St is the Steinberg representation of Gv, i.e., +the unique irreducible quotient of the parabolic induction of the trivial representation of Bv. +Proof. The cases (r) and (c) are explained in [Yu21b, 6.2, 5.2.4]. The case (s) is deduced from the +unramified case where dim πKv +v += 1. In fact, it is enough to tensor Rv by a rank 1 local system to +make it trivial on Iv, which corresponds in the automorphic side to twist the representation by a +character. Similarly, we can tensor a character to make the case (u) into the unipotent case, which +corresponds to the Steinberg representation under Langlands correspondence. +□ +Let π = ⊗′πv be a cuspidal automorphic representation of G(A). We will say that πv has the +correct ramification type (for our counting problem) if πv ∈ IrrR(Gv). +3.2. Automorphic representations. Let Ccusp(G(A)) be the space of cuspidal automorphic forms. +Recall that a cuspidal automorphic form is a complex-valued function ϕ over G(F)\G(A) which +generates a finite-dimensional vector space under G(O)-right translation and ZG(A) translation, +such that the following cuspidality condition is satisfied +� +N(F)\N(A) ϕ(nx)dn, +∀x ∈ G(A). +Note that the above integration is a finite sum because of G(O)-finiteness. The right translation by +G(A) makes Ccusp(G(A)) into a G(A)-representation which is semisimple. The multiplicity one +theorem of Jacquet & Langlands and Piatetski-Shapiro says that Ccusp(G(A)) is multiplicity free. +Its irreducible summands are called cuspidal automorphic representations. The decomposition +of Flath decomposes a cuspidal automorphic representation π as a restricted tensor product π = +⊗′πv for representations πv of Gv which are called local components of π. +Let A2(F) be the set of isomorphic classes of cuspidal automorphic representations of G(A). +We call two cuspidal automorphic representations π1 and π2 are inertially equivalent π1 ∼ π2 if +there are is a character λ : G(A) +deg ◦ det +−−−−−→ Z −→ C× such that π1 ∼= π2 ⊗ λ. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +11 +Theorem 3.3. The set E2(R)Frob∗ is in bijection with the subset of A2(F)/ ∼ consisting of inertial equiv- +alent classes of cuspidal automorphic representations π such that for any character λ : G(A) +deg ◦ det +−−−−−→ +Z −→ C×, +π ⊗ λ ∼= π =⇒ λ = 1, +and πv ∈ IrrR(Gv) for all v ∈ S. +Proof. Applying global Langlands correspondence and the fact that it is compatible with local +Langlands correspondence, this is a corollary of Proposition 3.1 and Theorem 3.2. +□ +4. SPECTRAL SIDE OF THE TRACE FORMULA +We will use equality provided by the noninvariant Arthur-Selberg trace formula (a similar but +slightly different formula is obtained first by Jacquet-Langlands for GL2) established by Lafforgue +[Laf97]. Indeed the noninvariant Arthur-Selberg trace formula over a function field is an equality +for each e ∈ Z between two distributions on C∞ +c (G(A)): +Je +geom( f) = Je +spec( f). +We will construct a function f ∈ C∞ +c (G(A)) using Theorem 3.2 and do explicit calculation for +J1spec( f). The result is summarized by Theorem 4.3 from which we will see that it is always a sum +of |E2(R)Frob∗| and an explicit error term. In a later section, we will use a geometric method to +study Je +geom( f), which gives a relation with the number of Fq-points of Hitchin moduli spaces. +4.1. Explicit spectral decomposition of J1spec( f). Let M be either T or G. Let XM be the group of +characters of M(A) to C× which is trivial on M(A)0. Let XG +M ⊆ XM be the subgroup consisting +of those characters which are trivial on ZG(A), i.e., +XG +M = Hom(M(A)0\M(A)/ZG(A), C×). +We have +XG +G = {1, ǫ}, +where ǫ(x) = (−1)deg(det x) is the sign character of G(A). We identify XG +G with {±1} ⊆ C×. We +also have an identification +XG +T ∼= C×, +where for any λ ∈ C× we associate a character +λ( +� +a +0 +0 +b +� +) = λ− deg(a)+deg(b). +We will use this isomorphism in our calculations. Let ImXG +T be the subgroup of XG +T consisting of +unitary characters. Therefore, it is formed by elements λ ∈ C× of absolute value 1. We endow a +Haar measure on XG +G and ImXG +T so that the total volume is 1. +Following Jacquet-Langlands, we have the spectral decomposition: +L2(G(F)\G(A)) ∼= L2 +cusp ⊕ L2 +res ⊕ L2 +cont, +where L2 +cusp ⊕ L2 +res is the largest semisimple subspace, L2 +cont is its orthogonal complement, and +L2cusp is the completion of the space of cuspidal automorphic forms. The residual spectrum is +decomposed as +L2 +res ∼= � +� +χ +χ, +where χ are compositions of Hecke characters with the determinant morphism. The sum here is +the Hilbert direct sum. For continuous spectrum L2 +cont, we have a decomposition: +L2 +cont = � +� +ψ +L2 +[B,ψ], + +12 +HONGJIE YU +where the sum is taken over the set of inertial equivalent classes of pairs (B, ψ) with ψ being a +Hecke character of T(A) ∼= (A×)2. The explicit construction of L2 +[B,ψ] is given by the theory of +Eisenstein series, which we do not need in this work. +We fix an id`ele a ∈ A× of degree 1, viewed as a scalar matrix. Let f ∈ C∞ +c (G(A)), it acts +via the regular representation on L2(G(F)\G(A)/aZ), equivalently by convolution from right +by ˘f := (x �→ f(x−1)), which is an integral operator. We denote this action by the function +R( f). Therefore its trace, if it exists, will be the integration of the kernel function on the diago- +nal. However, the trace of f on the whole space L2(G(F)\G(A)/aZ) does not exist in general. +Arthur defines a truncated kernel function so that its integration on the diagonal will contain the +information Tr( f|L2 +cusp) and expresses this integration in two ways: a geometric expansion and +a spectral expansion so that we have an identity. The spectral expansion contains a piece that +gives the most interesting part Tr( f|L2 +cusp), and we usually hope to obtain information from the +geometric expansion and an understanding of the error terms in the spectral expansion. +Over a function field, we have a decomposition +G(A) = ∐ G(A)e, +and G(F)\G(A)e has finite volume. The two different ways to express Arthur’s truncated integral +over the diagonal in G(F)\G(A)e × G(F)\G(A)e give an identity: +Je +geom( f) = Je +spec( f). +It is slightly simpler to consider an odd integer e or simply that e = 1. The following result is a +special case of the formula obtained by L. Lafforgue. +Let A1 be the set of Hecke characters of F×\A×/aZ. Let +Acont := A1 × A1/S2, +where S2 acts by permutation. An element [(ψ1, ψ2)] ∈ Acont is called regular if ψ1 ̸= ψ2 and is +called non-regular otherwise. Let +Ares +be the inertial equivalent classes of 1-dimensional representations of G(A) trivial on aZG(F). Let +A0 +be the set inertial equivalent classes of cuspidal automorphic representations of G(A) whose +central characters are trivial on aZ. +Let AB,ψ be the space of complex valued functions ϕ over G(A) which satisfy that for any +k ∈ G(O), there is a constant ck ∈ C so that for any n ∈ N(A), t ∈ T(A) we have +ϕ(ntk) = ckρB(t)ψ(t), +where ρB( +� +a +0 +0 +c +� +) = |a| +1 +2 +|c| +1 +2 . Equivalently, it is the space of those ϕ such that for any x ∈ G(A), +n ∈ N(A) and t ∈ T(A) we have ϕ(ntx) = ρB(t)ψ(t)ϕ(x). Let w be the non-trivial element in the +Weyl group of (G, T) and λ ∈ XG +T . We have the intertwining operator AB,ψ −→ AB,w(ψ) defined +by analytic continuation of the integral below which converges when |Reλ| >> 0, +(6) +(M(w, λ)ϕ)(x) := λ(x) +� +N(A) ϕ(w−1nx)λ(w−1nx)dn, +where we view λ ∈ XG +T as a function over G(A) using Iwasawa decomposition, i.e., if x = ntk +with n ∈ N(A), t ∈ T(A) and k ∈ K, we define λ(x) := λ(t). +Theorem 4.1 (Arthur, Lafforgue). The spectral expansion is the identity +J1 +spec( f) = ∑ +[π]∈A0 +Jπ( f) + +∑ +[χ]∈Ares +Jχ( f) + +∑ +[ψ]∈Acont +Jψ( f), + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +13 +where each sum is taken over a set of representatives, and the terms are defined as follows. We denote the +three sums by respectively J1 +cusp( f), J1 +res( f) and J1 +cont( f). +For π ∈ A0, if π ⊗ ǫ ∼= π, then +Jπ( f) = 1 +2(Tr(R( f)|π) − Tr(R( f) ◦ ǫ|π)), +and if π ⊗ ǫ ̸∼= π, then +Jπ( f) = Tr(R( f)|π). +For χ ∈ Ares, we have +Jχ( f) = Tr(R( f)|χ). +For any λ ∈ XG +T , let R( f, λ) be the twisted action on AB,ψ: +R( f, λ)ϕ = (R( f)(ϕλ))λ−1, +where we view λ ∈ XG +T as a function over G(A) by Iwasawa decomposition: λ(x) = λ(t) if x = ntk for +n ∈ N(A), t ∈ T(A) and k ∈ G(O). For ψ ∈ Acont, if ψ is regular, then +Jψ( f) = +� +ImXG +T +lim +µ−→1 TrAB,ψ((− +1 +µ−1 − µM(w, λ)−1 ◦ M(w, λ/µ) + +1 +µ−1 − µ) ◦ R( f, λ))dλ, +and if ψ is not-regular, then +(7) +Jψ( f) = 1 +2 +� +ImXG +T +lim +µ−→1 TrAB,ψ((− +1 +µ−1 − µM(w, λ)−1 ◦ M(w, λ/µ) + +1 +µ−1 − µ) ◦ R( f, λ))dλ ++ 1 +8 +∑ +λG∈{±1} +∑ +λw∈ImXG +T +λ2w=λ−1 +G +λGTrAB,ψ(M(w, w−1(λw)) ◦ R( f, λ)). +Note that if f is supported in G(O), then R( f, λ) = R( f). +Proof. The theorem is established by L. Lafforgue in [Laf97], and we refer the reader to [Yu18, Th. +5.2.2, Co. 5.2.3]. For the group G = GL2, we can make the result more explicit. In [Yu18, 5.2.1] +we have calculate explicitly the functions � +11 +G, � +11 +B and � +11 +B on XG +T , which appears in the spectral +expansion of the trace formula. They’re given by the following formula: +� +11 +G(1) = 1 +and +� +11 +G(ǫ) = −1, +i.e., +� +11 +G(λ) = λ, +∀λ ∈ XG +G, +and +� +11 +B(λ) = − +1 +λ − λ−1 , +� +11 +B(λ) = − +1 +λ−1 − λ. +□ +4.2. Summary of main results of this section. The following proposition provides a specific +function f ∈ C∞ +c (G(A)) and we will be interested in the computation of J1spec( f). +Proposition 4.2. We use notations of Theorem 3.2. For each v ∈ S, we define the following functions +following the ramification type of Rv. +(r) The function f (r) +v +∈ C∞ +c (Gv) is defined by +f (r) +v (x) = +� +0, +x /∈ Kv; +Tr(ρv(x−1)), +x ∈ Kv. +where x is the image of x in G(κv) under the projection Kv −→ G(κv). + +14 +HONGJIE YU +(c) The function f (c) +v +∈ C∞ +c (Gv) is defined by +f (r) +v (x) = +� +0, +x /∈ Kv; +Tr(ρv(x−1)), +x ∈ Kv. +(s) The function f (s) +v +∈ C∞ +c (Gv) is supported in Kv such that for any x ∈ Kv, we have +f (s) +v (x) = θv(det x−1); +(u) The function f (u) +v +∈ C∞ +c (Gv) is defined by +f (u) +v += +� +1 +vol(Iv) +1Iv(x) − 2 1Kv(x) +� +θv(det x−1). +Let +f = ⊗ f (?) +v +∈ C∞ +c (G(A)), +where f (?) +v +is the function defined above if v ∈ S(?) for ? = r, c, s, u and f (?) +v += +1Kv if v /∈ S. Then for any +cuspidal automorphic representation π of G(A) which is not one dimensional, the condition +Tr( f|π) ̸= 0 +holds if and only if π has correct ramification type. If it is the case, we have +Tr( f|π) = 1. +Proof. Suppose π = ⊗′πv is a cuspidal automorphic representation, we have +Tr( f|π) = ∏ +v +Tr( fv|πv). +The statement is, therefore, of local nature. +For v /∈ S, the function +1Kv acts as a projection to Kv-fixed part of πv, which is non-zero if and +only if πv is unramified by definition. If this is the case, πKv +v +is an 1 dimensional vector space and +the trace of +1Kv is one. For v ∈ Ss, it is similar. In fact, suppose that πv = π′v ⊗ η with π′v being +unramified. Then +Tr( f (s) +v |πv) = Tr( +1Kv|π′ +v). +The trace is 1 or 0 depending on whether or not πv has the correct ramification. +For v ∈ Sc and v ∈ Sr, this result has been proved in [Yu21b, 5.2.4]. +Finally, we consider the case v ∈ Su. As above, up to a twist, we are reduced to the case that χ′ +is trivial. Note that if +Tr( f (u) +v +|πv) ̸= 0, +then πv contains a non-zero fixed vector under Iv. By a result of Casselman [Lau96, Th. 7.4.4], πv +must be an irreducible subrepresentation of a parabolic induction IndGv +Bv θv for a character θv of Tv +that is trivial on T(Ov). If the representation IndGv +Bv θv is irreducible, then it is unramified. Hence +its Iv-fixed subspace has dimension 2. We deduce that +Tr( f (u) +v +|πv) = 0. +If IndGv +Bv θv is not irreducible, then it is of length 2 whose irreducible quotients are a 1-dimensional +representation and an unramified twist of the Steinberg representation. Since π is cuspidal, πv +cannot be 1-dimensional, therefore πv is an unramified twist of the Steinberg representation +whose Iv-fixed subspace is of 1 dimensional and Kv-fixed subspace is 0. This completes the +proof. +□ +The primary purpose of this section is to prove the following theorem which calculates J1spec( f) +of the Theorem 4.1. The result is deduced from Lemma 4.7, Corollary 4.9, Proposition 4.10, and +Proposition 4.11. Notations are those of the introduction. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +15 +Theorem 4.3. For a finite set of places V of F, we define deg V := ∑v∈V deg v = |V|. Let Su,even be the +subset of Su consisting of places v such that deg v is even. +The expression J1spec( f) equals the following numbers depending on the cases: +(1) Sc ̸= ∅, and Su,even ̸= ∅. We have +J1 +spec( f) = |E2(R)Frob∗|. +(2) Sc ̸= ∅, Su,even = ∅ but Su ̸= ∅. We have +J1 +spec( f) = |E2(R)Frob∗| + (−1)|Su|+1bR(1)2|Su|−2(−1)deg ScPic(2). +(3) Sc ̸= ∅, and Su = ∅. We have +J1 +spec( f) = |E2(R)Frob∗| + bR(1) +4 +(1 − (−1)deg Sc)Pic(2). +(4) Sc = ∅, Sr ̸= ∅, and Su = ∅. We have +J1 +spec( f) = |E2(R)Frob∗| + 1 +2cR(1)Pic(1)2(2g − 2 + deg Sr) +(5) Scr = Su = ∅. We have +J1 +spec( f) = |E2(R)Frob∗| + cR(1)Pic(1)2(g − 1) + cR(1)Pic(1) +(6) Scr = ∅, Su = {v} and 2 | deg v. Then J1 +spec( f) equals +|E2(R)Frob∗| − cR(1)Pic(1) + deg v +2 +cR(1)Pic(1)2 +(7) Scr = ∅, Su = {v} and 2 ∤ deg v. Then J1 +spec( f) equals +|E2(R)Frob∗| + cR(1) +2 +Pic(2)−cR(1)Pic(1) + cR(1)deg v +2 +Pic(1)2. +(8) Scr = ∅, |Su| ⩾ 2, and Su,even ̸= ∅. Then J1 +spec( f) equals +|E2(R)Frob∗| + cR(1)(−1)|Su|Pic(1). +(9) Scr = ∅, |Su| ⩾ 2, and Su,even = ∅. Then J1 +spec( f) equals +|E2(R)Frob∗| + cR(1)(−1)|Su|Pic(1) + cR(1)(−1)|Su|+12|Su|−2Pic(2) +(10) Sc = ∅, Sr ̸= ∅, Su = {v}, and 2 | deg v. We have +J1 +spec( f) = |E2(R)Frob∗| + 1 +2cR(1)Pic(1)2 deg v. +(11) Sc = ∅, Sr ̸= ∅, Su = {v}, and 2 ∤ deg v, +J1 +spec( f) = |E2(R)Frob∗| + 1 +2cR(1)Pic(1)2 deg v + bR(1) +2 +Pic(2). +(12) Sc = ∅, Sr ̸= ∅, |Su| ⩾ 2, and Su,even ̸= ∅. We have +J1 +spec( f) = |E2(R)Frob∗|. +(13) Sc = ∅, Sr ̸= ∅, |Su| ⩾ 2, and Su,even = ∅. We have +J1 +spec( f) = |E2(R)Frob∗| + (−1)|Su|+1bR(1)2|Su|−2Pic(2). + +16 +HONGJIE YU +4.3. Counting ℓ-adic local systems in rank 1. We need to discuss the cases in rank 1 first, not +only for completeness but also because these results will be needed when calculating the cases in +rank 2. It has been dealt with in [De15, Section 6]. +A difference between a number field and a function field is that the function field F, hence all +its local fields, is an Fq-algebra. Given a character χv : Fv −→ Q× +ℓ , we can consider its restriction +to F× +q . Let χ be a Hecke character +χ = ∏ +v +χv : F×\A× −→ Q× +ℓ . +It is a character of A× which is trivial on F×, in particular on F× +q , therefore necessarily we have +(8) +∏ +v +χv|F× +q = 1. +Suppose that R1 is a set of rank 1 ℓ-adic local systems over (X∗ +(x))x∈S fixed by Frob∗. Suppose +that they are tame and εx are eigenvalues of a tame generator. By similar discussion as in 3.1, but +using local class field theory, we obtain for each v ∈ S a character θv of O× +v which is trivial on +1 + ℘v if it is tame. The condition +(9) +∏ +x∈S +εx = 1, +is equivalent to +(10) +∏ +v +θv|F× +q = 1. +Let A1(R1) be the set of inertial equivalent classes of Hecke characters χ of F×\A× such that χv +extends θv. The set A1(R1) is in bijection with E1(R1)Frob∗. +Lemma 4.4. The condition (10) is satisfied is and only if A1(R1) is non-empty. +Proof. By (8), we have seen that it is a necessary condition for A1(R) to be non-empty. Conversely, +note that F×\A× is an extension of F×\A×/O× ∼= PicX(Fq) by O×/F× +q , the condition (8) ensures +that ∏v θv defines a character of O×/F× +q . Taking (ℓ-adic) Pontryagin dual, we see immediately +that A1(R) is non-empty. +□ +If the condition (9) is satisfied then we have +(11) +|E1(R)Frob∗| = |Pic0(X)(Fq)∨| = Pic(1), +otherwise +E1(R)Frob∗ = ∅. +In fact, as long as E1(R)Frob∗ is non-empty, it is a principal homogenous space under E1(∅)Frob∗ +which has cardinality Pic(1) = |Pic0(X)(Fq)|. +4.4. Eulerian decomposition and calculations on Whittaker functions. Let f = ⊗ fv ∈ C∞ +c (G(A)) +be the function defined in Proposition 4.2. Let π ∼= ⊗′πv be a cuspidal automorphic representa- +tion of G(A). +Suppose that π ∼= π ⊗ ǫ as representations of G(A), we need to consider +Tr(ǫ ◦ R( f)|π). +Note that this trace is not well-defined from a pure representation theoretical point of view be- +cause the action of ǫ on π relies on the isomorphism π ∼= π ⊗ ǫ. For G = GL2, this isomorphism is +furnished by the multiplicity one theorem, which says that π and π ⊗ ǫ have the same underlying +space of cusp forms. +A similar problem arises if we want to get an Eulerian decomposition of the trace Tr(ǫ ◦ +R( f)|π). We need to choose isomorphisms πv ∼= πv ⊗ ǫ so that their tensor product is com- +patible with the global isomorphism (we also need to assume that the isomorphisms can be glued + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +17 +together to a restricted tensor product isomorphism, i.e., they fix the implicit chosen Kv-invariant +vector for almost all places v). A natural way to do so is by using Whittaker models. +Let ψ = ⊗ψv : A −→ C× be an additive character of A which can be viewed as a character of +N(A), where ψv are additive characters of Fv. Suppose that ψv has conductor ℘−nv +v +, i.e., is trivial +on ℘−nv +v +but not on ℘−nv−1 +v +. We have +∑ +v +nv deg v = 2g − 2, +recall that g is the genus of the curve X. Let W(π) be the global Whittaker model of π with respect +to ψ, i.e., the space of smooth functions ϕ over G(A) such that +ϕ(ux) = ψ(u)ϕ(x), +∀u ∈ N(A), ∀x ∈ G(A), +and W(πv) the local Whittaker model of πv with respect to ψv (space of functions over Gv +similarly defined). Then we have a natural decomposition W(π) = ⊗′vW(πv) and W(πv) = +W(πv ⊗ ǫ). Therefore, we have +(12) +Tr(ǫ ◦ R( f)|π) = ∏ +v∈|X| +Tr(ǫv ◦ R( fv)|W(πv)), +where ǫv(xv) = (−1)deg v deg(det(xv)) for v ∈ Gv. +Theorem 4.5 (Paskunas-Stevens). Let ρ be a representation of Kv which inflates a cuspidal represen- +tation of G(κv). Let πv be an irreducible representation of Gv that contains ρ. Let ψ′ +v be an additive +character of Fv of conductor ℘v, i.e., is trivial on ℘v and is non-trivial on Ov. It defines a character of Nv +by +� +1 +x +0 +1 +� +�→ ψ′ +v(x). Let W be the space of Whittaker functions of πv with respect to ψ′ +v. Let Wρ be the +ρ isotypic subspace of W. Then every function in Wρ is supported in N(Fv)ZvKv. +Proof. This is a corollary of Paskunas-Stevens’ result [PS08, Theorem 5.8]. Let’s explain their +notations from type theory which we need to apply to our specific case. The group J is Kv, the +group J is ZvKv, the group U is Nv, the representation Λ is our ρ, and the character Ψα is obtained +by restriction of ψ′ +v to N(Ov) then extends to N(Ov)(1 + ℘vM2(Ov)). +Let X ∈ πv and Y ∈ π∨ +v , where π∨ +v is the contragredient representation of πv. Let ΦX,Y be the +matrix coefficient of πv defined by ΦX,Y(x) = ⟨πv(x)X, Y⟩, for any x ∈ Gv. If there is an X such +that ΦX,Y ̸= 0, then the map X �→ ΦX,Y embeds πv into C∞ +c (Gv). +The result [PS08, Th. 5.8] shows that π∨v contains a special vector Y∨ +α so that the linear map +from πv to the space of Whittaker functions +X �→ +� +x �→ +� +Nv +ψ′ +v(u)ΦX,Y∨ +α (u−1x)du +� +, +is non-zero. Moreover, Paskunas and Stevens show that πv contains a vector Yα, which is con- +tained in ρ-isotypic part (πv)ρ of πv, so that ΦYα,Y∨ +α has support in ZvKv ([PS08, Prop. 5.7]) and +its associated Whittaker function +x �→ +� +Nv +ψ′ +v(u)ΦYα,Y∨ +α (u−1x)du, +is supported in ZvNvKv and extends the function ΦYα,Y∨ +α supported on ZvKv. Note that since +the irreducible representation ρ is contained with multiplicity one in πv ([Yu21b, 5.2.4]), every +X ∈ (πv)ρ is generated by ⟨kYα : k ∈ Kv⟩. Therefore, every Whittaker function associated to +ΦX,Y∨ +α for X ∈ (πv)ρ has support in ZvNvKv. +□ +Corollary 4.6. Suppose we’re in the situation of Theorem 4.5. Let W(πv) be the Whittaker model for the +character ψv of conductor ℘−nv +v +, then every function in W(πv)ρ is supported in +{x ∈ Gv|v(det(x)) ∈ −nv − 1 + 2Z}, +where v is the valuation of Fv normalized to be surjective to Z. + +18 +HONGJIE YU +Proof. Let tv be a uniformizer of ℘v. Let ψ′v := (y �→ ψv(t−nv−1 +v +y)). Then ψ′v has conductor ℘v. Let +b = +� +t−nv−1 +v +0 +0 +1 +� +, we have an Gv-equivariant isomorphism: +W(πv, ψv) −→ W(πv, ψ′ +v), +ϕ �→ (x �→ ϕ(bx)). +By Theorem 4.5, we deduce that functions in W(πv, ψv)ρ are supported in bZvNvKv, which im- +plies the result needed. +□ +4.5. Cuspidal terms. We apply the previous preparation works to compute the cuspidal terms +Jπ( f) in the spectral expansion. In fact, it’s the case that π ⊗ ǫ ∼= π that is non-trivial. +Let σ be an involution on PR (see (2) for its definition) that sends (εx(ix))x∈S to (εx(3 − ix))x∈S. +Define +bR(k) := |Pσ=Frob∗k +R +| +as the cardinality of the set of fixed points of the action of σ ◦ Frob∗k. +Lemma 4.7. If Scr = ∅, then bR(k) = cR(k) = cR(1) is either 0 or 1 for all k ⩾ 1. +If either Sc contains a point of even degree or Sr contains a point of odd degree, we have bR(1) = 0. +If Sc ̸= ∅, we have cR(1) = 0. +Proof. The first statement is trivial because, in this case, PR is at most a singleton. +Suppose that Sr contains a point of odd degree, meaning that Frob∗ has an orbit of odd length +on Scr. Suppose that a ∈ 2Z + 1 is the length of such an orbit and x0 ∈ Sr ⊗ Fq is a point in it. We +have εx0(1) ̸= εx0(2). Suppose that Frob∗a((εx(1x))x) = (ε′ +x)x. Note that by definition, we have +ε′x0 = εx0(1x0). In particular, +Frob∗a((εx(ix))x) ̸= σa((εx(ix))x), +for any (ix)x ∈ {1, 2}S, since a is odd. This implies that bR(a) = 0 hence bR(1) = 0. +Similarly, we can prove the case when Sc contains a point of even degree. Indeed, note that +σa = Id if a is even. +□ +Theorem 4.8. Let π be a cuspidal automorphic representation of G(A). Recall that ǫ is the sign character +of G(A) that factors through deg ◦ det. Suppose that π ⊗ ǫ ∼= π. Then +Tr(ǫ ◦ R( f)|π) = 0, +except if the following conditions are satisfied: 1. bR(1) ̸= 0; 2. every place in Su has odd degree; 3. +πv ∈ IrrR(Gv) for v ∈ |X| − Su and πv has scalar ramification determined by semisimplification of Rv +for v ∈ Su. If this is the case, we have +Tr(ǫ ◦ R( f)|π) = (−1)|Su|2|Su|(−1)deg Sc. +Proof. By Langlands correspondence, if π ∼= π ⊗ ǫ, no local component of π can be a twisted +Steinberg representation. In fact, since a Hecke character of F×\A× is of finite order if it sends +an element of degree 1 to a root of unity, if necessary, by replacing π by an inertially equivalent, +we may assume that the central character of π is of finite order. Suppose that L is the ℓ-adic local +system over X − S that corresponds to π. If π ⊗ ǫ ∼= π, then +L|X−S ∼= L1 ⊕ L2, +and Frob∗ permutes L1 and L2 ([Yu21b, Prop. 2.1.3]). In particular, the ramification of L at every +x ∈ S is semisimple. Therefore, π does not have a twisted Steinberg component by Theorem 3.2. +Moreover, it is clear that if +Tr(ǫ ◦ R( f)|π) ̸= 0, + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +19 +then πv has the desired has the desired ramification type for v ∈ |X| − Su, and the (Iv, ι ◦ χ)- +isotypic subspace (πv)(Iv,ι◦χ) is non-trivial (notations as in Theorem 3.2). In particular, πv is +either a twisted Steinberg representation in IrrR(Gv) or a twisted unramified principal series in +IrrRss(Gv) where Rss is the semisimplification of R. We have seen that πv can not be a twisted +Steinberg representation. Moreover, the product of all eigenvalues of ramifications of L1 and L2 +should be 1, and they are permuted by Frobenius action. This implies that bR(1) ̸= 0. +We need to calculate +Tr(ǫ ◦ R( f)|π) +when π ∼= π ⊗ ǫ and πv has the above described property. The equation (12) allows us to calculate +it locally. +We note that Theorem 3.2 says if +Tr(R( fv)|πv) ̸= 0 +then πv ∈ IrrR(Gv). Let v ∈ Su. If deg(v) is even, then ǫv equals the trivial character of Gv and +we have +Tr(ǫv ◦ R( fv)|W(πv)) = Tr(R( fv)|πv) ̸= 0. +This is impossible as πv is not a twisted Steinberg representation. +Suppose now that v ∈ Su and deg(v) is odd. Up to a twist, we may assume that πv is unrami- +fied and the eigenvalues of Rv at v are 1. We take a Whittaker function ϕv ∈ W(πv)Kv. Note that +since both ǫvϕv and ϕv are contained in W(πv)Kv = W(πv ⊗ ǫv)Kv, they’re differed by a scalar: +ǫvϕv = cϕv. +Let xv be any element in the support of ϕv, we deduce that c = ǫv(xv). Let ℘c(ψv) +v +be the conductor +of ψv. We have shown in [Yu18] that the support of ϕv contains an element of valuation c(ψv). +Therefore c = (−1)deg(v)c(ψv). Let ϕ′ +v ∈ W(πv) be the function defined by +x �→ ϕv(x +� +̟v +0 +0 +1 +� +). +We have +ǫvϕ′ +v = (−1)deg(v)(c(ψv)+1)ϕ′ +v. +Then {ϕv, ϕ′v} is a basis of W(πv)Iv. The endomorphisms on W(πv), +ǫv ◦ R( +1 +vol(Iv) +1Iv) and ǫv ◦ R( +1Kv) +are composition of a projection onto πIv +v together with a linear map represented respectively by +the matrix +� +(−1)deg(v)c(ψv) +0 +0 +(−1)deg(v)(c(ψv)+1) +� +, +� +(−1)deg(v)c(ψv) +0 +0 +0 +� +. +We deduce that +Tr(ǫv ◦ R( fv)|W(πv)) = (−1)deg(v)(c(ψv)+1) − (−1)deg(v)c(ψv). +Since deg(v) is odd, it equals +Tr(ǫv ◦ R( fv)|W(πv)) = −2(−1)c(ψv). +Similarly, for v ∈ |X| − Scr − Su, we have +Tr(ǫv ◦ R( fv)|W(πv)) = (−1)c(ψv) deg(v). +For v ∈ Sc, we deduce from Corollary 4.6 that: +Tr(ǫv ◦ R( fv)|W(πv)) = (−1)deg(v)(c(ψv)+1)Tr(R( fv)|W(πv)) = (−1)deg(v)(c(ψv)+1). + +20 +HONGJIE YU +For v ∈ Sr, as we have seen in Lemma 4.7 that v must have even degree otherwise bR(1) = 0, we +have +Tr(ǫv ◦ R( fv)|W(πv)) = Tr(R( fv)|W(πv)) = 1. +In conclusion, by (12) we have +Tr(ǫ ◦ R( f)|π) = (−1)|Su|2|Su|(−1)∑v∈|X|−Sc deg(v)c(ψv)+∑v∈Sc deg(v)(c(ψv)+1). +We have +∑ +v +c(ψv) deg(v) = −(2g − 2). +Therefore, +Tr(ǫ ◦ R( f)|π) = (−1)|Su|2|Su|(−1)deg Sc. +□ +Corollary 4.9. If Su ̸= ∅ and at least one place in it has even degree, then +J1 +cusp( f) = |E2(R)Frob∗|. +If Su ̸= ∅ and every place in it has an odd degree, then +J1 +cusp( f) = |E2(R)Frob∗| + +� +(−1)|Su|+1bR(1)2|Su|−2(Pic(2) − Pic(1)), +Scr = ∅; +(−1)|Su|+1bR(1)2|Su|−2(−1)deg ScPic(2), +Scr ̸= ∅. +. +If Su = ∅, then +J1 +cusp( f) = |E2(R)Frob∗| + +� +0, +Scr = ∅; +bR(1) +4 +(1 − (−1)deg Sc)Pic(2), +Scr ̸= ∅. +. +Proof. In fact, the sum of Jπ( f) over inertial equivalent classes of cuspidal automorphic represen- +tations π such that π ⊗ ǫ ̸∼= π gives |E2(R)Frob∗| after Theorem 3.3. +We need to consider the sum +1 +2 ∑ +π +Tr(R( f)|π) − 1 +2 ∑ +π +Tr(ǫ ◦ R( f)|π), +where the sums over π are taken over inertial equivalent classes of cuspidal automorphic repre- +sentations π such that π ⊗ ǫ ∼= π. The Langlands correspondence (see the proof of Theorem 4.8) +shows that no such cuspidal automorphic π can have a twisted Steinberg component. Therefore, +if Su ̸= ∅, +1 +2 ∑ +π +Tr(R( f)|π) = 0. +If Su = ∅, we need to know the number of equivalence classes of such π. By [Yu18, 2.1.3] and the +first paragraph of the proof of Theorem 4.8, the number of such π are in bijections withe the set of +non-ordered pairs (L1, L2) of rank 1 ℓ-adic systems over X − S such that Frob∗Li ∼= L3−i, L1 ̸∼= L2, +and that the local monodromies of the direct sum L1 ⊕ L2 is given by Rss, the semisimplification +of R. If Scr = ∅, then bR(1) equals 0 or 1 and there are bR(1) +Pic(2)−Pic(1) +2 +such pairs. If Scr ̸= ∅, +then there are +bR(1) +2 +Pic(2) such pairs. We have +1 +2 ∑ +π +Tr(R( f)|π) = +� bR(1) +4 +(Pic(2) − Pic(1)), +Scr = ∅; +bR(1) +4 +Pic(2), +Scr ̸= ∅. +By Theorem 4.8, if bR(1) = 0 or if Su ̸= ∅ and contains a place of even degree, then the sum +1 +2 ∑ +π +Tr(ǫ ◦ R( f)|π) + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +21 +is 0. Otherwise if Su ̸= ∅ and does not contain any place of even degree or Su = ∅, then by +Theorem 4.8, we have +1 +2 ∑ +π +Tr(ǫ ◦ R( f)|π) = (−1)|Su| +� +bR(1)2|Su|−2(Pic(2) − Pic(1)), +Scr = ∅; +bR(1)2|Su|−2(−1)deg ScPic(2), +Scr ̸= ∅. +□ +4.6. Residuel terms. The proposition below describes the contributions to the trace formula from +the residual spectrum. +Proposition 4.10. If Scr is non-empty, then +J1 +res( f) = 0. +If Scr = ∅, then +J1 +res( f) = cR(1)(−1)|Su|Pic(1). +Proof. For any v ∈ Scr and any character µv of Gv, we have +Tr(R( fv)|µv) = 0. +Therefore, the first statement holds, and it suffices to consider the case that Scr = ∅. +It’s clear that if v /∈ S, then for a character µv of Gv we have +Tr(R( +1Kv)|µv) = +� +1, +µv|Kv = 1; +0, +µv|Kv ̸= 1. +Let v ∈ Ss, we have +Tr(R( fv)|µv) = +� +1, +µv|Kv = θv; +0, +µv|Kv ̸= θv; +where θv is the character defined in Theorem 3.2. Similarly, let v ∈ Su, we have +Tr(R( fv)|µv) = +� +−1, +µv|Kv = θv; +0, +µv|Kv ̸= θv. +We deduce that if +(13) +∏ +v +θv|F× +q ̸= 1, +then +J1 +res( f) = 0. +Otherwise, following our discussions in rank 1 in 4.3, there are Pic(1) such equivalent classes of +µ. Therefore, we have +J1 +res( f) = (−1)|Su|Pic(1). +The result is thus proved since (13) is equivalent to cR(1) ̸= 0 (cf. [De15, 3.2]). +□ +4.7. Continuous terms. The proposition below describes the contributions to the trace formula +from the continuous spectrum. +Proposition 4.11. If Sc ̸= ∅ then +J1 +cont( f) = 0. +If Sc = ∅, then we have the following results. +(1) Sr ̸= ∅. +J1 +cont( f) = + + + + + + + +1 +2cR(1)Pic(1)2(2g − 2 + deg Sr), +Su = ∅; +1 +2cR(1)Pic(1)2 deg v, +Su = {v}; +0, +|Su| ⩾ 2. + +22 +HONGJIE YU +(2) Sr = ∅. +J1 +cont( f) = + + + + + + + + + + + + + + + + + +cR(1)Pic(1)2(g − 1), +Su = ∅; +cR(1)Pic(1)2 deg v +2 ++ 1 +2cR(1)Pic(1), +Su = {v} and 2 ∤ deg v; +cR(1)Pic(1)2 deg v +2 +, +Su = {v} and 2 | deg v; +cR(1)Pic(1)(−1)|Su|+12|Su|−2, +|Su| ⩾ 2, and Su,even = ∅; +0, +|Su| ⩾ 2, and Su,even ̸= ∅. +The proof is given in 4.7.6. We need to do some calculations about L-functions and intertwining +operators. +The first thing to remark is that if Sc ̸= ∅, then for any ψ ∈ Acont, the action of R( f) on AB,ψ +must be the 0 map. In fact, at a place v ∈ Sc, we have AB,ψ ∼= IB(ψv), the induced representation +of ψv, and by Theorem 3.2, it has no ρv, which is cuspidal, isotypic subspace. Therefore we assume +that Sc = ∅ in the following. +4.7.1. L-functions of Hecke characters. We’ll need information on L-functions of Hecke characters +when dealing with the continuous part of the trace formula. Over a function field, we have a +complete understanding of them. +Let χ : A×/F× −→ C× be a Hecke character of finite order. Let χ = ⊗v∈|F|χv be its factorisa- +tion. Recall that the L-function L(χ, z) can be defined by the formal power series +L(χ, z) := ∏ +v∈|X| +Lv(χ, z), +where +Lv(χ, z) = + + + +1 +1−χv(̟v)zdeg v , +if χv is unramified ; +1 +otherwise. +The infinite product is absolutely convergent and is holomorphic if |z| < q−1, which admits a +meromorphic continuation to the whole C. It is, in fact, a rational function in z. Let R be the set of +places of ramifications of χ, identified with a subset of closed points of X. We fix an isomorphism +between C and Qℓ. Let Lχ be an ℓ-adic local systems over X − R corresponding to χ obtained by +global class field theory. The L-function of Lχ equals that of χ: +L(Lχ, z) = L(χ, z). +Moreover, we know from Grothendieck’s cohomological interpretation (cf., for example, [Laf02, +Th´eor`eme VI.1]) that +L(Lχ, z) = +det(1 − zFq|H1 +c (X − R, Lχ)) +det(1 − zFq|H2c (X − R, Lχ)) det(1 − zFq|H1c (X − R, Lχ)). +where Fq is a geometric Frobenius element which acts on the cohomology with compact supports. +Proposition 4.12. (Riemann hypothesis) Let χ be a Hecke character on F×\A× of finite order so that the +set of ramified places is R. If χ is inertially equivalent to the trivial character, then +L(χ, z) = +P(z) +(1 − z)(1 − qz) +where P(z) is a polynomial of 2g. +If χ is not inertially equivalent to the trivial character, then its L-function L(χ, z) is a polynomial in z +of degree 2g − 2 + deg R. +In any case, all of the zeros of L(χ, z) satisfy |z| = q− 1 +2 . + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +23 +Proof. If R is empty, then there are two cases. If Lχ|X is trivial, up to a twist of a rank 1 sheaf over +Spec(Fq), we may assume that Lχ ∼= Qℓ is the constant sheaf. We have +L(Lχ, z) = ζX(z) = +P(z) +(1 − z)(1 − qz), +deg P(z) = 2g and all of the zeros of P(z) satisfy |z| = q− 1 +2 . If Lχ|X is non-trivial then L(Lχ, z) is +a polynomial of degree 2g − 2. For reference, see [Yu18, Prop. 6.1.1]. +We consider the case that R is non-empty. We know that +(14) +H0 +c (X − R, Lχ) = 0, +as Lχ has no nonzero properly supported section over X − R. By Poincar´e duality +(15) +H2 +c (X − R, Lχ)∨ ∼= H0(X − R, L∨ +χ)(1) ∼= HomX−R(L∨ +χ, Qℓ)(1), +from which we deduce that H2 +c (X − R, Lχ) is 0. The dimension of H1 +c (X − R, Lχ) can then be +derived from the Euler-Poincar´e characteristic: +dim H1 +c (X − R, Lχ) = −χc(X − R, Lχ), +which can be calculated by the Grothendieck-Ogg-Shafarevich formula, see [Ra95, Th´eor`eme 1, +p.133]. In fact, since every local monodromy of Lχ is tamely ramified, the Swan conductor is zero. +We deduce from loc. cit. that χc(X − R, Lχ) = χc(X − R) = 2 − 2g − deg R. +The assertion about the positions of zeros is the Riemann hypothesis for rank 1 local systems +(see [Laf02, Th´eor`eme VI.10] for a general statement). +□ +4.7.2. Eulerian expansions of intertwining operators. Let ψ ∈ Acont. Let +M(w, λ) : AB,ψ −→ AB,w(ψ) +be an intertwining operator. It is a G(A)-morphism. +Let IB(ψv) be the space of functions ϕ over Gv satisfying ϕ(ntx) = ρB(t)ψv(t)ϕ(x) for any +n ∈ Nv, t ∈ Tv and x ∈ Gv. By definition of intertwining operator, we have an Eulerian expansion. +Indeed, let Mv(w, λ) : IB(ψv) −→ IB(w(ψv)) be an operator defined by analytic continuation of +the integral which converges when |Reλ| >> 0, +(16) +(Mv(w, λ)ϕ)(x) = λ(x) +� +Nv +ϕ(w−1nx)λ(w−1nx)dn. +Choosing isomorphisms cψ : AB,ψ −→ ⊗vIB(ψv) and cw(ψ) : AB,w(ψ) −→ ⊗vIB(w(ψv)), there is a +constant c depending only on these isomorphisms such that the following diagram is commuta- +tive: +AB,ψ +M(w,λ) +−−−−→ +AB,w(ψ) +cψ +� +cw(ψ) +� +⊗vIB(ψv) +c⊗vMv(w,λ) +−−−−−−−→ ⊗vIB(w(ψv)) +. +In the special case that ψ is non-regular, i.e., w(ψ) = ψ, we have +c = q1−g. +It comes from the different normalization of the Haar measures on Nv and N(A). + +24 +HONGJIE YU +4.7.3. Intertwining operator on (Kv, θv)-isotypical subspace. In this part, we treat the local intertwin- +ing operator when v ∈ Ss or v /∈ S. +Let θv be a character: +θv : Kv +det +−→ O× +v −→ κ× +v −→ C×. +We have +dim IB(ψv)(Kv,θv) ⩽ 1. +The space IB(ψv)(Kv,θv) is non-zero if and only if ψv = (ψv,1, ψv,2) with ψv,1|O× +v = ψv,2|O× +v = θv. In +this case, there is a µ ∈ C× such that for any y ∈ F× +v , we have ψv1(y)/ψv2(y) = µdeg y. Moreover, +IB(ψv)(Kv,θv) is generated by the function ϕψv which satisfies b ∈ Bv and k ∈ Kv: +ϕψv(bk) = ρB(b)ψv(b)θv(det k). +For x ∈ Gv and λ ∈ XG +T ∼= C× (see subsection 4.1 for definition), let +ϕψv,λ(x) := ϕψv(x)λ(x). +By dimension one, we have +Mv(w, λ)ϕψv = cλϕw(ψv), +for some constant cλ ∈ C×. It suffices to evaluate the above equation at x = 1 to find the value c: +cλ = +� +Nv +ϕψv,λ(w−1n)dn. +The integral can be decomposed: +� +N(Ov) ϕψv,λ(w−1n)dn + +� +Fv−Ov +ϕψv,λ( +� +1 +y−1 +0 +1 +� � +y−1 +0 +0 +y +� � +−1 +0 +y−1 +−1 +� +)dy. +Note that +� +N(Ov) ϕψv,λ(w−1n)dn = vol(N(Ov)) = 1. +Besides, for y ∈ Fv − Ov, we have +ϕψv,λ( +� +1 +y−1 +0 +1 +� � +y−1 +0 +0 +y +� � +−1 +0 +y−1 +−1 +� +) = µ− deg yλ−2 deg v deg y. +Under our additive Haar measure on Ov, we know that vol(̟n +vO× +v ) = q−n +v (1 − q−1 +v ). We deduce +that +cλ = +� +N(Fv) ϕψv,λ(w−1n)dn = 1 − q−1 +v µλ2 deg v +1 − µλ2 deg v += +Lv(ψ1ψ−1 +2 , λ2) +Lv(ψ1ψ−1 +2 , q−1λ2) +. +4.7.4. Intertwining operator on (Iv, χv)-typical subspace: regular cases. In this part, we treat the case +that v ∈ Sr. +If χv is regular, then we have +dim IB(ψv)(Iv,χv) ⩽ 1. +In fact, we have Gv = BvIv ∐ BvwIv. Any function ϕ ∈ IB(ψv)(Iv,χv) is entirely determined +by ϕ(1) and ϕ(w). Moreover, for such a ϕ, using the definition of IB(ψv) and (Iv, χv)-typical +condition, we have +ϕ(tw) = ψv(t)ϕ(w) = ϕ(w)χv(w−1tw) +and +ϕ(t) = χv(t)ϕ(1) = ϕ(1)ψv(t). +Therefore if ψv|T(Ov) ̸= χv and ψv|T(Ov) ̸= w(ψv) then ϕ must be zero. If ψ|T(Ov) = χv, then ϕ is +supported in BvIv and if ψv|T(Ov) = w(χv) then ϕ is supported in BvwIv. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +25 +Suppose that we have ψv|T(Ov) = χv. The space IB(ψv)(Iv,χv) is generated by the function ϕψv +which is defined in such a way that for any n ∈ Nv, t ∈ Tv and k ∈ Kv: +ϕψv(ntk) = +� +ρB(t)ψv(t)ψv(k), +if k ∈ Iv; +0, +if k ∈ IvwIv. +The space IB(w(ψv))(Iv,χv) is generated by ϕw(ψv) which is defined in such a way that for any +n ∈ Nv, t ∈ Tv and k ∈ Kv: +ϕw(ψv)(ntk) = +� +ρB(t)ψv(t)ψ(b), +if k = nwb ∈ Iv+wIv; +0, +if k ∈ Iv. +The local intertwining operator Mv(w, λ) is a linear map from IB(ψv)(Iv,χv) to IB(w(ψv))(Iv,χv). +By dimension 1, there is a constant cλ ∈ C such that +Mv(w, λ)ϕψv = cλϕw(ψv). +Evaluating at the point x = w, we see that +cλ = +� +Nv +ϕψv(w−1nw)λ(w−1nw)dn. +We break the integral into two parts following the union: +w−1Nvw = +� +1 +0 +℘v +1 +� +∪ +� +1 +0 +Fv − ℘v +1 +� +. +The first part is included in Iv. For the second part, we need +� +1 +0 +x +1 +� += +� +x−1 +0 +0 +x +� � +1 +x +0 +1 +� � +0 +−1 +1 +x−1 +� +. +Since +� +0 +−1 +1 +x−1 +� +∈ IvwIv, the integral over +� +1 +0 +Fv − ℘v +1 +� +vanishes and cλ = vol(℘v) = q−1 +v . As +the local L-factor Lv(ψ1ψ−1 +2 , λ2) is trivial, we can present the result as +cλ = q−1 +v +Lv(ψ1ψ−1 +2 , λ2) +Lv(ψ1ψ−1 +2 , q−1λ2) +. +4.7.5. Intertwining operator on (Iv, χv)-typical subspace: non-regular cases. In this subsection, we +calculate intertwining operator on (Iv, χv)-typical subspace of IB(ψv) when ψv is non-regular. +Such calculations have already been done in [Fl15, Lemma 4.7]. In fact, the calculations are similar +to the cases we have already treated. Therefore, we’ll briefly recall the results. +If χv is a character of Iv that factors through determinant, we have +dim IB(ψv)(Iv,χv) = 2 or 0. +The dimension is non-zero if and only if ψ|T(Ov) lifts χv. In fact, the double quotient Bv\Gv/Iv ∼= +Iv\Kv/Iv has cardinality 2. We know that +dim IB(ψv)(Iv,χv) ⩽ 2. +If ψv|T(Ov) lifts χv, the space IB(ψv)(Iv,χv) is generated by the basis (ϕψv,1, ϕψv,w), where for any +n ∈ Nv, t ∈ Tv and k ∈ Kv, we have: +ϕψ,1(ntk) = +� +ρB(t)ϕ(t)ψv(det(k)), +if k ∈ Iv; +0, +if k ∈ IvwIv. +ϕψ,w(ntk) = +� +ρB(t)ψ(t)ψv(det(k)), +if k ∈ IvwIv; +0, +if k ∈ Iv. + +26 +HONGJIE YU +If ψv|T(Ov) does not extend χv, then IB(ψv)(Iv,χv) is zero since for any ϕ ∈ IB(ψv)(Iv,χv), we have +ϕ(t) = ψv(t)ϕ(1) = ϕ(1.t) = χv(t) +for any t ∈ T(Ov). +The local intertwining operator Mv(w, λ) is a linear map from IB(ψv)(Iv,χv) to IB(w(ψv))(Iv,χv). +In the basis (ϕψv,1, ϕψv,w) and (ϕw(ψv),1, ϕw(ψv),w), we have (see [Fl15, Lemma 4.7]), +Mv(w, λ)(ϕψv,1, ϕψv,w) = (ϕw(ψv),1, ϕw(ψv),w) + +(1 − q−1 +v ) +µλ2 +1−µλ2 +1 +q−1 +v +(1 − q−1 +v ) +1 +1−µλ2 + + . +We need to present the result in the form +Mv(w, λ)(ϕψv,1, ϕψv,w) = (ϕw(ψv),1, ϕw(ψv),w) +Lv(ψ1ψ−1 +2 , λ2) +Lv(ψ1ψ−1 +2 , q−1λ2) + + +(1−q−1 +v )µλ2 deg v +1−q−1 +v µλ2 deg v +1−µλ2 deg v +1−q−1 +v µλ2 deg v +q−1 +v (1−µλ2 degv) +1−q−1 +v µλ2 deg v +1−q−1 +v +1−q−1 +v µλ2 deg v + + . +In particular, +det(Mv(w, λ)) = −q−1 +v +1 − qvµλ2 deg v +1 − q−1 +v µλ2 deg v ( +Lv(ψ1ψ−1 +2 , λ2) +Lv(ψ1ψ−1 +2 , q−1λ2))2. +and +Tr(Mv(w, λ)) = +Lv(ψ1ψ−1 +2 , λ2) +Lv(ψ1ψ−1 +2 , q−1λ2) +(1 − q−1 +v )(1 + µλ2 deg v) +1 − q−1 +v µλ2 deg v +. +4.7.6. Continuous terms. Now we come to the calculations of contributions from continuous spec- +trum using the previous preparations. +We are going to consider J1 +ψ( f) for ψ ∈ Acont. +If Sr ̸= ∅, then J1 +ψ( f) ̸= 0 implies that ψ is regular. In this case, there are +cR(1) +2 +Pic(1)2 +equivalent classes of ψ such that J1 +ψ( f) can be non-zero. +If Sr = ∅, then there are +cR(1)1 +2Pic(1)(Pic(1) − 1) +regular classes of ψ such that J1 +ψ( f) can be non-zero, and +cR(1)Pic(1) +non-regular classes. +Recall that for each place v ∈ Su, we have set in Proposition 4.2 that +fv = +� +1 +vol(Iv) +1Iv(x) − 2 1Kv(x) +� +θv(det x−1). +We have +J1 +ψ( f) = ∑ +S0⊆Su +(−1)|S0|2|S0|J1 +ψ( fS0). +where +fS0 = ⊗v fS0,v ∈ C∞ +c (G(A)) +is the function that fS0,v = fv for all places v outside Su and is equal to x �→ +1Kv(x)θv(det x−1) if +v ∈ S0 and is equal to +1 +vol(Iv) +1Iv(x)θv(det x−1) for v ∈ S1 = Su − S0. +We need a lemma for our calculations. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +27 +Lemma 4.13. Let V be a finite-dimensional C-linear space. Let m be a meromorphic function over C with +values in GL(V). Suppose that m is holomorphic at any point in the unit circle. We use Z|λ|<1(h) for the +integer defined as the number of zeros (with multiplicity) minus the number of poles (with multiplicity) in +the region |λ| < 1 of a meromorphic function h over C. Then +� +ImXG +T +lim +µ−→1 TrV( +1 +µ−1 − µ m(λ)−1 ◦ m(λ/µ) − +1 +µ−1 − µ Id)dλ = 1 +2 Z|λ|<1(det(m(λ))). +Proof. We have +lim +µ−→1TrV( +1 +µ−1 − µm(λ)−1 ◦ m(λ/µ) − +1 +µ−1 − µ Id) = λ +2 TrV(m(λ)−1 ◦ m′(λ)), +where m(λ) is the C-linear endomorphism V defined as the derivative of m(λ). We use Jacobi’s +formula: +TrV(m(λ)−1 ◦ m′(λ)) = +d +dλ det(m(λ)) +det(m(λ)) +. +Finally since the volume of ImXG +T is normalized to be 1, by definition of contour integration and +argument principle, the integral +� +ImXG +T +d +dλ det(m(λ)) +det(m(λ)) +λdλ +equals Z|λ|<1(det(m(λ))). +□ +We are going to apply this result to intertwining operators. Although, the operator M(w, λ) +is a morphism from AB,ψ to AB,w(ψ), the representation structures of AB,ψ and AB,w(ψ) are the +same. Let V be the parabolic induction of ψ from B(A) to G(A). We fix a G(A)-equivariant +isomorphism: ι1 : AB,ψ −→ V and ι2 : AB,w(ψ) −→ V. Then +TrAB,ψ((− +1 +µ−1 − µM(w, λ)−1 ◦ M(w, λ/µ) + +1 +µ−1 − µ) ◦ R( fS0)), +equals +TrV((− +1 +µ−1 − µ(ι2M(w, λ)ι−1 +1 )−1 ◦ ι2M(w, λ/µ)ι−1 +1 ++ +1 +µ−1 − µ) ◦ ι1R( fS0)ι−1 +1 ), +We apply Lemma 4.13. Note that given a finite family of complex vector spaces Vi of dimension +ni and endomorphisms φi ∈ End(Vi), we have +det(⊗iφi) = ∏ +i +det(φi)∏j̸=i ni. +By Eulerian expansion in 4.7.2, and local calculations in 4.7.3, 4.7.4, 4.7.5, we deduce that the +integral +� +ImXG +T +lim +µ−→1 TrAB,ψ((− +1 +µ−1 − µ M(w, λ)−1 ◦ M(w, λ/µ) + +1 +µ−1 − µ) ◦ R( fS0))dλ, +equals +(17) +2|S1| +2 Z|λ|<1( +L(ψ1ψ−1 +2 , λ2) +L(ψ1ψ−1 +2 , q−1λ2)) + 2|S1|−1 +2 +(2 ∑ +v∈S1 +deg v). +If ψ ∈ Acont is regular which has the correct ramification, (17) equals +2|S1|(2g − 2 + deg Sr) + (2|S1|−1) deg S1. +We have +(18) +J1 +ψ( f) = ∑ +S0⊆Su +(−1)|S0|2|S0| � +2|S1|(2g − 2 + deg Sr) + (2|S1|−1) deg S1 +� +. + +28 +HONGJIE YU +If ψ ∈ Acont is non-regular, then ψ can not have correct ramification if Sr is non-empty. Suppose +that Sr = ∅ and ψ has correct ramification, then J1 +ψ( f) is the sum of +(19) +1 +2 ∑ +S0⊆Su +(−1)|S0|2|S0| � +2|S1|(2g − 1) + (2|S1|−1) deg S1 +� +with +(20) +1 +8 ∑ +S0⊆Su +(−1)|S0|2|S0| +� +∑ +λG∈{±1} +∑ +λw∈ImXG +T +λ2w=λ−1 +G +λGTrAB,ψ(M(w, w−1(λw)) ◦ R( fS0)) +� +. +Lemma 4.14. We have +∑ +I⊆S +(−1)|I| = +� +1, +if S = ∅; +0, +if S ̸= ∅. +We also have +∑ +I⊆S +(−1)|S|−|I| deg I = + + + + + + + +0, +if |S| ⩾ 2; +deg v, +if S = {v}; +0, +if S = ∅. +Proof. The first equality is trivial. Let’s prove the second equality by induction. We may suppose +|S| ⩾ 1; otherwise, the equality is trivial. Let v ∈ S, we have +∑ +I⊆S +(−1)|S|−|I| deg I = +∑ +I⊆S−{v} +(−1)|S|−|I| deg I + ∑ +v∈I⊆S +(−1)|S|−|I| deg I += +∑ +I⊆S−{v} +(−1)|S|−|I|(− deg v). +Therefore, the result follows from the first equality. +□ +After this lemma, for regular ψ, the expression (18) J1 +ψ( f) vanishes if |Su| ⩾ 2. When Su = {v}, +it equals deg v. When Su = ∅, it equals 2g − 2 + deg Sr. +Now suppose that Sr = ∅. For non-regular ψ, the expression (19) vanishes if |Su| ⩾ 2. When +Su = {v}, it equals 1 +2 deg v. When Su = ∅, it equals 1 +2(2g − 1). +Let ψ ∈ Acont be non-regular, we’re going to consider +(21) +1 +8 +∑ +λG∈{±1} +∑ +λw∈ImXG +T +λ2w=λ−1 +G +λGTrAB,ψ(M(w, w−1(λw)) ◦ R( fS0)). +Note that in this case, L(ψ1ψ−1 +2 , z) = ζX(z) where ζX is the zeta function of the curve X. From the +local calculations in 4.7.3 and 4.7.5, we know that +TrAB,ψ(M(w, w−1(λw)) ◦ R( fS0)) = q1−g +ζX(λ−2 +w ) +ζX(q−1λ−2 +w ) ∏ +v∈S1 +(1 − q−1 +v )(λ−2 degv +w ++ 1) +1 − q−1 +v λ−2 deg v +w +. +Here we should regard +ζX(z) +ζX(q−1z) as a rational function so that the pole at z = 1 of the denominator +and numerator are canceled. For each λG, there are two λw such that λ2w = λ−1 +G . The expression +(21) equals: +(22) +∑ +λG∈{±1} +1 +4q1−gλG +ζX(λG) +ζX(q−1λG) ∏ +v∈S1 +(1 − q−1 +v )(λdeg v +G ++ 1) +1 − q−1 +v λdeg v +G +. +There is a polynomial P(z) of degree 2g such that +ζX(z)/ζX(zq−1) = P(z)(1 − q−1z) +P(zq−1)(1 − qz). + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +29 +By functional equation of zeta function, we have +P(1)(1 − q−1) +P(q−1)(1 − q) = −qg−1, +and +P(−1)(1 + q−1) +P(−q−1)(1 + q) = qg−1. +The corresponding summand for λG = 1 in the expression (22) equals +1 +4q1−g2|S1| P(1)(1 − q−1) +P(q−1)(1 − q) = −2|S1|−2. +If there is a place of odd degree in S1, then the corresponding summand for λG = −1 in +the expression (22) vanishes. If all the places in S1 are of even degree, then the corresponding +summand for λG = −1 in the expression (22) equals +1 +4q1−g(−1) P(−1)(1 + q−1) +P(−q−1)(1 + q)2|S1| = −2|S1|−2. +We decompose Su as the union of the set of places of odd degree and those of even degree: +Su = Su,odd ∪ Su,even. +Following the above discussions, the expression (20) equals +(23) +∑ +S0⊆Su +(−1)|S0|2|S0|(−2|S1|−2) + +∑ +Su,odd⊆S0⊆Su +(−1)|S0|2|S0| � +−2|S1|−2� ++ ∑ +S0∈S +0, +where S = {S0|S0 ⊆ Su} − {S0|Su,odd ⊆ S0 ⊆ Su}. The first sum in (23) is − 1 +4 if Su = ∅ and is 0 if +Su ̸= ∅. The second sum in (23) equals 0 if Su,even ̸= ∅, and equals (−1)|Su|+12|Su|−2 if Su,even = ∅. +In conclusion, if Su = ∅, then the expression (20) equals − 1 +2. If Su ̸= ∅ but Su,even = ∅, then +(20) equals (−1)|Su|+12|Su|−2; if Su,even ̸= ∅, (20) equals 0. +To conclude the proof of Proposition 4.11, we summarize that if Sr ̸= ∅ then there is no non- +regular ψ with correct ramification and J1 +cont( f) is equal to cR(1) +2 +Pic(1)2 times (18); if Sr = ∅, +J1 +cont( f) is equal to cR(1) 1 +2Pic(1)(Pic(1) − 1) times (18) plus cR(1)Pic(1) times ((19)+(20)). +5. GEOMETRIC SIDE OF THE TRACE FORMULA AND HITCHIN MODULI SPACES +In this section, we treat J1 +geom( f), whose definition is given in Section 5.4. The goal is to prove +Theorem 5.6. We need to introduce a Lie algebra analog of J1geom( f), which was studied first in +[Ch15] where one can also find relations between this Lie algebra trace formula and the groupoid +cardinality of the category of semistable Higgs bundles over X. +Now we introduce the moduli space of semistable Hitchin bundles with parabolic structures. +It is constructed by Yokogawa ([Yo93]) using GIT theory. Let +R = Scr ∪ Su. +If we can identify R with a subset of X(Fq), then Yokogawa’s construction perfectly suits our +needs. Otherwise, we meet a problem since a point in R can be split into several points over an +algebraically closed field, and we have to make extra arguments about how to treat with (semi)- +stability. To remedy it, we work over an algebraically closed field or a large enough extension of +Fq where we can apply Yokogawa’s construction to obtain a moduli space. Then we define an +Fq-structure on it. + +30 +HONGJIE YU +5.1. Moduli of quasi-parabolic Hitchin bundles over Fq. Let D be a divisor over X and R be a +set of closed points of X. We identify R := R ×Spec(Fq) Spec(Fq) with a subset of X(Fq) and view +D also as a divisor over X. +A quasi-parabolic Hitchin triple (or quasi-parabolic Hitchin bundle) over X is a triple +(E, ϕ, (Lx)x∈R) +where (E, ϕ) is a Hitchin pair for the divisor D, i.e., a vector bundle E together with a bundle +morphism +ϕ : E −→ E(D) := E ⊗ OX(D), +and for each point x ∈ R, Lx is a line in the Fq-vector space Ex, the fiber over x of the bundle E, +such that +ϕx(Lx) ⊆ Lx +where we view Lx also as a line in E(D)x. A quasi-parabolic Hitchin bundle is called (strictly) +parabolic if ϕx(Lx) = 0 and Imϕx ⊆ Lx, i.e., ϕx is nilpotent for 0 ⊆ Lx ⊆ Ex. +If D = KX + ∑v∈R v, we will call them quasi-parabolic Higgs bundles. +For each x ∈ R, let ξx := (ξx,1, ξx,2) ∈ Q2 such that ξx,1 ⩾ ξx,2 ⩾ ξx,1 − 1. Let ξ = (ξx)x∈R. Let +(E, ϕ, (Lx)x∈R) be a quasi-parabolic Hitchin bundle over X. Let L be a sub-line bundle of E, we +define the parabolic degree p-deg(L) by +p-deg(L) := deg(L) + ∑ +x∈R +� +ξx,1, if Lx = Lx; +ξx,2, if Lx ̸= Lx. +We say that (E, ϕ, (Lx)x∈R) is ξ-semistable if for any sub-line bundle L of E satisfying ϕ(L) ⊆ +L(D), we have +p-deg(L) ⩽ deg E + ∑x∈R(ξx,1 + ξx,2) +2 +. +Note that if +deg(E) + ∑ +x∈R +±(ξx,1 − ξx,2) /∈ 2Z, +then the equality can never be achieved. We say that such cases are in general position. +Yokogawa has constructed a moduli space which is a variety defined over Fq that classifies +isomorphism classes of ξ-stable quasi-parabolic Hitchin bundles (E, ϕ, (Lx)x∈R) with E being of +rank 2 and degree e. We denote the variety by +qMe,ξ +2,R(D). +If we are considering the case of parabolic ones instead of being only quasi-parabolic, we will +denote it by Me,ξ +2,R(D). +Remark 5.1. Yokogawa’s results apply under the assumption (under our terminology) that ξx,1 > ξx,2 > +ξx,1 − 1. In particular, it does not include the case that ξx,1 = ξx,2. However, this is not an issue because +when ξx,1 and ξx,2 are close enough, semistability of quasi-parabolic Hitchin bundles coincide with the case +that ξx,1 = ξx,2. +5.2. Fq-points. We have defined a coarse moduli space qMe,ξ +2,R(D) which is a variety over Fq con- +structed by Yokogawa. Suppose n is a divisible enough integer such that every place (closed +point) in R totally splits over Fqn, i.e., the residue field of a place in R can be embedded in Fqn. +Yokogawa’s construction works if the curve is X ⊗ Fqn and the parabolic structures are imposed +at each point of S ⊗ Fqn and in this way qMe,ξ +2,R(D) is a variety defined over Fqn. In this subsection, +we are going to endow qMe,ξ +2,R(D) with a Fq-structure when ξx are the same for points x ∈ R ⊗ Fqn +lying over each v ∈ R. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +31 +For a variety Y defined over Fqn, let FY/Fqn be the arithemetic Frobenius morphism of Y ⊗Fqn +Fq. Recall that it is a morphism of schemes Y ⊗Fqn Fq which is identity on Y and is x �→ xqn over +Spec(Fq). In particular, it is not a morphism of Fqn-schemes. +The pullback by FX/Fq defines an action on the set of isomorphism classes of Higgs bundles. To +define an action on the parabolic structure, we need to use an equivalent definition that identifies +vector bundles with locally free sheaves. Suppose (E, ϕ, (Lx)x∈R) is a quasi-parabolic Higgs bun- +dle over X, For each x ∈ R, Lx defines a unique rank 2 coherent sub-sheaf E x of E such that E/E x +is a skyscraper sheaf of degree 1 supported in {x} and the inclusion E x −→ E induces a mor- +phism of their fibers at x which has image Lx in Ex. Then F∗ +X/FqE x defines a parabolic structure of +F∗ +X/FqE at Frob(x). +Proposition-Definition 5.2. Suppose that the family (ξv)v∈R ∈ (Q2)R satisfies that for any v ∈ R, +0 ⩽ ξv,1 − ξv,2 ⩽ [κv : Fq]. +Let +ξx = +1 +[κv : Fq]ξv, +for any point x ∈ R lying over v ∈ R. +Let σ be the Frobenius element in Gal(Fq|Fq). We define an action of Gal(Fq|Fq) on qMe,ξ +2,R(D)(Fq) +so that σ sends a parabolic Higgs bundle (E, ϕ, (Lx)x∈R) over X to +F∗ +X/Fq(E, ϕ, (Lx)x∈R). +There is a unique variety defined over Fq whose Fqk-points are exactly those in qMe,ξ +2,R(D)(Fq) fixed by σk +and whose base change to Fq is isomorphic to qMe,ξ +2,R(D). We denote the variety by qMe,ξ +2,R(D). +Proof. First, we need to show that the action is well-defined, i.e., a stable quasi-parabolic Higgs +bundle is sent to a stable one. This follows from two observations: 1. Let L be a sub-line bundle of +E such that ϕ(L) ⊆ L(D), then the parabolic degree on F∗ +X/FqL equals that of L. 2. For a divisible +enough k ⩾ 1, we have +F∗k +X/Fq(E, ϕ, (Lx)x∈R) ∼= (E, ϕ, (Lx)x∈R). +Note that qMe,ξ +2,R(D) is quasi-projective. Varieties over Fq whose base change to Fq are isomor- +phic to qMe,ξ +2,R(D) are in bijection with Galois descent data on qMe,ξ +2,R(D). It suffices to prove that +the above action defines a Galois descent data ([BLR90, Example B, p.139]). +We note that the action is induced by an algebraic morphism. This is clear, because the action of +(F∗ +X/Fq)n = F∗ +X⊗Fqn/Fqn coincides with the action of FqMe,ξ +2 /Fqn on qMe,ξ +2,R(D)(Fq) which is algebraic. +This also shows that the action is continuous. +□ +Next, we are going to study qMe,ξ +2,R(D)(Fq), the fixed points of F∗ +X/Fq on qMe,ξ +2,R(D)(Fq). We +define a rank 2 quasi-parabolic Higgs bundle as a tuple (E, ϕ, (Lx)x∈R) over X consisting of a +vector bundle E over X, a bundle morphism ϕ : E −→ E(D) and for each v ∈ R a 1 dimensional +κv sub-vector space Lv of E(v) such that ϕv(Lv) ⊆ Lv. Let (e, ξ) ∈ Z × (Q2)R. Let L be a sub-line +bundle of E such that ϕ(L) ⊆ L(D), we define the parabolic degree L by +p-deg(L) := deg(L) + ∑ +v∈R +� +ξv,1, if Lv = Lv; +ξv,2, if Lv ̸= Lv. +The quasi-parabolic Higgs bundle (E, ϕ, (Lx)x∈R) is semistable if for all such L, we have +p-deg(L) ⩽ deg E + ∑v∈R(ξv,1 + ξv,2) +2 +. + +32 +HONGJIE YU +Proposition 5.3. Under the hypothesis of Proposition-Definition 5.2, the set of stable quasi-parabolic +Higgs bundles (E, ϕ, (Lx)x∈R) over X which are fixed by F∗ +X/Fq is in bijection with the set of stable quasi- +parabolic Higgs bundles (E0, ϕ0, (Lv)v∈R) over X. +Proof. Let σ be the Frobenius element that generates Gal(Fq|Fq). By Galois descent (see [BLR90, +Example B. p.139]), the category of vector bundles over X is equivalent to the category of Gal(Fq|Fq)- +equivariant vector bundles over X, that is the category of vector bundles E over X together with +an isomorphism for each i ⩾ 1, +φσi : (E, ϕ, (Lx)x∈R) −→ F∗i +X/Fq(E, ϕ, (Lx)x∈R), +such that σi �→ φσi satisfies the cocycle condition: +σi(φσj) ◦ φσi = φσi+j, +and for some d ∈ N∗, φσd is the identity. Note that the category of vector bundles is equivalent +to the category of locally free sheaves, hence every vector bundle defined over X is automatically +defined over X ⊗ Fqn for some n ∈ N∗ and the requirement that φσd is the identity map makes +sense when d is divisible by n. +By Galois descent for morphisms of quasi-coherent sheaves ([BLR90, Proposition 1, p.130] +and [BLR90, Example B. p.139]), the above equivalence extends to an equivalence between the +category of quasi-parabolic Higgs bundles over X is equivalent to the category of Gal(Fq|Fq)- +equivariant quasi-parabolic Higgs bundles over X. Indeed, the parabolic structure is determined +by a sub-sheaf F ⊆ E such that E/F is a skyscraper sheaf supported in R and that the inclusion +F → E induces a morphism Fx → Ex of their fibers at each point x which has image Lx in Ex. If +both F and E come from X, i.e. F = F0|X and E = E0|X then F0 ⊆ E0 is a subsheaf such that +the quotient E0/F0 is a skyscraper sheaf supported in R and the map of the fibers Fv → Ev has a +1-dimensional image as κv-vector space. +We need to show that if (E, ϕ, (Lx)x∈R) is stable and its isomorphism class is fixed by Frobenius, +then it defines a unique descent datum (up to isomorphism). +By our assumption, there is an isomorphism +φσ : (E, ϕ, (Lx)x∈R) −→ F∗ +X/Fq(E, ϕ, (Lx)x∈R). +We can define simply for every i ⩾ 1, +φσi := σi−1(φσ) ◦ · · · ◦ σ(φσ) ◦ φσ. +Note that by cocycle condition, this is the unique possible way to extend φσ to a 1-cocycle. +We will prove that we can choose φσ so that for some n, the isomorphism φσn defined above is +the identity. Suppose that (E, ϕ, (Lx)x∈R) is defined over X ⊗ Fqn. Then +F∗n +X/Fq(E, ϕ, (Lx)x∈R) = (E, ϕ, (Lx)x∈R). +Any φσn is an automorphism of (E, ϕ, (Lx)x∈R) which must be a scalar multiplication because +(E, ϕ, (Lx)x∈R) is stable. Hence there is a λ′ ∈ F× +q such that +φσn = λ′.id(E,ϕ,(Lx)x∈R). +Suppose λ′ ∈ F× +qm for some m ⩾ 1. We deduce that φσnm = λ.id(E,ϕ,(Lx)x∈R) with +λ = (λ′)1+qn+···+qn(m−1) ∈ F× +qm ⊆ F× +qnm. +Note that this implies that +(24) +σnm−1(φσ) ◦ · · · ◦ σ(φσ) ◦ φσ = λ.id(E,ϕ,(Lx)x∈R) +Applying σ to both sides, we obtain that +(25) +σnm(φσ) ◦ · · · ◦ σ2(φσ) ◦ σ(φσ) = σ(λ).idF∗ +X/Fq(E,ϕ,(Lx)x∈R). + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +33 +Since σnm(φσ) = φσ, we deduce that σ(λ) = λ and hence λ ∈ F× +q . As the norm map from Fqnm to +Fq is surjective, we conclude that by modifying φσ by a scalar, we can get a φσ such that λ = 1. +It is easy to see that such descent data are unique up to isomorphism. In fact, any two iso- +morphisms φσ and φ′ +σ are differed by a scalar because these are the only automorphisms of +(E, ϕ, (Lx)x∈R). As the map λ �→ λ/σ(λ) from Fq to Fq is surjective (Hilbert’s Theorem 90), +we conclude that φσ is isomorphic to φ′ +σ. +It remains to prove that the quasi-parabolic Higgs bundle (E0, ϕ0, (Lv)v∈R) determined by a +descent datum attached to (E, ϕ, (Lx)x∈R) is stable and reversely if (E0, ϕ0, (Lv)v∈R) is stable then +the quasi-parabolic Higgs bundle (E, ϕ, (Lx)x∈R) over X is stable. The first statement is trivial. For +the second statement, we need to use the fact that semistability coincides with stability since the +parameter ξ is in general position. We can argue by contradiction. Suppose (E, ϕ, (Lx)x∈R) is not +semistable, then it admits a maximal destabilizing sub line bundle over X preserved by ϕ. Such +a line bundle over X is unique, hence is fixed by Galois action. It then descends to a line bundle +over X, which, by our definition of parabolic degree, is again destabilizing. This contradicts the +fact that (E0, ϕ0, (Lv)v∈R) is semistable. +□ +5.3. Residue morphism. Next, we are going to define and study residue morphism. Let V be a +subset of Su. We will be interested in the case that R = V ∪ Scr. We suppose that +DR = KX + ∑ +v∈Scr +v + ∑ +v∈V +v, +and we will be interested in M1 +2,V(DR) = M1,0 +2,V(DR), the moduli space of strictly parabolic Hitchin +bundles with parabolic structures in V where we set parabolic weights to be trivial. +Let (E, ϕ) be a Hitchin bundle. The morphism ϕ is equivalent to a morphism OX −→ End(E)(DR). +Its characteristic polynomial t2 + at + b gives a section (a, b) in +H0(X, OX(DR)) ⊕ H0(X, OX(2DR)). +Let AR be the affine space H0(X, OX(DR)) ⊕ H0(X, OX(2DR)) defined over Fq. Let +RR = ∏ +v∈R +Rv, +where Rv = OX(DR)v ⊕ OX(2DR)v, the fiber OX(DR) ⊕ OX(2DR) in v which is an Fq-vector +space by forgetting its κv-vector space structure (i.e., the restriction of scalars). Note that we have +Rv(Fq) ∼= {t2 + at + b|a, b ∈ κv}. +We have a morphism +AR −→ RR, +sending (a, b) to ((av, bv))v∈R. +Proposition 5.4. Suppose that deg R = |R ⊗Fq Fq| > 2 − 2g. The morphism AR −→ RR is linear of +codimension 1. The image consists of elements (t2 + avt + bv)v∈S such that +(26) +∑ +v∈S +Trκv|Fq(av) = 0. +Let R1 +R be the linear sub-scheme of RR of elements satisfying the residue condition (26). +Proof. By Riemann-Roch theorem, it’s easy to verify that when deg S > 2 − 2g, we have +H0(X, OX(2D)) −→ ∏ +v∈S +OX(2D)v +is surjective. The kernel of the map +H0(X, OX(D)) −→ ∏ +v∈S +OX(D)v, +is H0(X, ωX). Therefore, for dimension reasons, the image is a linear subspace of codimension 1. +The last thing to observe is that the condition (26) is necessary due to the residue theorem. +□ + +34 +HONGJIE YU +The Hitchin fibration is the morphism that sends a Hitchin bundle to the characteristic polyno- +mial of the Higgs field: +M1 +2(DR) −→ AR. +By forgetting the parabolic structure, we obtain a morphism of Fq-schemes +qM1 +2,V(DR) −→ AR. +We define the residue morphism as the composition of the natural morphism AR −→ R1 +R with +Hitchin fibration: +res : qM1 +2,V(DR) −→ AR −→ R1 +R, +and +res : M1 +2,V(DR) −→ AR −→ R1 +R, +It sends a triple (E, ϕ, (Lv)v∈R) to the family of characteristic polnomial of ϕv, v ∈ R. Let +R1 +Scr +be the linear subspace of R1 +R whose components in V are zero. The residue morphism factors +through the morphism +res : M1 +2,V(DR) −→ R1 +Scr. +Given a point o ∈ R1 +Scr(Fq), we will denote +M1 +2,V(o) := res−1(o). +5.4. Geometric side of trace formula. We introduce a variant of the trace formulas with an extra +parameter ξ ∈ (Q2)R. +Let HB : B(A) −→ Q2 be the function defined by +HB( +� +a +b +0 +d +� +) = (− deg a, − deg d). +The Harish-Chandra’s map is the extension of HB to the whole G(A) by Iwasawa decomposition, +i.e., if x = bk ∈ G(A) with b ∈ B(A) and k ∈ G(O), we have HB(x) = HB(b). +Let �τB be the characteristic function over Q2 of the subset +{(x, y) ∈ Q2|x > y}. +For any x ∈ G(A), let x = bk be its Iwasawa decomposition with b ∈ B(A) and k = (kv)v∈|X| ∈ +G(O). For every v ∈ R, we define sx,v to be the identity if kv ∈ Iv and to be the non-trivial +permutation in S2 otherwise. Therefore, given (ξ1, ξ2) ∈ Q2, we have +sx,v(ξ1, ξ2) = +� +(ξ1, ξ2), +if kv ∈ Iv; +(ξ2, ξ1), +if kv /∈ Iv. +Let f ∈ C∞ +c (g(A)), and ξ = (ξx)x∈|S| ∈ (Q2)R. The ξ-variant of truncated trace for Lie algebra +is defined by the integral +Jg,e,ξ( f) := +� +G(F)\G(A)e kg,ξ(x)dx, +where kg,ξ(x) equals +∑ +γ∈g(F) +f(ad(x)−1γ) +− +∑ +δ∈B(F)\G(F) +�τB(HB(δx) + ∑ +v∈R +sδx,vξv) ∑ +γ∈t(F) +� +n(A) f(ad(δx)−1(γ + U))dU. +Let +Eg = {t2 + at + b ∈ F[t]}, + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +35 +be the set of rank 2 unitary polynomials. Let +EG = {t2 + at + b ∈ F[t]|b ̸= 0}, +be the subset of Eg consisting of polynomials whose constant term is non-zero. +For any element γ ∈ g(F), we define χγ ∈ Eg as the characteristic polynomial of γ. Given +χ ∈ Eg, we define kg,ξ +χ (x) for x ∈ G(A) by +∑ +γ∈g(F),χγ=χ +f(ad(x)−1γ) +− +∑ +δ∈B(F)\G(F) +�τB(HB(δx) + ∑ +v∈R +sδx,vξv) +∑ +γ∈t(F),χγ=χ +� +n(A) f(ad(δx)−1(γ + U))dU. +We define +Jg,e,ξ +χ +( f) := +� +G(F)\G(A)e kg,ξ +χ (x)dx. +The integral converges, and it is non-zero for only finitely many χ ∈ Eg. Therefore, we have +Jg,e,ξ( f) = ∑ +χ∈Eg +Jg,e,ξ +χ +( f). +If we set ξ = 0, we’ll omit ξ from the notation. +We also define a group version. Let χ ∈ EG and f ∈ C∞ +c (G(A)), then we define +kG,ξ +χ (x) = +∑ +γ∈G(F),χγ=χ ∑ +i∈Z +f(x−1γaix) +− +∑ +δ∈B(F)\G(F) +�τB(HB(δx) + ∑ +v∈R +sδx,vξv) +∑ +γ∈T(F),χγ=χ ∑ +i∈Z +� +N(A) f((δx)−1(γn)δaix)dn, +where a ∈ A× is a fixed degree 1 id`ele. Note that if the support of f is contained in G(A)0 (for +example in G(O)), then only i = 0 in the sums over i ∈ Z contributes. We define kG,ξ(x) as the +sum of kG,ξ +χ (x) over χ ∈ EG. We define JG,e,ξ +χ +( f) (resp. JG,e,ξ( f)) as the integral of kG,ξ +χ (x) (resp. +kG,ξ(x)) over x ∈ G(F)\G(A)e. We have proved in [Yu21a] that the integrals converge. We have +then, by definition +JG,e,ξ( f) = ∑ +χ∈EG +JG,e,ξ +χ +( f). +The sum is again a finite sum. If we take e = 1 and ξ = 0, we get the geometric side of the trace +formula, which is Arthur’s original definition adapted to a function field: +(27) +J1 +geom( f) = JG,1,0( f). +Remark 5.5. We call JG,e,ξ( f) truncated trace since the main term in the integrand: +∑ +γ∈G(F)aZ +f(x−1γx) +is the diagonal evaluation of the kernel function of the regular action R( f) on L2(G(F)\G(A)/aZ). The +extra term contains the function in x: +∑ +γ∈T(F)aZ +� +N(A) f(x−1(γn)x)dn, +which is the diagonal evaluation of the regular action R( f) on L2(T(F)N(A)\G(A)/aZ). + +36 +HONGJIE YU +5.5. A geometric interpretation of J1geom( f). Suppose that f is the function constructed in Propo- +sition 4.2. We have the following result. +Theorem 5.6. Let o ∈ R1 +Scr(Fq) so that every polynomial ov has distinct roots and is split over κv if +v ∈ Sr, and is irreducible if v ∈ Sc. Then we have +J1 +geom( f) = ∑ +V⊆Su +(−1)|Su−V|2|Su−V|q−(4g−3+|R|)|M1 +2,V(o)(Fq)|. +Proof. Similar results are obtained in [Yu21b]. The main new ingredient of the proof is the treat- +ment of the local components for v ∈ Su. +Since J1 +geom is a linear in C∞ +c (G(A)), we have +J1 +geom( f) = ∑ +V⊆Su +(−1)|Su−V|2|Su−V|J1 +geom( f V), +where f V = ⊗ f V +v is the tensor product of functions f V +v ∈ C∞ +c (Gv) such that +f V +v = fv +at a place v /∈ Su, for v ∈ Su − V it is defined so that for x ∈ Gv: +f V +v (x) = +1Kv(x)θv(det x), +and for v ∈ V it is defined so that for x ∈ Gv: +f V +v (x) = +1 +vol(Iv) +1Iv(x)θv(det x). +We apply [Ch15, Th. 6.2.1] (see also [Yu21b, Th. 3.2.5]). It says that, as the support of f V ∈ +C∞ +c (G(A)) is contained in G(O), we have +J1 +χ( f V) = 0, +except if χ = (t − a)2 with a ∈ F× +q . Therefore for any V ⊆ Su, we have +J1 +geom( f V) = +∑ +χ=(t−α)2,α∈F× +q +J1 +χ( f V). +Since the eigenvalues of ramifications R satisfy +∏ +x∈S +εx,1εx,2 = 1, +by our construction of f, it implies that +f V(zx) = f V(x), +∀x ∈ G(A), z ∈ Z(Fq). +In particular, we may use the identity to z = α, and we obtain +J1 +(t−α)2( f V) = J1 +(t−1)2( f V). +Therefore, +(28) +J1 +geom( f) = (q − 1) ∑ +V⊆Su +(−1)|Su−V|2|Su−V|J1 +unip( f V), +where we denote J1 +unip( f V) = J1 +(t−1)2( f V) because the elements whose characteristic polynomial +is (t − 1)2 are exactly those unipotent elements. +We are going to pass to the Lie algebra version. For every V ⊆ Su, we will define a function +ϕV = ⊗v∈|X|ϕV +v ∈ C∞ +c (g(A)), +whose support is in g(O), and that the map x �→ x + 1 from nilpotent elements in g(F) to unipo- +tent elements in G(F) induces an identity +Jg,1 +nilp(ϕV) = J1 +unip( f V). + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +37 +Indeed, we are more interested in the Fourier transform of ϕV. We need the following proposition +5.7, obtained by direct calculations except Springer’s hypothesis proved by Kazhdan, to construct +this ϕV. More details can be found in [Yu21b, Prop 5.3.2]. +Let +KX = ∑ +v +dvv. +We have +A/(F + ∏ +v +℘−dv +v +) ∼= H1(X, Ω1 +X) ∼= H0(X, OX)∗ ∼= Fq, +by Serre duality and the fact that X is geometrically connected. We fix a non-trivial additive +character ψ of Fq. Via the above isomorphisms, ψ can be viewed as a character of A/F. Let ⟨, ⟩ be +the bilinear form on g defined for any two x, y ∈ g by ⟨x, y⟩ := Tr(xy), where the product is the +product of matrices. Then ⟨, ⟩ is non-degenerate and G-adjoint invariant. We define the Fourier +transform of any ϕ ∈ C∞ +c (g(A)) by +�ϕ(x) := +� +g(A) f(y)ψ(⟨x, y⟩)dy. +Based on the Poisson summation formula, we have an identity ([Yu21a, Theorem 5.7]): +(29) +Jg,e,ξ(ϕ) = q4−4gJg,e,ξ( �ϕ). +Proposition 5.7. ([Yu21b, Prop. 5.3.2]) We have the following results on Fourier transformation. +(1) For any place v, the Fourier transform of the characteristic function of g(Ov) can be calculated by +the following formula: +� +1g(Ov) = +1℘−dv +v +g(Ov). +(2) Let Iv be the Iwahori subalgebra of gv consisting of matrices in +� +Ov +Ov +℘v +Ov +� +and Iv+ its open +subset consisting of matrices in +� +℘v +Ov +℘v +℘v +� +. Then we have +� +1Iv = q−1 +v +1℘−dv +v +Iv+. +(3) (Springer’s hypothesis [Ka77][KV06].) Let U be a maximal torus of Gκv and θ any character of +U(κv), t a regular element in uv(κv) and ρ = ǫRG +Uθ the Deligne-Lusztig virtual representation of +G(κv) induced from (Tv, θ), where ǫ = 1 if U = T and ǫ = −1 if U is not split. Let +eρ := +� +Tr(ρ(x−1)), +x ∈ Kv; +0, +x /∈ Kv; +where x denotes the image of x under the map Kv −→ G(κv). Let Ωt ⊆ g(κv) the Ad(G(κv))- +orbits of t and Ωt ⊆ g(Ov) be the preimage of Ωt of the map g(Ov) −→ g(κv). +Then for any unipotent element u ∈ Kv ∩ Gv,unip, we have +eρ(u) = q−(4dv+1) +v +� +1℘−dv +v +Ωt(u − 1). +Now we come to the construction of ϕV = ⊗vϕV +v . The local components ϕV +v are defined so that +with the notations of Proposition 5.7, we have the follows: +• If v /∈ Scr ∪ V, +ϕV +v = +1g(Ov); +• If v ∈ Scr, +� +ϕVv = q−1 +v � +1℘−dv +v +Ωtv; +here, tv ∈ T(κv) for v ∈ Sr and tv ∈ U(κv) for v ∈ Sc is a regular element; + +38 +HONGJIE YU +• If v ∈ V, +� +ϕVv = +1 +vol(Iv)q−1 +v +1℘−dv +v +Iv+. +For the Fourier transform, we have +��fv(X) = q4dv +v +fv(−X), +it is then direct to see that +(30) +Jg,1 +unip( f V) = Jg,1 +nilp(ϕV). +We require that +∑ +v∈Scr +Trκv|FqTr(tv) = 0. +This ensures that +ϕV(z + x) = ϕV(x) +for any x ∈ g(A) and z ∈ zG(Fq). Note that the support of ϕ is contained in g(O), we deduce by +[Ch15, Th. 6.2.1] that +Jg,1(ϕV) = qJg,1 +nilp(ϕV). +Combining with the trace formula for Lie algebra (29) and (30), we have +(31) +J1( f V) = (q − 1)q3−4g−degV−deg Scr(∏ +v∈V +vol(Iv)−1)Jg,1( � +ϕV). +The result is then deduced by a geometric interpretation of Jg,1( � +ϕV) using Weil’s dictionary. We +refer the reader to [Yu21b] for details (the parameter ξ is set to be zero): the treat for components +v ∈ V are proved in [Yu21b, Prop. B.3], the treat for components v ∈ Scr is proved in [Yu21b, Cor. +5.3.9]. In fact, for GL2, we can obtain a simpler proof by slightly modifying the proof of [Yu21b, +Prop. B.3] to avoid using [Yu21b, Cor. 5.3.9]. +□ +5.6. Independence of parabolic residues and parabolic weights. We come back to assumptions +in Section 5.3 that V ⊆ Su, R = V ∪ Scr and DR = KX + ∑v∈R v. We consider the case that +the parameter ξ = (ξv)v∈R of parabolic weights is not necessarily trivial. Recall that we use +ξ = (ξx)x∈R to denote the family such that ξx = +1 +[κv:Fq]ξv for any point x ∈ R lying over v ∈ R. +Let t be the Lie algebra of T, which is a vector scheme of rank 2 defined over Fq. We have a +residue morphism over Fq: +qres : qMe,ξ +2,R(DR) −→ qRR, +where qRR = ∏x∈R tFq. It sends a quasi-parabolic Higgs bundle (E, ϕ, (Lx)x∈R) to the image of +ϕx in End(Lx) × End(E(x)/Lx) ∼= tFq for every x ∈ R. We define qR1 +R := ∏v∈R Rκv|Fqtκv as a +scheme over Fq where Rκv|Fq is the restriction of scalars from κv to Fq. We have +qRR ∼= qRR ⊗ Fq. +There is an obvious Gal(Fq|Fq)-action on qRR and qres is Gal(Fq|Fq)-equivariant. Therefore qres +defines a morphism of Fq-schemes which we denote still by qres: +qres : qMe,ξ +2,R(DR) −→ qRR. +The residue theorem (see [Ta68]) tells us that the morphism qres factors through the linear +subscheme of ”residue 0” qR1 +R: i.e., elements (tv)v∈R ∈ ∏v t(κv) such that +∑ +v∈R +Trκv|Fq(tv) = 0. +We have defined in Section 5.3 a residue morphism +res : M1 +2,V(DR) −→ R1 +Scr. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +39 +We view qR1 +Scr as the linear subspace of qR1 +R of those elements whose coordinates in V are zero. +Let qR1,rs +Scr ⊆ qR1 +Scr be the open sub-variety of elements (t1,x, t2,x)x∈R ∈ qR1 +R so that t1,x ̸= t2,x at +every point x ∈ R. It is an Fq-scheme. We have a finite morphism qR1 +R −→ R1 +R which is ´etale +when restricting to R1,rs +R . +Recall that we have defined +AR = H0(X, OX(DR)) ⊕ H0(X, OX(2DR)) +as a vector scheme over Fq. Yokogawa studied the Hitchin fibration for quasi-parabolic Higgs +bundles: +qMe,ξ +2,R −→ AR, +and he shows that it is a projective morphism ([Yo93, Co.5.12]). This morphism is Gm-equivariant. +Since Gm has only positive weights on AR, (qMe,ξ +2,R)Gm is contained in the zero fiber of the Hitchin +fibration, hence is projective. It is smooth since it is the Gm-fixed subvariety of a smooth variety. +Therefore (qMe,ξ +2,R)Gm is smooth projective (smoothness is proved in [Yo95, Th. 5.2]). In particular, +it has pure cohomologies. We remark that +(32) +(qMe,ξ +2,R)Gm = (Me,ξ +2,R)Gm. +Theorem 5.8. +(1) For any qo ∈ qR1(Fq), and parabolic weights (e, ξ) in general position (in partic- +ular if e = 1 and ξ = 0), we have +|qres−1(qo)(Fq)| = q3−4g−deg R|(Me,ξ +2,R)Gm(Fq)|. +(2) Suppose that Fq ̸= F2. Suppose that qo ∈ qR1(Fq). Varying the parabolic weights (e, ξ) but +remaining in general position and +ξv,1 ⩾ ξv,2 ⩾ ξv,1 − deg v +for any v ∈ R, the cardinality of the set qres−1(qo)(Fq) remains the same. +Proof. For the proof, we need first to introduce an ad`elic formula, the Lie algebra version of the +Arthur-Selberg trace formula, to express the number of Fq-points of the varieties concerned. +Suppose that e is an odd number. It is proved in [Ch15, Th. 6.2.1] that if the support of f is +contained in g(O), we have +Jg,e +χ ( f) = 0, +except if χ = (t − a)2 with a ∈ Fq. To generalize this result, we showed in [Yu21b, Theorem 3.2.6] +that if the support of f is contained in g(OR) ∏v∈R Iv (where OR = ∏v/∈R Ov) and if (e, ξ) is in +general position in the sense that +e + ∑ +v∈R +ǫv(ξv,1 − ξv,2) /∈ 2Z, +for any (ǫv)v∈R ∈ {1, −1}R, we have +(33) +Jg,e,ξ +χ +( f) = 0, +except if χ = (t − a)2 with a ∈ Fq. +Recall that we have fixed a divisor +D = KX + ∑ +v∈R +v = ∑ +v +nvv +on the curve X. Let qo ∈ qR1(Fq). We may suppose qo = (tv)v∈R with tv ∈ t(κv). Let +1R(qo) be +the function defined as the tensor product +� +v/∈R +1℘−nvg(Ov) ⊗ +� +v∈R +( +1 +vol(Iv) +1℘−nv +v +(tv+Iv+)), + +40 +HONGJIE YU +where Iv+ consists of elements in g(Ov) whose reduction modd-℘v belongs to n(κv). We have +shown in [Yu21a, Appendix B] that if ξv,1 ⩾ ξv,2 ⩾ ξv,1 − [κv : Fq] and (e, ξ) is in general position +(hence semistability coincides with stability), we have +(34) +Jg,e,ξ( +1R(qo)) = +1 +q − 1|qres−1(qo)(Fq)|. +The factor +1 +q−1 comes from the fact that there are q − 1 automorphisms for stable quasi-parabolic +Higgs bundles. +(1) Now we go back to the proof of (1). Note that if qo = 0 then +qMe,ξ +2,R(qo) ∼= Me,ξ +2,R(DR). +In this case, we can argue as in [Yu21b, Theorem 6.3.9]. Note that our argument in [Yu21b, Theo- +rem 6.3.9] needs qMe,ξ +2,R(DR) to be a fine moduli space to calculate its tangent sheaf. However, this +is not necessary because qMe,ξ +2,R(DR) −→ qMe,ξ +2,R(DR) is a gerbe. ´Etale locally, it is a neutral gerbe, +hence admits a section. Therefore, the coarse moduli scheme qMe,ξ +2,R(DR) admits a Poincar´e family +´etale locally, and the proof of [Yu21b, Theorem 6.3.9] can be modified slightly to work. +To complete the proof it is enough to show that Jg,e,ξ( +1R(qo)) is independent of qo ∈ R1(Fq). +We use the trace formula for Lie algebra (which we have discussed in the proof Theorem 5.6): +Jg,e,ξ( +1R(qo)) = q4−4gJg,e,ξ( � +1R(qo)). +The Fourier transform � +1R(qo) can be calculated by Proposition 5.7. We have +1g(OR) ⊗ +� +v∈R +( +q−1 +v +vol(Iv) +1Ivλv), +where λv is the composition of the maps Iv → t(κv) λv +−→ C× and λv : t(κv) → C× is the character +defined by +λv(t) = ψ(Trκv|FqTr(ttv)). +Since qo ∈ R1(Fq), the characters λv satisfy the condition +∏ +v∈R +λv|z(Fq) = 1, +where we have used the inclusion z(Fq) ⊆ z(κv) ⊆ Iv. It implies that for any a ∈ Fq and +χ = (t − a)2 we have +(35) +Jg,e,ξ +χ +( � +1R(qo)) = Jg,e,ξ +nilp ( � +1R(qo)), +where Jg,e,ξ +nilp ( � +1R(qo)) = Jg,e,ξ +t2 +( � +1R(qo)). As the function � +1R(qo) is contained in g(OR) ∏v Iv, after +(33) and (35), we deduce that +(36) +Jg,e,ξ( � +1R(qo)) = qJg,e,ξ +nilp ( � +1R(qo)). +Note that Jg,e,ξ +unip( � +1R(qo)) is independent of (λv)v∈R by definition. This finishes the proof of the part +(1). +Next, we prove the part (2) of the Theorem. We need a lemma first. +Lemma 5.9. Suppose that Fq ̸= F2. There exists a family of characters (χv)v∈R, χv : T(κv) → C×, so +that +∏ +v∈R +χv|Z(Fq) = 1, +and the following properties are satisfied. Let ρ be the representation of ∏v Iv obtained by the tensor +product of the representations Iv −→ Tv(κv) +χv +−→ C×. For any automorphic representation π of G(A), if + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +41 +π contains ρ, i.e., the ρ-isotypic part πρ ̸= 0, then π is cuspidal. Moreover, for any character λ of G(A) +which factors through deg ◦ det, we have +π ⊗ λ ∼= π =⇒ λ = 1. +Proof. Let v0 ∈ R. Let χ1 be a primitive character of κ× +v0 −→ C×, i.e., an injective homomorphism. +Since Fq ̸= F2,the character χ1 is non-trivial on F× +q . We set χv = 1 for any v ̸= v0, and χv0 = +(χ1, χ−1 +1 ). +We prove that the family (χv)v∈R satisfies all the properties we need. +We have clearly +∏ +v∈R +χv|F× +q = 1. +Given an automorphic representation π, it is either cuspidal, or it is a sub-quotient of a par- +abolic induction IndG(A) +B(A)µ, for a Hecke character µ = (µ1, µ2) of T(A)/T(F). The latter case is +impossible. In fact, the condition πρ ̸= 0 implies (IndG(A) +B(A)µ)ρ ̸= 0. This, in turn, implies that µ is +unramified outside {v0} and +HomT(Ov0)(χv0, µ|T(Ov0)) ̸= 0, or HomT(Ov0)(χw +v0, µ|T(Ov0)) ̸= 0, +here we review χv0 as a character of T(Ov0), and χwv0 = (χ−1 +1 , χ1). In particular, +µ1|F× +q = χ1|F× +q , or µ1|F× +q = χ−1 +1 |F× +q . +This implies that µ1|F× +q ̸= 1 which contradicts the fact that µ1 is a Hecke character: it is trivial on +F×. +For the last assertion of the lemma, we use Langlands correspondence to prove it. Suppose that +(σ, i : σ ∼ +−→ F∗ +X/Fqσ) is the Weil sheaf that corresponds to the cuspidal automorphic representation +π. The sheaf σ has a rank 2 and is smooth over (X − {v0}) ⊗ Fq. The local monodromies of +σ over punctured discs X(∗) +x +(defined in the Introduction) centered at points x in {v0} ⊗ Fq are +semisimple tame local systems so that a tame generator has as eigenvalues (ζqi, ζ−qi)i=1,2,...,dv0, +where ζ is a (qdv0 − 1)th primitive root of unity. If the assertion is not correct, then σ = σ1 ⊕ σ2 +and the Frobenius action F∗ +X/Fq exchanges isomorphism classes of σ1 and σ2. Hence, σ1 is fixed +by F∗2 +X/Fq. Note that the ramifications of σ1 at points in {v0} ⊗ Fq must be multiplication by +(ζǫiqi)i=1,2,...,dv0, where ǫi ∈ {1, −1}. If dv0 is odd, then {v0} ⊗ Fq is cyclically permuted by F∗2 +X/Fq, +hence ǫi has the same sign and their product is +ζ±(qdv0 −1/q−1) ̸= 1, +which is impossible. If dv0 is even, then ǫ2i (resp. ǫ2i+1) have the same sign. This is also not +possible because the product of eigenvalues of ramifications of σ1 is one of the following +ζ(qdv0 −1/q−1), ζ−(qdv0 −1/q−1), ζ(qdv0 −1/q+1), ζ−(qdv0 −1/q+1). +This is again not possible because none of them is 1. +□ +We choose a family of characters (χv)v∈R as in the Lemma 5.9. Let h be the function +1G(OR) ⊗ +� +v∈R +( +1 +vol(Iv) +1Ivχv). +Since the support of h is contained in G(OR) ∏v Iv, we have +JG,e,ξ +χ +(h) = 0, +except if χ = (t − a)2 ∈ EG. Since +∏ +v∈R +χv|Z(Fq) = 1, + +42 +HONGJIE YU +we deduce, similar to the Lie algebra case that +JG,e,ξ(h) = (q − 1)JG,e,ξ +unip (h), +where JG,e,ξ +unip (h) = JG,e,ξ +(t−1)2(h). It is direct to see that the map X �→ 1 + X from the set of nilpotent +elements in g(F) to that of unipotent elements gives us an identity: +JG,e,ξ +unip (h) = q−degRJg,e,ξ +nilp ( � +1R(qo)). +Therefore we have +(37) +JG,e,ξ(h) = q− deg R+3−4g|qres−1(qo)(Fq)|. +Recall that a ∈ A× is a degree 1 id`ele, viewed as a scalar matrix. By Lemma 5.9, we know that +the regular action R(h) on L2(G(F)\G(A)/aZ) is a projection whose image lies inside the space +of cuspidal automorphic forms and the regular action R(h) on L2(T(F)N(A)\G(A)/aZ) is zero. +It shows that for any x ∈ G(A), +0 = ∑ +γ∈T(F) ∑ +i∈Z +� +N(A) h((δx)−1(γn)δaix)dn +and hence +JG,e,ξ(h) = 1 +2(Tr(R(h)|L2 +cusp(G(F)\G(A)/aZ)) + (−1)eTr(R(h)ǫ|L2 +cusp(G(F)\G(A)/aZ))) += 1 +2(Tr(R(h)|L2 +cusp(G(F)\G(A)/aZ)). +which is independent of (e, ξ). Recall that ǫ is the sign character on G(A) that factors through +deg ◦ det. By (37), this finishes the proof. +□ +6. PROOF OF THE MAIN THEOREMS +We will prove our main result Theorem 1.1. The main ingredient of the proof is Theorem 4.3 +and Theorem 5.6. These two results give an expression for |E2(R)Frob∗|. It remains to apply this +result to the curve X ⊗ Fqk over Fqk and study how |E2(R)Frob∗| varies for k ∈ N∗. +6.1. Functions of Lefschetz type. We continue to use our notations in the introduction. Let’s +prove first the following proposition. +Proposition 6.1. Let A be a set with a permutation τ acting on it. We use O(τ|A) to denote the number +of orbits of τ acting on A. +(1) Let k �→ αA(k) be the function that αA(k) = |A| if τk is a cyclic permutation on A and αA(k) = 0 +otherwise. It is of Lefschetz type. +(2) We define βA(k) by βA(k) = 0 if τk has an orbit of even length, βA(k) = 2O(τk|A)−1 if all orbits +are of odd length. It is of Lefschetz type. +(3) We define γA(k) by +γA(k) = (−1)O(τk|A). +It is of Lefschetz type. +(4) We define ωA = 1 +2(αA + (−1)|A|βA). It is of Lefschetz type. +(5) If Scr ̸= ∅, then cR/2 + bR/2 is of Lefscehtz type. +(6) The function bR is of Lefschetz type. +(7) If Scr ̸= ∅, then the functions +k �→ cR(k)αSu(k)/2 + bR(k)βSu(k)/2 +are of Lefschetz type. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +43 +Proof. (1) If τ is not a cyclic permutation on A, then neither is τk for every k ⩾ 1. In this case, αA +is constantly 0, and the assertion is trivial. +We suppose in the following that τ is a cyclic permutation. +Let |A| = pa1 +1 · · · pass be a prime decomposition of n with pi being different prime numbers and +ai > 0. Let ζpi be a primitive pi-th roots of unity. Then +αA(k) = |A|, +if pi ∤ k for all pi, otherwise +αA(k) = 0. +It’s direct to verify that we have the following identity: +(38) +αA(k) = ∏ +i +(pai +i − pai−1 +i +pi +∑ +j=1 +ζjk +pi). +Since roots of unity are q-Weil integers of weight 0, the statement follows. +(2) Let n be an odd integer. We first prove the following assertion by induction on the number +of prime divisors of n (counting multiplicities). +(∗) For any odd integer m such that (m, n) = 1, the function +k �→ 1 +m(2φ(m)(n,k)−1 − 1) +is a function of Lefschetz type. +We only need the case that m = 1 of the assertion (∗), but this stronger assertion is easier to +prove by induction. The case that n = 1 is trivial since the function is constant in k, and it is an +integer by Fermat’s little theorem. +Let l be any prime number, not dividing nm. Let β ∈ N. Suppose that the assertion (∗) holds +for n, nl, . . ., nlβ−1. Note that we have the following identity +1 +m(2φ(m)(nlβ,k)−1 − 1) − 1 +m(2φ(m)(n,k)−1 − 1) = +β +∑ +j=1 +1 +m(2φ(m)(nlj,k)−1 − 2φ(m)(nlj−1,k)−1) += +β +∑ +j=1 +2φ(m)(nlj−1,k)−1 +m +(2φ(m)((nlj,k)−(nlj−1,k)) − 1) += +β +∑ +j=1 +2φ(m)(nlj−1,k)−1 +mlj +(2φ(mlj)(n,k) − 1) +lj +∑ +s=1 +ζsk +lj . +In the last equality, we have used the fact if lj ∤ k, then (nlj, k) − (nlj−1, k) = 0 and if lj | k, then +(nlj, k) − (nlj−1, k) = (n, k)φ(lj). We have also used the fact that φ(m)φ(lj) = φ(mlj) since l ∤ m. +Since the product of Lefschetz type functions and integral multiple of Lefschetz type functions +are of Lefschetz type, we deduce that the assertion (∗) is correct for nlβ as well. By induction, we +obtain the result needed. +To prove that βA is of Lefschetz type, it is sufficient to prove that τ is a cyclic permutation on +A by multiplicativity on orbits and the fact that the integral multiple of the function of Lefschetz +type is again of Lefschetz type. Suppose that |A| = 2al with l being an odd integer. If a = 0, then +βA is the function +k �→ 2(|A|,k)−1 +which is of Lefschetz type by the above result (by setting m = 1). Suppose a ⩾ 1. In this case, we +have +βA(k) = +� +22a(l,k)−1, +2a | k; +0, +2a ∤ k. + +44 +HONGJIE YU +Since a ⩽ 2a − 1, the function βA is of Lefschetz type, because we have +(39) +βA(k) = 22a−a−1(2(l,k)−1)2a 2a +∑ +j=1 +ζk +2a. +We have shown that k �→ 2(l,k)−1 is of Lefschetz type, therefore so is βA. +(3) Since the product of Lefschetz type function is still of Lefschetz type, it suffices to consider +the case that τ is a cyclic permutation of A. If A has odd cardinality, then (−1)O(τk|A) = −1. +Suppose that |A| = 2am with m being m odd and a ⩾ 1. Then O(τk|A) = (2am, k), and +(−1)O(τk|A) = +� +−1, +2 ∤ k; +1, +2 | k. +Therefore in this case +(−1)O(τk|A) = (−1)k. +This is a function of Lefschetz type, and we have proved the assertion. +(4) Let’s consider ωA. If τ is not a cyclic permutation on A, then αA = 0. Let A = A1 ∪ A2 be a +partition of A into non-empty τ-stable subsets. Then +ωA = (−1)|A|βA1βA2. +Therefore ωA is of Lefschetz type. +In the following, we suppose that τ is a cyclic permutation. It’s sufficient to consider αA−βA +2 +. +Let f : N∗ −→ Z be a periodic function of period n. Then we have +f(k) = +n +∑ +i=1 +∑n +j=1 f(j)ζ−ij +n +n +ζki +n . +Therefore, f is of Lefschetz type if and only if +(40) +∑n +j=1 f(j)ζ−ij +n +n +∈ Z +for i = 1, . . . , n. We will use this criterion to prove that ωA is of Lefschetz type. +If n := |A| is an odd integer. By (40) and the fact that βA is of Lefschetz type, we know that for +any i, the number +n +∑ +j=1 +2(n,j)−1ζ−ij +n +is an integer that is divisible by n. Moreover, +n +∑ +j=1 +αA(k)ζ−ij +n += ncn(i), +where cn(i) is the sum ith power of primitive nth roots of unity, the Ramanujan’s function. The +function i �→ cn(i) takes an integral value (since αA is of Lefschetz type). We need to prove that +the number +(41) +2 +n +∑ +j=1 +ωA(j)ζ−ij +n += − +n +∑ +j=1 +2(n,j)−1ζ−ij +n ++ ncn(i) +is divisible by 2n. Since we’re in the case that n is an odd number, and (41) is divisible by n, we +need to show that it is divisible by 2. Note that we have +n +∑ +j=1 +2(n,j)−1ζ−ij +n += ∑ +d|n +2d−1cn/d(i). +Modulo 2, the expression (41) equals (n − 1)cn(i). Since n is odd, this is 0 modulo 2. We’re done. +Now suppose that n = |A| is an even integer. If 4 | n, then clearly both 1 +2αA(k) and 1 +2 βA(k) are +of Lefschetz type. We can see it from equations (38) and (39). + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +45 +If 4 ∤ n, we need to prove that for any i +2 +n +∑ +j=1 +ωA(j)ζ−ij +n += ncn(i) − +n/2 +∑ +j=1 +2(n,2j)−1ζ−2ij +n +is divisible by 2n. As we have shown before, it is divisible by n. Therefore, it remains to show +that it is divisible by 4. Note that +n/2 +∑ +j=1 +2(n,2j)−1ζ−2ij +n += ∑ +d| n +2 +22d−1cn/2d(i). +Modulo 4, we need to prove that 2cn(i) − 2cn/2(i) is divisible by 4. By M¨obius inversion formula, +we have +cn(i) = ∑ +d|(n,i) +µ(n/d)d, +where µ is the M¨obius function. Note that if the divisor d of n is odd, we have +µ(n/d) = −µ(n/2d). +Therefore, if i is odd, then +cn(i) = −cn/2(i). +If i is even, we have +cn(i) − cn/2(i) = ∑ +d|(n,i) +µ(n/d)d − +∑ +d|(n/2,i) +µ(n/2d)d. +The sum over d | (n, i) can be decomposed into two parts following d is odd or d is even. We +deduce that +cn(i) = +∑ +d|(n/2,i) +µ(n/d)d + +∑ +d|(n/2,i) +µ(n/2d)2d += − +∑ +d|(n/2,i) +µ(n/2d)d + +∑ +d|(n/2,i) +µ(n/d)2d += cn/2(i). +We conclude that in either case, the number 2cn(i) − 2cn/2(i) is divisible by 4. +(5) Let PR/S2 be the quotient of PR by the action of S2 = {1, σ}. Since Scr is non-empty, every +point in PR/S2 has a preimage of cardinality 2 in PR. The action of Frob∗ defines an action of +PR/S2 since it commutes with σ. For any e = (a, σ(a)) ∈ PR/S2, if +Frob∗k(e) = e, +then we have either Frob∗k(e) = e or we have Frob∗k(e) = σ(e). Therefore, +cR(k) + bR(k) = 2|(PR/S2)Frob∗k|, +∀k ⩾ 1. +This proves the result. +(6) If Scr = ∅, then bR(k) = |PR| is either 1 or 0. If Scr ̸= ∅, then it follows from (5), because +bR = 2 cR+bR +2 +− cR. +(7) The function under consideration equals +cR + bR +2 +αSu − bR +αSu − βSu +2 +. +It results from (4), (5), and (6). +□ + +46 +HONGJIE YU +6.2. HiggR is of Lefschetz type. +Theorem 6.2. Let o = (ov)v∈Scr ∈ R1 +Scr(Fq) so that every polynomial ov has distinct roots and is split +over κv if v ∈ Sr, and is irreducible if v ∈ Sc. +(1) The function over N∗: +HiggR(k) = +∑ +V⊆Su⊗Fqk +(−1)|Su⊗Fqk−V|2|V|q−k(4g−3+|V|+|Scr|)|M1 +2,V(o)(Fqk)|. +is of Lefschetz type in k. +(2) The number HiggR(k) is divisible by Pic(k) and the quotient function +k �→ HiggR(k)/Pic(k) +is still of Lefschetz type. +Proof. (1) Let Fq : a �→ a1/q be the geometric Frobenius element in Gal(Fq|Fq) and Φq its inverse, +the arithemetic Frobenius element. Let +U(1/q) := U ×Spec(Fqn),Fq Spec(Fqn), +and +U(q) := U ×Spec(Fqn),Φq Spec(Fqn). +Let’s prove a lemma first. +Lemma 6.3. Let V ⊆ Su ⊗ Fqk. Let d be the least positive integer such that Frobd(V) = V. Then +M1 +2,V(o) is defined over Fqd-structure and +M1 +2,Frob(V)(o) ∼= M1 +2,V(o)(1/q). +In particular, we obtain a linear map: +F∗ +V : H∗ +c (M1 +2,V(o), Qℓ) −→ H∗ +c (M1 +2,Frob(V)(o), Qℓ) +whose d-times composition coincides with the action of geometric Frobenius element Fqd ∈ Gal(Fq|Fqd). +Proof. Let R = (Scr ⊗ Fqd) ∪ V, and R = Scr ∪ V, where V = V ⊗Fqd Fq. +Note that the functor (·)(q) is an equivalence of categories from (Sch/Fq), the category of +schemes over Fq, to itself. Its d-fold iterate is the identity functor. Suppose that (E, ϕ, (Lx)x∈V) +is a quasi-parabolic Hitchin bundle over X, then E (1/q) is again a vector bundle over X, but the +parabolic structures are set at points in Frob(V). The association E �→ E (1/q) is a bijection between +qM1 +2,Frob(V)(Frob(DR))(Fq) and qM1 +2,V(DR)(1/q)(Fq). Moreover since the relative Frobenius mor- +phism, denoted by FV: +FV : qM1 +2,V(DR)(1/q) −→ qM1 +2,V(DR) +induces a bijection of geometric points. We obtain a bijection θ between qM1 +2,V(DR)(1/q)(Fq) and +the set of quasi-parabolic Hitchin bundles over X with parabolic structures in Frob(V) for the +divisor Frob(DR). +With the bijection θ above, we can apply [Yo93, Th. 4.6, (4.6.5)]: both qM1 +2,Frob(V)(DFrob(R)) +and qM1 +2,V(DR)(1/q) solve the same moduli problem so they’re isomorphic. In fact, let T be a +scheme defined over Fq and (E, ϕ, (Lx)x∈Frob(V)T be a flat family of quasi-parabolic Hitchin bun- +dle over XT with parabolic structures in Frob(V). The associated quasi-parabolic Hitchin bundle +(E, ϕ, (Lx)x∈Frob(V)(q) +T ) is a falt family of quasi-parabolic Hitchin bundles over XT(q) with par- +abolic structures in V. By [Yo93, Th. 4.6, (4.6.4)], we obtain a morphism T(q) → qM1 +2,V(DR). +Applying the functor (·)(1/q), we obtain T → qM1 +2,V(DR)(1/q). By the definition of θ, we see that +it has the desired property to apply [Yo93, Th. 4.6, (4.6.5)]. + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +47 +It’s clear that (R1 +R)(1/q) ∼= R1 +Frob(R). We have the relative Frobenius morphism: +(R1 +R)(1/q) −→ (R1 +R). +Since o ∈ R1 +Scr(Fq), its embedding in R1 +R(Fq) is sent to o viewed as a point in R1 +R(Fq) via the +above morphism. These imply the first assertion. +We still denote the induced relative Frobenius morphism by FV: +FV : M1 +2,Frob(V)(o) −→ M1 +2,V(o). +Since FV is a universal homeomorphism, it induces an isomorphism of ℓ-adic cohomology with +compact support. Note that by definition, the composition +FV ◦ FFrob(V) ◦ · · · ◦ FFrobd−1(V) +coincides with the Frobenius endomorphism deduced as base change to Fq of the qd-Frobenius +morphism of M1 +2,V(o) (it is defined over Fqd). On ´etale cohomology, its action coincides with +the geometric Frobenius element Fqd of the Galois group Gal(Fq|Fqd). The last assertion hence +follows. +□ +Let’s choose a total order on Su. Note that Frob acts on Su. For any V ⊆ Su, let +inv(Frob|V) +be the inversion number of Frob on V, i.e., the number of pairs (x1, x2) of points in V such that +x1 < x2 and Frob(x1) > Frob(x2). +For any subset V of Su, let’s consider H∗(P1, Qℓ)⊗V, where the tensor product is understood as +the tensor product of graded vector space. Let α be a generator in H0(P1, Qℓ) and β be a generator +in H2(P1). +We set +τ : H∗(P1, Qℓ)⊗V −→ H∗(P1, Qℓ)⊗Frob(V) +as the map of graded vector spaces which sends an element in H∗(P1, Qℓ)⊗V represented by +(axαx + bxβ)x∈V to (axαFrob(x) + bxβFrob(x))x∈Frob(V): +τ((axαx + bxβx)x∈V) = (axαFrob(x) + bxβFrob(x))x∈Frob(V). +Let +H∗ +V := H∗ +c (M1 +2,V(o), Qℓ) ⊗ H∗(P1, Qℓ)⊗V. +Let ς be a linear endomorphism: +ς : +� +V⊆Su +Hi +V −→ +� +V⊆Su +Hi +V, +defined by +ς = ⊕V(−1)inv(Frob|V)q3−4g−|V|−|Scr|F∗ +V ⊗ τ. +We will show that: +(∗) the eigenvalues of ς are q-Weil integers. +(∗∗) +HiggR(k) = ∑ +i +(−1)iTr(ςk| +� +V⊆Su +Hi +V). +These two properties suffice to prove the theorem. +For (1), since the eigenvalues of Frobenius action on ℓ-adic cohomology are q-Weil integers, +the only non-trivial point is to show that the eigenvalues of ς are divisible by +q4g−3+|V|+|Scr|. + +48 +HONGJIE YU +It suffices to show that, for some k, the eigenvalues of ςk are divisible by (qk)4g−3+|V|+|Scr|. We +need a lemma. Recall that we have a finite morphism qR1 +R −→ R1 +R and a residue morphism +qres : qM1 +2,R −→ qR1 +R. +Lemma 6.4. Let V be a subset of Su. For any qo ∈ qR1 +R(Fq) with image o = (ov)v∈R ∈ R1 +Scr(Fq) ⊆ +R1 +R(Fq). Suppose that ov has distinct roots for v ∈ Scr. +We have an isomorphism of Fq-schemes: +(42) +qres−1(qo) ∼= M1 +2,V(o). +Proof. Note that by forgetting the parabolic structure, we have a commutative diagram: +qM1 +2,R(DR) +qres +−−−−→ qR1 +R +� +� +qM1 +2,V(DR) +res +−−−−→ R1 +R +. +It’s sufficient to prove that it is a Cartesian diagram restricting to R1,rs +Scr . Using the modular de- +scription and [Yun11, 2.1.2], we’re reduced to prove that the following diagram is Cartesian +(43) +[brs/B] −−−−→ brs � B ∼= trs +� +� +[grs/G] −−−−→ +grs � G +. +In fact, we can use Grothendieck’s simultaneous resolution ˜g −→ g, where ˜g = {(x, b)|x ∈ +g, b ∋ x}, then we have +[˜g/G] ∼= [b/B]. +We obtain the following Cartesian diagram, +˜grs +π2 +−−−−→ brs � B ∼= trs +π1 +� +πt +� +grs +πg +−−−−→ +grs � G +, +where π1 and π2 are G-equivariant. Suppose we are given an Fq-scheme S, a G-torsor E over S +with a G-equivariant map α : E −→ grs, and a morphism β : E −→ trs such that πg ◦ α = πt ◦ β, +we obtain a unique morphism γ : E −→ ˜grs whose composition with π1 (resp. π2) equals with α +(resp. β). By uniqueness, we see that γ must be G-equivariant otherwise, we can replace γ by its +G-conjugations. Therefore, the diagram 43 is Cartesian. +□ +We come back to the assumption of the theorem. Note that there is a point qo ∈ qR1 +Scr(Fq2) such +that o is the image of qo via the morphism qR1 +Scr → R1 +Scr. Applying the Lemma 6.4, we deduce +that +|M1 +2,V(o)(Fq2kdV)| = |qM1 +2,V∪(Scr⊗FqdV )(qo)(Fq2kdV)|, +∀k ⩾ 1, +where dV is the least integer such that V is defined over FqdV . Note that qo is viewed as a point +in RV∪(Scr⊗FqdV )(Fq2dV ). Then (∗) is a corollary of the part (1) of Theorem 5.8 and Grothendieck- +Lefscehtz fixed point formula. +For (∗∗), note that for any k, only those V fixed by ςk will contribute a non-trivial trace. +These are exactly those coming from subsets of Su ⊗ Fqk. For such a V, the alternative trace +∑i(−1)iTr(ςk|Hi +V) equals +(−1)O(Frobk|V)−|V|q3−4g−|V|−|Scr| + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +49 +times +∑ +i +(−1)iTr(Frob∗k/dV|Hi +c(M1 +2,V(o), Qℓ))∑ +i +(−1)iTr(τk|H∗(P1, Qℓ)⊗V). +Recall that O(Frobk|V) means the number of orbits of Frobk on V. +We have +∑ +i +(−1)iTr(Frob∗k/dV|Hi +c(M1 +2(o)Fq, Qℓ)) = |M1 +2,V(o)(Fqk)|. +It reduces to prove that +∑ +i +(−1)iTr(τk|H∗(P1, Qℓ)⊗V) = 2O(Frobk|V). +By multiplicativity of the two sides, it suffices to consider the case that Frobk has only one orbit +in V, in which case we can do the calculation by choosing a basis: (δxα + (1 − δx)β)x∈I where +(δx)x∈I ∈ {0, 1}I. It is clear that in this basis, τk is a permutation matrix, and there are exactly two +elements in the basis fixed by τk: those given by δx = δx′ for all x, x′ ∈ I. Therefore the left hand +side is 2 as well. +(2) For the second assertion, Deligne ([De15, 6.4, 6.5]) has proven a result that can be applied to +show that |M1 +2,V(o)(Fqk)| is divisible by |Pic0 +X(Fqk)|. Let A = Pic0 +X ⊗ FqdV and A its base change +to Fq. Note that A acts on M1 +2,V(o) by tensor on the vector bundles, and we have a morphism +f : M1 +2,V(o) −→ A which sends a Higgs bundle to the determinant of its underlying vector +bundle. This morphism is clearly equivariant for the action of Pic0 +X defined above and the action +of Pic0 +X on itself by L0 : L1 −→ L1 ⊗ L⊗2 +0 . In [De15, 6.4, 6.5], Deligne proves that under these +hypotheses, the sheaf Ri f!Qℓ is smooth and semisimple, and there is a sheaf Hi over Spec(Fq) +such that the invariant sub-sheaf of Ri f!Qℓ under the action of π1(A) satisfies: +(Ri f !Qℓ)π1(A⊗Fq) ∼= a∗Hi|A, +where a : A −→ Spec(FqdV ) is the structure morphism. The Grothendick-Lefschetz fixed points +formula implies that: +|M1 +2,V(o)(Fqk)| = |Pic0 +X(Fqk)|Tr(F∗(k/dV) +qd +|Hi|Spec(Fq)). +Note that |Pic0 +X(Fqk)| = ∏ +2g +i=1(1 − σk +i ) where σi are q-Weil numbers of the curve X which, for any +embedding in C, have absolute value q +1 +2 . Suppose that the eigenvalues of F∗ +qdV on Hi|Spec(Fq) are +βi, then βi are also q-Weil numbers of M1 +2,V(o). In particular, their quotients by q4g−3+|V|+|Scr| are +q-Weil integers. This finishes the proof. +□ +6.3. Proof of Theorem 1.1. The proof is based on Theorem 4.3 where we have computed the +number of cuspidal automorphic representations which corresponds to E2(R)Frob∗ (Theorem 3.3). +We need to have an expression for E2(R)Frob∗k for k ⩾ 1. Since +X ⊗Fq Fq ∼= (X ⊗Fq Fqk) ⊗Fqk Fq, +and the Frobenius endomorphism of X obtained from X ⊗Fq Fqk is the kth power of the Frobe- +nius obtained from X, we can apply this theorem to the function field F ⊗Fq Fqk. The only dif- +ficulty remains that the ramification type on the automorphic side may change when k varies. +For example, a place can split into several places, and a supercuspidal representation can become +non-supercuspidal after base change. +Let’s explain how ramification types on the automorphic side change when k varies. First, a +place v ∈ S of degree n corresponds to an orbit of length n of Frobenius endomorphism on S. +A place of F of degree n splits into (n, k)-points of degree n/(n, k) of F ⊗Fq Fqk for k ⩾ 1. For +ramification types, suppose that +S ⊗Fq Fqk = Sr(k)∐ Sc(k)∐ Ss(k)∐ Su(k), + +50 +HONGJIE YU +is a decomposition following the ramification type furnished by Theorem 3.2. Then we have +Su(k) = Su(1) ⊗Fq Fqk, +and +Ss(k) = Ss(1) ⊗Fq Fqk. +The sets Sr(k) and Sc(k) behave differently. If k is an odd number, we have +Sr(k) = Sr(1) ⊗Fq Fqk, +and +Sc(k) = Sc(1) ⊗Fq Fqk. +However, if k is an even number, we have +Sr(k) = (Sr(1) ⊗Fq Fqk) ∪ (Sc(1) ⊗Fq Fqk), +and +Sc(k) = ∅. +Note that we have +S2Pic0 +X(Fq) = 1 +2 +� +|Pic0 +X(Fq2)| + |Pic0 +X(Fq)|2 +� +. +We obtain the cardinality |E2(R)Frob∗| by comparing Theorem 4.3 and Theorem 5.6 by the trace +formula: +J1 +spec( f) = J1 +geom( f). +The theorem is then just a reformulation of the results using definitions of the functions HiggR, +αSu, βSu, ηSu and ωSu in Proposition 6.1 and the vanishing on cR and bR in Lemma 4.7. +It is tedious but direct and easy to verify. Let us be satisfied to explain how to verify the most +complicated case that Scr ̸= ∅ and Su ̸= ∅. It should be divided into some sub-cases. If Sc = ∅, +then |E2(R)Frob∗k| is given by Higg(k) minus the error terms in one of the cases (13), (14), (15), (16) +of Theorem 4.3. If Sc ̸= ∅, then for k odd, |E2(R)Frob∗k| is given by Higg(k) minus the error terms +in one of the cases (1), (2), (4) (5) of Theorem 4.3 but for k even, it is given by Higg(k) minus the +error terms in one of the cases (13), (14), (15), (16). We use the definition of αSu and βSu to write +the result in a uniform formula. One needs to note that by Lemma 4.7, bR(2k) = 0 if deg Sc is odd +and cR(2k + 1) = 0 if Sc ̸= ∅. +7. CASE THAT g = 0 +7.1. Case that g = 0. We are going to give another expression for HiggR(k) when g = 0. Let +R = Scr ∪ Su and DR = KX + ∑v∈R v. +Suppose (e, ξ) ∈ Z × (Q2)R is in general position. We have a Gm-action on Me,ξ +2,R = Me,ξ +2,R(DR) +given by dilation on the Higgs field. We have a modular description for the fixed points variety +due to Hitchin and Simpson. +Suppose that (E, θ, (Lx)x∈R) is a parabolic Higgs bundle fixed by Gm-action, then (E, θ) ∼= +(E, tθ) for any t ∈ Gm. By arguments of [Si92, Lemma 4.1], either θ = 0 and the underlying +parabolic bundle (E, (Lx)x∈R) is ξ-semistable or if θ ̸= 0, E is decomposed as a direct sum of line +bundles +E = L1 ⊕ L2, +and θ is obtained by +θ : L2 −→ L1(KX + ∑ +x∈R +x). +Note that if g = 0, the first case does not happen as there are no semistable parabolic Higgs bun- +dles of rank 2 when the stability is in general position. Let f : (E, θ, (Lx)x∈R) ∼ +−→ (E, tθ, (Lx)x∈R) + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +51 +be an isomorphism of parabolic Higgs bundles. Then f has constant coefficient in Fq. Then we +have +� +f θ = tθ f; +f(Lx) = Lx, +∀x ∈ R. +Let λ be an eigenvalue of f, then Eλ := ker( f − λ)2 is a subbundle of E and θ sends Eλ to Etλ. If θ +is non-zero, then Eλ and Etλ are non-zero and therefore E = Eλ ⊕ Etλ. In this case, either +Lx = Eλ,x +or +Lx = Etλ,x. +Therefore (Me,ξ +2,R)Gm consists of so-called graded parabolic Higgs bundles, which we will denote +by grMe,ξ +2,R. +Theorem 7.1. Suppose that Fq ̸= F2, ξv,1 = ξv,2 for v ∈ Su and ξ is in general position. Let grMe,ξ +2,R(Su) +be the open subvariety of grMe,ξ +2,R = (Me,ξ +2,R)Gm consisting of those graded semistable parabolic Higgs +bundles (E, θ, (Lx)x∈R) such that θx ̸= 0 for any x lying over points in Su. +Suppose that g = 0, and Sc = ∅, then we have +|grMe,ξ +2,R(Su)(Fqk)| = HiggR(k), +∀k ⩾ 1. +Proof. From Theorem 5.8 and Lemma 6.4, for any k ⩾ 1, we have +HiggR(k) = +∑ +V⊆Su⊗Fqk +(−1)|Su⊗Fqk−V|2|V||grMe,ξ +2,V∪(Sr⊗Fqk)(Fqk)|. +It suffices to verify the Theorem for the case k = 1. +Let (E, θ, (Lx)x∈R) be a graded parabolic Higgs bundles. For each x ∈ R, +θx : Ex −→ E(KX + ∑ +x∈R +x)x, +preserves the parabolic structure. It means that Imθx ⊆ Lx and θx(Lx) = 0. Suppose that θx is +zero, then it is possible that Lx = L1,x or Lx = L2,x. If θx is non-zero, then we can only have +Lx = L1,x. We obtain a stratification for any T ⊆ Su and x ∈ Su − T, +grMe,ξ +2,R(T) = N(1x) ∪ N(2x) ∪ grMe,ξ +2,R(T ∪ {x}). +where N(ix) consists of those (E, θ, (Lx)x∈R) in grMe,ξ +2,R(T) such that θx = 0 and Lx = Li,x. Note +that as a variety over Fq, Nix ∼= grMe,ξ +2,R−{x}(T). Let v ∈ Su − T be a closed point in Su. Repeat the +procedure above, we obtain a decomposition of grMe,ξ +2,R(T) as a disjoint union by locally closed +subvarieties: +grMe,ξ +2,R(T ∪ {v}) ∪ +� +(ax)x∈{v}∈{1,2}{v} +N((ax)x∈{v}), +where N((ax)x∈{v}) consists of those (E, θ, (Lx)x∈R) in grMe,ξ +2,R(T) such that so that θx = 0 for all +x ∈ {v} and Lx = Lax,x. +Now we must consider how to descend to Fq. Since for an Fq-sub-variety Z defined over Fq +of grMe,ξ +2,R = (grMe,ξ +2,R)Fq, it comes from a variety defined over Fq if it is fixed by Frobenius, i.e., + +52 +HONGJIE YU +Z(q) = Z as sub-varieties of grMe,ξ +2,R where Z(q) is defined by the Cartesian diagram: +Z(q) +−−−−→ +Z +� +� +Spec(Fq) +x�→xq +−−−−→ Spec(Fq) +. +Here it is important that Z(q) = Z instead of just isomorphism; otherwise, we don’t have a descent +datum. The equality here is another way to express the commutativity of the following diagram: +(Me,ξ +2,R)(q) +∼ +−−−−→ (Me,ξ +2,R) +� +� +Z(q) +∼ +−−−−→ +Z +. +Suppose T comes from a subset of Su. We have +N((ax)x∈{v}) = N((aFrob(x))x∈{v})(q). +Therefore, we see that N((1x)x∈{v}) and N((2x)x∈{v}) are defined over Fq and the variety +� +(ax)x∈{v}∈{1,2}{v}−{(1x)x∈{v},(2x)x∈{v}} +N((ax)x∈{v}) +is defined over Fq which has no Fq-points, since F∗q permutes its components without any fixed +component. Since ξv,1 = ξv,2 for v ∈ Su, as varieties, we have +N((1x)x∈{v}) ∼= grMe,ξ′ +2,R−{v}(T), +and +N((1x)x∈{v}) ∼= grMe,ξ′ +2,R−{v}(T), +where ξ′ = (ξv)v∈R−{v} ∈ (Q2)R−{v} (it is still in general position because of our assumption). +We deduce that for any T ⊆ Su and v ∈ Su − T, +|grMe,ξ +2,R(T)(Fq)| = |grMe,ξ +2,R(T ∪ {v})(Fq)| + 2|grMe,ξ +2,R−{v}(T)(Fq)|. +By repeating this equality, we obtain the desired identity. +□ +7.2. An example. Let’s consider an example that X = P1 = P1 +Fq. Note that in this case, Ω1 +P1 ∼= +OP1(−2). +Suppose S = {x1, . . . , xn} ⊆ P1 is a finite set of closed points of degree 1. Namely, we can +identify S with a subset of P1(Fq). Let’s consider those ℓ-adic local systems over P1 − S fixed +by Frob∗ whose local monodromies around xi (i < n) are tame and are at most unipotent, i.e., +they’re either trivial at xi or are unipotent with one Jordan block at xi, and they’re at most quasi- +unipotent with eigenvalues −1 at xn. We can deduce either from Theorem 1.4 or from its proof +directly that they’re in bijection with equivalent classes of semistable graded parabolic Higgs +bundles composed of the following data (simply because they have the same number): +E = OP1(m) ⊕ OP1(1 − m), +θ : E −→ OP1(m) −→ OP1(−m + n − 1) −→ E(n − 2), +and parabolic structures +Lxi = OP1(1 − m)xi, +1 ⩽ i ⩽ n. +We choose parabolic weights to be zero. Then the semistability says that m is an integer such that +m > 1 − m. +Note that θ exists if and only if +m ⩽ −m + n − 1, + +RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE +53 +and when such a θ exists, the pair (E, θ) is semistable if and only if θ is non-zero. Therefore we +have +1 ⩽ m ⩽ [n − 1 +2 +], +where [ n−1 +2 ] is the largest integer smaller or equal to n−1 +2 . +Two graded parabolic Higgs bundles are isomorphic if and only if m is the same and θ are +differed by a non-zero scalar. In fact, two isomorphic graded parabolic Higgs bundles have iso- +morphic underlying vector bundles. Therefore m should be the same. Suppose that (E, θ1) and +(E, θ2) are graded parabolic Higgs bundles with E = OP1(m) ⊕ OP1(1 − m). It is clear that θ1 and +θ2 are differed by a constant if and only if there is a ϕ ∈ Aut(E) such that +(ϕ ⊗ idΩ1 +P1 ) ◦ θ1 ◦ ϕ−1 = θ2, +i.e. (E, θ1) and (E, θ2) are isomorphic. +We conclude from the above discussion that the moduli space of semistable graded parabolic +Higgs bundles of rank 2, degree 1, and for the zero parabolic weight is +∐ +m +PHom(OP1(m), OP1(−m + n − 1)) ∼= +[ n−1 +2 ] +∐ +m=1 +Pn−1−2m. +In particular, the number of its Fq-points equals +(44) +n +∑ +i=3 +[i − 1 +2 +]qn−i. +This number is also the number of ℓ-adic local systems over P1 +Fq − S fixed by Frob∗ whose local +monodromies around xi (i < n) are tame and are at most unipotent, i.e., they’re either trivial at xi +or are unipotent with one Jordan block at xi, and they’re at most quasi-unipotent with eigenvalues +−1 at xn. We can not provide a natural bijection between these objects from our method, but I get +to know from Kang Zuo that in his work under preparation joint with Jinbang Yang, when n = 4, +they can construct a natural injective map from graded parabolic Higgs bundles to ℓ-adic local +systems, which then is bijective. +REFERENCES +[BM02] Breuil, C., M´ezard, A.. Multiplicit´es modulaires et repr´esentations de GL2(Zp) et de Gal(Qp/Qp) en l = p. with an +appendix by Guy Henniart. Duke Math. J. 115 (2002), no. 2, 205-310. +[BLR90] Bosch, S.; L¨utkebohmert, W.; Raynaud, M. N´eron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), +21. Springer-Verlag, Berlin, 1990. x+325 pp. +[Ch15] Chaudouard, P.-H. Sur le comptage des fibr´es de Hitchin. Ast´erisque No. 369 (2015), 223-284. +[De80] Deligne, P. La conjecture de Weil. II. Inst. Hautes ´etudes Sci. Publ. Math. No. 52 (1980), 137-252. +[De13] Deligne, P. Syst`emes locaux l-adiques sur une vari´et´e sur un corps fini Cours d’Arithm´etique et de G´eom´etrie +Alg´ebrique at IHES 2013. https://www.ihes.fr/~abbes/CAGA/deligne.html. +[De15] Deligne, P. Comptage de faisceaux l-adiques. Ast´erisque No. 369 (2015), 285-312. +[DF13] Deligne, P.; Flicker, Y. Z. Counting local systems with principal unipotent local monodromy. Ann. of Math. (2) 178 (2013), +no. 3, 921-982. +[Dr81] Drinfel’d, V. G. The number of two-dimensional irreducible representations of the fundamental group of a curve over a finite +field. (Russian) Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 75-76. +[Fl15] Flicker, Y. Counting rank two local systems with at most one, unipotent, monodromy. Amer. J. Math. 137 (2015), no. 3, +739-763. +[Hei10] Heinloth, J. Lectures on the moduli stack of vector bundles on a curve. Affine flag manifolds and principal bundles, +123-153, Trends Math., Birkh¨auser Springer Basel AG, Basel, 2010. +[Hi87] Hitchin, N. Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114. +[Ka77] Kazhdan, D. Proof of Springer’s hypothesis. Israel J. Math. 28 (1977), no. 4, 272-286. +[KV06] Kazhdan, D.; Varshavsky, Y. Endoscopic decomposition of certain depth zero representations. Studies in Lie theory, +223-301, Progr. Math., 243, Birkh¨auser Boston, Boston, MA, 2006. +[Ko09] Kontsevich, M. Notes on motives in finite characteristic. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. +Vol. II, 213-247, Progr. Math., 270, Birkh¨auser Boston, Boston, MA, 2009. + +54 +HONGJIE YU +[Lau96] Laumon, G. Cohomology of Drinfeld modular varieties. Part I. Geometry, counting of points and local harmonic analysis. +Cambridge Studies in Advanced Mathematics, 41. Cambridge University Press, Cambridge, 1996. xiv+344 pp. +[Laf97] Lafforgue, L. Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson.Ast´erisque No. 243 (1997), ii+329 pp. +[Laf02] Lafforgue, L. Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147 (2002), no. 1, 1-241. +[MW95] Moeglin, C.; Waldspurger, J.-L. Spectral decomposition and Eisenstein series. Une paraphrase de l’ ´Ecriture [A paraphrase +of Scripture]. Cambridge Tracts in Mathematics, 113. Cambridge University Press, Cambridge, 1995. xxviii+338 pp. +[MP96] Moy, A.; Prasad, G. Jacquet functors and unrefined minimal K-types. Comment. Math. Helv. 71 (1996), no. 1, 98-121. +[Ni91] Nitsure, N. Moduli space of semistable pairs on a curve. Proc. London Math. Soc. (3) 62 (1991), no. 2, 275-300. +[PS08] Paskunas, V.; Stevens, S. On the realization of maximal simple types and epsilon factors of pairs. Amer. J. Math. 130 +(2008), no. 5, 1211-1261. +[Ra95] Raynaud, M. Caract´eristique d’Euler-Poincar´e d’un faisceau et cohomologie des vari´et´es ab´eliennes. S´eminaire Bourbaki, +Vol. 9, Exp. No. 286, 129-147, Soc. Math. France, Paris, 1995. +[Sc16] Schiffmann, O. Indecomposable vector bundles and stable Higgs bundles over smooth projective curves. Ann. of Math. (2) +183 (2016), no. 1, 297-362. +[ST68] Serre, J.-P.; Tate, J. Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968), 492-517. +[Si92] Simpson, C. T. Higgs bundles and local systems. Inst. Hautes ´Etudes Sci. Publ. Math. No. 75 (1992), 5-95. +[Si90] Simpson, C. T. Harmonic bundles on noncompact curves. J. Amer. Math. Soc. 3 (1990), no. 3, 713-770. +[Ta68] Tate, J. Residues of differentials on curves. Ann. Sci. ´Ecole Norm. Sup. (4) 1 (1968), 149-159. +[Vi05] Vistoli, A. Grothendieck topologies, fibered categories and descent theory. Fundamental algebraic geometry, 1-104, Math. +Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005. +[Yo93] Yokogawa, K. Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves. J. Math. Kyoto Univ. +33 (1993), no. 2, 451-504. +[Yo95] Yokogawa, K. Infinitesimal deformation of parabolic Higgs sheaves. Internat. J. Math. 6 (1995), no. 1, 125-148. +[Yu18] Yu, H. Comptage des syst`emes locaux ℓ-adiques sur une courbe. to appear in Ann. of Math. +[Yu21a] Yu, H. A coarse geometric expansion of a variant of Arthur’s truncated traces. to appear in Pac. J. Math. +[Yu21b] Yu, H. +Number of cuspidal automorphic representations and Hitchin moduli spaces preprint. arXiv:2110.13858v2. +https://arxiv.org/abs/2110.13858 +[Yun11] Yun, Z. Global Springer theory. Adv. Math. 228 (2011), no. 1, 266-328. +DEPARTMENT OF MATHEMATICS, WEIZMANN INSTITUTE OF SCIENCE, HERZL ST 234, REHOVOT, ISRAEL. +Email address: hongjie.yu@weizmann.ac.il + diff --git a/ZtFPT4oBgHgl3EQfuzVj/content/tmp_files/load_file.txt b/ZtFPT4oBgHgl3EQfuzVj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0009fba602316ac57b19ecde73a2906e7271a21c --- /dev/null +++ b/ZtFPT4oBgHgl3EQfuzVj/content/tmp_files/load_file.txt @@ -0,0 +1,2052 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf,len=2051 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='13157v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='AG] 30 Jan 2023 RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE HONGJIE YU Abstract Let X be a smooth, projective, and geometrically connected curve defined over a finite field Fq and S ⊆ X a subset of closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let X and S be their base changes to an algebraic closure of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We study the number of ℓ-adic local systems in rank 2 over X − S with prescribed tame local monodromies fixed by k- fold iterated action of Frobenius endomorphism for every k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We confirm some conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, in all cases, our counting results are expressed in terms of the numbers of some Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Notations 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Global and local Langlands correspondence for GL2 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Spectral side of the trace formula 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Geometric side of the trace formula and Hitchin moduli spaces 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof of the main theorems 42 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Case that g = 0 50 References 53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' INTRODUCTION Let X be a smooth, projective, and geometrically connected curve defined over a finite field Fq of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In the two pages article [Dr81] of Drinfeld, he counts the number of two-dimensional geometrically irreducible ℓ-adic (in Qℓ-coefficients with ℓ ∤ q) representations of π1(X ⊗ Fq) which can be extended to a representation of π1(X) (here we ignore the base point in the notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' These numbers behave as if they were expressed by a Lefschetz fixed-point formula on an algebraic variety over the finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, they’re independent of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is equivalent to consider ℓ-adic local systems (smooth Qℓ-sheaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Although the Langlands correspondence established by Drinfeld and Lafforgue shows the motivic nature of ℓ-adic local systems counted by Drinfeld, their definition depends very much on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We don’t know how to construct a moduli space of ℓ-adic local systems in a reasonable sense that can explain these counting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Deligne has made some conjectures ([De15]) on counting ℓ-adic local systems over curves over finite fields with prescribed local monodromies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', with prescribed ramification types, to ex- tend and understand Drinfeld’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We mention that Kontsevich [Ko09] has also made some proposals toward understanding Drinfeld’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Some progress has been made since Deligne raised his conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Indeed, when the ramifications are split semisimple and in general position (which ensures that an ℓ-adic local system is automatically irreducible), Arinkin has verified that in these cases, similar results hold ([De15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When the ramifications are unipotent with one Jor- dan block, and there are at least two such ramifications, Deligne’s conjecture has been verified by Deligne-Flicker [DF13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The case in rank 2 with one unipotent ramification is verified by Flicker [Fl15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have generalized Drinfeld’s result to a higher rank in [Yu18], and Arinkin’s result to allow semisimple regular in general position but possibly non-split ramifications in [Yu21b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1 2 HONGJIE YU This article aims to verify some of Deligne’s predictions on counting of ℓ-adic local systems in rank 2 for all possible tame ramifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We show that the number is always related to the number of Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The results show an interesting analogy with Simpson’s non-abelian Hodge theory, especially when g = 0 and the ramifications are in general position and the par- abolic weights of the parabolic Higgs bundles are also in general position (these two conditions correspond in Simpson’s theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will discuss it in more detail at the end of the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s recall Deligne’s conjectures which will be treated in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We follow Deligne’s presentation in [De15], but we restrict to the rank 2 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let X be a smooth, projective, and geometrically connected curve defined over a finite field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let S ⊆ X be a subset of closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix an algebraic closure Fq of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let X := X ⊗ Fq and S := S ⊗ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For each point x ∈ S, let X∗ x = Xx − {x} be a punctured disc in x (Xx is defined as either the Henselization or the completion of X in x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix a rank 2 ℓ-adic local system (Qℓ- smooth sheaf) Rx over X∗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let E2(R) be the set isomorphism classes of irreducible rank 2 ℓ-adic local systems over X − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Frob be the Frobenius endomorphism of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the base change to Fq of the morphism induced by the map a �→ aq on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If (1) Frob∗(RFrob(x)) ∼= Rx for every x ∈ S, then the pullback of Frob permutes E2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let E2(R)Frob∗k be the set of fixed elements of k-iterated action of Frob∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Deligne conjectured that if all Rx are tamely ramified, then there are q-Weil integers α and integers mα such that |E2(R)Frob∗k| = ∑ α mααk, ∀k ⩾ 1, where |E2(R)Frob∗k| is the cardinality of subset of the fixed points by k-fold iterated action of Frob∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To formalize this property, let’s introduce some integral valued functions on N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We say that a function k �→ h(k) form N∗ to Z is of Lefschetz type if there are q-Weil integers α and integers mα ∈ Z such that h(k) = ∑ α mααk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, the conjecture is to prove that k �→ |E2(R)Frob∗k| is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A typical example is a function k �→ |V(Fqk)| for a variety V defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, given a permutation σ on a finite set P, the function k �→ |Pσk| is a periodic function of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that not all integral valued periodic functions are Lefschetz type as the integrality of mα is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The tame ´etale fundamental group of X∗ x is topologically generated by one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There- fore, an isomorphism class of tame local system of rank 2 over X∗ x corresponds bijectively to con- jugacy classes in GL2(Qℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The set S ⊆ X(Fq) is fixed by Frob and its orbits correspond bijectively to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Following the types of prescribed local monodromies, we can define a partition on S hence S into a disjoint union of subsets: S = Ss ∪ Su ∪ Scr, where Rx has different eigenvalues for x ∈ Scr, Rx induces a scalar matrix in GL2(Qℓ) for x ∈ Ss and Rx induces a quasi-unipotent conjugacy class with non-trivial Jordan block for x ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As each of these sets is stable under Frob, we have a partition S = Ss ∪ Su ∪ Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 3 Let x1 ∈ Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose x1 Frob −−→ x2 Frob −−→ · · · Frob −−→ xd+1 = x1 be the orbit containing x1 of the Frobenius action (xi ̸= x1 for any 1 < i ⩽ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There are two non-isomorphic rank 1 ℓ-adic local systems L1 and L2 over X∗ x1 such that Rx1 ∼= L1 ⊕ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The condition (1) implies that Frob∗dL1 ∼= Li, for i = 1 or i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This allows us to further subdivide Scr so that Scr = Sc ∪ Sr, where Sr is the set of points such that i = 1 and Sc consists of those points such that i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Again, we deduce a partition Scr = Sc ∪ Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we need to introduce some functions of Lefschetz type that are used to express the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R be a collection of tame local monodromies as above so that the condition (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Its eigenvalues for each x ∈ S define a couple of numbers (εx(1), εx(2)) ∈ Q× ℓ which could be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let S = {x1, · · · , xr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define a set PR by (2) PR := {(εx1(i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' , εxr(ir)) | r ∏ j=1 εx(ij) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ij ∈ {1, 2}, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Frob∗ be a permutation on PR defined so that for any (εx)x∈S ∈ (Q× ℓ )S, we have Frob∗((εx)x∈S) = (ε′ x)x∈S, with ε′ x = εq Frob(x), ∀x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The relation (1) tells us that it is a well-defined permutation, since εFrob(x)(1)q equals either εx(1) or εx(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define a function cR : N∗ −→ Z by cR(k) := |PFrob∗k R |, the number of the fixed points of Frob∗k on PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let σ be an involution on PR that sends (εx(ix))x∈S to (εx(3 − ix))x∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Define bR(k) := |Pσ=Frob∗k R | as the cardinality of the fixed point set of the action of σ ◦ Frob∗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We prove in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 that it is also of Lefschetz type when Scr ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we introduce some functions of Lefschetz type coming from points count of Hitchin bun- dles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that k ∈ N∗, and V ⊆ Su ⊗ Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let V = V ⊗Fqk Fq and D = KX + ∑ x∈V∪Scr x be a divisor over X where KX is a canonical divisor on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', a divisor associated to the canonical line bundle Ω1 X/Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A parabolic Hitchin bundle of rank 2 and degree 1 with parabolic structures in V for the divisor D is a triple (E, ϕ, (Lx)x∈V) consisting of a vector bundle of rank 2 and degree 1 over X, a bundle morphism ϕ : E → E ⊗ OX(D), 4 HONGJIE YU and a family of one dimensional Fq-subspace Lx of Ex (x ∈ V), the fiber of E in x, such that ϕx(Lx) = 0 and Im(ϕx) ⊆ Lx, for any x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We say that (E, ϕ, (Lx)x∈V) is semistable if for any subline bundle L of E satisfying ϕ(L) ⊆ L ⊗ OX(D), we have deg(L) ⩽ deg E 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote M1 2,V(D) the moduli space of these parabolic Hitchin bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is a variety defined over Fq (more details are given in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We show in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2 that it admits canonical Fqk-structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' comeing from the base change of a variety defined over Fqk) whose Fqk-points classify semistable parabolic Hitchin bundles over X ⊗ Fqk, which we denote by M1 2,V(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For each v ∈ Scr, we choose ov ∈ κv[t] a unitary polynomial of degree 2 with coefficients in κv (the residue field of the point v), so that ov is irreducible for v ∈ Sc and ov has distinct roots in κv if v ∈ Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It defines a polynomial in κx[t], for every closed point x of X lying over v via the isomorphism: κv[t] ⊗Fq Fq ∼= ∏ x�→v Fq[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define M1 2,V(o) as the closed sub-variety of M1 2,V(D) over Fqk consisting of those parabolic Hitchin bundles so that the characteristic polynomial of ϕx at x ∈ Scr is given by ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We suppose that the sum of roots of ox (x ∈ V) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We refer to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 for a more precise and detailed definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define for each k ⩾ 1, HiggR(k) = ∑ V⊆Su⊗Fqk (−1)|Su⊗Fqk−V|2|V|q−k(4g−3+|V|+|Scr|)|M1 2,V(o)(Fqk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We show in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2 that this is a function of Lefschetz type in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Pic0 X be the Jacobian variety of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We also define for every k ⩾ 1 Pic(k) := |Pic0 X(Fqk)|, and Pic(2)(k) := |S2Pic0 X(Fqk)|, where S2Pic0 X := (Pic0 X)2/S2 is the symmetric square of Pic0 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' They’re surely also functions of Lefschetz type in k ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The following theorem proves Deligne’s conjectures [De15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='15 (i)(iii)] when n = 2 and ramifications are tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (1) is satisfied, so that Frob∗ acts on E2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (3) ∏ x∈S εx(1)εx(2) = 1, otherwise E2(R) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The function k �→ |E2(R)Frob∗k| is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' More precisely, we have the following explicit identities that express |E2(R)Frob∗k| following different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) Scr = Su = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then |E2(R)Frob∗k| equals HiggR(k) − cR(k) � Pic(k)2(g − 1) + Pic(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) Scr = ∅, Su ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then |E2(R)Frob∗k| equals HiggR(k) − cR(k) � βSu(k)(−1)|Su|+1Pic(2)(k) + γSu(k)Pic(k) + ωSuPic(k)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 5 (3) Scr ̸= ∅, Su = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If |Sc| is even, then |E2(R)Frob∗k| equals HiggR(k) − cR(k)(2g − 2 + |Scr|) 2 Pic(k)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (4) Scr ̸= ∅, Su = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If |Sc| is odd, then |E2(R)Frob∗k| equals HiggR(k)− � cR(k)2g − 1 + |Scr| 2 − cR(k) + bR(k) 2 � Pic(k)2 − bR(k)Pic(2)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (5) Scr ̸= ∅, Su ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If |Sc| is even, then |E2(R)Frob∗k| equals HiggR(k)− �cR(k)αSu(k) 2 + (−1)|Su| bR(k)βSu(k) 2 � Pic(k)2 + (−1)|Su|bR(k)βSu(k)Pic(2)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (6) Scr ̸= ∅, Su ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If |Sc| is odd, then |E2(R)Frob∗k| equals HiggR(k)− �cR(k)αSu(k) 2 − (−1)|Su| bR(k)βSu(k) 2 � Pic(k)2 − (−1)|Su|bR(k)βSu(k)Pic(2)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In the above expressions αSu, βSu, γSu and ωSu are periodic functions of Lefschetz type (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 for their explicit expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When Scr is non-empty, cR/2 + bR/2 and cRαSu/2 ± bRβSu/2 are of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If |Sr| is odd, then cR/2 is of Lefschetz type and bR is constantly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) The necessity of (3) for E2(R) to be non-empty is explained in [De15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It can be checked by passing to characteristic 0 and then passing to C, where one can use an explicit presentation of the topological fundamental group of a punctured Riemann surface (see the proof of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7 of [DF13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Using Langlands correspondence, one can prove that E2(R) has no Frob∗k fixed points for k ⩾ 1 if (3) does not hold as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) The ramifications Rx for x ∈ Ss only affects cR(k) and bR(k) and is not involved in any other term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (3) In the case of general position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', when the set PR = ∅, we have |E2(R)Frob∗k| = HiggR(k), ∀k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that the extra terms (those different from HiggR(k)) appear only if PR ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This phe- nomenon is possibly related to the singularity of the moduli space of (S-equivalent classes of) semistable parabolic Higgs bundles in the non-coprime cases or the cases not in general position in the terminology of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The following corollary confirms Deligne’s conjectures [De15, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The cardinality |E2(R)Frob∗k| is divisible by Pic(k) and k �→ |E2(R)Frob∗k|/Pic(k) is still a function of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Although we deal only with the tame local monodromies, the method of this article allows us to treat some wild ramified cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For example, we can allow some places to give the so-called simple supercuspidal representation on the automorphic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There could be a similar result involving wild Hitchin bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In a private note by Zhiwei Yun, he has a simple geometric method to deal with some cases with wild ramifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 6 HONGJIE YU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that g = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the curve X is P1, and Sc = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will present an analogy with Simpson’s non-abelian Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It does not hold in a more general case which we hope to understand in the spirit of conjectures [De15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We are interested in the case that R is in general position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' when PR is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We assume that Fq ̸= F2 to ensure that such cases are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R = Sr ∪ Su, and D = KX + ∑ v∈R v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We view D also as a line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ξ = (ξx)x∈R ∈ (Q2)R such that ξx,1 ⩾ ξx,2 ⩾ ξx,1 − 1 and ξx = ξy for any x, y lying over the same closed point v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' These vectors serve as parabolic weights (stability parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let (E, ϕ, (Lx)x∈R) be a parabolic Higgs bundle (we call a parabolic Hitchin bundle a Higgs bundle if D = KX + ∑v v and parabolic structures are imposed in R) over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let L be a sub-line bundle of E, we define the parabolic degree p-deg(L) by p-deg(L) := deg(L) + ∑ x∈R � ξx,1, if Lx = Lx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ξx,2, if Lx ̸= Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We say that (E, ϕ, (Lx)x∈R) is ξ-semistable if for any sub-line bundle L of E satisfying ϕ(L) ⊆ L ⊗ OX(D), we have p-deg(L) ⩽ deg E + ∑x∈R(ξx,1 + ξx,2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if deg(E) + ∑ x∈R ±(ξx,1 − ξx,2) /∈ 2Z, then the equality can never be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We say that such cases are in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Choose ξ as above and suppose that it is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The moduli space of ξ-semistable parabolic Higgs bundles of rank 2 and of degree e which are semistable with parabolic weights (ξx)x∈R over X has a canonical Fq-structure (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote the moduli space by Me,ξ 2,R = Me,ξ 2,R(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We show in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8 that |Me,ξ 2,R(Fq)| is independent of the choice of the parabolic weights as long as ξ is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The space Me,ξ 2,R has a Gm-action via dilation of the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let grMe,ξ 2,R := (Me,ξ 2,R)Gm, and grMe,ξ 2,R(Su) be its open subvariety consisting of those parabolic Higgs bundles whose Higgs field does not vanish at x ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that g = 0 and Sc = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (e, ξ) is in general position and that ξx,1 = ξx,2 for x ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that Fq ̸= F2 and R is in general position in the sense that PR = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that ∏ x∈S εx(1)εx(2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have |grMe,ξ 2,R(Su)(Fqk)| = |E2(R)Frob∗k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given a compact Riemann surface Σ and a finite set of points R ⊆ Σ, Simpson has estab- lished a correspondence between semistable C-local systems over Σ − R of degree 0 and semistable quasi-parabolic Higgs bundles over Σ with parabolic structures in R of parabolic degree 0 (we re- fer to Simpson’s original article [Si90] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' On the local system side, Simpson defines residual data for each x ∈ R using the local monodromy and stability weight at the puncture x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Similarly, Simpson defines residual data for each x ∈ R using the Higgs field and the parabolic weight in the Higgs bundle side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' His correspondence preserves the nilpotent part of the resid- ual data and permutes the stability weights and eigenvalues of the residual datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4 RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 7 presents an analogy with Simpson’s theory if we choose the stability weights of the local systems to be trivial and we choose (e, ξ) in accordance with Simpson’s correspondence (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' the diagram in [Si90, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='720]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' An interesting phenomenon is that R being in general position corresponds to that (e, ξ) being in general position under Simpson’s correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The dominant term in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 when k varies is (q4g−3+|Su|+|Scr|)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is half of the dimen- sion of the moduli space of parabolic Higgs bundles of the relevant complex analogy in Simpson’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This may be related to the motivic nature of ℓ-adic local systems over a curve over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can not expect a naive generalization of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4 to the cases g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' However, we expect it to be a special case of (a possible modification of) Deligne’s conjecture in [De15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Indeed, suppose that all ramifications are split regular semisimple (for n = 2, it is the case that Ss = Sc = Su = ∅), in the cases of in general position, we have ([Yu21b, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4] for any rank) |En(R)Frob∗k| = ∑ i (−1)iTr(V∗k|Hi c((Me,ξ n,Sr)Fq, Qℓ)), for any endomorphism V∗ which is conjugate to q− 1 2 (n2(g−1)+|Sr|)F∗ q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can expect to generalize it to the cases where ramifications are only supposed to be semisimple but remain in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The more demanding question is to generalize it to allow non-trivial quasi-unipotent ramifications or even cases not in general positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we may ask if V∗ is induced from a morphism of (Me,ξ n,Sr)Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This does not seem to be the case if we consider only algebraic varieties over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We likely have to consider a lifting of the curve to characteristic 0 and consider p-adic geometry which I’m not competent to comment on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Instead, we refer the reader to Deligne’s course at IHES [De13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this article, we only need semistable parabolic Higgs bundles with parabolic weights in gen- eral position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is not necessary to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' However, with our method, it is more natural to consider the algebraic stack version of the moduli of semistable parabolic Higgs bundles when the para- bolic weights are not in general position, and we should expect a more complicated expression for the point counting problem in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This work continues the project that started from my thesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' I thank Pierre- Henri Chaudouard for giving this project to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' I thank Kang Zuo for the fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The work is finished during my stay at IST Austria and Weizmann Institute of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' I thank both institutes for providing me with excellent work conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Part of the work is finished with the support of the BSF grant 2019274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' NOTATIONS We gather some notations that will be used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Other notations will be defined where they appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' F, |X|, Fv, Ov, ℘v, κv, qv, A, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let F = Fq(X) be the global function field of the curve X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let |X| be the set of closed points of X, which is identified with the set of places of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For every v ∈ |X|, let Fv be the local field in v, Ov the ring of integers in Fv and κv the residue field of Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ℘v be the maximal ideal in Ov, and we choose a uniformizer ̟v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that κv has cardinality qv, therefore κv ∼= Fqv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A be the ring of ad`eles of F and O be the sub-ring of integral ad`eles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' G, B, N, T, B, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If not specified otherwise, we use G for GL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let B be the Borel subgroup of G consisting of upper triangular matrices and T be the torus consisting of diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let N be the unipotent radical of B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the group of upper triangular matrices with 1 on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let B be the Borel subgroup that is opposite to B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', consisting of lower triangular matrices, and N be the unipotent radical of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' g, b, n, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let g, b, n, and t be respectively the Lie algebra of G, B, N, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Gv, Bv, Kv, Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given a variety V defined over Fq, we will use Vv to denote V(Fv) for any places v ∈ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This notation applies in particular to Gv, Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will denote G(Ov) by Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Iv be the Iwahori subgroup consisting of matrix in Kv whose reduction modulo ℘v lies in B(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 8 HONGJIE YU G(A)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any e ∈ Z, let G(A)e = {x ∈ G(A)| deg det x = e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix Haar measures on G(A), N(A) so that G(O) and N(F)\\N(A) (with counting mea- sure on N(F)) have volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The local Haar measures on Gv, Bv and Nv are defined so that respectively the volumes of Kv, B(Ov) and N(Ov) are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Ev, ϕv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given a vector bundle E over X, and a place v ∈ |X| identified as a κv-point of X, we use Ev to denote the fiber over v which is a κv-vector scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose ϕ : E −→ F be a bundle morphism over X, then it induces a κv-linear map ϕv : Ev −→ Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' GLOBAL AND LOCAL LANGLANDS CORRESPONDENCE FOR GL2 We are going to reduce the calculation of the cardinality of E2(R)Frob∗k to a question of counting certain automorphic representations of GL2 with the help of global Langlands correspondence in rank 2 established by Drinfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that X = (X ⊗Fq Fqk) ⊗Fqk ⊗Fq, and the Frobenius endomorphism of X deduced from X ⊗Fq Fqk is Frobk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, we can do the calculation for k = 1 and apply the results to the curves X ⊗Fq Fqk over Fqk (k ⩾ 1) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Galois representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It has been explained by Deligne [De15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9] how to pass to the automorphic side, and the reader is invited there for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This section aims to give precise information on the ramifications of automorphic representations determined by the Frobenius ac- tion on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The data on the local monodromies are carried over to the automorphic side, described by local Langlands correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We continue to use notations in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let v ∈ S and x ∈ S that lies over v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix an algebraic closure F of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then η := Spec(F) is a geometric point lying over the generic point of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let X(x) be the Henselization of X in x and X∗ (x) = X(x) − {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If we choose an embedding of Fq(X∗ (x)) in F, then the ´etale fundamental group π1(X∗ x, η) (we will ignore the choice of a base point in the notation in what follows) is canonically isomorphic to the inertial group Ix = Gal(F|Fq(X∗ (x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' An ℓ-adic local system over X − S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' X∗ (x)) is equivalent to an ℓ-adic representation of π1(X − S, η) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Ix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let It x = π1(X∗ (x), η)t be the tame fundamental group of X∗ (x) which is the prime-to-p quotient of π1(X∗ (x), η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A tame ℓ-adic local system of X∗ (x) is equivalent to an ℓ-adic representation of It x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For an algebraic closed field k, let �Zp′(1)(k) := lim µn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote �Zp′(1) for �Zp′(1)(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let κx be the residue field of the point x, we have a canonical isomorphism It x ∼= �Zp′(1)(κx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Choose an embedding κv ֒→ Fq, we deduce from κv ⊗Fq Fq ∼= ∏ x�→v Fq, isomorphisms κx ∼= Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Which gives us isomorphisms It x ∼= �Zp′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The morphism Frob : X∗ x −→ X∗ Frob(x), induces an isomorphism between tame fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is the multiplication by q map on �Zp′(1) via the above isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To make the set E2(R) Frob∗-stable, the ℓ-adic local systems (Rx)x�→v have to satisfy the com- patibility condition (4) Frob∗(RFrob(x)) ∼= Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 9 Let Iv and Dv be, respectively, the inertial subgroup and decomposition group of F at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The condition (4) implies that (Rx)x�→v come from a representation ρv of Dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Grothendick’s local monodromy theorem, ρv|Iv is quasi-unipotent in the sense that it becomes unipotent on an open subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will use Rv to denote ρv|Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let WF be the Weil group of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then π1(X − S, η) is a quotient of the degree 0 part of WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let G2(F) be the set of isomorphism classes of ℓ-adic representation of WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We call two ℓ-adic representations σ1 and σ2 in G2(F) inertially equivalent: σ1 ∼ σ2 if there is a character λ : WF deg −−→ Z −→ Q× ℓ such that σ1 ∼= σ2 ⊗ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The set E2(R)Frob∗k is in bijection with the subset of inertially equivalent classes in G2(F ⊗ Fqk)/ ∼ consisting of σ such that (5) σ ⊗ λ ∼= σ =⇒ λ = 1, σ|Iv is trivial for v /∈ S and σ|Iv ∼= Rv for v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It has been explained in [De15, Section 2], see also [Yu18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3] for condition (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Let Wv be the local Weil group at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We choose an isomorphism ι : Qℓ ∼ −→ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that the local Langlands correspondence is a canonical bijection between the set of smooth irreducible C-representations of Gv and the set of rank 2, Frobenius semisimple ℓ-adic (continuous) represen- tations of Wv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any v ∈ S, let IrrR(Gv) be the set of irreducible representations of Gv whose associated ℓ-adic representation of the local Weil group Wv under local Langlands correspondence extends Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For a place v /∈ S, we define IrrR(Gv) to be the set of unramified representations of Gv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', those representations whose associated ℓ-adic representation of Wv under local Langlands correspondence is trivial when restricting to Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have the following theorem that characterizes the set IrrR(Gv) purely by their representa- tion theoretic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Rv be tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have one of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (r) We say that Rv is (split) regular if Rv ∼= χ1 ⊕ χ2, is the direct sum of two distinct characters χ1, χ2 of Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Each χi (i = 1, 2) has exponent qv − 1 and can be factored as Iv −→ κ× v χ′ i −→ Q× ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case for any irreducible smooth representation π of Gv, we have π ∈ IrrR(Gv) if and only if HomIv(χv, π) ∼= HomKv(ρv, π) ̸= 0, where χv : Iv −→ B(κv) \uf8eb \uf8eda b 0 d \uf8f6 \uf8f8�→ι(χ′ 1(a)χ′ 2(d)) −−−−−−−−−−−−−−→ C×, and ρv is the induced representation of χv to Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, dim HomIv(χv, π) = 1 for any π ∈ IrrR(Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (c) We say that Rv is cuspidal (or anisotropic regular), if Rv ∼= χ1 ⊕ χ2, is the direct sum of two distinct characters of Iv such that χqv 1 = χ2 (necessarily we also have χqv 2 = χ1), where χi can be factored as Iv −→ F× q2v χ′ i −→ Q× ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 10 HONGJIE YU In this case for any irreducible smooth representation π of Gv, we have π ∈ IrrR(Gv) if and only if dim HomKv(ρv, π) ̸= 1, where ρv is the irreducible representation of Kv inflated from the Deligne-Lusztig induced representation −ι(RG(κv) U(κv)χ′ 1), with U being any non-split maximal subtorus of G defined over κv so that we have U(κv) ∼= F× q2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, for any π ∈ IrrR(Gv), we have dim HomKv(ρv, π) = 1 and π is supercuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (s) We say that Rv is scalar if Rv ∼= χ⊕2, where χ is a character of Iv that can be factored as Iv −→ κ× v χ′ −→ Q× ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case for any irreducible smooth representation π of Gv, we have π ∈ IrrR(Gv) if and only if HomKv(θv, π) ̸= 0 where θv : Kv det −→ O× v −→ κ× v ι◦χ′ −−→ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, we have dim HomKv(θv, π) = 1 for any π ∈ IrrR(Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' And the set IrrR(Gv) consists of the one dimensional representation η of Gv which extends θv and the twist by η of irreducible unramified representations of Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (u) We say that Rv is principal quasi-unipotent if Rv ∼= χ ⊗ ν, is a quasi-unipotent with one principal Jordan block: χ is a character of Iv which can be factored as Iv −→ κ× v χ′ −→ Q× ℓ and ν is a non-trivial unipotent representation of Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case π ∈ IrrR(Gv) if and only if π = St ⊗ λ for some character λ of the form Gv det −→ F× v λ′ −→ C× so that θv = λ′|O× v inflates ι ◦ χ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Here St is the Steinberg representation of Gv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the unique irreducible quotient of the parabolic induction of the trivial representation of Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The cases (r) and (c) are explained in [Yu21b, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The case (s) is deduced from the unramified case where dim πKv v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, it is enough to tensor Rv by a rank 1 local system to make it trivial on Iv, which corresponds in the automorphic side to twist the representation by a character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Similarly, we can tensor a character to make the case (u) into the unipotent case, which corresponds to the Steinberg representation under Langlands correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Let π = ⊗′πv be a cuspidal automorphic representation of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will say that πv has the correct ramification type (for our counting problem) if πv ∈ IrrR(Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Automorphic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Ccusp(G(A)) be the space of cuspidal automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that a cuspidal automorphic form is a complex-valued function ϕ over G(F)\\G(A) which generates a finite-dimensional vector space under G(O)-right translation and ZG(A) translation, such that the following cuspidality condition is satisfied � N(F)\\N(A) ϕ(nx)dn, ∀x ∈ G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that the above integration is a finite sum because of G(O)-finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The right translation by G(A) makes Ccusp(G(A)) into a G(A)-representation which is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The multiplicity one theorem of Jacquet & Langlands and Piatetski-Shapiro says that Ccusp(G(A)) is multiplicity free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Its irreducible summands are called cuspidal automorphic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The decomposition of Flath decomposes a cuspidal automorphic representation π as a restricted tensor product π = ⊗′πv for representations πv of Gv which are called local components of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A2(F) be the set of isomorphic classes of cuspidal automorphic representations of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We call two cuspidal automorphic representations π1 and π2 are inertially equivalent π1 ∼ π2 if there are is a character λ : G(A) deg ◦ det −−−−−→ Z −→ C× such that π1 ∼= π2 ⊗ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 11 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The set E2(R)Frob∗ is in bijection with the subset of A2(F)/ ∼ consisting of inertial equiv- alent classes of cuspidal automorphic representations π such that for any character λ : G(A) deg ◦ det −−−−−→ Z −→ C×, π ⊗ λ ∼= π =⇒ λ = 1, and πv ∈ IrrR(Gv) for all v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Applying global Langlands correspondence and the fact that it is compatible with local Langlands correspondence, this is a corollary of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' SPECTRAL SIDE OF THE TRACE FORMULA We will use equality provided by the noninvariant Arthur-Selberg trace formula (a similar but slightly different formula is obtained first by Jacquet-Langlands for GL2) established by Lafforgue [Laf97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Indeed the noninvariant Arthur-Selberg trace formula over a function field is an equality for each e ∈ Z between two distributions on C∞ c (G(A)): Je geom( f) = Je spec( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will construct a function f ∈ C∞ c (G(A)) using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2 and do explicit calculation for J1spec( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The result is summarized by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 from which we will see that it is always a sum of |E2(R)Frob∗| and an explicit error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In a later section, we will use a geometric method to study Je geom( f), which gives a relation with the number of Fq-points of Hitchin moduli spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Explicit spectral decomposition of J1spec( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let M be either T or G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let XM be the group of characters of M(A) to C× which is trivial on M(A)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let XG M ⊆ XM be the subgroup consisting of those characters which are trivial on ZG(A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', XG M = Hom(M(A)0\\M(A)/ZG(A), C×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have XG G = {1, ǫ}, where ǫ(x) = (−1)deg(det x) is the sign character of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We identify XG G with {±1} ⊆ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We also have an identification XG T ∼= C×, where for any λ ∈ C× we associate a character λ( � a 0 0 b � ) = λ− deg(a)+deg(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will use this isomorphism in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ImXG T be the subgroup of XG T consisting of unitary characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, it is formed by elements λ ∈ C× of absolute value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We endow a Haar measure on XG G and ImXG T so that the total volume is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Following Jacquet-Langlands, we have the spectral decomposition: L2(G(F)\\G(A)) ∼= L2 cusp ⊕ L2 res ⊕ L2 cont, where L2 cusp ⊕ L2 res is the largest semisimple subspace, L2 cont is its orthogonal complement, and L2cusp is the completion of the space of cuspidal automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The residual spectrum is decomposed as L2 res ∼= � � χ χ, where χ are compositions of Hecke characters with the determinant morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The sum here is the Hilbert direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For continuous spectrum L2 cont, we have a decomposition: L2 cont = � � ψ L2 [B,ψ], 12 HONGJIE YU where the sum is taken over the set of inertial equivalent classes of pairs (B, ψ) with ψ being a Hecke character of T(A) ∼= (A×)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The explicit construction of L2 [B,ψ] is given by the theory of Eisenstein series, which we do not need in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix an id`ele a ∈ A× of degree 1, viewed as a scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let f ∈ C∞ c (G(A)), it acts via the regular representation on L2(G(F)\\G(A)/aZ), equivalently by convolution from right by ˘f := (x �→ f(x−1)), which is an integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote this action by the function R( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore its trace, if it exists, will be the integration of the kernel function on the diago- nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' However, the trace of f on the whole space L2(G(F)\\G(A)/aZ) does not exist in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Arthur defines a truncated kernel function so that its integration on the diagonal will contain the information Tr( f|L2 cusp) and expresses this integration in two ways: a geometric expansion and a spectral expansion so that we have an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The spectral expansion contains a piece that gives the most interesting part Tr( f|L2 cusp), and we usually hope to obtain information from the geometric expansion and an understanding of the error terms in the spectral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Over a function field, we have a decomposition G(A) = ∐ G(A)e, and G(F)\\G(A)e has finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The two different ways to express Arthur’s truncated integral over the diagonal in G(F)\\G(A)e × G(F)\\G(A)e give an identity: Je geom( f) = Je spec( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is slightly simpler to consider an odd integer e or simply that e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The following result is a special case of the formula obtained by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lafforgue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A1 be the set of Hecke characters of F×\\A×/aZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Acont := A1 × A1/S2, where S2 acts by permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' An element [(ψ1, ψ2)] ∈ Acont is called regular if ψ1 ̸= ψ2 and is called non-regular otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Ares be the inertial equivalent classes of 1-dimensional representations of G(A) trivial on aZG(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A0 be the set inertial equivalent classes of cuspidal automorphic representations of G(A) whose central characters are trivial on aZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let AB,ψ be the space of complex valued functions ϕ over G(A) which satisfy that for any k ∈ G(O), there is a constant ck ∈ C so that for any n ∈ N(A), t ∈ T(A) we have ϕ(ntk) = ckρB(t)ψ(t), where ρB( � a 0 0 c � ) = |a| 1 2 |c| 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Equivalently, it is the space of those ϕ such that for any x ∈ G(A), n ∈ N(A) and t ∈ T(A) we have ϕ(ntx) = ρB(t)ψ(t)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let w be the non-trivial element in the Weyl group of (G, T) and λ ∈ XG T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have the intertwining operator AB,ψ −→ AB,w(ψ) defined by analytic continuation of the integral below which converges when |Reλ| >> 0, (6) (M(w, λ)ϕ)(x) := λ(x) � N(A) ϕ(w−1nx)λ(w−1nx)dn, where we view λ ∈ XG T as a function over G(A) using Iwasawa decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', if x = ntk with n ∈ N(A), t ∈ T(A) and k ∈ K, we define λ(x) := λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 (Arthur, Lafforgue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The spectral expansion is the identity J1 spec( f) = ∑ [π]∈A0 Jπ( f) + ∑ [χ]∈Ares Jχ( f) + ∑ [ψ]∈Acont Jψ( f), RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 13 where each sum is taken over a set of representatives, and the terms are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote the three sums by respectively J1 cusp( f), J1 res( f) and J1 cont( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For π ∈ A0, if π ⊗ ǫ ∼= π, then Jπ( f) = 1 2(Tr(R( f)|π) − Tr(R( f) ◦ ǫ|π)), and if π ⊗ ǫ ̸∼= π, then Jπ( f) = Tr(R( f)|π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For χ ∈ Ares, we have Jχ( f) = Tr(R( f)|χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any λ ∈ XG T , let R( f, λ) be the twisted action on AB,ψ: R( f, λ)ϕ = (R( f)(ϕλ))λ−1, where we view λ ∈ XG T as a function over G(A) by Iwasawa decomposition: λ(x) = λ(t) if x = ntk for n ∈ N(A), t ∈ T(A) and k ∈ G(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For ψ ∈ Acont, if ψ is regular, then Jψ( f) = � ImXG T lim µ−→1 TrAB,ψ((− 1 µ−1 − µM(w, λ)−1 ◦ M(w, λ/µ) + 1 µ−1 − µ) ◦ R( f, λ))dλ, and if ψ is not-regular, then (7) Jψ( f) = 1 2 � ImXG T lim µ−→1 TrAB,ψ((− 1 µ−1 − µM(w, λ)−1 ◦ M(w, λ/µ) + 1 µ−1 − µ) ◦ R( f, λ))dλ + 1 8 ∑ λG∈{±1} ∑ λw∈ImXG T λ2w=λ−1 G λGTrAB,ψ(M(w, w−1(λw)) ◦ R( f, λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if f is supported in G(O), then R( f, λ) = R( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The theorem is established by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lafforgue in [Laf97], and we refer the reader to [Yu18, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2, Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For the group G = GL2, we can make the result more explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In [Yu18, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1] we have calculate explicitly the functions � 11 G, � 11 B and � 11 B on XG T , which appears in the spectral expansion of the trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' They’re given by the following formula: � 11 G(1) = 1 and � 11 G(ǫ) = −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', � 11 G(λ) = λ, ∀λ ∈ XG G, and � 11 B(λ) = − 1 λ − λ−1 , � 11 B(λ) = − 1 λ−1 − λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Summary of main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The following proposition provides a specific function f ∈ C∞ c (G(A)) and we will be interested in the computation of J1spec( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We use notations of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For each v ∈ S, we define the following functions following the ramification type of Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (r) The function f (r) v ∈ C∞ c (Gv) is defined by f (r) v (x) = � 0, x /∈ Kv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Tr(ρv(x−1)), x ∈ Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' where x is the image of x in G(κv) under the projection Kv −→ G(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 14 HONGJIE YU (c) The function f (c) v ∈ C∞ c (Gv) is defined by f (r) v (x) = � 0, x /∈ Kv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Tr(ρv(x−1)), x ∈ Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (s) The function f (s) v ∈ C∞ c (Gv) is supported in Kv such that for any x ∈ Kv, we have f (s) v (x) = θv(det x−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (u) The function f (u) v ∈ C∞ c (Gv) is defined by f (u) v = � 1 vol(Iv) 1Iv(x) − 2 1Kv(x) � θv(det x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let f = ⊗ f (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=') v ∈ C∞ c (G(A)), where f (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=') v is the function defined above if v ∈ S(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=') for ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' = r, c, s, u and f (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=') v = 1Kv if v /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then for any cuspidal automorphic representation π of G(A) which is not one dimensional, the condition Tr( f|π) ̸= 0 holds if and only if π has correct ramification type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If it is the case, we have Tr( f|π) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose π = ⊗′πv is a cuspidal automorphic representation, we have Tr( f|π) = ∏ v Tr( fv|πv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The statement is, therefore, of local nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For v /∈ S, the function 1Kv acts as a projection to Kv-fixed part of πv, which is non-zero if and only if πv is unramified by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If this is the case, πKv v is an 1 dimensional vector space and the trace of 1Kv is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For v ∈ Ss, it is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, suppose that πv = π′v ⊗ η with π′v being unramified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then Tr( f (s) v |πv) = Tr( 1Kv|π′ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The trace is 1 or 0 depending on whether or not πv has the correct ramification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For v ∈ Sc and v ∈ Sr, this result has been proved in [Yu21b, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Finally, we consider the case v ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As above, up to a twist, we are reduced to the case that χ′ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if Tr( f (u) v |πv) ̸= 0, then πv contains a non-zero fixed vector under Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By a result of Casselman [Lau96, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4], πv must be an irreducible subrepresentation of a parabolic induction IndGv Bv θv for a character θv of Tv that is trivial on T(Ov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If the representation IndGv Bv θv is irreducible, then it is unramified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Hence its Iv-fixed subspace has dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that Tr( f (u) v |πv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If IndGv Bv θv is not irreducible, then it is of length 2 whose irreducible quotients are a 1-dimensional representation and an unramified twist of the Steinberg representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since π is cuspidal, πv cannot be 1-dimensional, therefore πv is an unramified twist of the Steinberg representation whose Iv-fixed subspace is of 1 dimensional and Kv-fixed subspace is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ The primary purpose of this section is to prove the following theorem which calculates J1spec( f) of the Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The result is deduced from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='10, and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Notations are those of the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 15 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For a finite set of places V of F, we define deg V := ∑v∈V deg v = |V|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Su,even be the subset of Su consisting of places v such that deg v is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The expression J1spec( f) equals the following numbers depending on the cases: (1) Sc ̸= ∅, and Su,even ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) Sc ̸= ∅, Su,even = ∅ but Su ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗| + (−1)|Su|+1bR(1)2|Su|−2(−1)deg ScPic(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (3) Sc ̸= ∅, and Su = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗| + bR(1) 4 (1 − (−1)deg Sc)Pic(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (4) Sc = ∅, Sr ̸= ∅, and Su = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗| + 1 2cR(1)Pic(1)2(2g − 2 + deg Sr) (5) Scr = Su = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗| + cR(1)Pic(1)2(g − 1) + cR(1)Pic(1) (6) Scr = ∅, Su = {v} and 2 | deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then J1 spec( f) equals |E2(R)Frob∗| − cR(1)Pic(1) + deg v 2 cR(1)Pic(1)2 (7) Scr = ∅, Su = {v} and 2 ∤ deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then J1 spec( f) equals |E2(R)Frob∗| + cR(1) 2 Pic(2)−cR(1)Pic(1) + cR(1)deg v 2 Pic(1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (8) Scr = ∅, |Su| ⩾ 2, and Su,even ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then J1 spec( f) equals |E2(R)Frob∗| + cR(1)(−1)|Su|Pic(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (9) Scr = ∅, |Su| ⩾ 2, and Su,even = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then J1 spec( f) equals |E2(R)Frob∗| + cR(1)(−1)|Su|Pic(1) + cR(1)(−1)|Su|+12|Su|−2Pic(2) (10) Sc = ∅, Sr ̸= ∅, Su = {v}, and 2 | deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗| + 1 2cR(1)Pic(1)2 deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (11) Sc = ∅, Sr ̸= ∅, Su = {v}, and 2 ∤ deg v, J1 spec( f) = |E2(R)Frob∗| + 1 2cR(1)Pic(1)2 deg v + bR(1) 2 Pic(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (12) Sc = ∅, Sr ̸= ∅, |Su| ⩾ 2, and Su,even ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (13) Sc = ∅, Sr ̸= ∅, |Su| ⩾ 2, and Su,even = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 spec( f) = |E2(R)Frob∗| + (−1)|Su|+1bR(1)2|Su|−2Pic(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 16 HONGJIE YU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Counting ℓ-adic local systems in rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to discuss the cases in rank 1 first, not only for completeness but also because these results will be needed when calculating the cases in rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It has been dealt with in [De15, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A difference between a number field and a function field is that the function field F, hence all its local fields, is an Fq-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given a character χv : Fv −→ Q× ℓ , we can consider its restriction to F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let χ be a Hecke character χ = ∏ v χv : F×\\A× −→ Q× ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is a character of A× which is trivial on F×, in particular on F× q , therefore necessarily we have (8) ∏ v χv|F× q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that R1 is a set of rank 1 ℓ-adic local systems over (X∗ (x))x∈S fixed by Frob∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that they are tame and εx are eigenvalues of a tame generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By similar discussion as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1, but using local class field theory, we obtain for each v ∈ S a character θv of O× v which is trivial on 1 + ℘v if it is tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The condition (9) ∏ x∈S εx = 1, is equivalent to (10) ∏ v θv|F× q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A1(R1) be the set of inertial equivalent classes of Hecke characters χ of F×\\A× such that χv extends θv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The set A1(R1) is in bijection with E1(R1)Frob∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The condition (10) is satisfied is and only if A1(R1) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By (8), we have seen that it is a necessary condition for A1(R) to be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Conversely, note that F×\\A× is an extension of F×\\A×/O× ∼= PicX(Fq) by O×/F× q , the condition (8) ensures that ∏v θv defines a character of O×/F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Taking (ℓ-adic) Pontryagin dual, we see immediately that A1(R) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ If the condition (9) is satisfied then we have (11) |E1(R)Frob∗| = |Pic0(X)(Fq)∨| = Pic(1), otherwise E1(R)Frob∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, as long as E1(R)Frob∗ is non-empty, it is a principal homogenous space under E1(∅)Frob∗ which has cardinality Pic(1) = |Pic0(X)(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Eulerian decomposition and calculations on Whittaker functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let f = ⊗ fv ∈ C∞ c (G(A)) be the function defined in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let π ∼= ⊗′πv be a cuspidal automorphic representa- tion of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that π ∼= π ⊗ ǫ as representations of G(A), we need to consider Tr(ǫ ◦ R( f)|π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that this trace is not well-defined from a pure representation theoretical point of view be- cause the action of ǫ on π relies on the isomorphism π ∼= π ⊗ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For G = GL2, this isomorphism is furnished by the multiplicity one theorem, which says that π and π ⊗ ǫ have the same underlying space of cusp forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A similar problem arises if we want to get an Eulerian decomposition of the trace Tr(ǫ ◦ R( f)|π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to choose isomorphisms πv ∼= πv ⊗ ǫ so that their tensor product is com- patible with the global isomorphism (we also need to assume that the isomorphisms can be glued RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 17 together to a restricted tensor product isomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', they fix the implicit chosen Kv-invariant vector for almost all places v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A natural way to do so is by using Whittaker models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ψ = ⊗ψv : A −→ C× be an additive character of A which can be viewed as a character of N(A), where ψv are additive characters of Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that ψv has conductor ℘−nv v , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', is trivial on ℘−nv v but not on ℘−nv−1 v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have ∑ v nv deg v = 2g − 2, recall that g is the genus of the curve X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let W(π) be the global Whittaker model of π with respect to ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the space of smooth functions ϕ over G(A) such that ϕ(ux) = ψ(u)ϕ(x), ∀u ∈ N(A), ∀x ∈ G(A), and W(πv) the local Whittaker model of πv with respect to ψv (space of functions over Gv similarly defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we have a natural decomposition W(π) = ⊗′vW(πv) and W(πv) = W(πv ⊗ ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, we have (12) Tr(ǫ ◦ R( f)|π) = ∏ v∈|X| Tr(ǫv ◦ R( fv)|W(πv)), where ǫv(xv) = (−1)deg v deg(det(xv)) for v ∈ Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5 (Paskunas-Stevens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ρ be a representation of Kv which inflates a cuspidal represen- tation of G(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let πv be an irreducible representation of Gv that contains ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ψ′ v be an additive character of Fv of conductor ℘v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', is trivial on ℘v and is non-trivial on Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It defines a character of Nv by � 1 x 0 1 � �→ ψ′ v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let W be the space of Whittaker functions of πv with respect to ψ′ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Wρ be the ρ isotypic subspace of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then every function in Wρ is supported in N(Fv)ZvKv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This is a corollary of Paskunas-Stevens’ result [PS08, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s explain their notations from type theory which we need to apply to our specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The group J is Kv, the group J is ZvKv, the group U is Nv, the representation Λ is our ρ, and the character Ψα is obtained by restriction of ψ′ v to N(Ov) then extends to N(Ov)(1 + ℘vM2(Ov)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let X ∈ πv and Y ∈ π∨ v , where π∨ v is the contragredient representation of πv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ΦX,Y be the matrix coefficient of πv defined by ΦX,Y(x) = ⟨πv(x)X, Y⟩, for any x ∈ Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If there is an X such that ΦX,Y ̸= 0, then the map X �→ ΦX,Y embeds πv into C∞ c (Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The result [PS08, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8] shows that π∨v contains a special vector Y∨ α so that the linear map from πv to the space of Whittaker functions X �→ � x �→ � Nv ψ′ v(u)ΦX,Y∨ α (u−1x)du � , is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, Paskunas and Stevens show that πv contains a vector Yα, which is con- tained in ρ-isotypic part (πv)ρ of πv, so that ΦYα,Y∨ α has support in ZvKv ([PS08, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7]) and its associated Whittaker function x �→ � Nv ψ′ v(u)ΦYα,Y∨ α (u−1x)du, is supported in ZvNvKv and extends the function ΦYα,Y∨ α supported on ZvKv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that since the irreducible representation ρ is contained with multiplicity one in πv ([Yu21b, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4]), every X ∈ (πv)ρ is generated by ⟨kYα : k ∈ Kv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, every Whittaker function associated to ΦX,Y∨ α for X ∈ (πv)ρ has support in ZvNvKv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose we’re in the situation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let W(πv) be the Whittaker model for the character ψv of conductor ℘−nv v , then every function in W(πv)ρ is supported in {x ∈ Gv|v(det(x)) ∈ −nv − 1 + 2Z}, where v is the valuation of Fv normalized to be surjective to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 18 HONGJIE YU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let tv be a uniformizer of ℘v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ψ′v := (y �→ ψv(t−nv−1 v y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then ψ′v has conductor ℘v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let b = � t−nv−1 v 0 0 1 � , we have an Gv-equivariant isomorphism: W(πv, ψv) −→ W(πv, ψ′ v), ϕ �→ (x �→ ϕ(bx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5, we deduce that functions in W(πv, ψv)ρ are supported in bZvNvKv, which im- plies the result needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Cuspidal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We apply the previous preparation works to compute the cuspidal terms Jπ( f) in the spectral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, it’s the case that π ⊗ ǫ ∼= π that is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let σ be an involution on PR (see (2) for its definition) that sends (εx(ix))x∈S to (εx(3 − ix))x∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Define bR(k) := |Pσ=Frob∗k R | as the cardinality of the set of fixed points of the action of σ ◦ Frob∗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Scr = ∅, then bR(k) = cR(k) = cR(1) is either 0 or 1 for all k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If either Sc contains a point of even degree or Sr contains a point of odd degree, we have bR(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sc ̸= ∅, we have cR(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The first statement is trivial because, in this case, PR is at most a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that Sr contains a point of odd degree, meaning that Frob∗ has an orbit of odd length on Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that a ∈ 2Z + 1 is the length of such an orbit and x0 ∈ Sr ⊗ Fq is a point in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have εx0(1) ̸= εx0(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that Frob∗a((εx(1x))x) = (ε′ x)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that by definition, we have ε′x0 = εx0(1x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, Frob∗a((εx(ix))x) ̸= σa((εx(ix))x), for any (ix)x ∈ {1, 2}S, since a is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This implies that bR(a) = 0 hence bR(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Similarly, we can prove the case when Sc contains a point of even degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Indeed, note that σa = Id if a is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let π be a cuspidal automorphic representation of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that ǫ is the sign character of G(A) that factors through deg ◦ det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that π ⊗ ǫ ∼= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then Tr(ǫ ◦ R( f)|π) = 0, except if the following conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' bR(1) ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' every place in Su has odd degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' πv ∈ IrrR(Gv) for v ∈ |X| − Su and πv has scalar ramification determined by semisimplification of Rv for v ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If this is the case, we have Tr(ǫ ◦ R( f)|π) = (−1)|Su|2|Su|(−1)deg Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Langlands correspondence, if π ∼= π ⊗ ǫ, no local component of π can be a twisted Steinberg representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, since a Hecke character of F×\\A× is of finite order if it sends an element of degree 1 to a root of unity, if necessary, by replacing π by an inertially equivalent, we may assume that the central character of π is of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that L is the ℓ-adic local system over X − S that corresponds to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If π ⊗ ǫ ∼= π, then L|X−S ∼= L1 ⊕ L2, and Frob∗ permutes L1 and L2 ([Yu21b, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, the ramification of L at every x ∈ S is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, π does not have a twisted Steinberg component by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, it is clear that if Tr(ǫ ◦ R( f)|π) ̸= 0, RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 19 then πv has the desired has the desired ramification type for v ∈ |X| − Su, and the (Iv, ι ◦ χ)- isotypic subspace (πv)(Iv,ι◦χ) is non-trivial (notations as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, πv is either a twisted Steinberg representation in IrrR(Gv) or a twisted unramified principal series in IrrRss(Gv) where Rss is the semisimplification of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have seen that πv can not be a twisted Steinberg representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, the product of all eigenvalues of ramifications of L1 and L2 should be 1, and they are permuted by Frobenius action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This implies that bR(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to calculate Tr(ǫ ◦ R( f)|π) when π ∼= π ⊗ ǫ and πv has the above described property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The equation (12) allows us to calculate it locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2 says if Tr(R( fv)|πv) ̸= 0 then πv ∈ IrrR(Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let v ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If deg(v) is even, then ǫv equals the trivial character of Gv and we have Tr(ǫv ◦ R( fv)|W(πv)) = Tr(R( fv)|πv) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This is impossible as πv is not a twisted Steinberg representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose now that v ∈ Su and deg(v) is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Up to a twist, we may assume that πv is unrami- fied and the eigenvalues of Rv at v are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We take a Whittaker function ϕv ∈ W(πv)Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that since both ǫvϕv and ϕv are contained in W(πv)Kv = W(πv ⊗ ǫv)Kv, they’re differed by a scalar: ǫvϕv = cϕv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let xv be any element in the support of ϕv, we deduce that c = ǫv(xv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ℘c(ψv) v be the conductor of ψv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have shown in [Yu18] that the support of ϕv contains an element of valuation c(ψv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore c = (−1)deg(v)c(ψv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ϕ′ v ∈ W(πv) be the function defined by x �→ ϕv(x � ̟v 0 0 1 � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have ǫvϕ′ v = (−1)deg(v)(c(ψv)+1)ϕ′ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then {ϕv, ϕ′v} is a basis of W(πv)Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The endomorphisms on W(πv), ǫv ◦ R( 1 vol(Iv) 1Iv) and ǫv ◦ R( 1Kv) are composition of a projection onto πIv v together with a linear map represented respectively by the matrix � (−1)deg(v)c(ψv) 0 0 (−1)deg(v)(c(ψv)+1) � , � (−1)deg(v)c(ψv) 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that Tr(ǫv ◦ R( fv)|W(πv)) = (−1)deg(v)(c(ψv)+1) − (−1)deg(v)c(ψv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since deg(v) is odd, it equals Tr(ǫv ◦ R( fv)|W(πv)) = −2(−1)c(ψv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Similarly, for v ∈ |X| − Scr − Su, we have Tr(ǫv ◦ R( fv)|W(πv)) = (−1)c(ψv) deg(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For v ∈ Sc, we deduce from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6 that: Tr(ǫv ◦ R( fv)|W(πv)) = (−1)deg(v)(c(ψv)+1)Tr(R( fv)|W(πv)) = (−1)deg(v)(c(ψv)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 20 HONGJIE YU For v ∈ Sr, as we have seen in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7 that v must have even degree otherwise bR(1) = 0, we have Tr(ǫv ◦ R( fv)|W(πv)) = Tr(R( fv)|W(πv)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In conclusion, by (12) we have Tr(ǫ ◦ R( f)|π) = (−1)|Su|2|Su|(−1)∑v∈|X|−Sc deg(v)c(ψv)+∑v∈Sc deg(v)(c(ψv)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have ∑ v c(ψv) deg(v) = −(2g − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, Tr(ǫ ◦ R( f)|π) = (−1)|Su|2|Su|(−1)deg Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Su ̸= ∅ and at least one place in it has even degree, then J1 cusp( f) = |E2(R)Frob∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Su ̸= ∅ and every place in it has an odd degree, then J1 cusp( f) = |E2(R)Frob∗| + � (−1)|Su|+1bR(1)2|Su|−2(Pic(2) − Pic(1)), Scr = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (−1)|Su|+1bR(1)2|Su|−2(−1)deg ScPic(2), Scr ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Su = ∅, then J1 cusp( f) = |E2(R)Frob∗| + � 0, Scr = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' bR(1) 4 (1 − (−1)deg Sc)Pic(2), Scr ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, the sum of Jπ( f) over inertial equivalent classes of cuspidal automorphic represen- tations π such that π ⊗ ǫ ̸∼= π gives |E2(R)Frob∗| after Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to consider the sum 1 2 ∑ π Tr(R( f)|π) − 1 2 ∑ π Tr(ǫ ◦ R( f)|π), where the sums over π are taken over inertial equivalent classes of cuspidal automorphic repre- sentations π such that π ⊗ ǫ ∼= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The Langlands correspondence (see the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8) shows that no such cuspidal automorphic π can have a twisted Steinberg component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, if Su ̸= ∅, 1 2 ∑ π Tr(R( f)|π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Su = ∅, we need to know the number of equivalence classes of such π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By [Yu18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3] and the first paragraph of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8, the number of such π are in bijections withe the set of non-ordered pairs (L1, L2) of rank 1 ℓ-adic systems over X − S such that Frob∗Li ∼= L3−i, L1 ̸∼= L2, and that the local monodromies of the direct sum L1 ⊕ L2 is given by Rss, the semisimplification of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Scr = ∅, then bR(1) equals 0 or 1 and there are bR(1) Pic(2)−Pic(1) 2 such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Scr ̸= ∅, then there are bR(1) 2 Pic(2) such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have 1 2 ∑ π Tr(R( f)|π) = � bR(1) 4 (Pic(2) − Pic(1)), Scr = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' bR(1) 4 Pic(2), Scr ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8, if bR(1) = 0 or if Su ̸= ∅ and contains a place of even degree, then the sum 1 2 ∑ π Tr(ǫ ◦ R( f)|π) RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 21 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Otherwise if Su ̸= ∅ and does not contain any place of even degree or Su = ∅, then by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8, we have 1 2 ∑ π Tr(ǫ ◦ R( f)|π) = (−1)|Su| � bR(1)2|Su|−2(Pic(2) − Pic(1)), Scr = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' bR(1)2|Su|−2(−1)deg ScPic(2), Scr ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Residuel terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The proposition below describes the contributions to the trace formula from the residual spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Scr is non-empty, then J1 res( f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Scr = ∅, then J1 res( f) = cR(1)(−1)|Su|Pic(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any v ∈ Scr and any character µv of Gv, we have Tr(R( fv)|µv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, the first statement holds, and it suffices to consider the case that Scr = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It’s clear that if v /∈ S, then for a character µv of Gv we have Tr(R( 1Kv)|µv) = � 1, µv|Kv = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, µv|Kv ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let v ∈ Ss, we have Tr(R( fv)|µv) = � 1, µv|Kv = θv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, µv|Kv ̸= θv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' where θv is the character defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Similarly, let v ∈ Su, we have Tr(R( fv)|µv) = � −1, µv|Kv = θv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, µv|Kv ̸= θv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that if (13) ∏ v θv|F× q ̸= 1, then J1 res( f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Otherwise, following our discussions in rank 1 in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3, there are Pic(1) such equivalent classes of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, we have J1 res( f) = (−1)|Su|Pic(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The result is thus proved since (13) is equivalent to cR(1) ̸= 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [De15, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Continuous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The proposition below describes the contributions to the trace formula from the continuous spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sc ̸= ∅ then J1 cont( f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sc = ∅, then we have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) Sr ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' J1 cont( f) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 2cR(1)Pic(1)2(2g − 2 + deg Sr), Su = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1 2cR(1)Pic(1)2 deg v, Su = {v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, |Su| ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 22 HONGJIE YU (2) Sr = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' J1 cont( f) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 cR(1)Pic(1)2(g − 1), Su = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' cR(1)Pic(1)2 deg v 2 + 1 2cR(1)Pic(1), Su = {v} and 2 ∤ deg v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' cR(1)Pic(1)2 deg v 2 , Su = {v} and 2 | deg v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' cR(1)Pic(1)(−1)|Su|+12|Su|−2, |Su| ⩾ 2, and Su,even = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, |Su| ⩾ 2, and Su,even ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The proof is given in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to do some calculations about L-functions and intertwining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The first thing to remark is that if Sc ̸= ∅, then for any ψ ∈ Acont, the action of R( f) on AB,ψ must be the 0 map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, at a place v ∈ Sc, we have AB,ψ ∼= IB(ψv), the induced representation of ψv, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2, it has no ρv, which is cuspidal, isotypic subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore we assume that Sc = ∅ in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' L-functions of Hecke characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We’ll need information on L-functions of Hecke characters when dealing with the continuous part of the trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Over a function field, we have a complete understanding of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let χ : A×/F× −→ C× be a Hecke character of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let χ = ⊗v∈|F|χv be its factorisa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that the L-function L(χ, z) can be defined by the formal power series L(χ, z) := ∏ v∈|X| Lv(χ, z), where Lv(χ, z) = \uf8f1 \uf8f2 \uf8f3 1 1−χv(̟v)zdeg v , if χv is unramified ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The infinite product is absolutely convergent and is holomorphic if |z| < q−1, which admits a meromorphic continuation to the whole C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is, in fact, a rational function in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R be the set of places of ramifications of χ, identified with a subset of closed points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix an isomorphism between C and Qℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Lχ be an ℓ-adic local systems over X − R corresponding to χ obtained by global class field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The L-function of Lχ equals that of χ: L(Lχ, z) = L(χ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, we know from Grothendieck’s cohomological interpretation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', for example, [Laf02, Th´eor`eme VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1]) that L(Lχ, z) = det(1 − zFq|H1 c (X − R, Lχ)) det(1 − zFq|H2c (X − R, Lχ)) det(1 − zFq|H1c (X − R, Lχ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' where Fq is a geometric Frobenius element which acts on the cohomology with compact supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (Riemann hypothesis) Let χ be a Hecke character on F×\\A× of finite order so that the set of ramified places is R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If χ is inertially equivalent to the trivial character, then L(χ, z) = P(z) (1 − z)(1 − qz) where P(z) is a polynomial of 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If χ is not inertially equivalent to the trivial character, then its L-function L(χ, z) is a polynomial in z of degree 2g − 2 + deg R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In any case, all of the zeros of L(χ, z) satisfy |z| = q− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If R is empty, then there are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Lχ|X is trivial, up to a twist of a rank 1 sheaf over Spec(Fq), we may assume that Lχ ∼= Qℓ is the constant sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have L(Lχ, z) = ζX(z) = P(z) (1 − z)(1 − qz), deg P(z) = 2g and all of the zeros of P(z) satisfy |z| = q− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Lχ|X is non-trivial then L(Lχ, z) is a polynomial of degree 2g − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For reference, see [Yu18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We consider the case that R is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We know that (14) H0 c (X − R, Lχ) = 0, as Lχ has no nonzero properly supported section over X − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Poincar´e duality (15) H2 c (X − R, Lχ)∨ ∼= H0(X − R, L∨ χ)(1) ∼= HomX−R(L∨ χ, Qℓ)(1), from which we deduce that H2 c (X − R, Lχ) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The dimension of H1 c (X − R, Lχ) can then be derived from the Euler-Poincar´e characteristic: dim H1 c (X − R, Lχ) = −χc(X − R, Lχ), which can be calculated by the Grothendieck-Ogg-Shafarevich formula, see [Ra95, Th´eor`eme 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, since every local monodromy of Lχ is tamely ramified, the Swan conductor is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce from loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' that χc(X − R, Lχ) = χc(X − R) = 2 − 2g − deg R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The assertion about the positions of zeros is the Riemann hypothesis for rank 1 local systems (see [Laf02, Th´eor`eme VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='10] for a general statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Eulerian expansions of intertwining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ψ ∈ Acont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let M(w, λ) : AB,ψ −→ AB,w(ψ) be an intertwining operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is a G(A)-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let IB(ψv) be the space of functions ϕ over Gv satisfying ϕ(ntx) = ρB(t)ψv(t)ϕ(x) for any n ∈ Nv, t ∈ Tv and x ∈ Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By definition of intertwining operator, we have an Eulerian expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Indeed, let Mv(w, λ) : IB(ψv) −→ IB(w(ψv)) be an operator defined by analytic continuation of the integral which converges when |Reλ| >> 0, (16) (Mv(w, λ)ϕ)(x) = λ(x) � Nv ϕ(w−1nx)λ(w−1nx)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Choosing isomorphisms cψ : AB,ψ −→ ⊗vIB(ψv) and cw(ψ) : AB,w(ψ) −→ ⊗vIB(w(ψv)), there is a constant c depending only on these isomorphisms such that the following diagram is commuta- tive: AB,ψ M(w,λ) −−−−→ AB,w(ψ) cψ \uf8e6\uf8e6� cw(ψ) \uf8e6\uf8e6� ⊗vIB(ψv) c⊗vMv(w,λ) −−−−−−−→ ⊗vIB(w(ψv)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In the special case that ψ is non-regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', w(ψ) = ψ, we have c = q1−g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It comes from the different normalization of the Haar measures on Nv and N(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 24 HONGJIE YU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Intertwining operator on (Kv, θv)-isotypical subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this part, we treat the local intertwin- ing operator when v ∈ Ss or v /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let θv be a character: θv : Kv det −→ O× v −→ κ× v −→ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have dim IB(ψv)(Kv,θv) ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The space IB(ψv)(Kv,θv) is non-zero if and only if ψv = (ψv,1, ψv,2) with ψv,1|O× v = ψv,2|O× v = θv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case, there is a µ ∈ C× such that for any y ∈ F× v , we have ψv1(y)/ψv2(y) = µdeg y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, IB(ψv)(Kv,θv) is generated by the function ϕψv which satisfies b ∈ Bv and k ∈ Kv: ϕψv(bk) = ρB(b)ψv(b)θv(det k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For x ∈ Gv and λ ∈ XG T ∼= C× (see subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 for definition), let ϕψv,λ(x) := ϕψv(x)λ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By dimension one, we have Mv(w, λ)ϕψv = cλϕw(ψv), for some constant cλ ∈ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It suffices to evaluate the above equation at x = 1 to find the value c: cλ = � Nv ϕψv,λ(w−1n)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The integral can be decomposed: � N(Ov) ϕψv,λ(w−1n)dn + � Fv−Ov ϕψv,λ( � 1 y−1 0 1 � � y−1 0 0 y � � −1 0 y−1 −1 � )dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that � N(Ov) ϕψv,λ(w−1n)dn = vol(N(Ov)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Besides, for y ∈ Fv − Ov, we have ϕψv,λ( � 1 y−1 0 1 � � y−1 0 0 y � � −1 0 y−1 −1 � ) = µ− deg yλ−2 deg v deg y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Under our additive Haar measure on Ov, we know that vol(̟n vO× v ) = q−n v (1 − q−1 v ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that cλ = � N(Fv) ϕψv,λ(w−1n)dn = 1 − q−1 v µλ2 deg v 1 − µλ2 deg v = Lv(ψ1ψ−1 2 , λ2) Lv(ψ1ψ−1 2 , q−1λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Intertwining operator on (Iv, χv)-typical subspace: regular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this part, we treat the case that v ∈ Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If χv is regular, then we have dim IB(ψv)(Iv,χv) ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, we have Gv = BvIv ∐ BvwIv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Any function ϕ ∈ IB(ψv)(Iv,χv) is entirely determined by ϕ(1) and ϕ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, for such a ϕ, using the definition of IB(ψv) and (Iv, χv)-typical condition, we have ϕ(tw) = ψv(t)ϕ(w) = ϕ(w)χv(w−1tw) and ϕ(t) = χv(t)ϕ(1) = ϕ(1)ψv(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore if ψv|T(Ov) ̸= χv and ψv|T(Ov) ̸= w(ψv) then ϕ must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If ψ|T(Ov) = χv, then ϕ is supported in BvIv and if ψv|T(Ov) = w(χv) then ϕ is supported in BvwIv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 25 Suppose that we have ψv|T(Ov) = χv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The space IB(ψv)(Iv,χv) is generated by the function ϕψv which is defined in such a way that for any n ∈ Nv, t ∈ Tv and k ∈ Kv: ϕψv(ntk) = � ρB(t)ψv(t)ψv(k), if k ∈ Iv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, if k ∈ IvwIv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The space IB(w(ψv))(Iv,χv) is generated by ϕw(ψv) which is defined in such a way that for any n ∈ Nv, t ∈ Tv and k ∈ Kv: ϕw(ψv)(ntk) = � ρB(t)ψv(t)ψ(b), if k = nwb ∈ Iv+wIv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, if k ∈ Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The local intertwining operator Mv(w, λ) is a linear map from IB(ψv)(Iv,χv) to IB(w(ψv))(Iv,χv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By dimension 1, there is a constant cλ ∈ C such that Mv(w, λ)ϕψv = cλϕw(ψv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Evaluating at the point x = w, we see that cλ = � Nv ϕψv(w−1nw)λ(w−1nw)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We break the integral into two parts following the union: w−1Nvw = � 1 0 ℘v 1 � ∪ � 1 0 Fv − ℘v 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The first part is included in Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For the second part, we need � 1 0 x 1 � = � x−1 0 0 x � � 1 x 0 1 � � 0 −1 1 x−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since � 0 −1 1 x−1 � ∈ IvwIv, the integral over � 1 0 Fv − ℘v 1 � vanishes and cλ = vol(℘v) = q−1 v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As the local L-factor Lv(ψ1ψ−1 2 , λ2) is trivial, we can present the result as cλ = q−1 v Lv(ψ1ψ−1 2 , λ2) Lv(ψ1ψ−1 2 , q−1λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Intertwining operator on (Iv, χv)-typical subspace: non-regular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this subsection, we calculate intertwining operator on (Iv, χv)-typical subspace of IB(ψv) when ψv is non-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Such calculations have already been done in [Fl15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, the calculations are similar to the cases we have already treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, we’ll briefly recall the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If χv is a character of Iv that factors through determinant, we have dim IB(ψv)(Iv,χv) = 2 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The dimension is non-zero if and only if ψ|T(Ov) lifts χv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, the double quotient Bv\\Gv/Iv ∼= Iv\\Kv/Iv has cardinality 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We know that dim IB(ψv)(Iv,χv) ⩽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If ψv|T(Ov) lifts χv, the space IB(ψv)(Iv,χv) is generated by the basis (ϕψv,1, ϕψv,w), where for any n ∈ Nv, t ∈ Tv and k ∈ Kv, we have: ϕψ,1(ntk) = � ρB(t)ϕ(t)ψv(det(k)), if k ∈ Iv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, if k ∈ IvwIv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ϕψ,w(ntk) = � ρB(t)ψ(t)ψv(det(k)), if k ∈ IvwIv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, if k ∈ Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 26 HONGJIE YU If ψv|T(Ov) does not extend χv, then IB(ψv)(Iv,χv) is zero since for any ϕ ∈ IB(ψv)(Iv,χv), we have ϕ(t) = ψv(t)ϕ(1) = ϕ(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='t) = χv(t) for any t ∈ T(Ov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The local intertwining operator Mv(w, λ) is a linear map from IB(ψv)(Iv,χv) to IB(w(ψv))(Iv,χv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In the basis (ϕψv,1, ϕψv,w) and (ϕw(ψv),1, ϕw(ψv),w), we have (see [Fl15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7]), Mv(w, λ)(ϕψv,1, ϕψv,w) = (ϕw(ψv),1, ϕw(ψv),w) \uf8eb \uf8ed(1 − q−1 v ) µλ2 1−µλ2 1 q−1 v (1 − q−1 v ) 1 1−µλ2 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to present the result in the form Mv(w, λ)(ϕψv,1, ϕψv,w) = (ϕw(ψv),1, ϕw(ψv),w) Lv(ψ1ψ−1 2 , λ2) Lv(ψ1ψ−1 2 , q−1λ2) \uf8eb \uf8ed (1−q−1 v )µλ2 deg v 1−q−1 v µλ2 deg v 1−µλ2 deg v 1−q−1 v µλ2 deg v q−1 v (1−µλ2 degv) 1−q−1 v µλ2 deg v 1−q−1 v 1−q−1 v µλ2 deg v \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, det(Mv(w, λ)) = −q−1 v 1 − qvµλ2 deg v 1 − q−1 v µλ2 deg v ( Lv(ψ1ψ−1 2 , λ2) Lv(ψ1ψ−1 2 , q−1λ2))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' and Tr(Mv(w, λ)) = Lv(ψ1ψ−1 2 , λ2) Lv(ψ1ψ−1 2 , q−1λ2) (1 − q−1 v )(1 + µλ2 deg v) 1 − q−1 v µλ2 deg v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Continuous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we come to the calculations of contributions from continuous spec- trum using the previous preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We are going to consider J1 ψ( f) for ψ ∈ Acont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sr ̸= ∅, then J1 ψ( f) ̸= 0 implies that ψ is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case, there are cR(1) 2 Pic(1)2 equivalent classes of ψ such that J1 ψ( f) can be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sr = ∅, then there are cR(1)1 2Pic(1)(Pic(1) − 1) regular classes of ψ such that J1 ψ( f) can be non-zero, and cR(1)Pic(1) non-regular classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that for each place v ∈ Su, we have set in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2 that fv = � 1 vol(Iv) 1Iv(x) − 2 1Kv(x) � θv(det x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have J1 ψ( f) = ∑ S0⊆Su (−1)|S0|2|S0|J1 ψ( fS0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' where fS0 = ⊗v fS0,v ∈ C∞ c (G(A)) is the function that fS0,v = fv for all places v outside Su and is equal to x �→ 1Kv(x)θv(det x−1) if v ∈ S0 and is equal to 1 vol(Iv) 1Iv(x)θv(det x−1) for v ∈ S1 = Su − S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need a lemma for our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 27 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let V be a finite-dimensional C-linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let m be a meromorphic function over C with values in GL(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that m is holomorphic at any point in the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We use Z|λ|<1(h) for the integer defined as the number of zeros (with multiplicity) minus the number of poles (with multiplicity) in the region |λ| < 1 of a meromorphic function h over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then � ImXG T lim µ−→1 TrV( 1 µ−1 − µ m(λ)−1 ◦ m(λ/µ) − 1 µ−1 − µ Id)dλ = 1 2 Z|λ|<1(det(m(λ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have lim µ−→1TrV( 1 µ−1 − µm(λ)−1 ◦ m(λ/µ) − 1 µ−1 − µ Id) = λ 2 TrV(m(λ)−1 ◦ m′(λ)), where m(λ) is the C-linear endomorphism V defined as the derivative of m(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We use Jacobi’s formula: TrV(m(λ)−1 ◦ m′(λ)) = d dλ det(m(λ)) det(m(λ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Finally since the volume of ImXG T is normalized to be 1, by definition of contour integration and argument principle, the integral � ImXG T d dλ det(m(λ)) det(m(λ)) λdλ equals Z|λ|<1(det(m(λ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ We are going to apply this result to intertwining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Although, the operator M(w, λ) is a morphism from AB,ψ to AB,w(ψ), the representation structures of AB,ψ and AB,w(ψ) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let V be the parabolic induction of ψ from B(A) to G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix a G(A)-equivariant isomorphism: ι1 : AB,ψ −→ V and ι2 : AB,w(ψ) −→ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then TrAB,ψ((− 1 µ−1 − µM(w, λ)−1 ◦ M(w, λ/µ) + 1 µ−1 − µ) ◦ R( fS0)), equals TrV((− 1 µ−1 − µ(ι2M(w, λ)ι−1 1 )−1 ◦ ι2M(w, λ/µ)ι−1 1 + 1 µ−1 − µ) ◦ ι1R( fS0)ι−1 1 ), We apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that given a finite family of complex vector spaces Vi of dimension ni and endomorphisms φi ∈ End(Vi), we have det(⊗iφi) = ∏ i det(φi)∏j̸=i ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Eulerian expansion in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2, and local calculations in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5, we deduce that the integral � ImXG T lim µ−→1 TrAB,ψ((− 1 µ−1 − µ M(w, λ)−1 ◦ M(w, λ/µ) + 1 µ−1 − µ) ◦ R( fS0))dλ, equals (17) 2|S1| 2 Z|λ|<1( L(ψ1ψ−1 2 , λ2) L(ψ1ψ−1 2 , q−1λ2)) + 2|S1|−1 2 (2 ∑ v∈S1 deg v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If ψ ∈ Acont is regular which has the correct ramification, (17) equals 2|S1|(2g − 2 + deg Sr) + (2|S1|−1) deg S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have (18) J1 ψ( f) = ∑ S0⊆Su (−1)|S0|2|S0| � 2|S1|(2g − 2 + deg Sr) + (2|S1|−1) deg S1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 28 HONGJIE YU If ψ ∈ Acont is non-regular, then ψ can not have correct ramification if Sr is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that Sr = ∅ and ψ has correct ramification, then J1 ψ( f) is the sum of (19) 1 2 ∑ S0⊆Su (−1)|S0|2|S0| � 2|S1|(2g − 1) + (2|S1|−1) deg S1 � with (20) 1 8 ∑ S0⊆Su (−1)|S0|2|S0| � ∑ λG∈{±1} ∑ λw∈ImXG T λ2w=λ−1 G λGTrAB,ψ(M(w, w−1(λw)) ◦ R( fS0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have ∑ I⊆S (−1)|I| = � 1, if S = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, if S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We also have ∑ I⊆S (−1)|S|−|I| deg I = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 0, if |S| ⩾ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' deg v, if S = {v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, if S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The first equality is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s prove the second equality by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We may suppose |S| ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' otherwise, the equality is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let v ∈ S, we have ∑ I⊆S (−1)|S|−|I| deg I = ∑ I⊆S−{v} (−1)|S|−|I| deg I + ∑ v∈I⊆S (−1)|S|−|I| deg I = ∑ I⊆S−{v} (−1)|S|−|I|(− deg v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, the result follows from the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ After this lemma, for regular ψ, the expression (18) J1 ψ( f) vanishes if |Su| ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When Su = {v}, it equals deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When Su = ∅, it equals 2g − 2 + deg Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now suppose that Sr = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For non-regular ψ, the expression (19) vanishes if |Su| ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When Su = {v}, it equals 1 2 deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' When Su = ∅, it equals 1 2(2g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ψ ∈ Acont be non-regular, we’re going to consider (21) 1 8 ∑ λG∈{±1} ∑ λw∈ImXG T λ2w=λ−1 G λGTrAB,ψ(M(w, w−1(λw)) ◦ R( fS0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that in this case, L(ψ1ψ−1 2 , z) = ζX(z) where ζX is the zeta function of the curve X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' From the local calculations in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5, we know that TrAB,ψ(M(w, w−1(λw)) ◦ R( fS0)) = q1−g ζX(λ−2 w ) ζX(q−1λ−2 w ) ∏ v∈S1 (1 − q−1 v )(λ−2 degv w + 1) 1 − q−1 v λ−2 deg v w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Here we should regard ζX(z) ζX(q−1z) as a rational function so that the pole at z = 1 of the denominator and numerator are canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For each λG, there are two λw such that λ2w = λ−1 G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The expression (21) equals: (22) ∑ λG∈{±1} 1 4q1−gλG ζX(λG) ζX(q−1λG) ∏ v∈S1 (1 − q−1 v )(λdeg v G + 1) 1 − q−1 v λdeg v G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There is a polynomial P(z) of degree 2g such that ζX(z)/ζX(zq−1) = P(z)(1 − q−1z) P(zq−1)(1 − qz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 29 By functional equation of zeta function, we have P(1)(1 − q−1) P(q−1)(1 − q) = −qg−1, and P(−1)(1 + q−1) P(−q−1)(1 + q) = qg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The corresponding summand for λG = 1 in the expression (22) equals 1 4q1−g2|S1| P(1)(1 − q−1) P(q−1)(1 − q) = −2|S1|−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If there is a place of odd degree in S1, then the corresponding summand for λG = −1 in the expression (22) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If all the places in S1 are of even degree, then the corresponding summand for λG = −1 in the expression (22) equals 1 4q1−g(−1) P(−1)(1 + q−1) P(−q−1)(1 + q)2|S1| = −2|S1|−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We decompose Su as the union of the set of places of odd degree and those of even degree: Su = Su,odd ∪ Su,even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Following the above discussions, the expression (20) equals (23) ∑ S0⊆Su (−1)|S0|2|S0|(−2|S1|−2) + ∑ Su,odd⊆S0⊆Su (−1)|S0|2|S0| � −2|S1|−2� + ∑ S0∈S 0, where S = {S0|S0 ⊆ Su} − {S0|Su,odd ⊆ S0 ⊆ Su}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The first sum in (23) is − 1 4 if Su = ∅ and is 0 if Su ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The second sum in (23) equals 0 if Su,even ̸= ∅, and equals (−1)|Su|+12|Su|−2 if Su,even = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In conclusion, if Su = ∅, then the expression (20) equals − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Su ̸= ∅ but Su,even = ∅, then (20) equals (−1)|Su|+12|Su|−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' if Su,even ̸= ∅, (20) equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To conclude the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='11, we summarize that if Sr ̸= ∅ then there is no non- regular ψ with correct ramification and J1 cont( f) is equal to cR(1) 2 Pic(1)2 times (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' if Sr = ∅, J1 cont( f) is equal to cR(1) 1 2Pic(1)(Pic(1) − 1) times (18) plus cR(1)Pic(1) times ((19)+(20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' GEOMETRIC SIDE OF THE TRACE FORMULA AND HITCHIN MODULI SPACES In this section, we treat J1 geom( f), whose definition is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The goal is to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to introduce a Lie algebra analog of J1geom( f), which was studied first in [Ch15] where one can also find relations between this Lie algebra trace formula and the groupoid cardinality of the category of semistable Higgs bundles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we introduce the moduli space of semistable Hitchin bundles with parabolic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is constructed by Yokogawa ([Yo93]) using GIT theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R = Scr ∪ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If we can identify R with a subset of X(Fq), then Yokogawa’s construction perfectly suits our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Otherwise, we meet a problem since a point in R can be split into several points over an algebraically closed field, and we have to make extra arguments about how to treat with (semi)- stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To remedy it, we work over an algebraically closed field or a large enough extension of Fq where we can apply Yokogawa’s construction to obtain a moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we define an Fq-structure on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 30 HONGJIE YU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moduli of quasi-parabolic Hitchin bundles over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let D be a divisor over X and R be a set of closed points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We identify R := R ×Spec(Fq) Spec(Fq) with a subset of X(Fq) and view D also as a divisor over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A quasi-parabolic Hitchin triple (or quasi-parabolic Hitchin bundle) over X is a triple (E, ϕ, (Lx)x∈R) where (E, ϕ) is a Hitchin pair for the divisor D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', a vector bundle E together with a bundle morphism ϕ : E −→ E(D) := E ⊗ OX(D), and for each point x ∈ R, Lx is a line in the Fq-vector space Ex, the fiber over x of the bundle E, such that ϕx(Lx) ⊆ Lx where we view Lx also as a line in E(D)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A quasi-parabolic Hitchin bundle is called (strictly) parabolic if ϕx(Lx) = 0 and Imϕx ⊆ Lx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', ϕx is nilpotent for 0 ⊆ Lx ⊆ Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If D = KX + ∑v∈R v, we will call them quasi-parabolic Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For each x ∈ R, let ξx := (ξx,1, ξx,2) ∈ Q2 such that ξx,1 ⩾ ξx,2 ⩾ ξx,1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ξ = (ξx)x∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let (E, ϕ, (Lx)x∈R) be a quasi-parabolic Hitchin bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let L be a sub-line bundle of E, we define the parabolic degree p-deg(L) by p-deg(L) := deg(L) + ∑ x∈R � ξx,1, if Lx = Lx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ξx,2, if Lx ̸= Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We say that (E, ϕ, (Lx)x∈R) is ξ-semistable if for any sub-line bundle L of E satisfying ϕ(L) ⊆ L(D), we have p-deg(L) ⩽ deg E + ∑x∈R(ξx,1 + ξx,2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if deg(E) + ∑ x∈R ±(ξx,1 − ξx,2) /∈ 2Z, then the equality can never be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We say that such cases are in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Yokogawa has constructed a moduli space which is a variety defined over Fq that classifies isomorphism classes of ξ-stable quasi-parabolic Hitchin bundles (E, ϕ, (Lx)x∈R) with E being of rank 2 and degree e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote the variety by qMe,ξ 2,R(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If we are considering the case of parabolic ones instead of being only quasi-parabolic, we will denote it by Me,ξ 2,R(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Yokogawa’s results apply under the assumption (under our terminology) that ξx,1 > ξx,2 > ξx,1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, it does not include the case that ξx,1 = ξx,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' However, this is not an issue because when ξx,1 and ξx,2 are close enough, semistability of quasi-parabolic Hitchin bundles coincide with the case that ξx,1 = ξx,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Fq-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have defined a coarse moduli space qMe,ξ 2,R(D) which is a variety over Fq con- structed by Yokogawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose n is a divisible enough integer such that every place (closed point) in R totally splits over Fqn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the residue field of a place in R can be embedded in Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Yokogawa’s construction works if the curve is X ⊗ Fqn and the parabolic structures are imposed at each point of S ⊗ Fqn and in this way qMe,ξ 2,R(D) is a variety defined over Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this subsection, we are going to endow qMe,ξ 2,R(D) with a Fq-structure when ξx are the same for points x ∈ R ⊗ Fqn lying over each v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 31 For a variety Y defined over Fqn, let FY/Fqn be the arithemetic Frobenius morphism of Y ⊗Fqn Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that it is a morphism of schemes Y ⊗Fqn Fq which is identity on Y and is x �→ xqn over Spec(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, it is not a morphism of Fqn-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The pullback by FX/Fq defines an action on the set of isomorphism classes of Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To define an action on the parabolic structure, we need to use an equivalent definition that identifies vector bundles with locally free sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose (E, ϕ, (Lx)x∈R) is a quasi-parabolic Higgs bun- dle over X, For each x ∈ R, Lx defines a unique rank 2 coherent sub-sheaf E x of E such that E/E x is a skyscraper sheaf of degree 1 supported in {x} and the inclusion E x −→ E induces a mor- phism of their fibers at x which has image Lx in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then F∗ X/FqE x defines a parabolic structure of F∗ X/FqE at Frob(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition-Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that the family (ξv)v∈R ∈ (Q2)R satisfies that for any v ∈ R, 0 ⩽ ξv,1 − ξv,2 ⩽ [κv : Fq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ξx = 1 [κv : Fq]ξv, for any point x ∈ R lying over v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let σ be the Frobenius element in Gal(Fq|Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define an action of Gal(Fq|Fq) on qMe,ξ 2,R(D)(Fq) so that σ sends a parabolic Higgs bundle (E, ϕ, (Lx)x∈R) over X to F∗ X/Fq(E, ϕ, (Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There is a unique variety defined over Fq whose Fqk-points are exactly those in qMe,ξ 2,R(D)(Fq) fixed by σk and whose base change to Fq is isomorphic to qMe,ξ 2,R(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We denote the variety by qMe,ξ 2,R(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' First, we need to show that the action is well-defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', a stable quasi-parabolic Higgs bundle is sent to a stable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This follows from two observations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let L be a sub-line bundle of E such that ϕ(L) ⊆ L(D), then the parabolic degree on F∗ X/FqL equals that of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For a divisible enough k ⩾ 1, we have F∗k X/Fq(E, ϕ, (Lx)x∈R) ∼= (E, ϕ, (Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that qMe,ξ 2,R(D) is quasi-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Varieties over Fq whose base change to Fq are isomor- phic to qMe,ξ 2,R(D) are in bijection with Galois descent data on qMe,ξ 2,R(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It suffices to prove that the above action defines a Galois descent data ([BLR90, Example B, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='139]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We note that the action is induced by an algebraic morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This is clear, because the action of (F∗ X/Fq)n = F∗ X⊗Fqn/Fqn coincides with the action of FqMe,ξ 2 /Fqn on qMe,ξ 2,R(D)(Fq) which is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This also shows that the action is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Next, we are going to study qMe,ξ 2,R(D)(Fq), the fixed points of F∗ X/Fq on qMe,ξ 2,R(D)(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define a rank 2 quasi-parabolic Higgs bundle as a tuple (E, ϕ, (Lx)x∈R) over X consisting of a vector bundle E over X, a bundle morphism ϕ : E −→ E(D) and for each v ∈ R a 1 dimensional κv sub-vector space Lv of E(v) such that ϕv(Lv) ⊆ Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let (e, ξ) ∈ Z × (Q2)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let L be a sub-line bundle of E such that ϕ(L) ⊆ L(D), we define the parabolic degree L by p-deg(L) := deg(L) + ∑ v∈R � ξv,1, if Lv = Lv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ξv,2, if Lv ̸= Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The quasi-parabolic Higgs bundle (E, ϕ, (Lx)x∈R) is semistable if for all such L, we have p-deg(L) ⩽ deg E + ∑v∈R(ξv,1 + ξv,2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 32 HONGJIE YU Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Under the hypothesis of Proposition-Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2, the set of stable quasi-parabolic Higgs bundles (E, ϕ, (Lx)x∈R) over X which are fixed by F∗ X/Fq is in bijection with the set of stable quasi- parabolic Higgs bundles (E0, ϕ0, (Lv)v∈R) over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let σ be the Frobenius element that generates Gal(Fq|Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Galois descent (see [BLR90, Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='139]), the category of vector bundles over X is equivalent to the category of Gal(Fq|Fq)- equivariant vector bundles over X, that is the category of vector bundles E over X together with an isomorphism for each i ⩾ 1, φσi : (E, ϕ, (Lx)x∈R) −→ F∗i X/Fq(E, ϕ, (Lx)x∈R), such that σi �→ φσi satisfies the cocycle condition: σi(φσj) ◦ φσi = φσi+j, and for some d ∈ N∗, φσd is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that the category of vector bundles is equivalent to the category of locally free sheaves, hence every vector bundle defined over X is automatically defined over X ⊗ Fqn for some n ∈ N∗ and the requirement that φσd is the identity map makes sense when d is divisible by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Galois descent for morphisms of quasi-coherent sheaves ([BLR90, Proposition 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='130] and [BLR90, Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='139]), the above equivalence extends to an equivalence between the category of quasi-parabolic Higgs bundles over X is equivalent to the category of Gal(Fq|Fq)- equivariant quasi-parabolic Higgs bundles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Indeed, the parabolic structure is determined by a sub-sheaf F ⊆ E such that E/F is a skyscraper sheaf supported in R and that the inclusion F → E induces a morphism Fx → Ex of their fibers at each point x which has image Lx in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If both F and E come from X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' F = F0|X and E = E0|X then F0 ⊆ E0 is a subsheaf such that the quotient E0/F0 is a skyscraper sheaf supported in R and the map of the fibers Fv → Ev has a 1-dimensional image as κv-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to show that if (E, ϕ, (Lx)x∈R) is stable and its isomorphism class is fixed by Frobenius, then it defines a unique descent datum (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By our assumption, there is an isomorphism φσ : (E, ϕ, (Lx)x∈R) −→ F∗ X/Fq(E, ϕ, (Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can define simply for every i ⩾ 1, φσi := σi−1(φσ) ◦ · · · ◦ σ(φσ) ◦ φσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that by cocycle condition, this is the unique possible way to extend φσ to a 1-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will prove that we can choose φσ so that for some n, the isomorphism φσn defined above is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (E, ϕ, (Lx)x∈R) is defined over X ⊗ Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then F∗n X/Fq(E, ϕ, (Lx)x∈R) = (E, ϕ, (Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Any φσn is an automorphism of (E, ϕ, (Lx)x∈R) which must be a scalar multiplication because (E, ϕ, (Lx)x∈R) is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Hence there is a λ′ ∈ F× q such that φσn = λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='id(E,ϕ,(Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose λ′ ∈ F× qm for some m ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that φσnm = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='id(E,ϕ,(Lx)x∈R) with λ = (λ′)1+qn+···+qn(m−1) ∈ F× qm ⊆ F× qnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that this implies that (24) σnm−1(φσ) ◦ · · · ◦ σ(φσ) ◦ φσ = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='id(E,ϕ,(Lx)x∈R) Applying σ to both sides, we obtain that (25) σnm(φσ) ◦ · · · ◦ σ2(φσ) ◦ σ(φσ) = σ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='idF∗ X/Fq(E,ϕ,(Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 33 Since σnm(φσ) = φσ, we deduce that σ(λ) = λ and hence λ ∈ F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As the norm map from Fqnm to Fq is surjective, we conclude that by modifying φσ by a scalar, we can get a φσ such that λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is easy to see that such descent data are unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, any two iso- morphisms φσ and φ′ σ are differed by a scalar because these are the only automorphisms of (E, ϕ, (Lx)x∈R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As the map λ �→ λ/σ(λ) from Fq to Fq is surjective (Hilbert’s Theorem 90), we conclude that φσ is isomorphic to φ′ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It remains to prove that the quasi-parabolic Higgs bundle (E0, ϕ0, (Lv)v∈R) determined by a descent datum attached to (E, ϕ, (Lx)x∈R) is stable and reversely if (E0, ϕ0, (Lv)v∈R) is stable then the quasi-parabolic Higgs bundle (E, ϕ, (Lx)x∈R) over X is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The first statement is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For the second statement, we need to use the fact that semistability coincides with stability since the parameter ξ is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose (E, ϕ, (Lx)x∈R) is not semistable, then it admits a maximal destabilizing sub line bundle over X preserved by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Such a line bundle over X is unique, hence is fixed by Galois action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It then descends to a line bundle over X, which, by our definition of parabolic degree, is again destabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This contradicts the fact that (E0, ϕ0, (Lv)v∈R) is semistable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Residue morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Next, we are going to define and study residue morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let V be a subset of Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will be interested in the case that R = V ∪ Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We suppose that DR = KX + ∑ v∈Scr v + ∑ v∈V v, and we will be interested in M1 2,V(DR) = M1,0 2,V(DR), the moduli space of strictly parabolic Hitchin bundles with parabolic structures in V where we set parabolic weights to be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let (E, ϕ) be a Hitchin bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The morphism ϕ is equivalent to a morphism OX −→ End(E)(DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Its characteristic polynomial t2 + at + b gives a section (a, b) in H0(X, OX(DR)) ⊕ H0(X, OX(2DR)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let AR be the affine space H0(X, OX(DR)) ⊕ H0(X, OX(2DR)) defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let RR = ∏ v∈R Rv, where Rv = OX(DR)v ⊕ OX(2DR)v, the fiber OX(DR) ⊕ OX(2DR) in v which is an Fq-vector space by forgetting its κv-vector space structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the restriction of scalars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that we have Rv(Fq) ∼= {t2 + at + b|a, b ∈ κv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have a morphism AR −→ RR, sending (a, b) to ((av, bv))v∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that deg R = |R ⊗Fq Fq| > 2 − 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The morphism AR −→ RR is linear of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The image consists of elements (t2 + avt + bv)v∈S such that (26) ∑ v∈S Trκv|Fq(av) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R1 R be the linear sub-scheme of RR of elements satisfying the residue condition (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Riemann-Roch theorem, it’s easy to verify that when deg S > 2 − 2g, we have H0(X, OX(2D)) −→ ∏ v∈S OX(2D)v is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The kernel of the map H0(X, OX(D)) −→ ∏ v∈S OX(D)v, is H0(X, ωX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, for dimension reasons, the image is a linear subspace of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The last thing to observe is that the condition (26) is necessary due to the residue theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 34 HONGJIE YU The Hitchin fibration is the morphism that sends a Hitchin bundle to the characteristic polyno- mial of the Higgs field: M1 2(DR) −→ AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By forgetting the parabolic structure, we obtain a morphism of Fq-schemes qM1 2,V(DR) −→ AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define the residue morphism as the composition of the natural morphism AR −→ R1 R with Hitchin fibration: res : qM1 2,V(DR) −→ AR −→ R1 R, and res : M1 2,V(DR) −→ AR −→ R1 R, It sends a triple (E, ϕ, (Lv)v∈R) to the family of characteristic polnomial of ϕv, v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R1 Scr be the linear subspace of R1 R whose components in V are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The residue morphism factors through the morphism res : M1 2,V(DR) −→ R1 Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given a point o ∈ R1 Scr(Fq), we will denote M1 2,V(o) := res−1(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Geometric side of trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We introduce a variant of the trace formulas with an extra parameter ξ ∈ (Q2)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let HB : B(A) −→ Q2 be the function defined by HB( � a b 0 d � ) = (− deg a, − deg d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The Harish-Chandra’s map is the extension of HB to the whole G(A) by Iwasawa decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', if x = bk ∈ G(A) with b ∈ B(A) and k ∈ G(O), we have HB(x) = HB(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let �τB be the characteristic function over Q2 of the subset {(x, y) ∈ Q2|x > y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any x ∈ G(A), let x = bk be its Iwasawa decomposition with b ∈ B(A) and k = (kv)v∈|X| ∈ G(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For every v ∈ R, we define sx,v to be the identity if kv ∈ Iv and to be the non-trivial permutation in S2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, given (ξ1, ξ2) ∈ Q2, we have sx,v(ξ1, ξ2) = � (ξ1, ξ2), if kv ∈ Iv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (ξ2, ξ1), if kv /∈ Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let f ∈ C∞ c (g(A)), and ξ = (ξx)x∈|S| ∈ (Q2)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The ξ-variant of truncated trace for Lie algebra is defined by the integral Jg,e,ξ( f) := � G(F)\\G(A)e kg,ξ(x)dx, where kg,ξ(x) equals ∑ γ∈g(F) f(ad(x)−1γ) − ∑ δ∈B(F)\\G(F) �τB(HB(δx) + ∑ v∈R sδx,vξv) ∑ γ∈t(F) � n(A) f(ad(δx)−1(γ + U))dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Eg = {t2 + at + b ∈ F[t]}, RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 35 be the set of rank 2 unitary polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let EG = {t2 + at + b ∈ F[t]|b ̸= 0}, be the subset of Eg consisting of polynomials whose constant term is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any element γ ∈ g(F), we define χγ ∈ Eg as the characteristic polynomial of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given χ ∈ Eg, we define kg,ξ χ (x) for x ∈ G(A) by ∑ γ∈g(F),χγ=χ f(ad(x)−1γ) − ∑ δ∈B(F)\\G(F) �τB(HB(δx) + ∑ v∈R sδx,vξv) ∑ γ∈t(F),χγ=χ � n(A) f(ad(δx)−1(γ + U))dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define Jg,e,ξ χ ( f) := � G(F)\\G(A)e kg,ξ χ (x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The integral converges, and it is non-zero for only finitely many χ ∈ Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, we have Jg,e,ξ( f) = ∑ χ∈Eg Jg,e,ξ χ ( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If we set ξ = 0, we’ll omit ξ from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We also define a group version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let χ ∈ EG and f ∈ C∞ c (G(A)), then we define kG,ξ χ (x) = ∑ γ∈G(F),χγ=χ ∑ i∈Z f(x−1γaix) − ∑ δ∈B(F)\\G(F) �τB(HB(δx) + ∑ v∈R sδx,vξv) ∑ γ∈T(F),χγ=χ ∑ i∈Z � N(A) f((δx)−1(γn)δaix)dn, where a ∈ A× is a fixed degree 1 id`ele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if the support of f is contained in G(A)0 (for example in G(O)), then only i = 0 in the sums over i ∈ Z contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define kG,ξ(x) as the sum of kG,ξ χ (x) over χ ∈ EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define JG,e,ξ χ ( f) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' JG,e,ξ( f)) as the integral of kG,ξ χ (x) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' kG,ξ(x)) over x ∈ G(F)\\G(A)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have proved in [Yu21a] that the integrals converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have then, by definition JG,e,ξ( f) = ∑ χ∈EG JG,e,ξ χ ( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The sum is again a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If we take e = 1 and ξ = 0, we get the geometric side of the trace formula, which is Arthur’s original definition adapted to a function field: (27) J1 geom( f) = JG,1,0( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We call JG,e,ξ( f) truncated trace since the main term in the integrand: ∑ γ∈G(F)aZ f(x−1γx) is the diagonal evaluation of the kernel function of the regular action R( f) on L2(G(F)\\G(A)/aZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The extra term contains the function in x: ∑ γ∈T(F)aZ � N(A) f(x−1(γn)x)dn, which is the diagonal evaluation of the regular action R( f) on L2(T(F)N(A)\\G(A)/aZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 36 HONGJIE YU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A geometric interpretation of J1geom( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that f is the function constructed in Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let o ∈ R1 Scr(Fq) so that every polynomial ov has distinct roots and is split over κv if v ∈ Sr, and is irreducible if v ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we have J1 geom( f) = ∑ V⊆Su (−1)|Su−V|2|Su−V|q−(4g−3+|R|)|M1 2,V(o)(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Similar results are obtained in [Yu21b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The main new ingredient of the proof is the treat- ment of the local components for v ∈ Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since J1 geom is a linear in C∞ c (G(A)), we have J1 geom( f) = ∑ V⊆Su (−1)|Su−V|2|Su−V|J1 geom( f V), where f V = ⊗ f V v is the tensor product of functions f V v ∈ C∞ c (Gv) such that f V v = fv at a place v /∈ Su, for v ∈ Su − V it is defined so that for x ∈ Gv: f V v (x) = 1Kv(x)θv(det x), and for v ∈ V it is defined so that for x ∈ Gv: f V v (x) = 1 vol(Iv) 1Iv(x)θv(det x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We apply [Ch15, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1] (see also [Yu21b, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It says that, as the support of f V ∈ C∞ c (G(A)) is contained in G(O), we have J1 χ( f V) = 0, except if χ = (t − a)2 with a ∈ F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore for any V ⊆ Su, we have J1 geom( f V) = ∑ χ=(t−α)2,α∈F× q J1 χ( f V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since the eigenvalues of ramifications R satisfy ∏ x∈S εx,1εx,2 = 1, by our construction of f, it implies that f V(zx) = f V(x), ∀x ∈ G(A), z ∈ Z(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, we may use the identity to z = α, and we obtain J1 (t−α)2( f V) = J1 (t−1)2( f V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, (28) J1 geom( f) = (q − 1) ∑ V⊆Su (−1)|Su−V|2|Su−V|J1 unip( f V), where we denote J1 unip( f V) = J1 (t−1)2( f V) because the elements whose characteristic polynomial is (t − 1)2 are exactly those unipotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We are going to pass to the Lie algebra version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For every V ⊆ Su, we will define a function ϕV = ⊗v∈|X|ϕV v ∈ C∞ c (g(A)), whose support is in g(O), and that the map x �→ x + 1 from nilpotent elements in g(F) to unipo- tent elements in G(F) induces an identity Jg,1 nilp(ϕV) = J1 unip( f V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 37 Indeed, we are more interested in the Fourier transform of ϕV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need the following proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7, obtained by direct calculations except Springer’s hypothesis proved by Kazhdan, to construct this ϕV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' More details can be found in [Yu21b, Prop 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let KX = ∑ v dvv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have A/(F + ∏ v ℘−dv v ) ∼= H1(X, Ω1 X) ∼= H0(X, OX)∗ ∼= Fq, by Serre duality and the fact that X is geometrically connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We fix a non-trivial additive character ψ of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Via the above isomorphisms, ψ can be viewed as a character of A/F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ⟨, ⟩ be the bilinear form on g defined for any two x, y ∈ g by ⟨x, y⟩ := Tr(xy), where the product is the product of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then ⟨, ⟩ is non-degenerate and G-adjoint invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define the Fourier transform of any ϕ ∈ C∞ c (g(A)) by �ϕ(x) := � g(A) f(y)ψ(⟨x, y⟩)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Based on the Poisson summation formula, we have an identity ([Yu21a, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7]): (29) Jg,e,ξ(ϕ) = q4−4gJg,e,ξ( �ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ([Yu21b, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2]) We have the following results on Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) For any place v, the Fourier transform of the characteristic function of g(Ov) can be calculated by the following formula: � 1g(Ov) = 1℘−dv v g(Ov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) Let Iv be the Iwahori subalgebra of gv consisting of matrices in � Ov Ov ℘v Ov � and Iv+ its open subset consisting of matrices in � ℘v Ov ℘v ℘v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we have � 1Iv = q−1 v 1℘−dv v Iv+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (3) (Springer’s hypothesis [Ka77][KV06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=') Let U be a maximal torus of Gκv and θ any character of U(κv), t a regular element in uv(κv) and ρ = ǫRG Uθ the Deligne-Lusztig virtual representation of G(κv) induced from (Tv, θ), where ǫ = 1 if U = T and ǫ = −1 if U is not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let eρ := � Tr(ρ(x−1)), x ∈ Kv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, x /∈ Kv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' where x denotes the image of x under the map Kv −→ G(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let Ωt ⊆ g(κv) the Ad(G(κv))- orbits of t and Ωt ⊆ g(Ov) be the preimage of Ωt of the map g(Ov) −→ g(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then for any unipotent element u ∈ Kv ∩ Gv,unip, we have eρ(u) = q−(4dv+1) v � 1℘−dv v Ωt(u − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we come to the construction of ϕV = ⊗vϕV v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The local components ϕV v are defined so that with the notations of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7, we have the follows: If v /∈ Scr ∪ V, ϕV v = 1g(Ov);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If v ∈ Scr, � ϕVv = q−1 v � 1℘−dv v Ωtv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' here, tv ∈ T(κv) for v ∈ Sr and tv ∈ U(κv) for v ∈ Sc is a regular element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 38 HONGJIE YU If v ∈ V, � ϕVv = 1 vol(Iv)q−1 v 1℘−dv v Iv+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For the Fourier transform, we have ��fv(X) = q4dv v fv(−X), it is then direct to see that (30) Jg,1 unip( f V) = Jg,1 nilp(ϕV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We require that ∑ v∈Scr Trκv|FqTr(tv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This ensures that ϕV(z + x) = ϕV(x) for any x ∈ g(A) and z ∈ zG(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that the support of ϕ is contained in g(O), we deduce by [Ch15, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1] that Jg,1(ϕV) = qJg,1 nilp(ϕV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Combining with the trace formula for Lie algebra (29) and (30), we have (31) J1( f V) = (q − 1)q3−4g−degV−deg Scr(∏ v∈V vol(Iv)−1)Jg,1( � ϕV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The result is then deduced by a geometric interpretation of Jg,1( � ϕV) using Weil’s dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We refer the reader to [Yu21b] for details (the parameter ξ is set to be zero): the treat for components v ∈ V are proved in [Yu21b, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3], the treat for components v ∈ Scr is proved in [Yu21b, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, for GL2, we can obtain a simpler proof by slightly modifying the proof of [Yu21b, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3] to avoid using [Yu21b, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Independence of parabolic residues and parabolic weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We come back to assumptions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 that V ⊆ Su, R = V ∪ Scr and DR = KX + ∑v∈R v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We consider the case that the parameter ξ = (ξv)v∈R of parabolic weights is not necessarily trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that we use ξ = (ξx)x∈R to denote the family such that ξx = 1 [κv:Fq]ξv for any point x ∈ R lying over v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let t be the Lie algebra of T, which is a vector scheme of rank 2 defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have a residue morphism over Fq: qres : qMe,ξ 2,R(DR) −→ qRR, where qRR = ∏x∈R tFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It sends a quasi-parabolic Higgs bundle (E, ϕ, (Lx)x∈R) to the image of ϕx in End(Lx) × End(E(x)/Lx) ∼= tFq for every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We define qR1 R := ∏v∈R Rκv|Fqtκv as a scheme over Fq where Rκv|Fq is the restriction of scalars from κv to Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have qRR ∼= qRR ⊗ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There is an obvious Gal(Fq|Fq)-action on qRR and qres is Gal(Fq|Fq)-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore qres defines a morphism of Fq-schemes which we denote still by qres: qres : qMe,ξ 2,R(DR) −→ qRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The residue theorem (see [Ta68]) tells us that the morphism qres factors through the linear subscheme of ”residue 0” qR1 R: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', elements (tv)v∈R ∈ ∏v t(κv) such that ∑ v∈R Trκv|Fq(tv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 a residue morphism res : M1 2,V(DR) −→ R1 Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 39 We view qR1 Scr as the linear subspace of qR1 R of those elements whose coordinates in V are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let qR1,rs Scr ⊆ qR1 Scr be the open sub-variety of elements (t1,x, t2,x)x∈R ∈ qR1 R so that t1,x ̸= t2,x at every point x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is an Fq-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have a finite morphism qR1 R −→ R1 R which is ´etale when restricting to R1,rs R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that we have defined AR = H0(X, OX(DR)) ⊕ H0(X, OX(2DR)) as a vector scheme over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Yokogawa studied the Hitchin fibration for quasi-parabolic Higgs bundles: qMe,ξ 2,R −→ AR, and he shows that it is a projective morphism ([Yo93, Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This morphism is Gm-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since Gm has only positive weights on AR, (qMe,ξ 2,R)Gm is contained in the zero fiber of the Hitchin fibration, hence is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is smooth since it is the Gm-fixed subvariety of a smooth variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore (qMe,ξ 2,R)Gm is smooth projective (smoothness is proved in [Yo95, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, it has pure cohomologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We remark that (32) (qMe,ξ 2,R)Gm = (Me,ξ 2,R)Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) For any qo ∈ qR1(Fq), and parabolic weights (e, ξ) in general position (in partic- ular if e = 1 and ξ = 0), we have |qres−1(qo)(Fq)| = q3−4g−deg R|(Me,ξ 2,R)Gm(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) Suppose that Fq ̸= F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that qo ∈ qR1(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Varying the parabolic weights (e, ξ) but remaining in general position and ξv,1 ⩾ ξv,2 ⩾ ξv,1 − deg v for any v ∈ R, the cardinality of the set qres−1(qo)(Fq) remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For the proof, we need first to introduce an ad`elic formula, the Lie algebra version of the Arthur-Selberg trace formula, to express the number of Fq-points of the varieties concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that e is an odd number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is proved in [Ch15, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1] that if the support of f is contained in g(O), we have Jg,e χ ( f) = 0, except if χ = (t − a)2 with a ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To generalize this result, we showed in [Yu21b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6] that if the support of f is contained in g(OR) ∏v∈R Iv (where OR = ∏v/∈R Ov) and if (e, ξ) is in general position in the sense that e + ∑ v∈R ǫv(ξv,1 − ξv,2) /∈ 2Z, for any (ǫv)v∈R ∈ {1, −1}R, we have (33) Jg,e,ξ χ ( f) = 0, except if χ = (t − a)2 with a ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that we have fixed a divisor D = KX + ∑ v∈R v = ∑ v nvv on the curve X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let qo ∈ qR1(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We may suppose qo = (tv)v∈R with tv ∈ t(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let 1R(qo) be the function defined as the tensor product � v/∈R 1℘−nvg(Ov) ⊗ � v∈R ( 1 vol(Iv) 1℘−nv v (tv+Iv+)), 40 HONGJIE YU where Iv+ consists of elements in g(Ov) whose reduction modd-℘v belongs to n(κv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have shown in [Yu21a, Appendix B] that if ξv,1 ⩾ ξv,2 ⩾ ξv,1 − [κv : Fq] and (e, ξ) is in general position (hence semistability coincides with stability), we have (34) Jg,e,ξ( 1R(qo)) = 1 q − 1|qres−1(qo)(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The factor 1 q−1 comes from the fact that there are q − 1 automorphisms for stable quasi-parabolic Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) Now we go back to the proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if qo = 0 then qMe,ξ 2,R(qo) ∼= Me,ξ 2,R(DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case, we can argue as in [Yu21b, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that our argument in [Yu21b, Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9] needs qMe,ξ 2,R(DR) to be a fine moduli space to calculate its tangent sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' However, this is not necessary because qMe,ξ 2,R(DR) −→ qMe,ξ 2,R(DR) is a gerbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ´Etale locally, it is a neutral gerbe, hence admits a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, the coarse moduli scheme qMe,ξ 2,R(DR) admits a Poincar´e family ´etale locally, and the proof of [Yu21b, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9] can be modified slightly to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To complete the proof it is enough to show that Jg,e,ξ( 1R(qo)) is independent of qo ∈ R1(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We use the trace formula for Lie algebra (which we have discussed in the proof Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6): Jg,e,ξ( 1R(qo)) = q4−4gJg,e,ξ( � 1R(qo)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The Fourier transform � 1R(qo) can be calculated by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have 1g(OR) ⊗ � v∈R ( q−1 v vol(Iv) 1Ivλv), where λv is the composition of the maps Iv → t(κv) λv −→ C× and λv : t(κv) → C× is the character defined by λv(t) = ψ(Trκv|FqTr(ttv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since qo ∈ R1(Fq), the characters λv satisfy the condition ∏ v∈R λv|z(Fq) = 1, where we have used the inclusion z(Fq) ⊆ z(κv) ⊆ Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It implies that for any a ∈ Fq and χ = (t − a)2 we have (35) Jg,e,ξ χ ( � 1R(qo)) = Jg,e,ξ nilp ( � 1R(qo)), where Jg,e,ξ nilp ( � 1R(qo)) = Jg,e,ξ t2 ( � 1R(qo)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As the function � 1R(qo) is contained in g(OR) ∏v Iv, after (33) and (35), we deduce that (36) Jg,e,ξ( � 1R(qo)) = qJg,e,ξ nilp ( � 1R(qo)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that Jg,e,ξ unip( � 1R(qo)) is independent of (λv)v∈R by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This finishes the proof of the part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Next, we prove the part (2) of the Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need a lemma first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that Fq ̸= F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' There exists a family of characters (χv)v∈R, χv : T(κv) → C×, so that ∏ v∈R χv|Z(Fq) = 1, and the following properties are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ρ be the representation of ∏v Iv obtained by the tensor product of the representations Iv −→ Tv(κv) χv −→ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any automorphic representation π of G(A), if RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 41 π contains ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the ρ-isotypic part πρ ̸= 0, then π is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, for any character λ of G(A) which factors through deg ◦ det, we have π ⊗ λ ∼= π =⇒ λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let v0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let χ1 be a primitive character of κ× v0 −→ C×, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', an injective homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since Fq ̸= F2,the character χ1 is non-trivial on F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We set χv = 1 for any v ̸= v0, and χv0 = (χ1, χ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We prove that the family (χv)v∈R satisfies all the properties we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have clearly ∏ v∈R χv|F× q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Given an automorphic representation π, it is either cuspidal, or it is a sub-quotient of a par- abolic induction IndG(A) B(A)µ, for a Hecke character µ = (µ1, µ2) of T(A)/T(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The latter case is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, the condition πρ ̸= 0 implies (IndG(A) B(A)µ)ρ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This, in turn, implies that µ is unramified outside {v0} and HomT(Ov0)(χv0, µ|T(Ov0)) ̸= 0, or HomT(Ov0)(χw v0, µ|T(Ov0)) ̸= 0, here we review χv0 as a character of T(Ov0), and χwv0 = (χ−1 1 , χ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, µ1|F× q = χ1|F× q , or µ1|F× q = χ−1 1 |F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This implies that µ1|F× q ̸= 1 which contradicts the fact that µ1 is a Hecke character: it is trivial on F×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For the last assertion of the lemma, we use Langlands correspondence to prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (σ, i : σ ∼ −→ F∗ X/Fqσ) is the Weil sheaf that corresponds to the cuspidal automorphic representation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The sheaf σ has a rank 2 and is smooth over (X − {v0}) ⊗ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The local monodromies of σ over punctured discs X(∗) x (defined in the Introduction) centered at points x in {v0} ⊗ Fq are semisimple tame local systems so that a tame generator has as eigenvalues (ζqi, ζ−qi)i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=',dv0, where ζ is a (qdv0 − 1)th primitive root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If the assertion is not correct, then σ = σ1 ⊕ σ2 and the Frobenius action F∗ X/Fq exchanges isomorphism classes of σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Hence, σ1 is fixed by F∗2 X/Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that the ramifications of σ1 at points in {v0} ⊗ Fq must be multiplication by (ζǫiqi)i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=',dv0, where ǫi ∈ {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If dv0 is odd, then {v0} ⊗ Fq is cyclically permuted by F∗2 X/Fq, hence ǫi has the same sign and their product is ζ±(qdv0 −1/q−1) ̸= 1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If dv0 is even, then ǫ2i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' ǫ2i+1) have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This is also not possible because the product of eigenvalues of ramifications of σ1 is one of the following ζ(qdv0 −1/q−1), ζ−(qdv0 −1/q−1), ζ(qdv0 −1/q+1), ζ−(qdv0 −1/q+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This is again not possible because none of them is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ We choose a family of characters (χv)v∈R as in the Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let h be the function 1G(OR) ⊗ � v∈R ( 1 vol(Iv) 1Ivχv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since the support of h is contained in G(OR) ∏v Iv, we have JG,e,ξ χ (h) = 0, except if χ = (t − a)2 ∈ EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since ∏ v∈R χv|Z(Fq) = 1, 42 HONGJIE YU we deduce, similar to the Lie algebra case that JG,e,ξ(h) = (q − 1)JG,e,ξ unip (h), where JG,e,ξ unip (h) = JG,e,ξ (t−1)2(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is direct to see that the map X �→ 1 + X from the set of nilpotent elements in g(F) to that of unipotent elements gives us an identity: JG,e,ξ unip (h) = q−degRJg,e,ξ nilp ( � 1R(qo)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore we have (37) JG,e,ξ(h) = q− deg R+3−4g|qres−1(qo)(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that a ∈ A× is a degree 1 id`ele, viewed as a scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='9, we know that the regular action R(h) on L2(G(F)\\G(A)/aZ) is a projection whose image lies inside the space of cuspidal automorphic forms and the regular action R(h) on L2(T(F)N(A)\\G(A)/aZ) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It shows that for any x ∈ G(A), 0 = ∑ γ∈T(F) ∑ i∈Z � N(A) h((δx)−1(γn)δaix)dn and hence JG,e,ξ(h) = 1 2(Tr(R(h)|L2 cusp(G(F)\\G(A)/aZ)) + (−1)eTr(R(h)ǫ|L2 cusp(G(F)\\G(A)/aZ))) = 1 2(Tr(R(h)|L2 cusp(G(F)\\G(A)/aZ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' which is independent of (e, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that ǫ is the sign character on G(A) that factors through deg ◦ det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By (37), this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' PROOF OF THE MAIN THEOREMS We will prove our main result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The main ingredient of the proof is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' These two results give an expression for |E2(R)Frob∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It remains to apply this result to the curve X ⊗ Fqk over Fqk and study how |E2(R)Frob∗| varies for k ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Functions of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We continue to use our notations in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s prove first the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A be a set with a permutation τ acting on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We use O(τ|A) to denote the number of orbits of τ acting on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) Let k �→ αA(k) be the function that αA(k) = |A| if τk is a cyclic permutation on A and αA(k) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) We define βA(k) by βA(k) = 0 if τk has an orbit of even length, βA(k) = 2O(τk|A)−1 if all orbits are of odd length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (3) We define γA(k) by γA(k) = (−1)O(τk|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (4) We define ωA = 1 2(αA + (−1)|A|βA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (5) If Scr ̸= ∅, then cR/2 + bR/2 is of Lefscehtz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (6) The function bR is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (7) If Scr ̸= ∅, then the functions k �→ cR(k)αSu(k)/2 + bR(k)βSu(k)/2 are of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 43 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) If τ is not a cyclic permutation on A, then neither is τk for every k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case, αA is constantly 0, and the assertion is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We suppose in the following that τ is a cyclic permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let |A| = pa1 1 · · · pass be a prime decomposition of n with pi being different prime numbers and ai > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ζpi be a primitive pi-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then αA(k) = |A|, if pi ∤ k for all pi, otherwise αA(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It’s direct to verify that we have the following identity: (38) αA(k) = ∏ i (pai i − pai−1 i pi ∑ j=1 ζjk pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since roots of unity are q-Weil integers of weight 0, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) Let n be an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We first prove the following assertion by induction on the number of prime divisors of n (counting multiplicities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (∗) For any odd integer m such that (m, n) = 1, the function k �→ 1 m(2φ(m)(n,k)−1 − 1) is a function of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We only need the case that m = 1 of the assertion (∗), but this stronger assertion is easier to prove by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The case that n = 1 is trivial since the function is constant in k, and it is an integer by Fermat’s little theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let l be any prime number, not dividing nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let β ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that the assertion (∗) holds for n, nl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', nlβ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that we have the following identity 1 m(2φ(m)(nlβ,k)−1 − 1) − 1 m(2φ(m)(n,k)−1 − 1) = β ∑ j=1 1 m(2φ(m)(nlj,k)−1 − 2φ(m)(nlj−1,k)−1) = β ∑ j=1 2φ(m)(nlj−1,k)−1 m (2φ(m)((nlj,k)−(nlj−1,k)) − 1) = β ∑ j=1 2φ(m)(nlj−1,k)−1 mlj (2φ(mlj)(n,k) − 1) lj ∑ s=1 ζsk lj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In the last equality, we have used the fact if lj ∤ k, then (nlj, k) − (nlj−1, k) = 0 and if lj | k, then (nlj, k) − (nlj−1, k) = (n, k)φ(lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have also used the fact that φ(m)φ(lj) = φ(mlj) since l ∤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since the product of Lefschetz type functions and integral multiple of Lefschetz type functions are of Lefschetz type, we deduce that the assertion (∗) is correct for nlβ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By induction, we obtain the result needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' To prove that βA is of Lefschetz type, it is sufficient to prove that τ is a cyclic permutation on A by multiplicativity on orbits and the fact that the integral multiple of the function of Lefschetz type is again of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that |A| = 2al with l being an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If a = 0, then βA is the function k �→ 2(|A|,k)−1 which is of Lefschetz type by the above result (by setting m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose a ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case, we have βA(k) = � 22a(l,k)−1, 2a | k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 0, 2a ∤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 44 HONGJIE YU Since a ⩽ 2a − 1, the function βA is of Lefschetz type, because we have (39) βA(k) = 22a−a−1(2(l,k)−1)2a 2a ∑ j=1 ζk 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have shown that k �→ 2(l,k)−1 is of Lefschetz type, therefore so is βA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (3) Since the product of Lefschetz type function is still of Lefschetz type, it suffices to consider the case that τ is a cyclic permutation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If A has odd cardinality, then (−1)O(τk|A) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that |A| = 2am with m being m odd and a ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then O(τk|A) = (2am, k), and (−1)O(τk|A) = � −1, 2 ∤ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1, 2 | k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore in this case (−1)O(τk|A) = (−1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This is a function of Lefschetz type, and we have proved the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (4) Let’s consider ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If τ is not a cyclic permutation on A, then αA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A = A1 ∪ A2 be a partition of A into non-empty τ-stable subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then ωA = (−1)|A|βA1βA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore ωA is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In the following, we suppose that τ is a cyclic permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It’s sufficient to consider αA−βA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let f : N∗ −→ Z be a periodic function of period n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we have f(k) = n ∑ i=1 ∑n j=1 f(j)ζ−ij n n ζki n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, f is of Lefschetz type if and only if (40) ∑n j=1 f(j)ζ−ij n n ∈ Z for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will use this criterion to prove that ωA is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If n := |A| is an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By (40) and the fact that βA is of Lefschetz type, we know that for any i, the number n ∑ j=1 2(n,j)−1ζ−ij n is an integer that is divisible by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover, n ∑ j=1 αA(k)ζ−ij n = ncn(i), where cn(i) is the sum ith power of primitive nth roots of unity, the Ramanujan’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The function i �→ cn(i) takes an integral value (since αA is of Lefschetz type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to prove that the number (41) 2 n ∑ j=1 ωA(j)ζ−ij n = − n ∑ j=1 2(n,j)−1ζ−ij n + ncn(i) is divisible by 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since we’re in the case that n is an odd number, and (41) is divisible by n, we need to show that it is divisible by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that we have n ∑ j=1 2(n,j)−1ζ−ij n = ∑ d|n 2d−1cn/d(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Modulo 2, the expression (41) equals (n − 1)cn(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since n is odd, this is 0 modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We’re done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now suppose that n = |A| is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If 4 | n, then clearly both 1 2αA(k) and 1 2 βA(k) are of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can see it from equations (38) and (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 45 If 4 ∤ n, we need to prove that for any i 2 n ∑ j=1 ωA(j)ζ−ij n = ncn(i) − n/2 ∑ j=1 2(n,2j)−1ζ−2ij n is divisible by 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' As we have shown before, it is divisible by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, it remains to show that it is divisible by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that n/2 ∑ j=1 2(n,2j)−1ζ−2ij n = ∑ d| n 2 22d−1cn/2d(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Modulo 4, we need to prove that 2cn(i) − 2cn/2(i) is divisible by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By M¨obius inversion formula, we have cn(i) = ∑ d|(n,i) µ(n/d)d, where µ is the M¨obius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if the divisor d of n is odd, we have µ(n/d) = −µ(n/2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, if i is odd, then cn(i) = −cn/2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If i is even, we have cn(i) − cn/2(i) = ∑ d|(n,i) µ(n/d)d − ∑ d|(n/2,i) µ(n/2d)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The sum over d | (n, i) can be decomposed into two parts following d is odd or d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that cn(i) = ∑ d|(n/2,i) µ(n/d)d + ∑ d|(n/2,i) µ(n/2d)2d = − ∑ d|(n/2,i) µ(n/2d)d + ∑ d|(n/2,i) µ(n/d)2d = cn/2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We conclude that in either case, the number 2cn(i) − 2cn/2(i) is divisible by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (5) Let PR/S2 be the quotient of PR by the action of S2 = {1, σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since Scr is non-empty, every point in PR/S2 has a preimage of cardinality 2 in PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The action of Frob∗ defines an action of PR/S2 since it commutes with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any e = (a, σ(a)) ∈ PR/S2, if Frob∗k(e) = e, then we have either Frob∗k(e) = e or we have Frob∗k(e) = σ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, cR(k) + bR(k) = 2|(PR/S2)Frob∗k|, ∀k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (6) If Scr = ∅, then bR(k) = |PR| is either 1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Scr ̸= ∅, then it follows from (5), because bR = 2 cR+bR 2 − cR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (7) The function under consideration equals cR + bR 2 αSu − bR αSu − βSu 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It results from (4), (5), and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 46 HONGJIE YU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' HiggR is of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let o = (ov)v∈Scr ∈ R1 Scr(Fq) so that every polynomial ov has distinct roots and is split over κv if v ∈ Sr, and is irreducible if v ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) The function over N∗: HiggR(k) = ∑ V⊆Su⊗Fqk (−1)|Su⊗Fqk−V|2|V|q−k(4g−3+|V|+|Scr|)|M1 2,V(o)(Fqk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' is of Lefschetz type in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) The number HiggR(k) is divisible by Pic(k) and the quotient function k �→ HiggR(k)/Pic(k) is still of Lefschetz type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (1) Let Fq : a �→ a1/q be the geometric Frobenius element in Gal(Fq|Fq) and Φq its inverse, the arithemetic Frobenius element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let U(1/q) := U ×Spec(Fqn),Fq Spec(Fqn), and U(q) := U ×Spec(Fqn),Φq Spec(Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s prove a lemma first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let V ⊆ Su ⊗ Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let d be the least positive integer such that Frobd(V) = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then M1 2,V(o) is defined over Fqd-structure and M1 2,Frob(V)(o) ∼= M1 2,V(o)(1/q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, we obtain a linear map: F∗ V : H∗ c (M1 2,V(o), Qℓ) −→ H∗ c (M1 2,Frob(V)(o), Qℓ) whose d-times composition coincides with the action of geometric Frobenius element Fqd ∈ Gal(Fq|Fqd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R = (Scr ⊗ Fqd) ∪ V, and R = Scr ∪ V, where V = V ⊗Fqd Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that the functor (·)(q) is an equivalence of categories from (Sch/Fq), the category of schemes over Fq, to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Its d-fold iterate is the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (E, ϕ, (Lx)x∈V) is a quasi-parabolic Hitchin bundle over X, then E (1/q) is again a vector bundle over X, but the parabolic structures are set at points in Frob(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The association E �→ E (1/q) is a bijection between qM1 2,Frob(V)(Frob(DR))(Fq) and qM1 2,V(DR)(1/q)(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Moreover since the relative Frobenius mor- phism, denoted by FV: FV : qM1 2,V(DR)(1/q) −→ qM1 2,V(DR) induces a bijection of geometric points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We obtain a bijection θ between qM1 2,V(DR)(1/q)(Fq) and the set of quasi-parabolic Hitchin bundles over X with parabolic structures in Frob(V) for the divisor Frob(DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' With the bijection θ above, we can apply [Yo93, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5)]: both qM1 2,Frob(V)(DFrob(R)) and qM1 2,V(DR)(1/q) solve the same moduli problem so they’re isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, let T be a scheme defined over Fq and (E, ϕ, (Lx)x∈Frob(V)T be a flat family of quasi-parabolic Hitchin bun- dle over XT with parabolic structures in Frob(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The associated quasi-parabolic Hitchin bundle (E, ϕ, (Lx)x∈Frob(V)(q) T ) is a falt family of quasi-parabolic Hitchin bundles over XT(q) with par- abolic structures in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By [Yo93, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4)], we obtain a morphism T(q) → qM1 2,V(DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Applying the functor (·)(1/q), we obtain T → qM1 2,V(DR)(1/q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By the definition of θ, we see that it has the desired property to apply [Yo93, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 47 It’s clear that (R1 R)(1/q) ∼= R1 Frob(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have the relative Frobenius morphism: (R1 R)(1/q) −→ (R1 R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since o ∈ R1 Scr(Fq), its embedding in R1 R(Fq) is sent to o viewed as a point in R1 R(Fq) via the above morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' These imply the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We still denote the induced relative Frobenius morphism by FV: FV : M1 2,Frob(V)(o) −→ M1 2,V(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since FV is a universal homeomorphism, it induces an isomorphism of ℓ-adic cohomology with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that by definition, the composition FV ◦ FFrob(V) ◦ · · · ◦ FFrobd−1(V) coincides with the Frobenius endomorphism deduced as base change to Fq of the qd-Frobenius morphism of M1 2,V(o) (it is defined over Fqd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' On ´etale cohomology, its action coincides with the geometric Frobenius element Fqd of the Galois group Gal(Fq|Fqd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The last assertion hence follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ Let’s choose a total order on Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that Frob acts on Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any V ⊆ Su, let inv(Frob|V) be the inversion number of Frob on V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', the number of pairs (x1, x2) of points in V such that x1 < x2 and Frob(x1) > Frob(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any subset V of Su, let’s consider H∗(P1, Qℓ)⊗V, where the tensor product is understood as the tensor product of graded vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let α be a generator in H0(P1, Qℓ) and β be a generator in H2(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We set τ : H∗(P1, Qℓ)⊗V −→ H∗(P1, Qℓ)⊗Frob(V) as the map of graded vector spaces which sends an element in H∗(P1, Qℓ)⊗V represented by (axαx + bxβ)x∈V to (axαFrob(x) + bxβFrob(x))x∈Frob(V): τ((axαx + bxβx)x∈V) = (axαFrob(x) + bxβFrob(x))x∈Frob(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let H∗ V := H∗ c (M1 2,V(o), Qℓ) ⊗ H∗(P1, Qℓ)⊗V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let ς be a linear endomorphism: ς : � V⊆Su Hi V −→ � V⊆Su Hi V, defined by ς = ⊕V(−1)inv(Frob|V)q3−4g−|V|−|Scr|F∗ V ⊗ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We will show that: (∗) the eigenvalues of ς are q-Weil integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (∗∗) HiggR(k) = ∑ i (−1)iTr(ςk| � V⊆Su Hi V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' These two properties suffice to prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For (1), since the eigenvalues of Frobenius action on ℓ-adic cohomology are q-Weil integers, the only non-trivial point is to show that the eigenvalues of ς are divisible by q4g−3+|V|+|Scr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 48 HONGJIE YU It suffices to show that, for some k, the eigenvalues of ςk are divisible by (qk)4g−3+|V|+|Scr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that we have a finite morphism qR1 R −→ R1 R and a residue morphism qres : qM1 2,R −→ qR1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let V be a subset of Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For any qo ∈ qR1 R(Fq) with image o = (ov)v∈R ∈ R1 Scr(Fq) ⊆ R1 R(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that ov has distinct roots for v ∈ Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have an isomorphism of Fq-schemes: (42) qres−1(qo) ∼= M1 2,V(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that by forgetting the parabolic structure, we have a commutative diagram: qM1 2,R(DR) qres −−−−→ qR1 R \uf8e6\uf8e6� \uf8e6\uf8e6� qM1 2,V(DR) res −−−−→ R1 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It’s sufficient to prove that it is a Cartesian diagram restricting to R1,rs Scr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Using the modular de- scription and [Yun11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2], we’re reduced to prove that the following diagram is Cartesian (43) [brs/B] −−−−→ brs � B ∼= trs \uf8e6\uf8e6� \uf8e6\uf8e6� [grs/G] −−−−→ grs � G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, we can use Grothendieck’s simultaneous resolution ˜g −→ g, where ˜g = {(x, b)|x ∈ g, b ∋ x}, then we have [˜g/G] ∼= [b/B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We obtain the following Cartesian diagram, ˜grs π2 −−−−→ brs � B ∼= trs π1 \uf8e6\uf8e6� πt \uf8e6\uf8e6� grs πg −−−−→ grs � G , where π1 and π2 are G-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose we are given an Fq-scheme S, a G-torsor E over S with a G-equivariant map α : E −→ grs, and a morphism β : E −→ trs such that πg ◦ α = πt ◦ β, we obtain a unique morphism γ : E −→ ˜grs whose composition with π1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' π2) equals with α (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By uniqueness, we see that γ must be G-equivariant otherwise, we can replace γ by its G-conjugations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, the diagram 43 is Cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ We come back to the assumption of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that there is a point qo ∈ qR1 Scr(Fq2) such that o is the image of qo via the morphism qR1 Scr → R1 Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Applying the Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4, we deduce that |M1 2,V(o)(Fq2kdV)| = |qM1 2,V∪(Scr⊗FqdV )(qo)(Fq2kdV)|, ∀k ⩾ 1, where dV is the least integer such that V is defined over FqdV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that qo is viewed as a point in RV∪(Scr⊗FqdV )(Fq2dV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then (∗) is a corollary of the part (1) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8 and Grothendieck- Lefscehtz fixed point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For (∗∗), note that for any k, only those V fixed by ςk will contribute a non-trivial trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' These are exactly those coming from subsets of Su ⊗ Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For such a V, the alternative trace ∑i(−1)iTr(ςk|Hi V) equals (−1)O(Frobk|V)−|V|q3−4g−|V|−|Scr| RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 49 times ∑ i (−1)iTr(Frob∗k/dV|Hi c(M1 2,V(o), Qℓ))∑ i (−1)iTr(τk|H∗(P1, Qℓ)⊗V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Recall that O(Frobk|V) means the number of orbits of Frobk on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have ∑ i (−1)iTr(Frob∗k/dV|Hi c(M1 2(o)Fq, Qℓ)) = |M1 2,V(o)(Fqk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It reduces to prove that ∑ i (−1)iTr(τk|H∗(P1, Qℓ)⊗V) = 2O(Frobk|V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By multiplicativity of the two sides, it suffices to consider the case that Frobk has only one orbit in V, in which case we can do the calculation by choosing a basis: (δxα + (1 − δx)β)x∈I where (δx)x∈I ∈ {0, 1}I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is clear that in this basis, τk is a permutation matrix, and there are exactly two elements in the basis fixed by τk: those given by δx = δx′ for all x, x′ ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore the left hand side is 2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (2) For the second assertion, Deligne ([De15, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5]) has proven a result that can be applied to show that |M1 2,V(o)(Fqk)| is divisible by |Pic0 X(Fqk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let A = Pic0 X ⊗ FqdV and A its base change to Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that A acts on M1 2,V(o) by tensor on the vector bundles, and we have a morphism f : M1 2,V(o) −→ A which sends a Higgs bundle to the determinant of its underlying vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This morphism is clearly equivariant for the action of Pic0 X defined above and the action of Pic0 X on itself by L0 : L1 −→ L1 ⊗ L⊗2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In [De15, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='5], Deligne proves that under these hypotheses, the sheaf Ri f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='Qℓ is smooth and semisimple, and there is a sheaf Hi over Spec(Fq) such that the invariant sub-sheaf of Ri f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='Qℓ under the action of π1(A) satisfies: (Ri f !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='Qℓ)π1(A⊗Fq) ∼= a∗Hi|A, where a : A −→ Spec(FqdV ) is the structure morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The Grothendick-Lefschetz fixed points formula implies that: |M1 2,V(o)(Fqk)| = |Pic0 X(Fqk)|Tr(F∗(k/dV) qd |Hi|Spec(Fq)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that |Pic0 X(Fqk)| = ∏ 2g i=1(1 − σk i ) where σi are q-Weil numbers of the curve X which, for any embedding in C, have absolute value q 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that the eigenvalues of F∗ qdV on Hi|Spec(Fq) are βi, then βi are also q-Weil numbers of M1 2,V(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, their quotients by q4g−3+|V|+|Scr| are q-Weil integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The proof is based on Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 where we have computed the number of cuspidal automorphic representations which corresponds to E2(R)Frob∗ (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We need to have an expression for E2(R)Frob∗k for k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since X ⊗Fq Fq ∼= (X ⊗Fq Fqk) ⊗Fqk Fq, and the Frobenius endomorphism of X obtained from X ⊗Fq Fqk is the kth power of the Frobe- nius obtained from X, we can apply this theorem to the function field F ⊗Fq Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The only dif- ficulty remains that the ramification type on the automorphic side may change when k varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For example, a place can split into several places, and a supercuspidal representation can become non-supercuspidal after base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s explain how ramification types on the automorphic side change when k varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' First, a place v ∈ S of degree n corresponds to an orbit of length n of Frobenius endomorphism on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' A place of F of degree n splits into (n, k)-points of degree n/(n, k) of F ⊗Fq Fqk for k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For ramification types, suppose that S ⊗Fq Fqk = Sr(k)∐ Sc(k)∐ Ss(k)∐ Su(k), 50 HONGJIE YU is a decomposition following the ramification type furnished by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we have Su(k) = Su(1) ⊗Fq Fqk, and Ss(k) = Ss(1) ⊗Fq Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The sets Sr(k) and Sc(k) behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If k is an odd number, we have Sr(k) = Sr(1) ⊗Fq Fqk, and Sc(k) = Sc(1) ⊗Fq Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' However, if k is an even number, we have Sr(k) = (Sr(1) ⊗Fq Fqk) ∪ (Sc(1) ⊗Fq Fqk), and Sc(k) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that we have S2Pic0 X(Fq) = 1 2 � |Pic0 X(Fq2)| + |Pic0 X(Fq)|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We obtain the cardinality |E2(R)Frob∗| by comparing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='6 by the trace formula: J1 spec( f) = J1 geom( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The theorem is then just a reformulation of the results using definitions of the functions HiggR, αSu, βSu, ηSu and ωSu in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1 and the vanishing on cR and bR in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is tedious but direct and easy to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let us be satisfied to explain how to verify the most complicated case that Scr ̸= ∅ and Su ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It should be divided into some sub-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sc = ∅, then |E2(R)Frob∗k| is given by Higg(k) minus the error terms in one of the cases (13), (14), (15), (16) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If Sc ̸= ∅, then for k odd, |E2(R)Frob∗k| is given by Higg(k) minus the error terms in one of the cases (1), (2), (4) (5) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='3 but for k even, it is given by Higg(k) minus the error terms in one of the cases (13), (14), (15), (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We use the definition of αSu and βSu to write the result in a uniform formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' One needs to note that by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='7, bR(2k) = 0 if deg Sc is odd and cR(2k + 1) = 0 if Sc ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' CASE THAT g = 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Case that g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We are going to give another expression for HiggR(k) when g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let R = Scr ∪ Su and DR = KX + ∑v∈R v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose (e, ξ) ∈ Z × (Q2)R is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have a Gm-action on Me,ξ 2,R = Me,ξ 2,R(DR) given by dilation on the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have a modular description for the fixed points variety due to Hitchin and Simpson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (E, θ, (Lx)x∈R) is a parabolic Higgs bundle fixed by Gm-action, then (E, θ) ∼= (E, tθ) for any t ∈ Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By arguments of [Si92, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1], either θ = 0 and the underlying parabolic bundle (E, (Lx)x∈R) is ξ-semistable or if θ ̸= 0, E is decomposed as a direct sum of line bundles E = L1 ⊕ L2, and θ is obtained by θ : L2 −→ L1(KX + ∑ x∈R x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that if g = 0, the first case does not happen as there are no semistable parabolic Higgs bun- dles of rank 2 when the stability is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let f : (E, θ, (Lx)x∈R) ∼ −→ (E, tθ, (Lx)x∈R) RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 51 be an isomorphism of parabolic Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then f has constant coefficient in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then we have � f θ = tθ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' f(Lx) = Lx, ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let λ be an eigenvalue of f, then Eλ := ker( f − λ)2 is a subbundle of E and θ sends Eλ to Etλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If θ is non-zero, then Eλ and Etλ are non-zero and therefore E = Eλ ⊕ Etλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In this case, either Lx = Eλ,x or Lx = Etλ,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore (Me,ξ 2,R)Gm consists of so-called graded parabolic Higgs bundles, which we will denote by grMe,ξ 2,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that Fq ̸= F2, ξv,1 = ξv,2 for v ∈ Su and ξ is in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let grMe,ξ 2,R(Su) be the open subvariety of grMe,ξ 2,R = (Me,ξ 2,R)Gm consisting of those graded semistable parabolic Higgs bundles (E, θ, (Lx)x∈R) such that θx ̸= 0 for any x lying over points in Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that g = 0, and Sc = ∅, then we have |grMe,ξ 2,R(Su)(Fqk)| = HiggR(k), ∀k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='8 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4, for any k ⩾ 1, we have HiggR(k) = ∑ V⊆Su⊗Fqk (−1)|Su⊗Fqk−V|2|V||grMe,ξ 2,V∪(Sr⊗Fqk)(Fqk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It suffices to verify the Theorem for the case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let (E, θ, (Lx)x∈R) be a graded parabolic Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' For each x ∈ R, θx : Ex −→ E(KX + ∑ x∈R x)x, preserves the parabolic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It means that Imθx ⊆ Lx and θx(Lx) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that θx is zero, then it is possible that Lx = L1,x or Lx = L2,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' If θx is non-zero, then we can only have Lx = L1,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We obtain a stratification for any T ⊆ Su and x ∈ Su − T, grMe,ξ 2,R(T) = N(1x) ∪ N(2x) ∪ grMe,ξ 2,R(T ∪ {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' where N(ix) consists of those (E, θ, (Lx)x∈R) in grMe,ξ 2,R(T) such that θx = 0 and Lx = Li,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that as a variety over Fq, Nix ∼= grMe,ξ 2,R−{x}(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let v ∈ Su − T be a closed point in Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Repeat the procedure above, we obtain a decomposition of grMe,ξ 2,R(T) as a disjoint union by locally closed subvarieties: grMe,ξ 2,R(T ∪ {v}) ∪ � (ax)x∈{v}∈{1,2}{v} N((ax)x∈{v}), where N((ax)x∈{v}) consists of those (E, θ, (Lx)x∈R) in grMe,ξ 2,R(T) such that so that θx = 0 for all x ∈ {v} and Lx = Lax,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Now we must consider how to descend to Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since for an Fq-sub-variety Z defined over Fq of grMe,ξ 2,R = (grMe,ξ 2,R)Fq, it comes from a variety defined over Fq if it is fixed by Frobenius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', 52 HONGJIE YU Z(q) = Z as sub-varieties of grMe,ξ 2,R where Z(q) is defined by the Cartesian diagram: Z(q) −−−−→ Z \uf8e6\uf8e6� \uf8e6\uf8e6� Spec(Fq) x�→xq −−−−→ Spec(Fq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Here it is important that Z(q) = Z instead of just isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' otherwise, we don’t have a descent datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' The equality here is another way to express the commutativity of the following diagram: (Me,ξ 2,R)(q) ∼ −−−−→ (Me,ξ 2,R) �\uf8e6\uf8e6 �\uf8e6\uf8e6 Z(q) ∼ −−−−→ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose T comes from a subset of Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We have N((ax)x∈{v}) = N((aFrob(x))x∈{v})(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore, we see that N((1x)x∈{v}) and N((2x)x∈{v}) are defined over Fq and the variety � (ax)x∈{v}∈{1,2}{v}−{(1x)x∈{v},(2x)x∈{v}} N((ax)x∈{v}) is defined over Fq which has no Fq-points, since F∗q permutes its components without any fixed component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Since ξv,1 = ξv,2 for v ∈ Su, as varieties, we have N((1x)x∈{v}) ∼= grMe,ξ′ 2,R−{v}(T), and N((1x)x∈{v}) ∼= grMe,ξ′ 2,R−{v}(T), where ξ′ = (ξv)v∈R−{v} ∈ (Q2)R−{v} (it is still in general position because of our assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We deduce that for any T ⊆ Su and v ∈ Su − T, |grMe,ξ 2,R(T)(Fq)| = |grMe,ξ 2,R(T ∪ {v})(Fq)| + 2|grMe,ξ 2,R−{v}(T)(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' By repeating this equality, we obtain the desired identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' An example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s consider an example that X = P1 = P1 Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that in this case, Ω1 P1 ∼= OP1(−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose S = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' , xn} ⊆ P1 is a finite set of closed points of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Namely, we can identify S with a subset of P1(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Let’s consider those ℓ-adic local systems over P1 − S fixed by Frob∗ whose local monodromies around xi (i < n) are tame and are at most unipotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', they’re either trivial at xi or are unipotent with one Jordan block at xi, and they’re at most quasi- unipotent with eigenvalues −1 at xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can deduce either from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='4 or from its proof directly that they’re in bijection with equivalent classes of semistable graded parabolic Higgs bundles composed of the following data (simply because they have the same number): E = OP1(m) ⊕ OP1(1 − m), θ : E −→ OP1(m) −→ OP1(−m + n − 1) −→ E(n − 2), and parabolic structures Lxi = OP1(1 − m)xi, 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We choose parabolic weights to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Then the semistability says that m is an integer such that m > 1 − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Note that θ exists if and only if m ⩽ −m + n − 1, RANK 2 ℓ-ADIC LOCAL SYSTEMS AND HIGGS BUNDLES OVER A CURVE 53 and when such a θ exists, the pair (E, θ) is semistable if and only if θ is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore we have 1 ⩽ m ⩽ [n − 1 2 ], where [ n−1 2 ] is the largest integer smaller or equal to n−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Two graded parabolic Higgs bundles are isomorphic if and only if m is the same and θ are differed by a non-zero scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In fact, two isomorphic graded parabolic Higgs bundles have iso- morphic underlying vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Therefore m should be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Suppose that (E, θ1) and (E, θ2) are graded parabolic Higgs bundles with E = OP1(m) ⊕ OP1(1 − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' It is clear that θ1 and θ2 are differed by a constant if and only if there is a ϕ ∈ Aut(E) such that (ϕ ⊗ idΩ1 P1 ) ◦ θ1 ◦ ϕ−1 = θ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (E, θ1) and (E, θ2) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We conclude from the above discussion that the moduli space of semistable graded parabolic Higgs bundles of rank 2, degree 1, and for the zero parabolic weight is ∐ m PHom(OP1(m), OP1(−m + n − 1)) ∼= [ n−1 2 ] ∐ m=1 Pn−1−2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' In particular, the number of its Fq-points equals (44) n ∑ i=3 [i − 1 2 ]qn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' This number is also the number of ℓ-adic local systems over P1 Fq − S fixed by Frob∗ whose local monodromies around xi (i < n) are tame and are at most unipotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', they’re either trivial at xi or are unipotent with one Jordan block at xi, and they’re at most quasi-unipotent with eigenvalues −1 at xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' We can not provide a natural bijection between these objects from our method, but I get to know from Kang Zuo that in his work under preparation joint with Jinbang Yang, when n = 4, they can construct a natural injective map from graded parabolic Higgs bundles to ℓ-adic local systems, which then is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' REFERENCES [BM02] Breuil, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', M´ezard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='. Multiplicit´es modulaires et repr´esentations de GL2(Zp) et de Gal(Qp/Qp) en l = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' with an appendix by Guy Henniart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 115 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 2, 205-310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [BLR90] Bosch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' L¨utkebohmert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Raynaud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' N´eron models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Springer-Verlag, Berlin, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' x+325 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [Ch15] Chaudouard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Sur le comptage des fibr´es de Hitchin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Ast´erisque No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 369 (2015), 223-284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [De80] Deligne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' La conjecture de Weil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Hautes ´etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 52 (1980), 137-252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [De13] Deligne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Syst`emes locaux l-adiques sur une vari´et´e sur un corps fini Cours d’Arithm´etique et de G´eom´etrie Alg´ebrique at IHES 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='ihes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='fr/~abbes/CAGA/deligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [De15] Deligne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Comptage de faisceaux l-adiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Ast´erisque No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 369 (2015), 285-312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [DF13] Deligne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Flicker, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Counting local systems with principal unipotent local monodromy.' metadata={'source': 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two-dimensional irreducible representations of the fundamental group of a curve over a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' (Russian) Funktsional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' i Prilozhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 15 (1981), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 4, 75-76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [Fl15] Flicker, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Counting rank two local systems with at most one, unipotent, monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 137 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 3, 739-763.' metadata={'source': 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bundles and integrable systems, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 54 (1987), 91-114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' [Ka77] Kazhdan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Proof of Springer’s hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Math.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Manin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' II, 213-247, Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=', 270, Birkh¨auser Boston, Boston, MA, 2009.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Global Springer theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 228 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' 1, 266-328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFPT4oBgHgl3EQfuzVj/content/2301.13157v1.pdf'} +page_content=' DEPARTMENT OF MATHEMATICS, WEIZMANN INSTITUTE OF SCIENCE, HERZL ST 234, REHOVOT, ISRAEL.' metadata={'source': 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+image-to- +semantics translation to mitigate learning difficulties in vision- +based robotics control agents. This problem assumes two envi- +ronments: a simulator environment with semantics, that is, low- +dimensional and essential information, as the state space, and +a real-world environment with images as the state space. By +learning mapping from images to semantics, we can transfer a +policy, pre-trained in the simulator, to the real world, thereby +eliminating real-world on-policy agent interactions to learn, +which are costly and risky. In addition, using image-to-semantics +mapping is advantageous in terms of the computational efficiency +to train the policy and the interpretability of the obtained policy +over other types of sim-to-real transfer strategies. To tackle the +main difficulty in learning image-to-semantics mapping, namely +the human annotation cost for producing a training dataset, we +propose two techniques: pair augmentation with the transition +function in the simulator environment and active learning. We +observed a reduction in the annotation cost without a decline +in the performance of the transfer, and the proposed approach +outperformed the existing approach without annotation. +Index Terms—deep reinforcement learning, policy transfer, +sim-to-real +I. INTRODUCTION +Deep reinforcement learning (DRL) has been actively stud- +ied for robot control applications in real-world environments +because of its ability to train vision-based agents; that is, the +robot control actions are output directly from the observed +images [1]–[4]. One of the major advantages of vision-based +agents in robotics is that camera-captured images can be +incorporated into the decision-making of the agent without +using a handcrafted feature extractor. +However, allowing vision-based robot control agents to +learn by reinforcement learning in the real-world is challeng- +ing in terms of risk and cost because it requires a large amount +of real-world interactions with unstable robots. Reinforcement +learning involves a learning policy interacting with the envi- +ronment, and it is theoretically and empirically known that the +length of the interaction required for training increases with +the dimension of the state space [5], [6]. +To address the difficulty associated with reinforcement +learning in a real-world environment, methods have been +proposed that pre-train a policy on a simulator environment +This research is partially supported by the JSPS KAKENHI Grant Number +19H04179, and based on a project, JPNP18002, commissioned by NEDO. +and transfer it to the real-world environment [7]–[17]. In +this methodology, policies are learned in a simulator, that +is, a reinforcement learning environment on a computer that +mimics the real-world environment. The policy pre-trained in +the simulator is expected to be the optimal policy in the real- +world environment. +However, developing a simulator that imitates the real-world +environment is not always an easy task. Particularly, because +the real world provides image observations, a simulator en- +vironment requires a renderer to generate images as states. +However, producing a renderer that can generate photorealistic +images is fraught with financial and technical difficulties. +In the case that a photorealistic renderer cannot be produced, +another style of observations must be adopted as states during +the pre-training of the policy in a simulator environment. Most +existing approaches substitute photorealistic observations for +non-photorealistic ones using transfer techniques [7]–[17]. +We investigated a type of transfer strategy called image- +to-semantics to deal with the absence of a photorealistic +renderer, which was created by [18]. In this approach, the +semantics—low-dimensional and essential information of a +state that represents an image—are employed as a form of state +observation instead of images in the simulator environment. +The transfer algorithm consists of two steps: pre-training a +policy on the simulator environment with semantics as its +observation, obtaining a mapping from photorealistic images +to their corresponding semantics, and using the image-to- +semantics mapping as a pre-processing component of the +policy in the real-world environment. A semantics-based pre- +trained policy can be operated in the real-world environment +using image observations. In addition to being a solution to +the case without a photorealistic renderer, image-to-semantics +mapping has advantages in terms of the computational cost +for policy pre-training in the simulator and the interpretability +of the acquired policy. +The crucial part of this approach is obtaining the image-to- +semantics translation mapping. To the best of our knowledge, +[18], [19] are the only studies that have dealt with learning +image-to-semantics translation. We highlight the remaining +problems of [18], [19]: (1) [19] used a paired dataset, that is, +multiple pairs of images and corresponding semantics, to train +the mapping. Considerable human effort is required to make +arXiv:2301.13343v1 [cs.LG] 31 Jan 2023 + +a paired dataset because human annotators provide seman- +tics that represent images. (2) Although the style translation +method without a paired dataset [18] aims at saving annotation +cost, its performance is not often satisfactory owing to the low +approximation quality of the image-to-semantics translation +mapping, as confirmed in our experiments. +In this study, we tackled learning image-to-semantics trans- +lation using a paired dataset; however, we reduced the cost of +creating a paired dataset using two strategies: pair augmenta- +tion and active learning. In our experiments, we confirmed the +following claims: first, compared to [19], we reduced the cost +of making a paired dataset while preserving the performance +of the policy transfer. Second, we achieved significantly higher +performance than [18], in which a paired dataset was not used, +by using a small paired dataset. For practicality, we conducted +experiments under the condition that only inaccurate paired +data can be obtained due to various errors, such as annotation +errors, and confirmed that the proposed method has a certain +robustness against errors. +Our +code +is +publicly +available +at +https://github.com/ +madoibito80/im2sem. +II. PROBLEM FORMULATION +A. Markov Decision Process (MDP) +We defined a vision-based robotics task in the real world; +that is, the real-world environment is a target MDP: Mτ = +(Sτ, A, pτ, rτ, γ), where Sτ is a state space, A is an action +space, pτ : Sτ ×A×Sτ → R is a transition probability density, +rτ : Sτ × A × Sτ → R is a reward function, and γ ∈ [0, 1] is +a discount factor. Because we assumed that the target MDP is +a vision-based task, Sτ consists of images, and each s ∈ Sτ +contains single or multiple image frames. In standard model- +free reinforcement learning (RL) settings, agents can interact +with the environment: they observe st+1 ∼ pτ(· | at, st) and +reward rt = rτ(st+1, at, st) by performing action at at state +st, which is internally preserved in the environment at timestep +t; after the transition, st+1 is stored in the environment. +However, there are concerns in terms of the risk and cost +associated with learning a policy through extensive interaction +with Mτ. +To reduce the risk and cost of training a policy in the target +MDP, we pre-trained a policy on a simulator environment, +called the source MDP: Mσ = (Sσ, A, pσ, rσ, γ). Note that +the action space A is the same between the two MDPs. +In contrast, the state space Sσ, the transition probability +density pσ : Sσ × A × Sσ → R, and the reward function +rσ : Sσ × A × Sσ → R are different from those of the target +MDP. We assumed that because we considered robotics tasks, +the deterministic transition function Trσ(s, a) = s′ ∼ pσ(· | +a, s) could be defined in the simulator environment and pσ +resembled a Dirac delta distribution. +The source state space Sσ corresponded to a semantic space, +that is, each s ∈ Sσ was semantic information. For example, +consider a robot-arm grasp task; each s ∈ Sτ is a single or +multiple image frame showing a robot arm and objects to be +Action Space +State Space +Simulator Env +(Source MDP) +Policy +Action Space +State Space +Real-World Env +(Target MDP) +Semantic Space +Image Space +(Photorealistic) +Image-to- +Semantics +Fig. 1. Illustration of transfer via image-to-semantics. We approximated the +image-to-semantics translation mapping F as ˆF. Because the action space was +common to both MDPs, we operated the composite of the source policy πσ +and approximated image-to-semantics translation mapping ˆF, that is, πσ ◦ ˆF +in the target MDP. +grasped. Each s ∈ Sσ consists of semantics such as xyz- +coordinates of the end-effector and target objects and angles +of joints. +The source MDP and target MDP are expected to have some +structural correspondence. Here, we describe our assumptions +regarding the relations of the two MDPs. We assumed the +existence of a function F : Sτ → Sσ satisfying the following +conditions: +Transition Condition: +For all (s′, a, s) ∈ Sτ × A × Sτ, +pσ(F(s′) | a, F(s)) = +� +¯s∈ ¯ +S pτ(¯s | a, s)d¯s, where ¯S = {¯s ∈ +Sτ | F(¯s) = F(s′)}. +Reward Condition: +For all (s′, a, s) ∈ Sτ × A × Sτ, +rσ(F(s′), a, F(s)) = rτ(s′, a, s). +In the above conditions, F is considered an oracle that takes +an image and outputs corresponding semantics; that is, F is the +true image-to-semantics translation mapping. In the transition +condition, ¯S is a set of images that has common semantics +F(s′). Imagine the transition from s ∈ Sτ to s′ ∈ Sτ with +action a ∈ A in the target MDP, the transition condition holds +F(s′) = Trσ(F(s), a). The reward condition indicates that +a reward for this transition rτ(s′, a, s) equals the one for a +transition from F(s) ∈ Sσ to F(s′) ∈ Sσ with the action a +in the source MDP. +B. Transfer via Image-to-Semantics +1) Policy Transfer: The objective of RL is the expectation +of the discounted cumulative reward: +J(π; p, r, γ, p0) = Eπ,p,p0 [�∞ +t=0 γtr(st+1, at, st)] +(1) +and maximizing it w.r.t. π. Here, π : S × A → R is a policy, +that is, a conditional distribution of at given st, and p0 is +the distribution of the initial state s0 over the state space. +Our objective was to obtain a well-trained policy on the target +MDP: πτ = arg max¯πτ J(¯πτ; pτ, rτ, γ, pτ +0). +Under the situation in which the transition and reward +conditions mentioned above hold for some F, we can replace +πτ by πσ ◦ F, where πσ is a well-trained policy on the +source MDP, that is, πσ = arg max¯πσJ(¯πσ; pσ, rσ, γ, pσ +0). +Solving this maximization by RL requires sole interaction +with Mσ instead of Mτ. As noted, interactions with Mτ + +require real-world operations; however, interactions with Mσ +are performed on the simulator, which is cost-effective. +Based on this property, we studied the following transfer +procedure: pre-train πσ on Mσ, approximate F as ˆF, and out- +put the target agent πσ◦ ˆF. This procedure was investigated by +[18]. Figure 1 illustrates the transfer via image-to-semantics. +2) Advantages: The above-mentioned transfer strategy, that +is, transfer via image-to-semantics, has the following three ad- +vantages over approaches using a renderer in the source MDP +shown in Table I. First, a renderer is not required. Existing +methods that use a renderer generally aim to transfer an agent +based on non-photorealistic images in a simulator to photoreal- +istic images in the real world [7]–[17]. Therefore, they require +the preparation of a renderer on the simulator to generate non- +photorealistic images as state observations. Transfer via image- +to-semantics performs similar transfer learning; however, it +does not require a renderer because the source MDP has +a semantic space as its state space. This can reduce the +development cost of the simulator for some tasks. Second, +because semantics are low-dimensional variables compared to +images, we can improve the sample efficiency required to train +the policy πσ on Mσ [5], [6]. Learning vision-based agents +are generally associated with large computational costs, even +on a simulator [20], but transfer via image-to-semantics is +relatively lightweight in this respect and occasionally allows +a human to design the policy. Third, using semantics as an +intermediate representation of the target agent contributes to +its high interpretability because of the low-dimensionality and +interpretability of semantics. Similar to [19], [21], because the +real-world agent πσ◦ ˆF can be separated into two components, +which are independently trained, it is easier to assess than one +trained in an end-to-end manner. +C. Resource Strategy +In this section, in addition to the two MDP environments, +we define resources that can be used to approximate F. +1) Transition Function: In the target MDP, the state transi- +tion result st+1 due to the selected action at can be observed +only for state st stored inside the environment. In contrast, in +the source MDP, we assumed that the state transition result +for any s ∈ Sσ could be observed, replacing the st stored +inside the environment with s. This is because the actual +state transition probability pτ in the target MDP is a physical +phenomenon in the real world, but the state transition rule Trσ +in the source MDP is a black-box function on the computer. +2) Offline Dataset: The offline dataset comprised observa- +tions of the target MDP, that is, T τ = {(st, at, 1end(st+1)) ∈ +Sτ × A × {0, 1}}t, where 1end(st+1) = 1 represents that +st+1 corresponding to a terminal state; otherwise, 0. Note +that successive indices in the offline dataset shared the same +context of the episode, except at the end of the episode. T τ can +be obtained before training starts and is collected by a behavior +policy. Because the offline dataset can be reused for any trial +and be obtained by a safety-guaranteed behavior policy, we +assumed it could be created at a relatively low cost. +TABLE I +RELATED POLICY TRANSFER METHODS FOR OBSERVATION STYLE SHIFT. +EACH METHOD REQUIRES DIFFERENT RESOURCES: RENDERER, OFFLINE +DATASET (OFF), AND PAIRED DATASET (PAIR). +Method +Renderer +OFF +PAIR +Tobin et al. [7] +✓ +RCAN [8] +✓ +DARLA [9] +✓ +Pinto et al. [10] +✓ +MLVR [11] +✓ +Tzeng et al. [12] +✓ +✓ +GraspGAN [13] +✓ +✓ +RL-CycleGAN [14] +✓ +✓ +RetinaGAN [15] +✓ +✓ +MDQN [16] +✓ +✓ +✓ +ADT [17] +✓ +✓ +✓ +Zhang et al. [18] +✓ +CRAR [19] +✓ +✓ +Ours +✓ +✓ +We solely used the offline dataset for supervised and unsu- +pervised learning purposes. If offline reinforcement learning +is executed, the vision-based agent can be trained directly +without approximating F. However, training a vision-based +agent using an offline dataset by reinforcement learning re- +quires large-scale trajectories in the scope of millions [22]. In +this study, we considered situations in which the total number +of timesteps in the offline dataset was limited, for example, +less than 100k timesteps. +We did not need to generate reward signals while collecting +the offline dataset. World models [23] have been studied for +the procedure: approximate MDP M as +ˆ +M using an offline +dataset of M; train a policy by reinforcement learning by +interacting with the approximated environment +ˆ +M instead +of interacting with the original environment M. One could +imagine that we could replace interactions with the target +MDP by interactions with the approximated one. However, to +accomplish this, we must observe signals regarding reward in +the real world while collecting the offline dataset, and we must +approximate a reward function that is often sparse; both of +these are not always easy [24]. Therefore, we did not consider +approximating the target MDP and did not assume the reward +was contained in T τ. +3) Paired Dataset: The paired dataset P consisted of mul- +tiple pairs of target state observations and their corresponding +source state observations. Let I denote the set of indices +that indicate the position of the offline dataset. Using the +true image-to-semantics translation mapping F, we can denote +P = {(F(si), si) | (si, ai, ei) ∈ T τ, i ∈ I}. Under prac- +tical situations, querying F equals annotating corresponding +semantics to the images of the indices I in the offline dataset +T τ by human annotators. Because of its annotation cost, we +assumed the size of the paired dataset |I| to be significantly +smaller than that of the offline dataset, for example, |I| ≤ 100. +III. RELATED WORK +We introduced some existing sim-to-real transfer methods +that use a non-photorealistic renderer on the simulator. Table I + +lists the transfer methods that do not require on-policy inter- +action in the target MDP, assuming vision-based agents. The +main difficulty tackled by these methods was the absence of +a photorealistic renderer on the simulator. In the real world, +images captured by a camera are input to the agent; however, +generating photorealistic images on the simulator is generally +difficult because it requires developing a high-quality renderer. +In [7]–[11], [25], the algorithms learned policies or interme- +diate representations that were robust to changes in image style +using a non-photorealistic renderer. Thus, these algorithms +were expected to perform well even when a photorealistic style +was applied in a real-world environment. In particular, the +domain randomization technique has been widely used [7]– +[10]. +A. Transfer via Image-to-Image Translation +In contrast to the above methods, [12]–[19] aimed to +perform style translation mapping among specific styles. To +accomplish this, these methods required an offline dataset of +the target MDP. Because these methods followed the principle +of collection without execution of on-policy interaction, the +offline dataset could be collected by a safety-guaranteed pol- +icy. Unsupervised style translation, such as domain adaptation +[26] and CycleGAN [27], are often used to change the styles +for state-of-the-art methods [13]–[15], [17], [18], [24], [28]. +Using this translation mapping as a pre-processing function of +the target agent, the pre-trained policy can determine actions +in the same image style as the source MDP in the target MDP. +However, domain adaptation and cycle-consistency [27] +only have a weak alignment ability [18], and some existing +methods use paired datasets to properly transfer styles [16], +[17], [19]. Therefore, these two datasets have been widely +employed in previous studies and can be assumed to be a +common setting. +The similarity of transfer via image-to-semantics and image- +to-image is that they train style translation mapping ˆF among +the source and target state spaces that preserves essential +information; furthermore, the agent is the composite π ◦ ˆF, +where π is a policy. +Again, the above methods use a non-photorealistic renderer +on the simulator. Thus, these methods cannot be compared +with transfer via image-to-semantics, as explained in Sec- +tion II-B2. +B. Learning Image-to-Semantics +Previous studies have used semantics in the source MDP +[10], [12], [16]–[18]. An important perspective on the appli- +cability of these methods to image-to-semantics is whether +they use a renderer on the simulator, as shown in Table I +and as discussed in Section II-B2. Because methods using a +renderer assume that the source state space is an image space, +image-to-semantics is beyond their scope, and it is not certain +that their mechanism will be successful in image-to-semantics. +For example, CycleGAN, which has been successfully used for +image-to-image learning, failed in image-to-semantics [18]. In +this regard, we refer to [18], an unpaired method that applies +the findings from image-to-image to image-to-semantics. In +addition, [19] is compared as a representative method that uses +a paired dataset as in this study. +1) CRAR: We refer to Section 4.4 of CRAR [19] as a +baseline of image-to-semantics learning. They described the +following policy transfer strategy: pre-train a source state +encoder Eσ : Sσ → Z, where Z is a latent space of the +encoder; train the source policy πσ : Z → A; and train a +target state encoder Eτ : Sτ → Z with regularization term +� +(sσ,sτ )∈P∥Eσ(sσ)−Eτ(sτ)∥2 +2, where P is a paired dataset. +Then, the target agent is the composite πσ ◦ Eτ : Sτ → A. +Here, Eτ can be regarded as a style translation mapping. Note +that they only performed this experiment in the setting where +Sσ and Sτ are both image spaces; however, it can be applied +easily where Sσ is the semantic space. +2) Zhang et al.: We referred to the cross-modality setting of +their experiment as our baseline for image-to-semantics [18]. +This setting is the same as the transfer via image-to-semantics. +There remain some challenges in [18], [19]. For [18], the +human annotation cost was eliminated because they did not +use a paired dataset. However, the loss function defined by +[18] for unpaired image-to-semantics style translation will not +necessarily provide a well-approximated F. Therefore, we +decided to use a paired dataset to efficiently supervise the +loss function as performed in [19], but with a paired dataset +smaller than [19]. +IV. METHODOLOGY +Our approach approximates the image-to-semantics transla- +tion F using an offline dataset T τ. Similar to [19], we used +a paired dataset P = {(F(si), si) | (si, ai, ei) ∈ T τ, i ∈ I}, +which was constructed by querying F(si) to human annotators +for an image observation of target MDP si ∈ Sτ included in +T τ. We incorporated two main ideas to reduce the annotation +cost. Pair augmentation generates an augmented paired dataset +P′ using an offline dataset T τ. Active learning defines I, that +is, it selects a subset of T τ to be annotated to construct P +(Algorithm 2). We present an overall procedure of our method +in Algorithm 1. +We assumed that we have an offline dataset T τ, which +comprises multiple episodes in the target MDP. Let O denote +the set of indices corresponding to the beginning of an episode +in T τ, that is, O = {0} ∪ {i | 0 < i < |T τ| and ei−1 = +1 for (sτ +i−1, ai−1, ei−1) ∈ T τ}, where ei is the indicator: +when timestep i is the end of an episode then ei = 1. For each +i ∈ O, let Ei = {t | 1 ≤ t ≤ min({k | k ≥ 1, ei+k = 1})}. +Then, for each i ∈ O, a subsequence of T τ starting from +timestep i and ending at i + |Ei| corresponds to an episode. +A. Pair Augmentation by Transition Function +The objective of pair augmentation is to construct artificial +paired data P′ such that sσ ≈ F(sτ) for (sσ, sτ) ∈ P′ and +sτ ∈ T τ. Using an augmented paired dataset, we aimed to +obtain ˆF that approximates F by minimizing the loss +L( ˆF, P ∪ P′) = +1 +|P ∪ P′| +� +(sσ,sτ )∈P∪P′ +∥sσ − ˆF(sτ)∥2 +2 . (2) + +action +Semantics +Offline +Dataset +action +action +action +F +Fig. 2. Illustration of pair augmentation. Oracle F generates semantics corresponding to a particular image in the offline dataset. The next state in semantics is +computed using the transition function Trσ with the current semantics along with the action taken while collecting the offline dataset. This allows us to obtain +semantics corresponding to the image at the next timestep in the offline dataset without any annotation costs. Augmented pairs with green dual directional +arrows were stored in P′. In this figure, note that rendered (non-photorealistic) images are shown in the offline dataset, but in reality, camera-captured +(photorealistic) images are contained. +Algorithm 1 Overall Procedure +Require: Source MDP Mσ, Offline dataset T τ, Oracle F +1: Train source MDP’s policy πσ on Mσ +2: Train VAE encoder Eτ using T τ +3: Determine indices I by active learning (Algorithm 2) +using Eτ, T τ +4: Create P for I, T τ by oracle (human annotator) F +5: Create augmented pairs P′ using P, T τ, Trσ ∈ Mσ +6: Train ˆF by minimizing Equation (2) +Ensure: Target MDP’s agent πσ ◦ ˆF +Note that CRAR [19] adopts L( ˆF, P) instead of L( ˆF, P∪P′). +Our principle is as follows. Let I ⊆ O be a subset of +indices corresponding to the beginning of the episodes in T τ. +Suppose we have a paired dataset P constructed by querying +semantics sσ +i = F(sτ +i ) corresponding to images sτ +i in T τ for +time index i ∈ I. Although semantics sσ +i+1 representing an +image of the next timestep sτ +i+1 in T τ is unknown, because of +the transition condition given in Section II-A and deterministic +transition, it equals sσ +i+1 = Trσ(sσ +i , ai), where ai is the action +taken at timestep i when collecting the offline dataset T τ +and is included in T τ. In reality, because human annotations +and state transition contain errors as compared to the truth, +the generated semantics sσ +i+1 do not exactly represent the +image sτ +i+1. However, even with errors in F and Trσ, it +is expected that the generation of the above semantics is a +valuable approximation. By recursively applying the above +generation, we obtained the augmented paired dataset P′. +Formally, P′ was constructed as follows: For each index i ∈ +I, we defined a sequence {�sσ +i+t}t∈Ei as �sσ +i = sσ +i (contained +in P) and �sσ +i+t = Trσ(�sσ +i+t−1, ai+t−1) for t ∈ Ei, where +ai+t−1 are contained in T τ. The augmented paired dataset is +then P′ = {(�sσ +i+t, sτ +i+t)}i∈I,t∈Ei, where sτ +i+t is contained in +T τ. Thus, we could construct an augmented paired dataset P′ +of size |P′| = � +i∈I|Ei| from the paired dataset P of size +|P| = |I|. +Figure 2 illustrates the pair augmentation scheme. +The reason why I was a subset of episode start indices +O rather than I ⊆ {j | 0 ≤ j < |T τ|, j ∈ Z} was to +maximize the size of augmented pairs |Ei|. In other words, +because we could augment sσ +i = F(sτ +i ) until the end of the +episode including sτ +i , to maximize |P∪P′|, human annotations +should be conducted at the beginning of an episode of T τ. +B. Active Learning for Pair Augmentation +To select episodes for annotation, that is, decide I, we +incorporated the idea of diversity-based active learning (AL) +[29]–[31]. Their motivation was to select dissimilar samples +to effectively reduce the approximation error. Intuitively, if +P ∪ P′ has many similar pairs, they might have a similar +effect on training ˆF; this may lead to a waste in annotation +cost. Therefore, we attempted to select episodes (indexed by +I ⊂ O) to be annotated to ensure the inclusion of diverse +pairs. +We successively selected the episode to annotate, and we +called each selection step the n-th round. For i ∈ O, let Bi = +{sτ +i+t}t∈{0}∪Ei be a set of target state observations present in +the episode starting at timestep i ∈ O. We referred to it as +batch. Let In−1 be the set of selected indices before the n-th +round, and let Sn−1 = � +k∈In−1 Bk be a set of all the state +vectors in the episodes selected before the n-th round. Let +d : Sτ × Sτ → R be some appropriate distance measure. In +the n-th round, a batch was selected based on the following +two diversity measures: The inter batch diversity +finter(Bi, Sn−1) = +� +sτ ∈Bi +min +sτ +j ∈Sn−1 d(sτ, sτ +j ) +(3) +can evaluate the dissimilarity of Bi and Sn−1. The batch with +the greatest finter was considered to be the most dissimilar + +Algorithm 2 Active Learning +Require: Trained VAE encoder Eτ, Offline dataset T τ +1: Initialize I0 = {c}|c∼Uniform(O) +2: for 1 ≤ n < N do +▷ n-th round +3: +Set Sn−1 = � +k∈In−1 Bk +4: +Measure finter(Bi, Sn−1) for all i ∈ O +5: +Pick top b% of indices in terms of finter as Q +6: +Measure fintra(Bi) for all i ∈ Q +7: +Pick the index c from Q with the greatest fintra +8: +Set In = {c} ∪ In−1 +9: end for +Ensure: Indices IN−1 (with the size of N) as I +batch against the pre-selected batches. The intra batch diver- +sity +fintra(Bi) = +� +sτp∈Bi +� +sτq ∈Bi +d(sτ +p, sτ +q) +(4) +can evaluate the dissimilarity of the states inside Bi. The batch +with the greatest fintra was considered to contain the most +diverse states. +We selected a batch that maximizes the above two diversity +measures; we performed a bi-objective optimization for selec- +tion. To avoid overemphasizing one measure over the other, +we employed two separate single-objective optimizations for +each measure. In each round, we picked up indices of batches +with finter in the top b% (b = 10 in our experiments) from +unselected episodes as Q, and subsequently, selected the batch +with the greatest fintra from Q. I0 was initialized with the +episode sampled from O uniformly at random. +C. Representation Learning Using Offline Dataset +For d : Sτ × Sτ → R to be a reasonable distance measure +in the image space, we employed a VAE encoder [32]: Eτ : +Sτ → Z. It stochastically outputs a latent vector z ∈ Z for +sτ ∈ Sτ. The distance between two states sτ +p ∈ Sτ and sτ +q ∈ +Sτ was given by the Euclidean distance between the mean +vectors for their latent representations, that is, d(sτ +p, sτ +q) = +∥E[Eτ(sτ +p)] − E[Eτ(sτ +q)]∥2. We trained Eτ using all states in +the offline dataset T τ before performing the active learning +procedure. +We used the states contained in � +i∈I Bi in training ˆF by +Equation (2); however, the remaining � +i∈O\I Bi were not +used. In order to use it, we included Eτ as a feature extractor +for ˆF by receiving the benefit of representation learning for +downstream tasks. We modeled ˆF = φ ◦ Eτ, and we trained +φ by Equation (2), whereas Eτ was fixed. +V. EXPERIMENTS +We aimed to verify the following two claims: (1) the +proposed paired augmentation and AL reduces the annotation +cost for approximating ˆF while maintaining its performance +level; and (2) the paradigm with the paired dataset performs +better than the method without paired datasets. +A. Evaluation Metrics +1) Policy Performance (PP): The most important evalua- +tion metric for ˆF is the expected cumulative reward of the +target agent using Equation (1): +PP( ˆF; πσ, Mτ) = J(πσ ◦ ˆF; pτ, rτ, γ, pτ +0) . +(5) +In our experiments, we approximated it by averaging the +cumulative reward of 50 episodes with γ = 1. This metric +was commonly used in [18], [19]. +2) Matching Distance (MD): Because our technical contri- +bution was mainly to approximate F, we used the following +empirical approximation error: +MD( ˆF; T , F) = +1 +|T | +� +(sτ ,a,e)∈T +∥F(sτ) − ˆF(sτ)∥2 +2, +(6) +where T is a trajectory collected by a behavior policy in the +target MDP, which is not used for learning ˆF. Unfortunately, in +a real-world environment, evaluating Equation (6) for a large +size of T is challenging because F requires human annotation. +To enable MD in our experiment, we performed experiments +using the simulator for both the source MDP and target MDP. +We adopted the rendered image space as the state space of +the target MDP. Because both semantics and images were +generated in the simulator, F was freely available to calculate +Equation (6). A similar metric to Equation (6) was used in +[18]. +B. Environment +We evaluated the proposed approach on three environments. +1) ViZDoom Shooting (Shooting): ViZDoom Shooting [33] +is a first-person view shooter task, in which an agent obtains +64×64 RGB images from the first-person perspective in the +target MDP. The agent can change its x-coordinate by moving +left and right in the room and attacking forward (|A| = 3). An +enemy spawns with a random x-coordinate on the other side +of the room at the start of the episode and does not move or +attack. The agent can destroy the enemy by moving to the front +of it and shooting it; time to destruction is directly related to +the reward. Semantics are the x-coordinates of the agent and +the enemy; hence, Sσ is a 2-dimensional space. The maximum +timesteps is 50 for each episode. The behavior policy to collect +the offline dataset T τ is a random policy, and T τ consists of +200 episodes, that is, 10k timesteps in total. +2) PyBullet KUKA Grasp (KUKA): This is a grasp task +using PyBullet’s KUKA iiwa robot arm [34]. Success is +achieved by manipulating the end-effector of the robot arm +and lifting a randomly placed cylinder. The semantics are the +xyz-coordinate and the 3-dimensional Euler angle of the end- +effector and the xyz-coordinate of the cylinder; hence, Sσ is a +9-dimensional space. We used the rendered 64×64 RGB im- +ages captured from three different viewpoints simultaneously +as the state observations in the target MDP. The total timesteps +per episode is fixed to 40. The behavior policy to collect T τ +is a random policy, and T τ comprised 250 episodes, that is, +10k timesteps in total. + +TABLE II +RESULTS OF SHOOTING. MD VALUES WERE SCALED TO 102 FOR +CONVENIENCE. πσ HAS PP=45.99, AND THE BEHAVIOR POLICY HAS +PP=16.39. +Method +MD +PP +|I| = 0 +Zhang et al. +37.42 ± 6.23 +22.46 ± 4.99 +|I| = 10 +CRAR +11.00 ± 1.72 +35.16 ± 4.31 +Ours w/o AL +3.44 ± 0.89 +43.99 ± 1.29 +Ours +0.16 ± 0.12 +44.73 ± 1.07 +|I| = 50 +CRAR +2.80 ± 0.70 +42.29 ± 2.12 +Ours w/o AL +0.06 ± 0.02 +46.02 ± 0.31 +Ours +0.02 ± 0.00 +45.66 ± 0.34 +3) PyBullet HalfCheetah-v0 (HalfCheetah): This is a Py- +Bullet version of the HalfCheetah, that is, a task in which a +2-dimensional cheetah is manipulated by continuous control +to run faster. The torque of the six joints can be controlled +(A = [−1, 1]6), and the semantic space is a 26-dimensional +space. We collected 64×64 images captured from three differ- +ent viewpoints for two consecutive timesteps and defined Sτ +as an image space containing a total of 6 frames. The total +timesteps per episode is fixed to 1000. The behavior policy +to collect T τ is a random policy, and T τ consists of 100 +episodes, which is 100k timesteps in total. +In our experiments, information such as xyz-coordinates +and velocity can be recovered from a combination of multiple +images by capturing images from multiple viewpoints at +consecutive times, and such a setup is necessary in practice. +C. Setting +We used a 7-layer convolutional neural network and a 4- +layer fully connected neural network for the VAE encoders +Eτ and φ, respectively, for both the proposed and existing +methods. We trained them in gradients using Adam [35]. +The dimensions of the latent space of VAE Z were set +to 32, 96, and 192 for Shooting, KUKA, and HalfCheetah, +respectively. For CRAR [19], we uniformly selected indices I +from {i | 0 ≤ i < |T τ|, i ∈ Z}. For our method, without an +AL setting, I was selected uniformly and randomly from O. +For Shooting and KUKA, we used a handcrafted policy instead +of one trained by RL as πσ. In HalfCheetah, we trained πσ +using PPO [36]. +D. Results +Tables II to IV show the results of the image-to-semantics +learning in the three environments. These tables show the +results on average±std over five trials. |I| denotes the number +of paired data, which is the annotation cost. Because of the +transition and reward conditions, the PP of πσ ◦ F on Mτ +assimilate to that of πσ on Mσ. +Note that most image-to-image methods shown in Table I +cannot be compared with image-to-semantics methods be- +cause some assumptions cannot be satisfied under image-to- +semantics settings. One way to speculate on the performance +of the image-to-image techniques in an image-to-semantics +CRAR +Ours w/o AL +Ours +Fig. 3. Scatter of the obtained semantics on ViZDoom Shooting with |P| = +|I| = 10: {F(sτ) | (sσ, sτ) ∈ P} for CRAR, and {F(sτ) | (sσ, sτ) ∈ P∪ +P′} for our method. Each square represents a 2-dimensional semantic space. +The semantic space shows that both pair augmentation and AL contribute to +expanding the coverage. +TABLE III +RESULTS OF KUKA. PP CORRESPONDS TO GRASP SUCCESS PROBABILITY. +πσ HAS PP=1.0, AND THE BEHAVIOR POLICY HAS PP=0.048. +Method +MD +PP +|I| = 0 +Zhang et al. +0.90 ± 0.12 +0.12 ± 0.08 +|I| = 10 +CRAR +0.59 ± 0.07 +0.24 ± 0.22 +Ours w/o AL +0.35 ± 0.05 +0.52 ± 0.19 +Ours +0.37 ± 0.03 +0.65 ± 0.07 +|I| = 100 +CRAR +0.32 ± 0.02 +0.52 ± 0.15 +Ours w/o AL +0.11 ± 0.01 +0.76 ± 0.09 +Ours +0.12 ± 0.01 +0.90 ± 0.04 +setting is to see Zhang et al. [18]. Zhang et al. used do- +main adaptation [26], which is commonly used in image-to- +image learning; thus, their method can be interpreted as a +representative example in which the techniques cultivated in +image-to-image are imported to image-to-semantics. Although +CycleGAN [27] is also widely employed in image-to-image +learning, along with domain adaptation, they confirmed in their +experiments that this method did not outperform their method +in the image-to-semantics setting [18]. +In all cases, compared with the approach of Zhang et al. +[18], our approaches with and without AL achieved a smaller +MD and a greater PP. Zhang et al.’s approach is designed to +learn ˆF without a paired dataset to eliminate the annotation +cost. However, learning without pairs does not necessarily +lead to the true image-to-semantics translation mapping, as +observed in the high MD and low PP in our results. This +result shows the effectiveness of the paradigm using paired +data when aiming for higher performance policy transfer while +compromising the annotation cost to prepare a small number +of paired data. +By comparing the results of our approaches with and +without AL and those of CRAR, we confirmed the efficacy +of pair augmentation in achieving a smaller MD and higher +PP. We achieved PP=44.73 ± 1.07 in Shooting with 10 pairs +using pair augmentation and AL, which is more than the +PP=42.29±2.12 achieved by CRAR with 50 pairs. In addition, +in KUKA, we achieved PP=0.65 ± 0.07 in 10 pairs, which +exceeds PP=0.52 ± 0.15 achieved by CRAR with 100 pairs. +This means that the annotation cost was reduced by more +than ×5 and ×10, respectively. This difference is even more + +TABLE IV +RESULTS OF HALFCHEETAH. πσ HAS PP=2735.73, AND THE BEHAVIOR +POLICY HAS PP=−1230.01. +Method +MD +PP +|I| = 0 +Zhang et al. +3.71 ± 0.32 +−1511.97 ± 192.51 +|I| = 10 +CRAR +1.80 ± 0.09 +−1411.83 ± 144.21 +Ours w/o AL +0.40 ± 0.04 +596.20 ± 121.43 +Ours +0.37 ± 0.05 +580.21 ± 71.95 +|I| = 50 +CRAR +0.88 ± 0.05 +−818.29 ± 300.08 +Ours w/o AL +0.12 ± 0.01 +878.79 ± 88.52 +Ours +0.07 ± 0.01 +968.49 ± 99.53 +pronounced in HalfCheetah. This may be because the ratio +|P ∪ P′|/|P| is the greatest in this environment: CRAR uses +|P| = |I| paired data, whereas our proposed approach used an +additional |P′| = 999|I| augmented paired data because the +number of timesteps per episode was 1000 in this environment. +A tendency of reduced MD and increased PP was observed +in the proposed approach with AL compared to that without +AL. Specifically, AL reduced MD except in KUKA, and +clearly improved PP in KUKA, while achieving competitive +PP in the other two environments. +In Figure 3, we present the effectiveness of our approach +in Shooting. The proposed AL maximized the diversity in the +latent space that represents the image space, but the diversity +was also maximized when this result was visualized in the +semantic space. +E. Experiments with Errors +In this section, we verify the robustness of the proposed +method against errors in annotation and state transitions. +1) Annotation Error: In our previous discussion and in +experiments of Section V-D, we assumed that we could query +oracle F, that is, true image-to-semantics mapping by human +annotations. However, because human annotation indicates the +process of assigning semantics to images by humans, errors +are expected to occur in the output semantics. Therefore, we +provided a new experimental setup here: for some sτ ∈ Sτ, +we can observe F(sτ) + ϵ instead of F(sτ) while creating +the paired dataset P, where ϵ ∈ Rdim(Sσ) is a random vector +representing the annotation error. +2) Transition Error: In reality, the state transition function +on the simulator Trσ is expected to contain modeling errors. +For example, environment parameters such as friction coeffi- +cients and motor torques in the real world cannot be accurately +estimated in the simulator, and thus, state transitions in reality +cannot be correctly imitated. Therefore, we provided a new +experimental setup here: for some (s, a) ∈ Sσ × A, we could +obtain Trσ(s, a) + ϵ instead of Trσ(s, a) while augmenting +a paired dataset, where ϵ ∈ Rdim(Sσ) is a random vector +representing the transition error. Note that when training πσ, +we used the one without errors in our experiments. +3) Error Generation: We generated two types of errors by +adding a random variable ϵ ∈ Rdim(Sσ). Here, we denoted a +TABLE V +RESULTS WITH ANNOTATION ERRORS. +Method +α +MD +PP +Shooting (|I| = 50) +CRAR +0.0 +2.80 ± 0.70 +42.29 ± 2.12 +Ours w/o AL +0.06 ± 0.02 +46.02 ± 0.31 +CRAR +0.04 +2.99 ± 0.93 +42.91 ± 1.71 +Ours w/o AL +0.12 ± 0.05 +45.90 ± 0.54 +CRAR +0.15 +3.50 ± 1.21 +43.51 ± 2.04 +Ours w/o AL +0.47 ± 0.07 +44.56 ± 0.59 +CRAR +0.3 +4.90 ± 0.85 +40.83 ± 2.78 +Ours w/o AL +1.42 ± 0.15 +42.47 ± 1.74 +KUKA (|I| = 100) +CRAR +0.0 +0.32 ± 0.02 +0.52 ± 0.15 +Ours w/o AL +0.11 ± 0.01 +0.76 ± 0.09 +CRAR +0.04 +0.33 ± 0.03 +0.58 ± 0.11 +Ours w/o AL +0.12 ± 0.01 +0.76 ± 0.18 +CRAR +0.15 +0.32 ± 0.03 +0.53 ± 0.15 +Ours w/o AL +0.14 ± 0.01 +0.68 ± 0.15 +HalfCheetah (|I| = 50) +CRAR +0.0 +0.88 ± 0.05 +−818.29 ± 300.08 +Ours w/o AL +0.12 ± 0.01 +878.79 ± 88.52 +CRAR +0.04 +0.91 ± 0.03 +−919.86 ± 328.14 +Ours w/o AL +0.12 ± 0.01 +787.0 ± 230.32 +CRAR +0.15 +0.99 ± 0.06 +−833.35 ± 464.76 +Ours w/o AL +0.15 ± 0.02 +616.63 ± 260.73 +value of the h-th dimension of x ∈ RH as x(h) ∈ R. We sam- +pled ϵ(h) ∼ N(h), where N(h) is a Gaussian distribution with +mean µ = 0 and standard deviation σ = α · std[sσ +(h)]sσ∈T σ. +Here, std[sσ +(h)]sσ∈T σ is the sample standard deviation of a +source trajectory T σ collected by a behavior policy in source +MDP, and α ≥ 0 is the noise scale. +For the annotation error, using the semantics sequence +of augmented pairs {�sσ +i+t}t∈Ei provided without error, we +provided the semantics as F(sτ +i )+¯ϵi for P and {�sσ +i+t+¯ϵi}t∈Ei +for P′, where ¯ϵi is a realized random vector with α > 0. +For the transition error, we defined {�sσ +i+t + �t +j=1 ¯ϵi,j}t∈Ei +for P′, where ¯ϵi,j is the realized random vector. Note that, +here, we approximated the error generation based on the +following assumption: Trσ(s, a) = s + Trσ +∆(a), that is, +Trσ(s+¯ϵ1, a)+¯ϵ2 = s+Trσ +∆(a)+¯ϵ1+¯ϵ2. This is an approxi- +mation simplifying implementation; however, for Shooting and +KUKA, the above assumption is actually satisfied for almost +all states and actions. +4) Results: Here, we analyze the effect of two types of +errors on P and P′, and understand how this affects the +approximation of F. Therefore, we do not experiment with +the method of Zhang et al., which does not utilize paired +datasets. In addition, to eliminate the effect of the choice of +I on the generation of P and P′ in the comparison between +the proposed method and CRAR, we conducted experiments +under the setting without AL. +The results with annotation errors are shown in Table V. +In both CRAR and the proposed method, the semantics of +the paired data deviated from the true data as the scale of +the annotation error α increased; thus, we observed that MD +tends to increase for both methods. Although PP tended to +decrease only for the proposed method, the proposed method +achieved better MD and PP than CRAR for the same error + +TABLE VI +RESULTS WITH TRANSITION ERRORS. +Method +α +MD +PP +Shooting (|I| = 50) +CRAR +0.0 +2.80 ± 0.70 +42.29 ± 2.12 +Ours w/o AL +0.06 ± 0.02 +46.02 ± 0.31 +Ours w/o AL +0.01 +0.12 ± 0.04 +45.90 ± 0.51 +Ours w/o AL +0.04 +0.67 ± 0.12 +44.78 ± 1.16 +Ours w/o AL +0.1 +3.43 ± 1.00 +43.44 ± 1.60 +KUKA (|I| = 100) +CRAR +0.0 +0.32 ± 0.02 +0.52 ± 0.15 +Ours w/o AL +0.11 ± 0.01 +0.76 ± 0.09 +Ours w/o AL +0.01 +0.12 ± 0.01 +0.80 ± 0.11 +Ours w/o AL +0.04 +0.14 ± 0.00 +0.67 ± 0.22 +HalfCheetah (|I| = 50) +CRAR +0.0 +0.88 ± 0.05 +−818.29 ± 300.08 +Ours w/o AL +0.12 ± 0.01 +878.79 ± 88.52 +Ours w/o AL +0.01 +0.18 ± 0.02 +426.61 ± 221.12 +Ours w/o AL +0.04 +1.21 ± 0.13 +−523.8 ± 336.51 +scale α. In addition, we confirmed that PP with an annotation +error for our method remains comparable to the case without +an annotation error for a certain degree of α. For example, +in KUKA experiments, the proposed method achieved PP = +0.76 ± 0.18 with α = 0.04, which was close to PP = 0.76 ± +0.09 without annotation error. We conclude that the proposed +pair augmentation is effective in image-to-semantics learning +even in the presence of annotation errors. +The results with transition errors are shown in Table VI. +Note that CRAR does not use Trσ; thus, the result did not +depend on transition error scale α; the result of CRAR with +α > 0 matched the result of α = 0. Because the proposed +pair augmentation scheme used Trσ to generate semantics, +for larger t ∈ Ei, the variance of error was expected to be +large; then, the augmented semantics in P′ were far from the +actual semantics. In fact, we observed an increase in MD and +a decrease in PP in the proposed method as the scale of α +increased. In contrast, both MD and PP were better than CRAR +up to α = 0.04 for Shooting and KUKA, and up to α = +0.01 for HalfCheetah. This indicates that the proposed pair +augmentation is effective in reducing the annotation costs up +to a certain level of transition errors. +F. Effect of Behavior Policy +To further reveal the behavior of image-to-semantics meth- +ods, we evaluated them on HalfCheetah by adopting a low +performance policy, rather than the random policy, as the +behavior policy. We pre-trained the low performance policy +with a small number of iterations using PPO. +We observed that, compared with Table IV and Table VII, +the performance of the behavior policy affected the PP of the +resulting target agent. Here, the PP of the random policy was +−1230.01 and that of the low performance policy was 822.39. +Therefore, the PP of our proposed method was improved from +968.49 ± 99.53 to 1527.37 ± 133.19 when |I| = 50. +These results indicate that owing to the low performance +of the random policy, faster-running states, that is, states with +high velocity, cannot be observed; in other words, the random +TABLE VII +RESULTS OF HALFCHEETAH WHEN T τ WAS COLLECTED BY THE LOW +PERFORMANCE POLICY. THE BEHAVIOR POLICY HAS PP=822.39 AND πσ +HAS PP=2735.73. THE TRAJECTORY FOR CALCULATING MD IS +COLLECTED BY THE LOW PERFORMANCE POLICY. +Method +MD +PP +|I| = 0 +Zhang et al. +1.03 ± 0.19 +−1241.31 ± 523.06 +|I| = 10 +CRAR +0.74 ± 0.06 +−1549.82 ± 106.04 +Ours w/o AL +0.09 ± 0.02 +1117.05 ± 127.93 +Ours +0.06 ± 0.00 +1145.77 ± 110.29 +|I| = 50 +CRAR +0.37 ± 0.04 +−1416.37 ± 194.31 +Ours w/o AL +0.03 ± 0.00 +1393.11 ± 226.39 +Ours +0.02 ± 0.00 +1527.37 ± 133.19 +policy can only observe a limited state. This limitation could +lead to an increase in the approximation error of ˆF. This +implied that image-to-semantics is affected by the performance +of the behavior policy in some tasks. +A promising result for the image-to-semantics framework +is that the target agents obtained by our approach outperform +the behavior policy. In particular, in Table VII, the PP of +the behavior policy is 822.39; furthermore, when image-to- +semantics was performed with 50 annotations (|I| = 50), +we obtained a PP of 1527.37 ± 133.19. In other words, we +could achieve a higher performance compared with that of the +behavior policy using a small number of annotations and the +image-to-semantics protocol. +In the previous discussion, we found that the PP achieved by +the image-to-semantics framework is affected by the quantity +and quality of paired data, and the region of state space +comprising the dataset for training ˆF. In fact, as an extreme +example, ˆF trained using |P|=100k with the trajectories col- +lected by the optimal policy, achieved PP=2624.65 ± 29.09, +which is almost identical to PP=2735.73, the performance of +the optimal source policy. Note that such a near-complete +policy transfer is already achieved in Shooting and KUKA, +as shown in Tables II and III. +VI. CONCLUSION +In this study, we investigated the image-to-semantics prob- +lem for vision-based agents in robotics. Using paired data for +learning image-to-semantics mapping is favorable for achiev- +ing high-performance policy transfer; however, the cost of +creating paired data cannot be ignored. This study contributes +to existing literature by reducing the annotation cost using +two techniques: pair augmentation and active learning. We +also confirmed the effectiveness of the proposed method in +our experiments. +In future work, we must address the following limitations: +(1) Experiments have not been conducted using actual robots; +therefore, it is not known how difficulties specific to actual +robots will affect the image-to-semantics performance; (2) +We cannot always freely query Trσ; therefore, it would be +beneficial to know if we can substitute the one learned using +source trajectory, similar to [18], [23]; (3) In some cases, the + +transition error is too large, and we would like to be able +to improve the approximation accuracy of ˆF by considering +performing pair augmentation for {t | t ∈ Ei, t ≤ K} rather +than Ei. This expectation is because augmented semantics with +larger t ∈ Ei are inaccurate. Furthermore, we would like to find +a way to automatically determine such a K. +REFERENCES +[1] D. Kalashnikov, A. Irpan, P. Pastor, J. Ibarz, A. Herzog, E. Jang, +D. Quillen, E. Holly, M. Kalakrishnan, V. 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Klimov, “Prox- +imal policy optimization algorithms,” CoRR, vol. arXiv:1707.06347, +2017. + diff --git a/a9FQT4oBgHgl3EQfgTbF/content/tmp_files/load_file.txt b/a9FQT4oBgHgl3EQfgTbF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ce762f77b80cd5d6feb82be9fff62f41d2db56e --- /dev/null +++ b/a9FQT4oBgHgl3EQfgTbF/content/tmp_files/load_file.txt @@ -0,0 +1,982 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf,len=981 +page_content='Few-Shot Image-to-Semantics Translation for Policy Transfer in Reinforcement Learning Rei Sato∗‡, Kazuto Fukuchi†‡, Jun Sakuma†‡ and Youhei Akimoto†‡ ∗ Graduate School of Science and Technology, University of Tsukuba, Tsukuba, Japan reisato@bbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='tsukuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='jp † Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan fukuchi@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='tsukuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='jp, jun@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='tsukuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='jp, akimoto@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='tsukuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='jp ‡ RIKEN Center for Advanced Intelligence Project Abstract—We investigate policy transfer using image-to- semantics translation to mitigate learning difficulties in vision- based robotics control agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This problem assumes two envi- ronments: a simulator environment with semantics, that is, low- dimensional and essential information, as the state space, and a real-world environment with images as the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' By learning mapping from images to semantics, we can transfer a policy, pre-trained in the simulator, to the real world, thereby eliminating real-world on-policy agent interactions to learn, which are costly and risky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In addition, using image-to-semantics mapping is advantageous in terms of the computational efficiency to train the policy and the interpretability of the obtained policy over other types of sim-to-real transfer strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To tackle the main difficulty in learning image-to-semantics mapping, namely the human annotation cost for producing a training dataset, we propose two techniques: pair augmentation with the transition function in the simulator environment and active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We observed a reduction in the annotation cost without a decline in the performance of the transfer, and the proposed approach outperformed the existing approach without annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Index Terms—deep reinforcement learning, policy transfer, sim-to-real I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' INTRODUCTION Deep reinforcement learning (DRL) has been actively stud- ied for robot control applications in real-world environments because of its ability to train vision-based agents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' that is, the robot control actions are output directly from the observed images [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' One of the major advantages of vision-based agents in robotics is that camera-captured images can be incorporated into the decision-making of the agent without using a handcrafted feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, allowing vision-based robot control agents to learn by reinforcement learning in the real-world is challeng- ing in terms of risk and cost because it requires a large amount of real-world interactions with unstable robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Reinforcement learning involves a learning policy interacting with the envi- ronment, and it is theoretically and empirically known that the length of the interaction required for training increases with the dimension of the state space [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To address the difficulty associated with reinforcement learning in a real-world environment, methods have been proposed that pre-train a policy on a simulator environment This research is partially supported by the JSPS KAKENHI Grant Number 19H04179, and based on a project, JPNP18002, commissioned by NEDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' and transfer it to the real-world environment [7]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In this methodology, policies are learned in a simulator, that is, a reinforcement learning environment on a computer that mimics the real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The policy pre-trained in the simulator is expected to be the optimal policy in the real- world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, developing a simulator that imitates the real-world environment is not always an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Particularly, because the real world provides image observations, a simulator en- vironment requires a renderer to generate images as states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, producing a renderer that can generate photorealistic images is fraught with financial and technical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In the case that a photorealistic renderer cannot be produced, another style of observations must be adopted as states during the pre-training of the policy in a simulator environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Most existing approaches substitute photorealistic observations for non-photorealistic ones using transfer techniques [7]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We investigated a type of transfer strategy called image- to-semantics to deal with the absence of a photorealistic renderer, which was created by [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In this approach, the semantics—low-dimensional and essential information of a state that represents an image—are employed as a form of state observation instead of images in the simulator environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The transfer algorithm consists of two steps: pre-training a policy on the simulator environment with semantics as its observation, obtaining a mapping from photorealistic images to their corresponding semantics, and using the image-to- semantics mapping as a pre-processing component of the policy in the real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A semantics-based pre- trained policy can be operated in the real-world environment using image observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In addition to being a solution to the case without a photorealistic renderer, image-to-semantics mapping has advantages in terms of the computational cost for policy pre-training in the simulator and the interpretability of the acquired policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The crucial part of this approach is obtaining the image-to- semantics translation mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To the best of our knowledge, [18], [19] are the only studies that have dealt with learning image-to-semantics translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We highlight the remaining problems of [18], [19]: (1) [19] used a paired dataset, that is, multiple pairs of images and corresponding semantics, to train the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Considerable human effort is required to make arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='13343v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='LG] 31 Jan 2023 a paired dataset because human annotators provide seman- tics that represent images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' (2) Although the style translation method without a paired dataset [18] aims at saving annotation cost, its performance is not often satisfactory owing to the low approximation quality of the image-to-semantics translation mapping, as confirmed in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In this study, we tackled learning image-to-semantics trans- lation using a paired dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, we reduced the cost of creating a paired dataset using two strategies: pair augmenta- tion and active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In our experiments, we confirmed the following claims: first, compared to [19], we reduced the cost of making a paired dataset while preserving the performance of the policy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Second, we achieved significantly higher performance than [18], in which a paired dataset was not used, by using a small paired dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For practicality, we conducted experiments under the condition that only inaccurate paired data can be obtained due to various errors, such as annotation errors, and confirmed that the proposed method has a certain robustness against errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Our code is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='com/ madoibito80/im2sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' PROBLEM FORMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Markov Decision Process (MDP) We defined a vision-based robotics task in the real world;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' that is, the real-world environment is a target MDP: Mτ = (Sτ, A, pτ, rτ, γ), where Sτ is a state space, A is an action space, pτ : Sτ ×A×Sτ → R is a transition probability density, rτ : Sτ × A × Sτ → R is a reward function, and γ ∈ [0, 1] is a discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because we assumed that the target MDP is a vision-based task, Sτ consists of images, and each s ∈ Sτ contains single or multiple image frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In standard model- free reinforcement learning (RL) settings, agents can interact with the environment: they observe st+1 ∼ pτ(· | at, st) and reward rt = rτ(st+1, at, st) by performing action at at state st, which is internally preserved in the environment at timestep t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' after the transition, st+1 is stored in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, there are concerns in terms of the risk and cost associated with learning a policy through extensive interaction with Mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To reduce the risk and cost of training a policy in the target MDP, we pre-trained a policy on a simulator environment, called the source MDP: Mσ = (Sσ, A, pσ, rσ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that the action space A is the same between the two MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In contrast, the state space Sσ, the transition probability density pσ : Sσ × A × Sσ → R, and the reward function rσ : Sσ × A × Sσ → R are different from those of the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We assumed that because we considered robotics tasks, the deterministic transition function Trσ(s, a) = s′ ∼ pσ(· | a, s) could be defined in the simulator environment and pσ resembled a Dirac delta distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The source state space Sσ corresponded to a semantic space, that is, each s ∈ Sσ was semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For example, consider a robot-arm grasp task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' each s ∈ Sτ is a single or multiple image frame showing a robot arm and objects to be Action Space State Space Simulator Env (Source MDP) Policy Action Space State Space Real-World Env (Target MDP) Semantic Space Image Space (Photorealistic) Image-to- Semantics Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Illustration of transfer via image-to-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We approximated the image-to-semantics translation mapping F as ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because the action space was common to both MDPs, we operated the composite of the source policy πσ and approximated image-to-semantics translation mapping ˆF, that is, πσ ◦ ˆF in the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' grasped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Each s ∈ Sσ consists of semantics such as xyz- coordinates of the end-effector and target objects and angles of joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The source MDP and target MDP are expected to have some structural correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Here, we describe our assumptions regarding the relations of the two MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We assumed the existence of a function F : Sτ → Sσ satisfying the following conditions: Transition Condition: For all (s′, a, s) ∈ Sτ × A × Sτ, pσ(F(s′) | a, F(s)) = � ¯s∈ ¯ S pτ(¯s | a, s)d¯s, where ¯S = {¯s ∈ Sτ | F(¯s) = F(s′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Reward Condition: For all (s′, a, s) ∈ Sτ × A × Sτ, rσ(F(s′), a, F(s)) = rτ(s′, a, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In the above conditions, F is considered an oracle that takes an image and outputs corresponding semantics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' that is, F is the true image-to-semantics translation mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In the transition condition, ¯S is a set of images that has common semantics F(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Imagine the transition from s ∈ Sτ to s′ ∈ Sτ with action a ∈ A in the target MDP, the transition condition holds F(s′) = Trσ(F(s), a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The reward condition indicates that a reward for this transition rτ(s′, a, s) equals the one for a transition from F(s) ∈ Sσ to F(s′) ∈ Sσ with the action a in the source MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Transfer via Image-to-Semantics 1) Policy Transfer: The objective of RL is the expectation of the discounted cumulative reward: J(π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' p, r, γ, p0) = Eπ,p,p0 [�∞ t=0 γtr(st+1, at, st)] (1) and maximizing it w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Here, π : S × A → R is a policy, that is, a conditional distribution of at given st, and p0 is the distribution of the initial state s0 over the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Our objective was to obtain a well-trained policy on the target MDP: πτ = arg max¯πτ J(¯πτ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' pτ, rτ, γ, pτ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Under the situation in which the transition and reward conditions mentioned above hold for some F, we can replace πτ by πσ ◦ F, where πσ is a well-trained policy on the source MDP, that is, πσ = arg max¯πσJ(¯πσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' pσ, rσ, γ, pσ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Solving this maximization by RL requires sole interaction with Mσ instead of Mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' As noted, interactions with Mτ require real-world operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, interactions with Mσ are performed on the simulator, which is cost-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Based on this property, we studied the following transfer procedure: pre-train πσ on Mσ, approximate F as ˆF, and out- put the target agent πσ◦ ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This procedure was investigated by [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Figure 1 illustrates the transfer via image-to-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2) Advantages: The above-mentioned transfer strategy, that is, transfer via image-to-semantics, has the following three ad- vantages over approaches using a renderer in the source MDP shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' First, a renderer is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Existing methods that use a renderer generally aim to transfer an agent based on non-photorealistic images in a simulator to photoreal- istic images in the real world [7]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, they require the preparation of a renderer on the simulator to generate non- photorealistic images as state observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Transfer via image- to-semantics performs similar transfer learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, it does not require a renderer because the source MDP has a semantic space as its state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This can reduce the development cost of the simulator for some tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Second, because semantics are low-dimensional variables compared to images, we can improve the sample efficiency required to train the policy πσ on Mσ [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Learning vision-based agents are generally associated with large computational costs, even on a simulator [20], but transfer via image-to-semantics is relatively lightweight in this respect and occasionally allows a human to design the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Third, using semantics as an intermediate representation of the target agent contributes to its high interpretability because of the low-dimensionality and interpretability of semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Similar to [19], [21], because the real-world agent πσ◦ ˆF can be separated into two components, which are independently trained, it is easier to assess than one trained in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Resource Strategy In this section, in addition to the two MDP environments, we define resources that can be used to approximate F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 1) Transition Function: In the target MDP, the state transi- tion result st+1 due to the selected action at can be observed only for state st stored inside the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In contrast, in the source MDP, we assumed that the state transition result for any s ∈ Sσ could be observed, replacing the st stored inside the environment with s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This is because the actual state transition probability pτ in the target MDP is a physical phenomenon in the real world, but the state transition rule Trσ in the source MDP is a black-box function on the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2) Offline Dataset: The offline dataset comprised observa- tions of the target MDP, that is, T τ = {(st, at, 1end(st+1)) ∈ Sτ × A × {0, 1}}t, where 1end(st+1) = 1 represents that st+1 corresponding to a terminal state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' otherwise, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that successive indices in the offline dataset shared the same context of the episode, except at the end of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' T τ can be obtained before training starts and is collected by a behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because the offline dataset can be reused for any trial and be obtained by a safety-guaranteed behavior policy, we assumed it could be created at a relatively low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' TABLE I RELATED POLICY TRANSFER METHODS FOR OBSERVATION STYLE SHIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' EACH METHOD REQUIRES DIFFERENT RESOURCES: RENDERER, OFFLINE DATASET (OFF), AND PAIRED DATASET (PAIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method Renderer OFF PAIR Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' [7] ✓ RCAN [8] ✓ DARLA [9] ✓ Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' [10] ✓ MLVR [11] ✓ Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' [12] ✓ ✓ GraspGAN [13] ✓ ✓ RL-CycleGAN [14] ✓ ✓ RetinaGAN [15] ✓ ✓ MDQN [16] ✓ ✓ ✓ ADT [17] ✓ ✓ ✓ Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' [18] ✓ CRAR [19] ✓ ✓ Ours ✓ ✓ We solely used the offline dataset for supervised and unsu- pervised learning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' If offline reinforcement learning is executed, the vision-based agent can be trained directly without approximating F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, training a vision-based agent using an offline dataset by reinforcement learning re- quires large-scale trajectories in the scope of millions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In this study, we considered situations in which the total number of timesteps in the offline dataset was limited, for example, less than 100k timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We did not need to generate reward signals while collecting the offline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' World models [23] have been studied for the procedure: approximate MDP M as ˆ M using an offline dataset of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' train a policy by reinforcement learning by interacting with the approximated environment ˆ M instead of interacting with the original environment M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' One could imagine that we could replace interactions with the target MDP by interactions with the approximated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, to accomplish this, we must observe signals regarding reward in the real world while collecting the offline dataset, and we must approximate a reward function that is often sparse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' both of these are not always easy [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, we did not consider approximating the target MDP and did not assume the reward was contained in T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 3) Paired Dataset: The paired dataset P consisted of mul- tiple pairs of target state observations and their corresponding source state observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Let I denote the set of indices that indicate the position of the offline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Using the true image-to-semantics translation mapping F, we can denote P = {(F(si), si) | (si, ai, ei) ∈ T τ, i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Under prac- tical situations, querying F equals annotating corresponding semantics to the images of the indices I in the offline dataset T τ by human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because of its annotation cost, we assumed the size of the paired dataset |I| to be significantly smaller than that of the offline dataset, for example, |I| ≤ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' RELATED WORK We introduced some existing sim-to-real transfer methods that use a non-photorealistic renderer on the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Table I lists the transfer methods that do not require on-policy inter- action in the target MDP, assuming vision-based agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The main difficulty tackled by these methods was the absence of a photorealistic renderer on the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In the real world, images captured by a camera are input to the agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, generating photorealistic images on the simulator is generally difficult because it requires developing a high-quality renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In [7]–[11], [25], the algorithms learned policies or interme- diate representations that were robust to changes in image style using a non-photorealistic renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Thus, these algorithms were expected to perform well even when a photorealistic style was applied in a real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In particular, the domain randomization technique has been widely used [7]– [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Transfer via Image-to-Image Translation In contrast to the above methods, [12]–[19] aimed to perform style translation mapping among specific styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To accomplish this, these methods required an offline dataset of the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because these methods followed the principle of collection without execution of on-policy interaction, the offline dataset could be collected by a safety-guaranteed pol- icy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Unsupervised style translation, such as domain adaptation [26] and CycleGAN [27], are often used to change the styles for state-of-the-art methods [13]–[15], [17], [18], [24], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Using this translation mapping as a pre-processing function of the target agent, the pre-trained policy can determine actions in the same image style as the source MDP in the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, domain adaptation and cycle-consistency [27] only have a weak alignment ability [18], and some existing methods use paired datasets to properly transfer styles [16], [17], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, these two datasets have been widely employed in previous studies and can be assumed to be a common setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The similarity of transfer via image-to-semantics and image- to-image is that they train style translation mapping ˆF among the source and target state spaces that preserves essential information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' furthermore, the agent is the composite π ◦ ˆF, where π is a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Again, the above methods use a non-photorealistic renderer on the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Thus, these methods cannot be compared with transfer via image-to-semantics, as explained in Sec- tion II-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Learning Image-to-Semantics Previous studies have used semantics in the source MDP [10], [12], [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' An important perspective on the appli- cability of these methods to image-to-semantics is whether they use a renderer on the simulator, as shown in Table I and as discussed in Section II-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because methods using a renderer assume that the source state space is an image space, image-to-semantics is beyond their scope, and it is not certain that their mechanism will be successful in image-to-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For example, CycleGAN, which has been successfully used for image-to-image learning, failed in image-to-semantics [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In this regard, we refer to [18], an unpaired method that applies the findings from image-to-image to image-to-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In addition, [19] is compared as a representative method that uses a paired dataset as in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 1) CRAR: We refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='4 of CRAR [19] as a baseline of image-to-semantics learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' They described the following policy transfer strategy: pre-train a source state encoder Eσ : Sσ → Z, where Z is a latent space of the encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' train the source policy πσ : Z → A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' and train a target state encoder Eτ : Sτ → Z with regularization term � (sσ,sτ )∈P∥Eσ(sσ)−Eτ(sτ)∥2 2, where P is a paired dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Then, the target agent is the composite πσ ◦ Eτ : Sτ → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Here, Eτ can be regarded as a style translation mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that they only performed this experiment in the setting where Sσ and Sτ are both image spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, it can be applied easily where Sσ is the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2) Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' : We referred to the cross-modality setting of their experiment as our baseline for image-to-semantics [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This setting is the same as the transfer via image-to-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' There remain some challenges in [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For [18], the human annotation cost was eliminated because they did not use a paired dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, the loss function defined by [18] for unpaired image-to-semantics style translation will not necessarily provide a well-approximated F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, we decided to use a paired dataset to efficiently supervise the loss function as performed in [19], but with a paired dataset smaller than [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' METHODOLOGY Our approach approximates the image-to-semantics transla- tion F using an offline dataset T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Similar to [19], we used a paired dataset P = {(F(si), si) | (si, ai, ei) ∈ T τ, i ∈ I}, which was constructed by querying F(si) to human annotators for an image observation of target MDP si ∈ Sτ included in T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We incorporated two main ideas to reduce the annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Pair augmentation generates an augmented paired dataset P′ using an offline dataset T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Active learning defines I, that is, it selects a subset of T τ to be annotated to construct P (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We present an overall procedure of our method in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We assumed that we have an offline dataset T τ, which comprises multiple episodes in the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Let O denote the set of indices corresponding to the beginning of an episode in T τ, that is, O = {0} ∪ {i | 0 < i < |T τ| and ei−1 = 1 for (sτ i−1, ai−1, ei−1) ∈ T τ}, where ei is the indicator: when timestep i is the end of an episode then ei = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For each i ∈ O, let Ei = {t | 1 ≤ t ≤ min({k | k ≥ 1, ei+k = 1})}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Then, for each i ∈ O, a subsequence of T τ starting from timestep i and ending at i + |Ei| corresponds to an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Pair Augmentation by Transition Function The objective of pair augmentation is to construct artificial paired data P′ such that sσ ≈ F(sτ) for (sσ, sτ) ∈ P′ and sτ ∈ T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Using an augmented paired dataset, we aimed to obtain ˆF that approximates F by minimizing the loss L( ˆF, P ∪ P′) = 1 |P ∪ P′| � (sσ,sτ )∈P∪P′ ∥sσ − ˆF(sτ)∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' (2) action Semantics Offline Dataset action action action F Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Illustration of pair augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Oracle F generates semantics corresponding to a particular image in the offline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The next state in semantics is computed using the transition function Trσ with the current semantics along with the action taken while collecting the offline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This allows us to obtain semantics corresponding to the image at the next timestep in the offline dataset without any annotation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Augmented pairs with green dual directional arrows were stored in P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In this figure, note that rendered (non-photorealistic) images are shown in the offline dataset, but in reality, camera-captured (photorealistic) images are contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Algorithm 1 Overall Procedure Require: Source MDP Mσ, Offline dataset T τ, Oracle F 1: Train source MDP’s policy πσ on Mσ 2: Train VAE encoder Eτ using T τ 3: Determine indices I by active learning (Algorithm 2) using Eτ, T τ 4: Create P for I, T τ by oracle (human annotator) F 5: Create augmented pairs P′ using P, T τ, Trσ ∈ Mσ 6: Train ˆF by minimizing Equation (2) Ensure: Target MDP’s agent πσ ◦ ˆF Note that CRAR [19] adopts L( ˆF, P) instead of L( ˆF, P∪P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Our principle is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Let I ⊆ O be a subset of indices corresponding to the beginning of the episodes in T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Suppose we have a paired dataset P constructed by querying semantics sσ i = F(sτ i ) corresponding to images sτ i in T τ for time index i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Although semantics sσ i+1 representing an image of the next timestep sτ i+1 in T τ is unknown, because of the transition condition given in Section II-A and deterministic transition, it equals sσ i+1 = Trσ(sσ i , ai), where ai is the action taken at timestep i when collecting the offline dataset T τ and is included in T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In reality, because human annotations and state transition contain errors as compared to the truth, the generated semantics sσ i+1 do not exactly represent the image sτ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, even with errors in F and Trσ, it is expected that the generation of the above semantics is a valuable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' By recursively applying the above generation, we obtained the augmented paired dataset P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Formally, P′ was constructed as follows: For each index i ∈ I, we defined a sequence {�sσ i+t}t∈Ei as �sσ i = sσ i (contained in P) and �sσ i+t = Trσ(�sσ i+t−1, ai+t−1) for t ∈ Ei, where ai+t−1 are contained in T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The augmented paired dataset is then P′ = {(�sσ i+t, sτ i+t)}i∈I,t∈Ei, where sτ i+t is contained in T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Thus, we could construct an augmented paired dataset P′ of size |P′| = � i∈I|Ei| from the paired dataset P of size |P| = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Figure 2 illustrates the pair augmentation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The reason why I was a subset of episode start indices O rather than I ⊆ {j | 0 ≤ j < |T τ|, j ∈ Z} was to maximize the size of augmented pairs |Ei|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In other words, because we could augment sσ i = F(sτ i ) until the end of the episode including sτ i , to maximize |P∪P′|, human annotations should be conducted at the beginning of an episode of T τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Active Learning for Pair Augmentation To select episodes for annotation, that is, decide I, we incorporated the idea of diversity-based active learning (AL) [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Their motivation was to select dissimilar samples to effectively reduce the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Intuitively, if P ∪ P′ has many similar pairs, they might have a similar effect on training ˆF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' this may lead to a waste in annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, we attempted to select episodes (indexed by I ⊂ O) to be annotated to ensure the inclusion of diverse pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We successively selected the episode to annotate, and we called each selection step the n-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For i ∈ O, let Bi = {sτ i+t}t∈{0}∪Ei be a set of target state observations present in the episode starting at timestep i ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We referred to it as batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Let In−1 be the set of selected indices before the n-th round, and let Sn−1 = � k∈In−1 Bk be a set of all the state vectors in the episodes selected before the n-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Let d : Sτ × Sτ → R be some appropriate distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In the n-th round, a batch was selected based on the following two diversity measures: The inter batch diversity finter(Bi, Sn−1) = � sτ ∈Bi min sτ j ∈Sn−1 d(sτ, sτ j ) (3) can evaluate the dissimilarity of Bi and Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The batch with the greatest finter was considered to be the most dissimilar Algorithm 2 Active Learning Require: Trained VAE encoder Eτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Offline dataset T τ 1: Initialize I0 = {c}|c∼Uniform(O) 2: for 1 ≤ n < N do ▷ n-th round 3: Set Sn−1 = � k∈In−1 Bk 4: Measure finter(Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Sn−1) for all i ∈ O 5: Pick top b% of indices in terms of finter as Q 6: Measure fintra(Bi) for all i ∈ Q 7: Pick the index c from Q with the greatest fintra 8: Set In = {c} ∪ In−1 9: end for Ensure: Indices IN−1 (with the size of N) as I batch against the pre-selected batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The intra batch diver- sity fintra(Bi) = � sτp∈Bi � sτq ∈Bi d(sτ p, sτ q) (4) can evaluate the dissimilarity of the states inside Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The batch with the greatest fintra was considered to contain the most diverse states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We selected a batch that maximizes the above two diversity measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' we performed a bi-objective optimization for selec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To avoid overemphasizing one measure over the other, we employed two separate single-objective optimizations for each measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In each round, we picked up indices of batches with finter in the top b% (b = 10 in our experiments) from unselected episodes as Q, and subsequently, selected the batch with the greatest fintra from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' I0 was initialized with the episode sampled from O uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Representation Learning Using Offline Dataset For d : Sτ × Sτ → R to be a reasonable distance measure in the image space, we employed a VAE encoder [32]: Eτ : Sτ → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' It stochastically outputs a latent vector z ∈ Z for sτ ∈ Sτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The distance between two states sτ p ∈ Sτ and sτ q ∈ Sτ was given by the Euclidean distance between the mean vectors for their latent representations, that is, d(sτ p, sτ q) = ∥E[Eτ(sτ p)] − E[Eτ(sτ q)]∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We trained Eτ using all states in the offline dataset T τ before performing the active learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We used the states contained in � i∈I Bi in training ˆF by Equation (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, the remaining � i∈O\\I Bi were not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In order to use it, we included Eτ as a feature extractor for ˆF by receiving the benefit of representation learning for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We modeled ˆF = φ ◦ Eτ, and we trained φ by Equation (2), whereas Eτ was fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' EXPERIMENTS We aimed to verify the following two claims: (1) the proposed paired augmentation and AL reduces the annotation cost for approximating ˆF while maintaining its performance level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' and (2) the paradigm with the paired dataset performs better than the method without paired datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Evaluation Metrics 1) Policy Performance (PP): The most important evalua- tion metric for ˆF is the expected cumulative reward of the target agent using Equation (1): PP( ˆF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' πσ, Mτ) = J(πσ ◦ ˆF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' pτ, rτ, γ, pτ 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' (5) In our experiments, we approximated it by averaging the cumulative reward of 50 episodes with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This metric was commonly used in [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2) Matching Distance (MD): Because our technical contri- bution was mainly to approximate F, we used the following empirical approximation error: MD( ˆF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' T , F) = 1 |T | � (sτ ,a,e)∈T ∥F(sτ) − ˆF(sτ)∥2 2, (6) where T is a trajectory collected by a behavior policy in the target MDP, which is not used for learning ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Unfortunately, in a real-world environment, evaluating Equation (6) for a large size of T is challenging because F requires human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' To enable MD in our experiment, we performed experiments using the simulator for both the source MDP and target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We adopted the rendered image space as the state space of the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because both semantics and images were generated in the simulator, F was freely available to calculate Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A similar metric to Equation (6) was used in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Environment We evaluated the proposed approach on three environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 1) ViZDoom Shooting (Shooting): ViZDoom Shooting [33] is a first-person view shooter task, in which an agent obtains 64×64 RGB images from the first-person perspective in the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The agent can change its x-coordinate by moving left and right in the room and attacking forward (|A| = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' An enemy spawns with a random x-coordinate on the other side of the room at the start of the episode and does not move or attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The agent can destroy the enemy by moving to the front of it and shooting it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' time to destruction is directly related to the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Semantics are the x-coordinates of the agent and the enemy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' hence, Sσ is a 2-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The maximum timesteps is 50 for each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The behavior policy to collect the offline dataset T τ is a random policy, and T τ consists of 200 episodes, that is, 10k timesteps in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2) PyBullet KUKA Grasp (KUKA): This is a grasp task using PyBullet’s KUKA iiwa robot arm [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Success is achieved by manipulating the end-effector of the robot arm and lifting a randomly placed cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The semantics are the xyz-coordinate and the 3-dimensional Euler angle of the end- effector and the xyz-coordinate of the cylinder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' hence, Sσ is a 9-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We used the rendered 64×64 RGB im- ages captured from three different viewpoints simultaneously as the state observations in the target MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The total timesteps per episode is fixed to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The behavior policy to collect T τ is a random policy, and T τ comprised 250 episodes, that is, 10k timesteps in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' TABLE II RESULTS OF SHOOTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' MD VALUES WERE SCALED TO 102 FOR CONVENIENCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' πσ HAS PP=45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='99, AND THE BEHAVIOR POLICY HAS PP=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method MD PP |I| = 0 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='42 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='23 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='46 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='99 |I| = 10 CRAR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='72 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='16 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='31 Ours w/o AL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='89 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='99 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 |I| = 50 CRAR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='70 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='31 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='34 3) PyBullet HalfCheetah-v0 (HalfCheetah): This is a Py- Bullet version of the HalfCheetah, that is, a task in which a 2-dimensional cheetah is manipulated by continuous control to run faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The torque of the six joints can be controlled (A = [−1, 1]6), and the semantic space is a 26-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We collected 64×64 images captured from three differ- ent viewpoints for two consecutive timesteps and defined Sτ as an image space containing a total of 6 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The total timesteps per episode is fixed to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The behavior policy to collect T τ is a random policy, and T τ consists of 100 episodes, which is 100k timesteps in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In our experiments, information such as xyz-coordinates and velocity can be recovered from a combination of multiple images by capturing images from multiple viewpoints at consecutive times, and such a setup is necessary in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Setting We used a 7-layer convolutional neural network and a 4- layer fully connected neural network for the VAE encoders Eτ and φ, respectively, for both the proposed and existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We trained them in gradients using Adam [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The dimensions of the latent space of VAE Z were set to 32, 96, and 192 for Shooting, KUKA, and HalfCheetah, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For CRAR [19], we uniformly selected indices I from {i | 0 ≤ i < |T τ|, i ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For our method, without an AL setting, I was selected uniformly and randomly from O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For Shooting and KUKA, we used a handcrafted policy instead of one trained by RL as πσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In HalfCheetah, we trained πσ using PPO [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Results Tables II to IV show the results of the image-to-semantics learning in the three environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' These tables show the results on average±std over five trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' |I| denotes the number of paired data, which is the annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because of the transition and reward conditions, the PP of πσ ◦ F on Mτ assimilate to that of πσ on Mσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that most image-to-image methods shown in Table I cannot be compared with image-to-semantics methods be- cause some assumptions cannot be satisfied under image-to- semantics settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' One way to speculate on the performance of the image-to-image techniques in an image-to-semantics CRAR Ours w/o AL Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Scatter of the obtained semantics on ViZDoom Shooting with |P| = |I| = 10: {F(sτ) | (sσ, sτ) ∈ P} for CRAR, and {F(sτ) | (sσ, sτ) ∈ P∪ P′} for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Each square represents a 2-dimensional semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The semantic space shows that both pair augmentation and AL contribute to expanding the coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' TABLE III RESULTS OF KUKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' PP CORRESPONDS TO GRASP SUCCESS PROBABILITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' πσ HAS PP=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0, AND THE BEHAVIOR POLICY HAS PP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method MD PP |I| = 0 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='08 |I| = 10 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='22 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='19 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 |I| = 100 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 setting is to see Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' used do- main adaptation [26], which is commonly used in image-to- image learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' thus, their method can be interpreted as a representative example in which the techniques cultivated in image-to-image are imported to image-to-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Although CycleGAN [27] is also widely employed in image-to-image learning, along with domain adaptation, they confirmed in their experiments that this method did not outperform their method in the image-to-semantics setting [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In all cases, compared with the approach of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' [18], our approaches with and without AL achieved a smaller MD and a greater PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Zhang et al.’s approach is designed to learn ˆF without a paired dataset to eliminate the annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, learning without pairs does not necessarily lead to the true image-to-semantics translation mapping, as observed in the high MD and low PP in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This result shows the effectiveness of the paradigm using paired data when aiming for higher performance policy transfer while compromising the annotation cost to prepare a small number of paired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' By comparing the results of our approaches with and without AL and those of CRAR, we confirmed the efficacy of pair augmentation in achieving a smaller MD and higher PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We achieved PP=44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 in Shooting with 10 pairs using pair augmentation and AL, which is more than the PP=42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 achieved by CRAR with 50 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In addition, in KUKA, we achieved PP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 in 10 pairs, which exceeds PP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 achieved by CRAR with 100 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This means that the annotation cost was reduced by more than ×5 and ×10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This difference is even more TABLE IV RESULTS OF HALFCHEETAH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' πσ HAS PP=2735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='73, AND THE BEHAVIOR POLICY HAS PP=−1230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method MD PP |I| = 0 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='32 −1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='97 ± 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='51 |I| = 10 CRAR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09 −1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='83 ± 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='21 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='20 ± 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='43 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='21 ± 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='95 |I| = 50 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 −818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 ± 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='08 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='79 ± 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='49 ± 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='53 pronounced in HalfCheetah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This may be because the ratio |P ∪ P′|/|P| is the greatest in this environment: CRAR uses |P| = |I| paired data, whereas our proposed approach used an additional |P′| = 999|I| augmented paired data because the number of timesteps per episode was 1000 in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A tendency of reduced MD and increased PP was observed in the proposed approach with AL compared to that without AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Specifically, AL reduced MD except in KUKA, and clearly improved PP in KUKA, while achieving competitive PP in the other two environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In Figure 3, we present the effectiveness of our approach in Shooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The proposed AL maximized the diversity in the latent space that represents the image space, but the diversity was also maximized when this result was visualized in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Experiments with Errors In this section, we verify the robustness of the proposed method against errors in annotation and state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 1) Annotation Error: In our previous discussion and in experiments of Section V-D, we assumed that we could query oracle F, that is, true image-to-semantics mapping by human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' However, because human annotation indicates the process of assigning semantics to images by humans, errors are expected to occur in the output semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, we provided a new experimental setup here: for some sτ ∈ Sτ, we can observe F(sτ) + ϵ instead of F(sτ) while creating the paired dataset P, where ϵ ∈ Rdim(Sσ) is a random vector representing the annotation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 2) Transition Error: In reality, the state transition function on the simulator Trσ is expected to contain modeling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For example, environment parameters such as friction coeffi- cients and motor torques in the real world cannot be accurately estimated in the simulator, and thus, state transitions in reality cannot be correctly imitated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, we provided a new experimental setup here: for some (s, a) ∈ Sσ × A, we could obtain Trσ(s, a) + ϵ instead of Trσ(s, a) while augmenting a paired dataset, where ϵ ∈ Rdim(Sσ) is a random vector representing the transition error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that when training πσ, we used the one without errors in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 3) Error Generation: We generated two types of errors by adding a random variable ϵ ∈ Rdim(Sσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Here, we denoted a TABLE V RESULTS WITH ANNOTATION ERRORS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method α MD PP Shooting (|I| = 50) CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='70 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='31 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='93 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='71 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='54 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='21 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='51 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='07 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='59 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='85 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='83 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='78 Ours w/o AL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='74 KUKA (|I| = 100) CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='11 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='18 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 HalfCheetah (|I| = 50) CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 −818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 ± 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='08 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='79 ± 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='03 −919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='86 ± 328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='14 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 ± 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='32 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 −833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='35 ± 464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='63 ± 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='73 value of the h-th dimension of x ∈ RH as x(h) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We sam- pled ϵ(h) ∼ N(h), where N(h) is a Gaussian distribution with mean µ = 0 and standard deviation σ = α · std[sσ (h)]sσ∈T σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Here, std[sσ (h)]sσ∈T σ is the sample standard deviation of a source trajectory T σ collected by a behavior policy in source MDP, and α ≥ 0 is the noise scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For the annotation error, using the semantics sequence of augmented pairs {�sσ i+t}t∈Ei provided without error, we provided the semantics as F(sτ i )+¯ϵi for P and {�sσ i+t+¯ϵi}t∈Ei for P′, where ¯ϵi is a realized random vector with α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For the transition error, we defined {�sσ i+t + �t j=1 ¯ϵi,j}t∈Ei for P′, where ¯ϵi,j is the realized random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that, here, we approximated the error generation based on the following assumption: Trσ(s, a) = s + Trσ ∆(a), that is, Trσ(s+¯ϵ1, a)+¯ϵ2 = s+Trσ ∆(a)+¯ϵ1+¯ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This is an approxi- mation simplifying implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, for Shooting and KUKA, the above assumption is actually satisfied for almost all states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 4) Results: Here, we analyze the effect of two types of errors on P and P′, and understand how this affects the approximation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, we do not experiment with the method of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=', which does not utilize paired datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In addition, to eliminate the effect of the choice of I on the generation of P and P′ in the comparison between the proposed method and CRAR, we conducted experiments under the setting without AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The results with annotation errors are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In both CRAR and the proposed method, the semantics of the paired data deviated from the true data as the scale of the annotation error α increased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' thus, we observed that MD tends to increase for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Although PP tended to decrease only for the proposed method, the proposed method achieved better MD and PP than CRAR for the same error TABLE VI RESULTS WITH TRANSITION ERRORS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method α MD PP Shooting (|I| = 50) CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='70 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='31 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='51 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='16 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='43 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='44 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='60 KUKA (|I| = 100) CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='15 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='11 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='22 HalfCheetah (|I| = 50) CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 −818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 ± 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='08 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='79 ± 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='52 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='61 ± 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='12 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='13 −523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='8 ± 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='51 scale α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In addition, we confirmed that PP with an annotation error for our method remains comparable to the case without an annotation error for a certain degree of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' For example, in KUKA experiments, the proposed method achieved PP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='18 with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04, which was close to PP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09 without annotation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We conclude that the proposed pair augmentation is effective in image-to-semantics learning even in the presence of annotation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' The results with transition errors are shown in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that CRAR does not use Trσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' thus, the result did not depend on transition error scale α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' the result of CRAR with α > 0 matched the result of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Because the proposed pair augmentation scheme used Trσ to generate semantics, for larger t ∈ Ei, the variance of error was expected to be large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' then, the augmented semantics in P′ were far from the actual semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In fact, we observed an increase in MD and a decrease in PP in the proposed method as the scale of α increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In contrast, both MD and PP were better than CRAR up to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 for Shooting and KUKA, and up to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 for HalfCheetah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This indicates that the proposed pair augmentation is effective in reducing the annotation costs up to a certain level of transition errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Effect of Behavior Policy To further reveal the behavior of image-to-semantics meth- ods, we evaluated them on HalfCheetah by adopting a low performance policy, rather than the random policy, as the behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We pre-trained the low performance policy with a small number of iterations using PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We observed that, compared with Table IV and Table VII, the performance of the behavior policy affected the PP of the resulting target agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Here, the PP of the random policy was −1230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='01 and that of the low performance policy was 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Therefore, the PP of our proposed method was improved from 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='49 ± 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='53 to 1527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='19 when |I| = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' These results indicate that owing to the low performance of the random policy, faster-running states, that is, states with high velocity, cannot be observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' in other words, the random TABLE VII RESULTS OF HALFCHEETAH WHEN T τ WAS COLLECTED BY THE LOW PERFORMANCE POLICY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' THE BEHAVIOR POLICY HAS PP=822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='39 AND πσ HAS PP=2735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' THE TRAJECTORY FOR CALCULATING MD IS COLLECTED BY THE LOW PERFORMANCE POLICY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Method MD PP |I| = 0 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='19 −1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='31 ± 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 |I| = 10 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 −1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='82 ± 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 1117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='05 ± 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='93 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 1145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='77 ± 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='29 |I| = 50 CRAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='04 −1416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='31 Ours w/o AL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 1393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='11 ± 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='39 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='00 1527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='19 policy can only observe a limited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This limitation could lead to an increase in the approximation error of ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This implied that image-to-semantics is affected by the performance of the behavior policy in some tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' A promising result for the image-to-semantics framework is that the target agents obtained by our approach outperform the behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In particular, in Table VII, the PP of the behavior policy is 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' furthermore, when image-to- semantics was performed with 50 annotations (|I| = 50), we obtained a PP of 1527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='37 ± 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In other words, we could achieve a higher performance compared with that of the behavior policy using a small number of annotations and the image-to-semantics protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In the previous discussion, we found that the PP achieved by the image-to-semantics framework is affected by the quantity and quality of paired data, and the region of state space comprising the dataset for training ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In fact, as an extreme example, ˆF trained using |P|=100k with the trajectories col- lected by the optimal policy, achieved PP=2624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='65 ± 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='09, which is almost identical to PP=2735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content='73, the performance of the optimal source policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Note that such a near-complete policy transfer is already achieved in Shooting and KUKA, as shown in Tables II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' CONCLUSION In this study, we investigated the image-to-semantics prob- lem for vision-based agents in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Using paired data for learning image-to-semantics mapping is favorable for achiev- ing high-performance policy transfer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' however, the cost of creating paired data cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This study contributes to existing literature by reducing the annotation cost using two techniques: pair augmentation and active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' We also confirmed the effectiveness of the proposed method in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' In future work, we must address the following limitations: (1) Experiments have not been conducted using actual robots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' therefore, it is not known how difficulties specific to actual robots will affect the image-to-semantics performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' (2) We cannot always freely query Trσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' therefore, it would be beneficial to know if we can substitute the one learned using source trajectory, similar to [18], [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' (3) In some cases, the transition error is too large, and we would like to be able to improve the approximation accuracy of ˆF by considering performing pair augmentation for {t | t ∈ Ei, t ≤ K} rather than Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' This expectation is because augmented semantics with larger t ∈ Ei are inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' Furthermore, we would like to find a way to automatically determine such a K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FQT4oBgHgl3EQfgTbF/content/2301.13343v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': 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0000000000000000000000000000000000000000..1b53d8e3b819014c083a45e5b4d5f2e506093f95 --- /dev/null +++ b/bNFJT4oBgHgl3EQf8y1j/content/tmp_files/2301.11685v1.pdf.txt @@ -0,0 +1,2105 @@ +arXiv:2301.11685v1 [math.FA] 27 Jan 2023 +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION +OPERATORS ON TWO DOMAINS +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND MICHAEL SPECKBACHER +Abstract. We derive eigenvalue estimates for concentration operators asso- +ciated with the discrete Fourier transform and two concentration domains sat- +isfying certain regularity conditions. These conditions are met, for example, +when the discrete domain, contained in a lattice, is obtained by discretization +of a suitably regular domain in the Euclidean space. As a limit, we obtain +eigenvalue estimates for Fourier concentration operators associated with two +suitably regular domains in the Euclidean space. The proof builds on Israel’s +work on one dimensional intervals: arXiv:1502.04404v1. +1. Introduction and results +Fourier concentration operators act by incorporating a spatial cut-off and a +subsequent frequency cut-off to the Fourier inversion formula. The chief example +concerns the Fourier transform on the Euclidean space F : L2(Rd) → L2(Rd), the +cut-offs are given by the indicator functions of two compact domains E, F ⊆ Rd, +and the concentration operator is +Sf = χFF −1χEFχFf, +f ∈ L2(Rd). +(1.1) +These operators, and their analogues defined with respect to the discrete Fourier +transform L2([−1/2, 1/2]d) → ℓ2(Zd) play a crucial role in many analysis problems +and fields of application [18, 13, 14, 7], such as imaging, where the shapes of E, F +are dictated by various acquisition constraints. +The basic intuition is that the concentration operator (1.1) is approximately a +projection with rank tr(S) = |E| · |F|. The error of such heuristic is encoded by +the so-called plunge region +Mε(S) = {λ ∈ σ(S) : ε < λ < 1 − ε}, +ε ∈ (0, 1/2), +(1.2) +consisting of intermediate eigenvalues. Asymptotics for the cardinality of Mε(S) +go back to Landau and Widom [15, 12] for the case of one dimensional intervals +E = [−a, a], F = [−b, b] and read +#Mε(S) = c · log(ab) · log( 1−ε +ε ) + o(log(ab)), +as ab → ∞, +(1.3) +for an explicit constant c that depends on the normalization of the Fourier trans- +form. The modern spectral theory of Wiener-Hopf operators gives similar asymp- +totics for concentration operators associated to rather general multi-dimensional +domains subject to increasing isotropic dilations. +2010 Mathematics Subject Classification. 47B35, 47A75, 42B35, 42C40. +Key words and phrases. Discrete Fourier transform, concentration operator, Hankel operator, +eigenvalue, spectrum. +The authors gratefully acknowledge support from the Austrian Science Fund (FWF): Y 1199. +1 + +2 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +While (1.3) precisely describes the cardinality of the set Mε(S) in the limit +ab → ∞, the asymptotic is often insufficient for many purposes because of the +quality of the error terms. Indeed, the error term in (1.3) depends in an unspecified +way on the spectral threshold ǫ, which precludes applications where ε is let to vary +with the domains E, F. Such limitations have motivated a great amount of work +aimed at deriving upper bounds for #Mε(S) that are threshold robust, that is, +bounds that are effective for concrete concentration domains and explicit in their +dependence on the spectral threshold [8, 10, 20, 11, 3, 17], significantly improving +on more classical results in this spirit [19]. +With the exception of [8], the mentioned articles on threshold-robust spectral +bounds for Fourier concentration operators concern only the one dimensional case, +because they exploit a connection with a Sturm–Liouville equation which is spe- +cific of that setting. On the other hand, while [8] studies Fourier concentration +operators associated with one dimensional intervals, the technique introduced by +Israel is very general, as it relies on an explicit almost diagonalization of the +concentration operator. In fact, as we were finishing this work, the preprint [9] +provided an extension of [8] to higher dimensions (see Sections 1.1 and 1.5). +In this article we derive upper bounds for the number of intermediate eigen- +values (1.2) associated with either the continuous or discrete Fourier transforms. +We obtain estimates that apply to two suitably regular multi-dimensional spatial +and frequency domains. The proofs build on Israel’s technique [8] and combine it +with arguments from geometric measure theory and operator theory. +1.1. The Euclidean space. Given two compact sets E, F ⊆ Rd, the Fourier +concentration operator S : L2(Rd) → L2(Rd) is defined by (1.1) where F denotes +the Fourier transform +Ff(ξ) = +� +Rd f(x)e−2πixξ dx. +(1.4) +A set E ⊆ Rd is said to have a maximally Ahlfors regular boundary if there exists +a constant κ∂E > 0 such that +Hd−1� +∂E ∩ Br(x) +� +≥ κ∂E · rd−1, +0 < r ≤ Hd−1(∂E)1/(d−1), +x ∈ ∂E. +Here, Hd−1 denotes the (d−1)-dimensional Hausdorff measure. The term maximal +in the definition refers to the range of r for which the estimate is required to hold. +See Section 2 for more context on Ahlfors regularity. In what follows, we denote +for short |∂E| = Hd−1� +∂E). +In this article we prove the following. +Theorem 1.1. Let E, F ⊆ Rd, d ≥ 2, be compact domains with maximally +Ahlfors regular boundaries with constants κ∂E, κ∂F respectively, and assume that +that |∂E||∂F| ≥ 1. Consider the concentration operator (1.1) and its eigenvalues +{λn : n ∈ N}. +Then for every α ∈ (0, 1/2), there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): +# +� +n ∈ N : λn ∈ (ε, 1 − ε) +� +≤ Aα,d · |∂E| +κ∂E +· |∂F| +κ∂F +· log +�|∂E||∂F| +κ∂E ε +�2d(1+α)+1 +. + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +3 +A closely related result is presented in the recent preprint [9]. For F = [0, 1]d and +E = rK, where r > 0 is a dilation parameter and K ⊆ Rd is a convex, coordinate +symmetric domain [9, Theorem 1.1] gives the following bound for ε ∈ (0, 1/2): +# +� +n ∈ N : λn ∈ (ε, 1 − ε) +� +≤ Cd · max{rd−1 log(r/ε)3, log(r/ε)3d}. +(1.5) +For large r, the right-hand side of (1.5) becomes Od +� +rd−1 log(r/ε)3) while Theo- +rem 1.1 gives the weaker bound Oα,d +� +rd−1 log(r/ε)2d(α+1)+1� +. On the other hand, +Theorem 1.1 applies to possibly non-convex, non-coordinate-symmetric and non- +dilated domains E, and other regular domains F besides cubes. (As pointed out +in [9], when E and F are both cubes, even slightly stronger estimates hold, c.f. +[9, Theorem 1.2].) +Our work is in great part motivated by applications where concentration do- +mains may be non-convex. For example, noise statistics are often estimated from +those pixels of a square image located outside a central disk, which is assumed to +contain the signal of interest (see, e.g., [2]). Thus, the need to sample pure noise +leads one to consider the complement of a disk within a two dimensional square +as concentration domain (or, more realistically, the set of grid points within that +domain; see below). Such a domain E is allowed by Theorem 1.1 (and Theorems +1.2 and 1.3 below) and has moreover a favorable regularity constant κ∂E. +1.2. Discretization of continuous domains. Theorem 1.1 is obtained by tak- +ing a limit on a more precise result concerning a discrete setting, which is our +main focus. +We consider a resolution parameter L > 0 and define the discrete Fourier +transform FL : L2((−L/2, L/2)d) → ℓ2(L−1Zd) by +FLf(k/L) = +� +(−L/2,L/2)d f(x)e−2πixk/Ldx, +k ∈ Zd. +(1.6) +We think of L as a discretization parameter for an underlying continuous problem. +Let us define the discretization at resolution L > 0 of a domain E ⊆ Rd by +EL = L−1Zd ∩ E. +Given two compact domains E ⊆ Rd and F ⊆ (−L/2, L/2)d, consider the dis- +cretized concentration operator T : L2(F) → L2(F) given by +T = χFF −1 +L χELFL. +(1.7) +Our second result reads as follows. +Theorem 1.2. Let E, F ⊆ Rd, d ≥ 2, be compact domains with maximally +Ahlfors regular boundaries with constants κ∂E, κ∂F respectively, and assume that +that |∂E||∂F| ≥ 1. +Fix a discretization resolution L ≥ |∂E|−1/(d−1) such that F ⊆ (−L/2, L/2)d +and consider the discretized concentration operator (1.7) and its eigenvalues {λn : +n ∈ N}. +Then for every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): +# +� +n ∈ N : λn ∈ (ε, 1 − ε) +� +≤ Aα,d · |∂E| +κ∂E +· |∂F| +κ∂F +· log +�|∂E||∂F| +κ∂E ε +�2d(1+α)+1 +. + +4 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +1.3. The discrete Fourier transform. Finally, we consider a discrete concen- +tration problem associated with the usual discrete Fourier transform, denoted +F1 : L2((−1/2, 1/2)d) → ℓ2(Zd) +for consistency with (1.6). +Given a finite set Ω ⊆ Zd and F ⊆ (−1/2, 1/2)d, the discrete Fourier concen- +tration operator T : L2(F) → L2(F) is defined as +T = χFF −1 +1 χΩF1. +(1.8) +The discrete boundary of a set Ω ⊆ Zd is given by +∂Ω = {k ∈ Ω : min{|j − k| : j ∈ Zd ∖ Ω} = 1}. +(1.9) +We say that Ω ⊆ Zd has a maximally Ahlfors regular boundary if there exists a +constant κ∂Ω such that +inf +k∈∂Ω # +� +∂Ω ∩ k + [−n/2, n/2)d� +≥ κ∂Ω · nd−1, +1 ≤ n ≤ (#∂Ω)1/(d−1), k ∈ ∂Ω. +(Note the slight notational abuse: though Ω ⊆ Zd ⊆ Rd, the notions of boundary +and boundary regularity are to be understood in the discrete sense.) +Our last result reads as follows. +Theorem 1.3. Let d ≥ 2, Ω ⊆ Zd a finite set with maximally Ahlfors regular +boundary and constant κ∂Ω. Let F ⊆ (−1/2, 1/2)d be compact with maximally +Ahlfors regular boundary and constant κ∂F. Assume that #∂Ω · |∂F| ≥ 1, and +consider the concentration operator (1.8) and its eigenvalues {λn : n ∈ N}. +Then for every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): +# +� +n ∈ N : λn ∈ (ε, 1 − ε) +� +≤ Aα,d · #∂Ω +κ∂Ω +· |∂F| +κ∂F +· log +�#∂Ω · |∂F| +κ∂Ω ε +�2d(1+α)+1 +. +1.4. One sided estimates. Finally, we remark that bounds on the number of +intermediate eigenvalues, as in Theorems 1.1, 1.2 and 1.3, can be equivalently +formulated in terms of the distribution function +Nε := {n ∈ N : λn > ε}, +ε ∈ (0, 1). +Remark 1.4. For example, for ε ∈ (0, 1) under the assumptions of Theorem 1.1 +we have +��#Nε(S) − |E| · |F| +�� ≤ Cα,d · |∂E| +κ∂E +· |∂F| +κ∂F +· log +� +|∂E||∂F| +κ∂E min{ε, 1 − ε} +�2d(1+α)+1 +. +(1.10) +See Section 8 for details. +1.5. Methods and related literature. We work for the most part with the +discrete Fourier transform and then obtain consequences for the continuous one +by a limiting argument. Theorem 1.2 is proved in two steps. We first revisit Israel’s +argument [8] and adapt it to prove eigenvalue estimates when one of the domains +is a rectangle and the other is a general multi-dimensional domain (Theorem 4.1 +below). These estimates are slightly stronger than those in Theorem 1.2, and + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +5 +the extra precision is exploited in the subsequent step. We follow the method of +almost diagonalization with wavepackets, which we achieve, unlike [8], through a +redundant system (frame) instead of an orthonormal basis. +The second step is a decomposition, rescaling, and dyadic approximation ar- +gument, implemented by means of p-Schatten norm estimates for certain Hankel +operators, and especially by quantifying those estimates as a function of p, as +p → 0+. +Our intermediate result, Theorem 4.1, is close in spirit to Theorem 1.2 in [9] +(which appeared as we were finishing this article). The estimates in [9], formulated +in the context of the continuous Fourier transform and concerning dilated convex +domains, are stronger than what follows from Theorem 4.1 in that regime, as +[9, Theorem 1.2] involves smaller powers of a certain logarithmic factor (see also +Section 1.1 and (1.5)). +On the other hand, Theorem 4.1 concerns sufficiently +regular, non-dilated and possibly non-convex domains, and covers the discrete +Fourier transform. +We also mention our recent work on concentration operators for the short-time +Fourier transform [16], that also makes use of Ahlfors regularity and Schatten +norm estimates. Though the goals and results are philosophically similar to those +in the present article, the settings are rather different from the technical point of +view. Indeed, the arguments used in [16] rely on the rapid off-diagonal decay of +the reproducing kernel of the range of the short-time Fourier transform, and do +not seem to be applicable to Fourier concentration operators. +The remainder of the article is organized as follows. +Section 2 sets up the +notation and provides background on boundary regularity. Section 3 revisits the +technique from [8] and implements certain adaptations. These are used in Section +4 to prove Theorem 4.1. Theorem 1.2 is proved in Section 5, Theorem 1.1 is proved +in Section 6, and Theorem 1.3 is proved in Section 7. Remark 1.4 is proved in +Section 8. +2. Preliminaries +2.1. Notation. We shall focus on Theorem 1.2 and set up the notation accord- +ingly. +Theorems 1.1 and 1.3 will be obtained afterwards as an application of +Theorem 1.2. +We denote cubes by Qa = [−a/2, a/2)d. The Euclidean norm on Rd is denoted +| · |. For two non-negative functions f, g we write f ≲ g if there exist a constant +C such that f(x) ≤ Cg(x), and write f ≍ g is f ≲ g and g ≲ f. The implied +constant is allowed to depend on the dimension d and the parameter α from +Theorems 1.1, 1.2 and 1.3, but not on other parameters. +We enumerate the eigenvalues of a compact self adjoint operator L : H → H +acting on a Hilbert space H as follows: +λk = inf{∥L − S∥ : S ∈ L(L2(Rd)), dim(Range(S)) < k}, +k ≥ 1. +(2.1) +Then {λk : k ≥ 1} ∖ {0} = σ(L) ∖ {0} as sets with multiplicities — see, e.g., [5, +Lemma 4.3]. + +6 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +For a set E ⊆ Rd, we write +Ec +L = L−1Zd ∖ EL +and ∂EL for the points in EL which are at distance L−1 of Ec +L. For L = 1 this is +consistent with (1.9). +We will work with the discrete Fourier transform FL : L2((−L/2, L/2)d) → +ℓ2(L−1Zd) given by (1.6) and reserve the notation Ff or �f for the continuous +Fourier transform (1.4). Note that if supp(f) ⊆ (−L/2, L/2)d, then FLf(k/L) = +Ff(k/L) for every k ∈ Zd. +We also write PE,L = F −1 +L χELFL. For F ⊆ (−L/2, L/2)d we define the operator +T = TE,F,L : L2(F) → L2(F) by +T = TE,F,L = χFPE,L +and let λn = λn(T) denote its eigenvalues as in (2.1). An easy computation shows +that +Tt−1E,tF,tL = Mt−1TE,F,LMt, +t > 0, +where Mt denotes the dilation operator +Mtf(x) = f(tx). +In particular, +λn(Tt−1E,tF,tL) = λn(TE,F,L), +n ∈ N. +(2.2) +2.2. Boundary regularity. Let us introduce regularity of sets in more generality +and discuss a few properties. +An Hd−1-measurable set X ⊆ Rd is said to be lower Ahlfors (d − 1)-regular +(regular for short) at scale ηX > 0 if there exists a constant κX > 0 such that +Hd−1� +X ∩ Br(x) +� +≥ κX · rd−1, +0 < r ≤ ηX, +x ∈ X. +Note that if X ⊆ Rd is regular at scale ηX > 0 with constant κX > 0 and +t > 0, then tX ⊆ Rd is regular at scale ηtX = tηX with constant κtX = κX. By +differentiation around a point of positive Hd−1-density, +κX ≤ cd, +(2.3) +for any regular X of positive Hd−1-measure. We also mention that if X is regular +with parameters ηX and κX, then choosing an arbitrary x ∈ X gives +Hd−1� +X +� +≥ Hd−1� +X ∩ BηX(x) +� +≥ κX · ηd−1 +X . +(2.4) +We shall use the following basic result, derived from [4]. +Lemma 2.1. There exists a universal constant Cd > 0 such that for every compact +set X ⊆ Rd that is regular at scale ηX > 0 with constant κX and every s > 0, +|X + Bs(0)| ≤ Cd +κX +· Hd−1(X) · s · +� +1 + sd−1 +ηd−1 +X +� +. +Proof. From [4, Theorems 5 and 6] it follows that +Hd−1� +{x : d(x, X) = r} +� +≤ Cd +κX +· Hd−1(X) · +� +1 + rd−1 +ηd−1 +X +� +, + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +7 +for almost every r > 0, and in addition, |∇d(x, X)| = 1, for almost every x ∈ Rd. +From this and the coarea formula, it follows that +|X + Bs(0)| = +� +Rd χ[0,s)(d(x, X))dx = +� +Rd χ[0,s)(d(x, X))|∇d(x, X)|dx += +� s +0 +Hd−1� +{x : d(x, X) = r} +� +dr ≤ Cd +κX +Hd−1(X) +� s +0 +� +1 + rd−1 +ηd−1 +X +� +dr +≤ Cd +κX +Hd−1(X)s +� +1 + sd−1 +ηd−1 +X +� +. +□ +Corollary 2.2. For E ⊆ Rd a compact domain with regular boundary at scale +η∂E ≥ 1 with constant κ∂E and a discretization resolution L ≥ 1, we have +L−d#EL ≲ |E| + |∂E| +κ∂EL. +In particular, for d ≥ 2 +L−d#EL ≲ max{|∂E|d/(d−1), 1} +κ∂E +. +Proof. Recall that QL−1 = L−1[−1/2, 1/2)d and define E′ +L = {m ∈ EL : +m + +QL−1 ⊆ E}. From Lemma 2.1, we get +L−d#EL = +��� +� +m∈E′ +L +m + QL−1 +��� + +��� +� +m∈EL∖E′ +L +m + QL−1 +��� ≤ |E| + |∂E + BL−1√ +d(0)| +≲ |E| + |∂E| +κ∂EL. +Finally, the second inequality follows from the isoperimetric inequality |E| ≲ +|∂E|d/(d−1) and (2.3). +□ +3. Israel’s argument revisited +We now revisit the core argument of [8] and make a few adaptations. +3.1. Israel’s lemma. We need a slight generalization of Lemma 1 in [8], phrasing +it in terms of frames rather than orthonormal bases. We include a proof for the +sake of completeness. +Recall that a frame for a Hilbert space H is a subset of vectors {φi}i∈I for which +the exist constants 0 < A, B < ∞ — called lower and upper frame bounds — +such that +A∥f∥2 ≤ +� +i∈I +|⟨f, φi⟩|2 ≤ B∥f∥2, +f ∈ H. +Lemma 3.1. Let T : H → H be a positive, compact, self-adjoint operator on a +Hilbert space H with ∥T∥ ≤ 1 and eigendecomposition T = � +n≥1 λn⟨·, fn⟩fn. +Let {φi}i∈I be a frame of unit norm vectors for H with lower frame bound A. +If I = I1 ∪ I2 ∪ I3, and +(3.1) +� +i∈I1 +∥Tφi∥2 + +� +i∈I3 +∥(I − T)φi∥2 ≤ A +2 ε2, + +8 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +then #Mε(T) ≤ 2 +A#I2, where Mε(T) is defined as in (1.2). +Proof. Let Sε = span{fn : +λn ∈ (ε, 1 − ε)} and let Pε : H → Sε denote the +orthogonal projection onto Sε. For f ∈ Sε one has ∥f∥ − ∥(I − T)f∥ ≤ ∥Tf∥ ≤ +(1 − ε)∥f∥, which shows +ε∥f∥ ≤ ∥Tf∥, +and +ε∥f∥ ≤ ∥(I − T)f∥, +f ∈ Sε. +Note that T and Pε commute since Sε is spanned by a collection of eigenvectors +of T. Therefore, by (3.1) we obtain +� +i∈I1∪I3 +ε2∥Pεφi∥2 ≤ +� +i∈I1 +∥TPεφi∥2 + +� +i∈I3 +∥(I − T)Pεφi∥2 ≤ A +2 ε2, +which implies +(3.2) +� +i∈I1∪I3 +∥Pεφi∥2 ≤ A +2 . +Using the frame property we get for f ∈ Sε: +A∥f∥2 ≤ +� +i∈I +|⟨f, φi⟩|2 = +� +i∈I +|⟨f, Pεφi⟩|2. +Now assume that dim(Sε) ≥ 1 (otherwise the result is trivial), take an orthonormal +basis {ψk}dim(Sε) +k=1 +of Sε, and sum the inequality above over all basis elements to +derive +A · #Mε(T) = A · dim(Sε) = A +dim(Sε) +� +k=1 +∥ψk∥2 ≤ +dim(Sε) +� +k=1 +� +i∈I +|⟨ψk, φi⟩|2 += +� +i∈I +∥Pεφi∥2 ≤ +� +i∈I2 +∥φi∥2 + A +2 = #I2 + A +2 ≤ #I2 + A +2 #Mε(T), +where in the second line we used (3.2). This shows that #Mε(T) ≤ 2 +A#I2. +□ +3.2. Local trigonometric frames. In this section, we construct a tight frame +that allows us to apply Lemma 3.1. +Let α > 0, and θ ∈ C∞(R) be such that +(i) θ(x) = 1, for x ≥ 1, and θ(x) = 0, for x ≤ −1, +(ii) θ(−x)2 + θ(x)2 = 1, for every x ∈ R, +(iii) |Dkθ(x)| ≤ Ck +αk(1+α)k, for all k ∈ N0, all x ∈ R, and a constant Cα > 0. +See, for example, [8, Proposition 1] or [6, Chapter 1] for the existence of such a +function. +Let W > 0. We decompose the interval +� +− W +2 , W +2 +� +into disjoint intervals +Ij = xj + +W +3 · 2|j|+1[−1, 1), +j ∈ Z, +where +xj = sign(j)W +2 +� +1 − 1 +2|j| +� +. + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +9 +Note that |Ij| = |I|j|| = 2|I|j|+1| for every j ∈ Z. We will also denote Dj = Ij∪Ij+1. +Now define +θj(x) = θ +�2(x − xj) +|Ij| +� +θ +� +−2(x − xj+1) +|Ij+1| +� +. +We have that θj(x) = 0 for x /∈ Dj, and furthermore by properties (i) and (ii) +∥θj∥2 +2 = +� +Ij +θ +�2(x − xj) +|Ij| +�2 +dx + +� +Ij+1 +θ +� +−2(x − xj+1) +|Ij+1| +�2 +dx += |Ij| +2 +� 1 +−1 +θ(x)2dx + |Ij+1| +2 +� 1 +−1 +θ(x)2dx = |Dj| +2 . +We define the set of vectors +(3.3) +φj,k(x) = +� +2 +|Dj| · θj(x) · exp +� +2πi xk +|Dj| +� +, +j, k ∈ Z, +and note that ∥φj,k∥2 = 1. +Lemma 3.2. The family {φj,k}j,k∈Z defined in (3.3) forms a tight frame for +L2(−W/2, W/2) with frame constants A = B = 2. +Proof. We write fj := f|Ij so that f = � +j∈Z fj. Since supp(θj) ⊆ Dj = Ij ∪ Ij+1, +we observe +� +j,k∈Z +|⟨f, φj,k⟩|2 = +� +j,k∈Z +|⟨fj + fj+1, φj,k⟩|2 . +As +� +|Dj|−1/2 exp (2πikx/|Dj|) +� +k∈Z is an orthonormal basis for L2(Dj), we find +that +� +k∈Z +|⟨fj + fj+1, φj,k⟩|2 = 2∥(fj + fj+1)θj∥2 +2 = 2∥fjθj∥2 +2 + 2∥fj+1θj∥2 +2. +Combining both identities and using property (ii), we conclude +� +j,k∈Z +|⟨f, φj,k⟩|2 = 2 +� +j∈Z +� +∥fjθj∥2 +2 + ∥fj+1θj∥2 +2 +� += 2 +� +j∈Z +� +∥fjθj∥2 +2 + ∥fjθj−1∥2 +2 +� += 2 +� +j∈Z +� +Ij +|f(x)|2 +� +θ +�2(x − xj) +|Ij| +�2 ++ θ +� +−2(x − xj) +|Ij| +�2� +dx += 2 +� +j∈Z +∥fj∥2 +2 = 2∥f∥2 +2. +□ +Let 0 < Wi ≤ L, i = 1, ..., d, and consider �d +i=1(−Wi/2, Wi/2). Set also +Wmax := max +i=1,...,dWi. +We define a frame for L2� �d +i=1(−Wi/2, Wi/2) +� +via the tensor product +Φj,k(x) = Φj1,...,jd,k1,...kd(x1, ..., xd) = φj1,k1(x1) · . . . · φjd,kd(xd), +where each family {φji,ki(xi)}ji,kk∈Z is the frame for L2(−Wi/2, Wi/2) given by +(3.3). This construction also yields a tight frame with frame bounds equal to 2d. + +10 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +3.3. Energy estimates. Consider +ψj(x) = θj +� +|Dj|x + xj − +W +3 · 2|j|+1 +� +, +x ∈ R, j ∈ Z. +A straightforward computation shows that ψj is supported on [0, 1] and satisfies +|Dkψj(x)| ≤ � +Cα +kk(1+α)k by property (iii). As shown in [8, Lemma 1] it thus follows +that |� +ψj(ξ)| ≤ Aα · exp +� +−aα|ξ|(1+α)−1� +. Since 1 − α ≤ (1 + α)−1, we derive that +t(1+α)−1 ≥ t1−α − 1 for t ≥ 0. Adjusting the constant Aα, we therefore get +(3.4) +|�θj(ξ)| ≤ Aα · |Dj| · exp +� +−aα(|Dj| · |ξ|)1−α� +, +ξ ∈ R. +With this at hand, we estimate the decay of +F(Φj,k)(ξ) = 2d/2 +d +� +i=1 +|Dji|−1/2 · � +θji +� +ξi − +ki +|Dji| +� +, +ξ ∈ Rd. +Define +Mj = diag(|Dj1|, ..., |Djd|) ∈ Rd×d. +By (3.4) (possibly enlarging Aα) and |ξ|1−α ≤ �d +i=1 |ξi|1−α, it follows +|F(Φj,k)(ξ)| ≤ Ad +α +d +� +i=1 +|Dji|1/2 · exp +� +−aα +��|Dji|ξi − ki +��1−α� +≤ Ad +α · det(Mj)1/2 · exp +� +−aα +��Mj(ξ − ξj,k) +��1−α� +, +(3.5) +where (ξj,k)i = ki|Dji|−1. +Consider now a compact domain E ⊆ Rd. Let s ≥ 1 be a parameter that will be +determined later. For j ∈ Zd fixed, we cover the index set Zd with three subsets +as follows +Llow +j +:= +� +k ∈ Zd : dist(k, MjEc +L) ≥ s +� +; +Lmed +j +:= +� +k ∈ Zd : dist(k, MjEL) < s, and dist(k, MjEc +L) < s}; +Lhigh +j +:= +� +k ∈ Zd : dist(k, MjEL) ≥ s +� +. +(3.6) +(Here, dist is associated with the usual Euclidean distance.) We claim that +Lmed +j +⊆ +� +k ∈ Zd : dist(k, Mj∂EL) < s +� +; +(3.7) +let us briefly sketch an argument. Fix k ∈ Lmed +j +and let k0 ∈ MjL−1Zd minimize +the distance to k. For any point x ∈ MjL−1Zd with |k − x| < s we can build a +path of adjacent points in MjL−1Zd from x to k0 such that the distance to k is +decreasing. In particular, choosing x1 ∈ MjEL and x2 ∈ MjEc +L at distance less +than s from k, we can connect x1 and x2 through a path of adjacent points in +MjL−1Zd that stays at distance less than s from k. Necessarily, one of the points +in the path must belong to Mj∂EL, which proves (3.7). +The indices (j, k) with k ∈ Lmed +j +, and j satisfying a condition specified below +(see (3.10)) will play the role of I2 in Lemma 3.1, so we need to estimate #(Lmed +j +). + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +11 +Lemma 3.3. Let E ⊆ Rd be a compact domain with regular boundary at scale +η∂E ≥ 1 and constant κ∂E. Let L ≥ Wmax and s ≥ 1. Then for all j ∈ Zd we have +# +� +k ∈ Zd : dist(k, Mj∂EL) < s +� +≲ max{Wmax, 1/η∂E}d−1 · |∂E| +κ∂E +· sd. +Proof. Since for every x ∈ Rd the cube x + Q1 contains one point in Zd, +# +� +k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · # +� +k ∈ Zd : k ∈ Mj∂EL + Q1}. +(3.8) +For x ∈ R, we denote the unique integer in x + [−1/2, 1/2) by x∗. For a set +X = {x1, ..., xN} ⊆ R and 0 < a ≤ 1, we write yi = axi. It is easy to check that if +x∗ +i = x∗ +i′, then |y∗ +i − y∗ +i′| ≤ 1. Using this, a straightforward argument shows that +# +� +k ∈ Z : k ∈ aX + [−1/2, 1/2)} ≤ 2# +� +k ∈ Z : k ∈ X + [−1/2, 1/2)}. +From (3.8), if we apply the inequality above componentwise (noting that (Mj)i,i ≤ +Wmax), we obtain for s ≥ 1 +# +� +k ∈ Zd : dist(k, Mj∂EL) < s} +≲ sd · # +� +k ∈ Zd : k ∈ Wmax∂EL + Q1}. +Since for every x ∈ ∂EL there exists x′ ∈ ∂E such that |x − x′| ≤ L−1, and +Wmax/L ≤ 1, it follows that +# +� +k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · # +� +k ∈ Zd : k ∈ Wmax∂E + Q3} +=: sd · #(KWmax). +(3.9) +Now let k ∈ KWmax. +There exists at least one point xk ∈ ∂E such that k ∈ +Wmaxxk + Q3. In particular, we have that xk ∈ W −1 +maxk + Q4/Wmax. Therefore, for +every k ∈ KWmax we get by regularity of ∂E +κ∂E · min{W −1 +max, η∂E}d−1 ≤ Hd−1� +∂E ∩ B1/Wmax(xk) +� +≤ Hd−1� +∂E ∩ xk + Q2/Wmax +� +≤ Hd−1� +∂E ∩ W −1 +maxk + Q6/Wmax +� +. +So, +κ∂E · min +� +W −1 +max, η∂E +�d−1 · #(KWmax) ≤ +� +k∈KWmax +Hd−1� +∂E ∩ W −1 +maxk + Q6/Wmax +� +≲ +� +k∈Zd +Hd−1� +∂E ∩ W −1 +maxk + Q1/Wmax +� += Hd−1(∂E). +Plugging this estimate into (3.9) completes the proof. +□ +Next, for a compact domain E ⊆ Rd and a parameter s ≥ 1 we recall the sets +(3.6), introduce a second auxiliary parameter 0 < δ < 1, and define the following +covering of Z2d: +Γlow := +� +(j, k) : min +i +|Dji| ≥ δ, k ∈ Llow +j +� +; +Γmed := +� +(j, k) : min +i +|Dji| ≥ δ, k ∈ Lmed +j +� +; +Γhigh := +� +(j, k) : min +i +|Dji| ≥ δ, k ∈ Lhigh +j +� +∪ +� +(j, k) : min +i +|Dji| < δ, k ∈ Zd� +. +(3.10) + +12 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +Lemma 3.4. Under the conditions of Lemma 3.3, let 0 < δ < 1 and consider the +set Γmed from (3.10). Then +#(Γmed) ≲ max{Wmax, 1/η∂E}d−1 · |∂E| +κ∂E +· max{log(Wmax/δ), 1}d · sd. +Proof. By (3.7) and Lemma 3.3 it follows +#(Γmed) = +� +j∈Zd +min |Dji|≥δ +#(Lmed +j +) ≲ +� +j∈Zd +min |Dji|≥δ +max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +sd. +In each coordinate, we have that the number of intervals Dji for which |Dji| ≥ δ +is bounded by C max{log(Wmax/δ), 1}. Hence, we arrive at +#(Γmed) ≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +max{log(Wmax/δ), 1}dsd. +□ +Lemma 3.5. Let d ≥ 2, L ≥ max{Wmax, 1}, and E ⊆ Rd be a compact domain +with regular boundary at scale η∂E ≥ 1 with constant κ∂E and such that |∂E| ≥ 1. +Let s ≥ 1 and δ ∈ (0, 1) be parameters and consider the sets from (3.10). Then +there exists a constant c = cα > 0 such that +� +(j,k)∈Γlow +L−d � +m∈Ec +L +|F(Φj,k)(m)|2 +≲ max{Wmax, 1/η∂E}d−1 · |∂E| +κ∂E +· exp +� +− cs1−α� +max{log(Wmax/δ), 1}d, +(3.11) +and +� +(j,k)∈Γhigh +L−d � +m∈EL +|F(Φj,k)(m)|2 ≲max{Wmax, 1/η∂E}d−1 +κ∂E +· +� +|∂E| +d +d−1 · δ ++ |∂E| · exp +� +− cs1−α� +· max{log(Wmax/δ), 1}d� +. +(3.12) +Proof. For j ∈ Zd and l ∈ N0 we set +Llow +j,l = +� +k ∈ Zd : dist(k, MjEc +L) ∈ [s2l, s2l+1) +� +, +and +Lhigh +j,l += +� +k ∈ Zd : dist(k, MjEL) ∈ [s2l, s2l+1) +� +. +Notice that +Llow +j,l ∪ Lhigh +j,l +⊆ +� +k ∈ Zd : dist(k, MjEL) < s2l+1, and dist(k, MjEc +L) < s2l+1} +⊆ +� +k ∈ Zd : dist(k, Mj∂EL) < s2l+1}, +where the last step follows as in (3.7). From Lemma 3.3 we get +(3.13) +#(Llow +j,l ), #(Lhigh +j,l ) ≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +sd2dl. +From (3.5) it follows that if k ∈ Llow +j,l +L−d � +m∈Ec +L +|F(Φj,k)(m)|2 ≤ L−d � +m∈Ec +L +A2d +α det(Mj) exp +� +− 2aα|Mj(m − ξj,k)|1−α� + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +13 +≤ A2d +α L−d det(Mj) +� +m′∈MjEc +L +exp +� +− 2aα|m′ − k|1−α� +≲ +� +{|x|≥s2l} +exp +� +− 2aα|x|1−α� +dx +≲ exp +� +− c(s2l)1−α� +, +(3.14) +where c can for example be chosen as aα. A similar argument also shows that for +k ∈ Lhigh +j,l , +L−d � +m∈EL +|F(Φj,k)(m)|2 ≲ exp +� +− c(s2l)1−α� +. +As Llow +j += � +l∈N0 Llow +j,l , it follows from (3.13) and (3.14) that +� +(j,k)∈Γlow +L−d � +m∈Ec +L +|F(Φj,k)(m)|2 = +� +j∈Zd +min |Dji|≥δ +� +l∈N0 +� +k∈Llow +j,l +L−d � +m∈Ec +L +|F(Φj,k)(m)|2 +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +� +j∈Zd +min |Dji|≥δ +� +l∈N0 +(s2l)d exp +� +− c(s2l)1−α� +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +� +j∈Zd +min |Dji|≥δ +exp +� +− c′s1−α� +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +exp +� +− c′s1−α� +max{log(Wmax/δ), 1}d, +which completes the proof of (3.11). Again, we can use an analogous reasoning +to show +� +j∈Zd +min |Dji|≥δ +� +k∈Lhigh +j +L−d � +m∈EL +|F(Φj,k)(m)|2 +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +exp +� +− c′s1−α� +max{log(Wmax/δ), 1}d. +Now suppose that j ∈ Zd is such that min1≤i≤d |Dji| < δ. For every m ∈ Zd we +can uniformly bound the subsequent series +� +k∈Zd +exp +� +− 2aα|Mj(m − ξj,k)|1−α� += +� +k∈Zd +exp +� +− 2aα|Mjm − k|1−α� +≤ C. +Since det(Mj) = |Dj|, where Dj = Dj1 × ... × Djd, we thus get by (3.5) +� +j∈Zd +min |Dji|<δ +� +k∈Zd +L−d � +m∈EL +|F(Φj,k)(m)|2 ≤ C +� +j∈Zd +min |Dji|<δ +L−d � +m∈EL +det(Mj) +≤ CL−d#EL +� +j∈Zd +min |Dji|<δ +|Dj| + +14 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +≲ |∂E|d/(d−1) +κ∂E +� +j∈Zd +min |Dji|<δ +|Dj|, +where in the last inequality we used Corollary 2.2. Finally, +� +j∈Zd +min |Dji|<δ +|Dj| ≤ +d +� +i=1 +� +j∈Zd +|Dji|<δ +|Dj| ≤ +d +� +i=1 +W d−1 +max4δ +≲ max{Wmax, 1/η∂E}d−1δ, +where we used that each interval Dji is at most at |Dji| < δ distance from the +boundary of (−Wi/2, Wi/2). This concludes the proof of (3.12). +□ +4. General domain vs. rectangle +In this section, we prove the following variant of Theorem 1.2 for F a rectangle. +Theorem 4.1. Let L ≥ 1 be a discretization resolution, d ≥ 2, and E ⊆ Rd be +a compact domain with regular boundary at scale η∂E ≥ 1 with constant κ∂E and +such that |∂E| ≥ 1. For 0 < Wi ≤ L, i = 1, ..., d, take F = �d +i=1(−Wi/2, Wi/2) +and denote Wmax = maxi Wi. For every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such +that for ε ∈ (0, 1/2): +# +� +n ∈ N : λn ∈ (ε, 1 − ε) +� +≤ Aα,d · max{Wmax, 1/η∂E}d−1 · |∂E| +κ∂E +· log +�max{Wmax, 1/η∂E}d−1|∂E| +κ∂E ε +�2d(1+α) +. +Proof. We adopt all the notation of Section 3. Fix parameters s ≥ 1, δ ∈ (0, 1) +and consider the sets from (3.10). +Observe that for f ∈ L2(F) one has +∥Tf∥2 +2 = ∥χFPE,Lf∥2 +2 ≤ ∥PE,Lf∥2 +2 = L−d � +m∈EL +�� �f(m) +��2. +and +∥f − Tf∥2 +2 = ∥χFf − χFPE,Lf∥2 +2 ≤ ∥(I − PE,L)f∥2 +2 = L−d � +m∈Ec +L +�� �f(m) +��2. +By Lemma 3.5 it thus follows +� +(j,k)∈Γlow +∥(I − T)Φj,k∥2 +2 + +� +(j,k)∈Γhigh +∥TΦj,k∥2 +2 +(4.1) +≤ C max{Wmax, 1/η∂E}d−1 +κ∂E +� +|∂E| exp +� +− cs1−α� +max{log(Wmax/δ), 1}d ++ |∂E|d/(d−1)δ +� +, +where the constants depend only on α and d. + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +15 +At last, we can now specify the parameters δ and s in order for the sets Γlow, Γmed +and Γhigh to play the role of I1, I2 and I3 in Lemma 3.1. We take +δ = +κ∂E ε2 +C max{Wmax, 1/η∂E}d−1|∂E|d/(d−1) . +This ensures that +C max{Wmax, 1/η∂E}d−1|∂E|d/(d−1) +κ∂E +δ ≤ ε2. +(4.2) +Also, we select s such that +C max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +exp +� +− cs1−α� +max{log(Wmax/δ), 1}d ≤ ε2. +(4.3) +This condition on s is equivalent to +s ≥ +�1 +c log +�C max{Wmax, 1/η∂E}d−1|∂E| max{log(Wmax/δ), 1}d +κ∂E ε2 +��1/(1−α) +, +and is satisfied if +s = Aα,d log +�max{Wmax, 1/η∂E}d−1|∂E| +κ∂E ε +�1/(1−α) +, +for an adequate constant Aα,d. Moreover, we can guarantee that s ≥ 1, since by +(2.4), the term inside the logarithm is ≥ 2. From (4.1), (4.2), (4.3), Lemma 3.1 +and Lemma 3.4, +#Mε(T) ≤ 21−d#(Γmed) +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +max{log(Wmax/δ), 1}dsd +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +log +�max{Wmax, 1/η∂E}d−1|∂E| +κ∂E ε +�d/(1−α)+d +≲ max{Wmax, 1/η∂E}d−1|∂E| +κ∂E +log +�max{Wmax, 1/η∂E}d−1|∂E| +κ∂E ε +�2d(1+α) +. □ +5. Eigenvalue estimates for two domains +5.1. Schatten quasi-norm estimates. For 0 < p ≤ 1, and ε > 0, define the +auxiliary function g = gp,ε : [0, 1] → R given by +g(t) = +� t − t2 +ε − ε2 +�p +. +Note that since χ(ε,1−ε) ≤ g, for a positive operator 0 ≤ S ≤ 1, +#Mε(S) = tr(χ(ε,1−ε)S) ≤ tr(g(S)) = ∥S − S2∥p +p +(ε − ε2)p , +where ∥ · ∥p, 0 < p ≤ 1, denotes the Schatten quasi-norm. The next lemma shows +that upper bounds for the left-hand side of the last inequality can be transferred +to the right-hand side without much loss. + +16 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +Lemma 5.1. Suppose that for a positive operator 0 ≤ S ≤ 1 there are constants +C, D, a > 0 such that for every ε ∈ (0, 1/2), +#Mε(S) ≤ C +� +D + log(ε−1) +�a. +Then, for every 0 < p ≤ 1 there is a constant Ca > 0 such that +∥S − S2∥p +p ≤ CaC +� +D + p−1�a. +Proof. By the symmetry of the function h(x) = x − x2 around 1/2, for 0 ≤ x ≤ 1, +h(x)p = +� min{x,1−x} +0 +(hp)′(t)dt = +� 1/2 +0 +χ(t,1−t)(x)php−1(t)h′(t)dt +≤ +� 1/2 +0 +χ(t,1−t)(x)ptp−1dt. +By a monotone convergence argument we get +∥S − S2∥p +p ≤ +� 1/2 +0 +Mt(S)ptp−1dt ≤ C +� 1 +0 +(D + log(t−1))aptp−1dt += C +� ∞ +0 +(D + u/p)ae−udu +≤ C(D + 1/p)a + C +� ∞ +1 +(D + u/p)ae−udu +≤ C(D + 1/p)a + C(D + 1/p)a +� ∞ +1 +uae−udu +≤ (1 + Γ(a + 1))C(D + 1/p)a. +□ +5.2. Decomposition of the domain and Hankel operators. In what follows, +we let F ⊆ (−L/2, L/2)d be a compact domain with regular boundary at scale +η∂F = |∂F|1/(d−1) ≥ 1 with constant κ∂F. We construct two auxiliary sets F − ⊆ +F ⊆ F + which will be dyadic approximations of F from above and below by cubes +of length at least 1. More precisely, let +F = +� +k∈Z +� +j∈Jk +Qk,j +be a dyadic decomposition of F in piecewise disjoint cubes of the form Qk,j = +Q2k + 2kj with k ∈ Z and j ∈ Jk ⊆ Zd, that are maximal (they are not contained +in a larger dyadic cube included in F). We define +F − = +� +k≥0 +� +j∈Jk +Qk,j. +For F + we add cubes of length 1 to fully cover F and intersect them with +(−L/2, L/2)d. The result is a covering of F that combines the cubes from F − +with rectangles of maximal side-length ≤ 1. More precisely, define +V = {v ∈ Zd : (F ∖ F −) ∩ (Q1 + v) ̸= ∅}, +and +F + = F − ∪ +� +v∈V +� +(Q1 + v) ∩ (−L/2, L/2)d� +=: F − ∪ +� +v∈V +Rv. + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +17 +Note that Theorem 4.1 can be applied to each rectangle in the decomposition +of F − and F +. This follows from a translation argument and the fact that the +boundaries of the rectangles have null measure, so we can replace them by their +interior. We write T ± for TE,F ±,L. +For a set K ⊆ (−L/2, L/2)d define the Hankel operator on L2((−L/2, L/2)d) +by +HK = (I − PE,L)χKPE,L +and write H± = HF ±. Note that +(HK)∗HK = PE,LχKPE,L − PE,LχKPE,LχKPE,L = PE,LχKPE,L − (PE,LχKPE,L)2. +Since PE,LχKPE,L and TK share the same non-zero eigenvalues, for p > 0, +∥TK − (TK)2∥p +p = ∥HK∥2p +2p. +Recall that for two operators S1, S2 in the p-Schatten class, 0 < p ≤ 1, one has +∥S1 + S2∥p +p ≤ ∥S1∥p +p + ∥S2∥p +p. +Lemma 5.2. Let L ≥ 1, d ≥ 2, and E, F ⊆ Rd be compact domains with regular +boundaries at scales η∂E ≥ 1, η∂F = |∂F|1/(d−1) ≥ 1, with constants κ∂E, κ∂F +respectively. Assume also that |∂E| ≥ 1 and F ⊆ (−L/2, L/2)d. For ε ∈ (0, 1/2), +we have +#Mε(T ±) ≲ |∂E| +κ∂E +· |∂F| +κ∂F +· log +�|∂E| max{|∂F|, 1} +κ∂E ε +�2d(1+α)+1 +. +Proof. If k ∈ Z is such that Jk ̸= ∅, then there is a cube of length 2k included in +F. In particular, the projection of ∂F onto the hyperplane {x1 = 0} contains a +(d−1)-dimensional cube of length 2k and therefore 2k(d−1) ≤ |∂F|. The maximality +of the dyadic decomposition of F implies that Qj,k ⊆ ∂F + B√ +d2k+1(0) for j ∈ Jk. +From Lemma 2.1 and the fact that η∂F = |∂F|1/(d−1), we thus derive +2dk#Jk ≤ |∂F + B√ +d2k+1(0)| ≲ 2k |∂F| +κ∂F +� +1 + 2k(d−1) +|∂F| +� +≲ 2k |∂F| +κ∂F +. +(5.1) +Similarly, +#V ≤ |∂F + B√ +d(0)| ≲ |∂F| +κ∂F +. +(5.2) +For 0 < 2p ≤ 1, and ε ∈ (0, 1/2), we thus get +#Mε(T +) ≤ ∥T + − (T +)2∥p +p +(ε − ε2)p += ∥H+∥2p +2p +(ε − ε2)p +≤ (2/ε)p � +k≥0 +� +j∈Jk +∥HQk,j∥2p +2p + (2/ε)p � +v∈V +∥HRv∥2p +2p +≲ ε−p � +k≥0 +� +j∈Jk +∥TQk,j − T 2 +Qk,j∥p +p + ε−p � +v∈V +∥TRv − T 2 +Rv∥p +p. +Theorem 4.1 shows that when applying Lemma 5.1 to TQk,j one can take +C ≲ 2k(d−1)|∂E| +κ∂E +, +and +D = log +� +2k(d−1)|∂E| +κ∂E +� +. + +18 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +Similarly, the same holds for TRv with k = 1. Choosing p = log(2) +� +2 log(ε−1) +�−1 +(which ensures that 2p ≤ 1 for every ε ∈ (0, 1/2)) thus yields +#Mε(T +) ≲ |∂E| +κ∂E +�� +k≥0 +� +j∈Jk +2k(d−1)log +�|∂E|2k(d−1) +κ∂E ε +�2d(1+α) ++#V ·log +� |∂E| +κ∂E ε +�2d(1+α)� +≲ |∂E| +κ∂E +|∂F| +κ∂F +� +k≥0 +2k(d−1)≤max{|∂F |,1} +log +�|∂E|2k(d−1) +κ∂E ε +�2d(1+α) += |∂E| +κ∂E +|∂F| +κ∂F +� +0≤k≤⌊ +log(max{|∂F |,1}) +log(2)(d−1) +⌋ +� +log +� |∂E| +κ∂E ε +� ++ (d − 1) log(2)k +�2d(1+α) +, +where in the second to last step we used (5.1), (5.2), and the fact that 2k(d−1) ≤ +|∂F| whenever Jk ̸= ∅. Finally, noting that for C, D, a > 0, +N +� +k=0 +(C + Dk)a ≤ +� N+1 +0 +(C + Dx)adx ≤ (C + D(N + 1))a+1 +D(a + 1) +, +we get, +#Mε(T +) ≲ |∂E| +κ∂E +|∂F| +κ∂F +log +�|∂E| max{|∂F|, 1} +κ∂E ε +�2d(1+α)+1 +. +The same argument works for #Mε(T −). +□ +5.3. The transition index. The following estimate is part of the proof of [1, +Theorem 1.5] (see also [16, Lemma 4.3]) and allows us to find the index where +eigenvalues cross the 1/2 threshold. We include a proof for the sake of complete- +ness. +Lemma 5.3. For any trace class operator 0 ≤ S ≤ 1, +(i) λn ≤ 1 +2, for every n ≥ ⌈tr(S)⌉ + max{2 tr(S − S2), 1}; +(ii) λn ≥ 1 +2, for every 1 ≤ n ≤ ⌈tr(S)⌉ − max{2 tr(S − S2), 1}. +Proof. First notice that if S is an orthogonal projection, then the result holds +trivially, so we can assume otherwise. In particular, we have that tr(S − S2) > 0. +Set K = ⌈tr(S)⌉ and write +tr(S) − tr(S2) = +∞ +� +n=1 +λn(1 − λn) += +K +� +n=1 +λn(1 − λn) + +∞ +� +n=K+1 +λn(1 − λn) +≥ λK +K +� +n=1 +(1 − λn) + (1 − λK) +∞ +� +n=K+1 +λn + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +19 += λKK − λK +K +� +n=1 +λn + (1 − λK) +� +tr(S) − +K +� +n=1 +λn +� += λKK + (1 − λK) tr(S) − +K +� +n=1 +λn += tr(S) − +K +� +n=1 +λn + λK(K − tr(S)). +Hence +(5.3) +∞ +� +n=K+1 +λn = tr(S) − +K +� +n=1 +λn ≤ tr(S) − tr(S2), +and +K−1 +� +n=1 +(1 − λn) = tr(S) − +K +� +n=1 +λn + λK(K − tr(S)) − (1 − λK)(1 + tr(S) − K) +≤ tr(S) − +K +� +n=1 +λn + λK(K − tr(S)) ≤ tr(S) − tr(S2). +(5.4) +Now let j ∈ N such that j ≥ 2(tr(S) − tr(S2)) and consider k = K + j . It follows +from (5.3) that +2(tr(S) − tr(S2)) · λk ≤ j · λK+j ≤ +∞ +� +n=K+1 +λn ≤ tr(S) − tr(S2), +which shows λk ≤ 1/2 as 0 < tr(S) − tr(S2) < ∞; this proves part (i). +For part (ii), if 1 ≤ k = K − j ≤ K − 2(tr(S) − tr(S2)) for j ∈ N, then (5.4) +implies +2(tr(S) − tr(S2)) · (1 − λk) ≤ j · (1 − λK−j) ≤ +K−1 +� +n=1 +(1 − λn) ≤ tr(S) − tr(S2), +yielding λk ≥ 1/2. This completes the proof. +□ +5.4. Proof of the main result. With all the preparatory work at hand, we are +ready to prove the main result. +Proof of Theorem 1.2. Recall from (2.2) that the eigenvalues of the concentration +operator remain the same if we replace E, F and L with t−1E, tF and tL respec- +tively. +We choose t = |∂E|1/(d−1) and notice that t−1E satisfies |∂t−1E| = 1, +η∂t−1E = t−1η∂E = 1, and κ∂t−1E = κ∂E. +Furthermore, we also have that tF +has regular boundary at scale η∂tF = tη∂F = (|∂E||∂F|)1/(d−1) ≥ 1 with constant +κ∂tF = κ∂F, and tL ≥ 1 by assumption on L. +Note that for F ′ ⊆ (−tL/2, tL/2)d, the operator T has integral kernel +K(x, y) = χF ′(x)χF ′(y) +1 +(tL)d +� +k∈(t−1E)tL +e−2πik(x−y). + +20 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +Thus, +tr(T) = +� +K(x, x)dx = +� +F ′ +1 +(tL)d +� +k∈(t−1E)tL +1dx = #(t−1E)tL +(tL)d +|F ′|. +On the other hand, from Lemmas 5.1 and 5.2 we have that +tr +� +T ± − (T ±)2� +≲ |∂t−1E| +κ∂t−1E +|∂tF| +κ∂tF +log +�e|∂t−1E||∂tF| +κ∂t−1E +�2d(1+α)+1 += |∂E| +κ∂E +|∂F| +κ∂F +log +�e|∂E||∂F| +κ∂E +�2d(1+α)+1 +=: CE,F. +So from Lemma 5.3, +λn(T +) ≤ 1 +2, +n ≥ +�#(t−1E)tL +(tL)d +|(tF)+| +� ++ 2CE,F; +λn(T −) ≥ 1 +2, +n ≤ +�#(t−1E)tL +(tL)d +|(tF)−| +� +− 2CE,F. +By Corollary 2.2 and |∂t−1E| = 1, +#{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} ≲ +1 +κ∂E +|(tF)+ ∖ (tF)−| + CE,F +≤ +1 +κ∂E +|∂tF + B√ +d(0)| + CE,F +≲ +1 +κ∂E +td−1|∂F| +κ∂F ++ CE,F ≲ CE,F, +where in the second to last step we used Lemma 2.1. Since λn(T −) ≤ λn(T) ≤ +λn(T +) for every n ∈ N, again by Lemma 5.2, +#Mε(T) ≤#{n ∈ N : 1/2 ≤ λn(T −) < 1 − ε} + #{n ∈ N : ε < λn(T +) ≤ 1/2} ++ #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} +≲#Mε(T −) + #Mε(T +) + #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} +≲|∂E| +κ∂E +|∂F| +κ∂F +log +�|∂E||∂F| +κ∂E ε +�2d(1+α)+1 +. +□ +6. The continuous Fourier transform +In this section we deduce Theorem 1.1 by taking L → ∞ in Theorem 1.2. +Proof of Theorem 1.1. Fix E and F as in the statement of Theorem 1.1. +We +consider a sufficiently large resolution such that L ≥ |∂E|−1/(d−1) and F ⊆ +(−L/2, L/2)d. +Let SL : L2(Rd) → L2(Rd) be the operator given by +SLf = TE,F,L(χFf) = χFF −1 +L χELFLχFf, +f ∈ L2(Rd). +An easy computation shows that SL and TE,F,L share the same non-zero eigenval- +ues. Also, recall the operator S from (1.1). + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +21 +Step 1. We show that +lim +L→∞ ∥SL − S∥ = 0. +(6.1) +Recall that QL−1 = L−1[−1/2, 1/2)d and define the auxiliary set +ΓL = +� +m∈EL +m + QL−1. +Note that the symmetric difference E∆ΓL is included in ∂E + BL−1√ +d(0). From +Lemma 2.1, +|E∆ΓL| ≤ |∂E + BL−1√ +d(0)| ≲ |∂E| +κL +� +1 + (Lη∂E)−(d−1)� L→∞ +−−−→ 0. +Using this and setting RL = χFF −1χΓLFχF, for f ∈ L2(Rd) we have +∥(RL − S)f∥2 +2 ≤ ∥(χΓL − χE)F(χFf)∥2 +2 ≤ |E∆ΓL|∥F(χFf)∥2 +∞ +≤ |E∆ΓL|∥χFf∥2 +1 ≤ |E∆ΓL||F|∥f∥2 +2 +L→∞ +−−−→ 0. +To prove (6.1), it only remains to show that +∥RL − SL∥ +L→∞ +−−−→ 0. +(6.2) +To this end, let f ∈ L2(Rd) and estimate +∥SLf − RLf∥2 +2 = +� +F +��� +� +ΓL +F(χFf)(w)e2πiwxdw − L−d � +m∈EL +F(χFf)(m)e2πimx��� +2 +dx += +� +F +��� +� +m∈EL +� +m+QL−1 +F(χFf)(w)e2πiwx − F(χFf)(m)e2πimxdw +��� +2 +dx += +� +F +��� +� +m∈EL +� +m+QL−1 +� +F +f(t) +� +e2πiw(x−t) − e2πim(x−t)� +dtdw +��� +2 +dx +≲ +� +F +� � +m∈EL +� +m+QL−1 +� +F +|f(t)||w − m||x − t|dtdw +�2 +dx +≲ +� +F +� � +m∈EL +L−(d+1) +� +F +|f(t)||x − t|dt +�2 +dx +≲ (#EL)2L−2(d+1) +� +F +∥f∥2 +2 +� +F +|x − t|2dtdx +≲ L−2max{|∂E|2d/(d−1), 1} +κ2 +∂E +|F|2 diam(F)2∥f∥2 +2, +where in the inequality step we used Corollary 2.2. Hence (6.2) holds. +Step 2. Since SL and TE,F,L share the same non-zero eigenvalues, the estimates in +Theorem 1.2 apply also to SL for all sufficiently large L. By the Fischer-Courant +formula, operator convergence of positive compact operators implies convergence +of their eigenvalues. Hence, by (6.1), the estimate satisfied by the spectrum of SL +extends to the spectrum of S. +□ + +22 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +7. The discrete Fourier transform +Proof of Theorem 1.3. Let us define E := Ω + Q1. Then Ω = EL for L = 1. Let +us apply Theorem 1.2 with L = 1 to E, F. We check the relevant hypotheses. +For each point k ∈ ∂Ω, there exist at least one face and at most 2d faces of the +cube k + Q1 that are contained in ∂E. Therefore, +(7.1) +#∂Ω ≤ +��∂E +�� ≤ 2d · #∂Ω, +and consequently +|∂E||∂F| ≥ #∂Ω · |∂F| ≥ 1. +Moreover, (7.1) shows that the choice L = 1 satisfies L ≥ |∂E +��−1/(d−1) as ∂Ω +contains at least one point. +Now fix 0 < r ≤ +√ +d·(#∂Ω)1/(d−1) and let us show that ∂E is regular at maximal +scale. If r < 2 +√ +d, and x ∈ ∂E we clearly have Hd−1� +∂E ∩ Br(x) +� +≳ rd−1 as E is +a union of cubes of length 1. If r ≥ 2 +√ +d, set n = ⌊r/ +√ +d⌋ and let x ∈ ∂E. There +exist kx ∈ ∂Ω such that |kx − x| ≤ +√ +d/2. Note that for y ∈ kx + Qn, +|y − x| ≤ +√ +dn +2 ++ +√ +d +2 +≤ r +2 + +√ +d +2 +< r. +Hence, kx + Qn ⊆ Br(x) and therefore, +Hd−1� +∂E ∩ Br(x) +� +≥ Hd−1� +∂E ∩ kx + Qn +� +≥ # +� +∂Ω ∩ kx + Qn +� +≥ κ∂Ωnd−1 ≳ κ∂Ωrd−1. +This shows that ∂E is regular at scale +√ +d · (#∂Ω)1/(d−1) with constant Cd · κ∂Ω. +Note that if a set X is regular at scale ηX and constant κX, then it is also regular +at scale αηX and constant min{1, α1−d}κX, for every α > 0. By (7.1) we therefore +see that ∂E is regular at scale η∂E = +��∂E +��1/(d−1) and constant κ∂E ≍ κ∂Ω. +The desired estimates now follow by applying Theorem 1.2 to E and F, with +L = 1. +□ +8. Proof of Remark 1.4 +First we combine Lemma 5.3, Lemma 5.1 (for p = 1) and Theorem 1.1 to +conclude that there exist a constant C = Cα,d > 0 such that if +n ≥ ⌈|E| · |F|⌉ + C |∂E| +κ∂E +|∂F| +κ∂F +· log +�e|∂E||∂F| +κ∂E +�2d(1+α)+1 +=: C1, +then λn ≤ 1/2, and if +n ≤ ⌈|E| · |F|⌉ − C |∂E| +κ∂E +|∂F| +κ∂F +· log +�e|∂E||∂F| +κ∂E +�2d(1+α)+1 +=: C2, +then λn ≥ 1/2. +For ε ∈ (0, 1), define ε0 := min{ε, 1 − ε} ≤ 1/2 and let 0 < τ < ε0. Observe +that +{1, ..., ⌊C2⌋} ∖ Mτ(S) ⊆ N1−ε0(S) ⊆ Nε(S) ⊆ Nε0(S) ⊆ {1, ..., ⌈C1⌉} ∪ Mτ(S), + +EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS +23 +where we understand {1, ..., ⌊C2⌋} to be ∅ if C2 < 1. Consequently, +C2 − 1 − #Mτ(S) ≤ #Nε(S) ≤ C1 + 1 + #Mτ(S). +Rearranging the last expression and using Theorem 1.1 for τ gives +��Nε(S) − |E| · |F| +�� ≲ |∂E| +κ∂E +· |∂F| +κ∂F +· log +�|∂E||∂F| +κ∂E τ +�2d(1+α)+1 +. +Letting τ ր ε0 yields (1.10). +References +[1] L. D. Abreu, J. M. Pereira, and J. L. Romero. Sharp rates of convergence for accumulated +spectrograms. Inverse Problems, 33(11):115008, 12, 2017. +[2] J. Anden and J. L. Romero. Multitaper estimation on arbitrary domains. SIAM J. Imaging +Sci., 13(3), 2021. +[3] A. Bonami, P. Jaming, and A. Karoui. Non-asymptotic behavior of the spectrum of the +sinc-kernel operator and related applications. J. Math. Phys., 62(3):Paper No. 033511, 20, +2021. +[4] D. G. Caraballo. Areas of level sets of distance functions induced by asymmetric norms. +Pacific J. Math., 218(1):37–52, 2005. +[5] J. Diestel, H. Jarchow, and A. Tonge. Absolutely summing operators, volume 43 of Cam- +bridge Studies in Advanced Mathematics. 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Anal., 35(2):309–340, 2013. +[18] D. Slepian and H. O. Pollak. Prolate spheroidal wave functions, Fourier analysis and un- +certainty. I. Bell System Tech. J., 40:43–63, 1961. +[19] H. Widom. Asymptotic behavior of the eigenvalues of certain integral equations. II. Arch. +Rational Mech. Anal., 17:215–229, 1964. + +24 +FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER +[20] Z. Zhu, S. Karnik, M. B. Wakin, M. A. Davenport, and J. Romberg. ROAST: Rapid +orthogonal approximate Slepian transform. IEEE Trans. Signal Process., 66(22):5887–5901, +2018. +Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, +A-1090 Vienna, Austria +Email address: felipe.marceca@univie.ac.at +Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, +A-1090 Vienna, Austria, and Acoustics Research Institute, Austrian Academy of +Sciences, Wohllebengasse 12-14, Vienna, 1040, Austria +Email address: jose.luis.romero@univie.ac.at +Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, +A-1090 Vienna, Austria +Email address: michael.speckbacher@univie.ac.at + diff --git a/bNFJT4oBgHgl3EQf8y1j/content/tmp_files/load_file.txt b/bNFJT4oBgHgl3EQf8y1j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7690989cd486eb7e58dfe5fba6c736ada6f9693c --- /dev/null +++ b/bNFJT4oBgHgl3EQf8y1j/content/tmp_files/load_file.txt @@ -0,0 +1,901 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf,len=900 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='11685v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='FA] 27 Jan 2023 EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS ON TWO DOMAINS FELIPE MARCECA, JOSÉ LUIS ROMERO, AND MICHAEL SPECKBACHER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We derive eigenvalue estimates for concentration operators asso- ciated with the discrete Fourier transform and two concentration domains sat- isfying certain regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' These conditions are met, for example, when the discrete domain, contained in a lattice, is obtained by discretization of a suitably regular domain in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' As a limit, we obtain eigenvalue estimates for Fourier concentration operators associated with two suitably regular domains in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The proof builds on Israel’s work on one dimensional intervals: arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='04404v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Introduction and results Fourier concentration operators act by incorporating a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The chief example concerns the Fourier transform on the Euclidean space F : L2(Rd) → L2(Rd), the cut-offs are given by the indicator functions of two compact domains E, F ⊆ Rd, and the concentration operator is Sf = χFF −1χEFχFf, f ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) These operators, and their analogues defined with respect to the discrete Fourier transform L2([−1/2, 1/2]d) → ℓ2(Zd) play a crucial role in many analysis problems and fields of application [18, 13, 14, 7], such as imaging, where the shapes of E, F are dictated by various acquisition constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The basic intuition is that the concentration operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) is approximately a projection with rank tr(S) = |E| · |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The error of such heuristic is encoded by the so-called plunge region Mε(S) = {λ ∈ σ(S) : ε < λ < 1 − ε}, ε ∈ (0, 1/2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) consisting of intermediate eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Asymptotics for the cardinality of Mε(S) go back to Landau and Widom [15, 12] for the case of one dimensional intervals E = [−a, a], F = [−b, b] and read #Mε(S) = c · log(ab) · log( 1−ε ε ) + o(log(ab)), as ab → ∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) for an explicit constant c that depends on the normalization of the Fourier trans- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The modern spectral theory of Wiener-Hopf operators gives similar asymp- totics for concentration operators associated to rather general multi-dimensional domains subject to increasing isotropic dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 47B35, 47A75, 42B35, 42C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Discrete Fourier transform, concentration operator, Hankel operator, eigenvalue, spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The authors gratefully acknowledge support from the Austrian Science Fund (FWF): Y 1199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1 2 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER While (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) precisely describes the cardinality of the set Mε(S) in the limit ab → ∞, the asymptotic is often insufficient for many purposes because of the quality of the error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Indeed, the error term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) depends in an unspecified way on the spectral threshold ǫ, which precludes applications where ε is let to vary with the domains E, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Such limitations have motivated a great amount of work aimed at deriving upper bounds for #Mε(S) that are threshold robust, that is, bounds that are effective for concrete concentration domains and explicit in their dependence on the spectral threshold [8, 10, 20, 11, 3, 17], significantly improving on more classical results in this spirit [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' With the exception of [8], the mentioned articles on threshold-robust spectral bounds for Fourier concentration operators concern only the one dimensional case, because they exploit a connection with a Sturm–Liouville equation which is spe- cific of that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' On the other hand, while [8] studies Fourier concentration operators associated with one dimensional intervals, the technique introduced by Israel is very general, as it relies on an explicit almost diagonalization of the concentration operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In fact, as we were finishing this work, the preprint [9] provided an extension of [8] to higher dimensions (see Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In this article we derive upper bounds for the number of intermediate eigen- values (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) associated with either the continuous or discrete Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We obtain estimates that apply to two suitably regular multi-dimensional spatial and frequency domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The proofs build on Israel’s technique [8] and combine it with arguments from geometric measure theory and operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Given two compact sets E, F ⊆ Rd, the Fourier concentration operator S : L2(Rd) → L2(Rd) is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) where F denotes the Fourier transform Ff(ξ) = � Rd f(x)e−2πixξ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4) A set E ⊆ Rd is said to have a maximally Ahlfors regular boundary if there exists a constant κ∂E > 0 such that Hd−1� ∂E ∩ Br(x) � ≥ κ∂E · rd−1, 0 < r ≤ Hd−1(∂E)1/(d−1), x ∈ ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Here, Hd−1 denotes the (d−1)-dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The term maximal in the definition refers to the range of r for which the estimate is required to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' See Section 2 for more context on Ahlfors regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In what follows, we denote for short |∂E| = Hd−1� ∂E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In this article we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let E, F ⊆ Rd, d ≥ 2, be compact domains with maximally Ahlfors regular boundaries with constants κ∂E, κ∂F respectively, and assume that that |∂E||∂F| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Consider the concentration operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) and its eigenvalues {λn : n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then for every α ∈ (0, 1/2), there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): # � n ∈ N : λn ∈ (ε, 1 − ε) � ≤ Aα,d · |∂E| κ∂E |∂F| κ∂F log �|∂E||∂F| κ∂E ε �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 3 A closely related result is presented in the recent preprint [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For F = [0, 1]d and E = rK, where r > 0 is a dilation parameter and K ⊆ Rd is a convex, coordinate symmetric domain [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1] gives the following bound for ε ∈ (0, 1/2): # � n ∈ N : λn ∈ (ε, 1 − ε) � ≤ Cd · max{rd−1 log(r/ε)3, log(r/ε)3d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5) For large r, the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5) becomes Od � rd−1 log(r/ε)3) while Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 gives the weaker bound Oα,d � rd−1 log(r/ε)2d(α+1)+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' On the other hand, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 applies to possibly non-convex, non-coordinate-symmetric and non- dilated domains E, and other regular domains F besides cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (As pointed out in [9], when E and F are both cubes, even slightly stronger estimates hold, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=') Our work is in great part motivated by applications where concentration do- mains may be non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For example, noise statistics are often estimated from those pixels of a square image located outside a central disk, which is assumed to contain the signal of interest (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Thus, the need to sample pure noise leads one to consider the complement of a disk within a two dimensional square as concentration domain (or, more realistically, the set of grid points within that domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Such a domain E is allowed by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 (and Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3 below) and has moreover a favorable regularity constant κ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Discretization of continuous domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 is obtained by tak- ing a limit on a more precise result concerning a discrete setting, which is our main focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We consider a resolution parameter L > 0 and define the discrete Fourier transform FL : L2((−L/2, L/2)d) → ℓ2(L−1Zd) by FLf(k/L) = � (−L/2,L/2)d f(x)e−2πixk/Ldx, k ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='6) We think of L as a discretization parameter for an underlying continuous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let us define the discretization at resolution L > 0 of a domain E ⊆ Rd by EL = L−1Zd ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Given two compact domains E ⊆ Rd and F ⊆ (−L/2, L/2)d, consider the dis- cretized concentration operator T : L2(F) → L2(F) given by T = χFF −1 L χELFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='7) Our second result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let E, F ⊆ Rd, d ≥ 2, be compact domains with maximally Ahlfors regular boundaries with constants κ∂E, κ∂F respectively, and assume that that |∂E||∂F| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Fix a discretization resolution L ≥ |∂E|−1/(d−1) such that F ⊆ (−L/2, L/2)d and consider the discretized concentration operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='7) and its eigenvalues {λn : n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then for every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): # � n ∈ N : λn ∈ (ε, 1 − ε) � ≤ Aα,d · |∂E| κ∂E |∂F| κ∂F log �|∂E||∂F| κ∂E ε �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 4 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The discrete Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Finally, we consider a discrete concen- tration problem associated with the usual discrete Fourier transform, denoted F1 : L2((−1/2, 1/2)d) → ℓ2(Zd) for consistency with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Given a finite set Ω ⊆ Zd and F ⊆ (−1/2, 1/2)d, the discrete Fourier concen- tration operator T : L2(F) → L2(F) is defined as T = χFF −1 1 χΩF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='8) The discrete boundary of a set Ω ⊆ Zd is given by ∂Ω = {k ∈ Ω : min{|j − k| : j ∈ Zd ∖ Ω} = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='9) We say that Ω ⊆ Zd has a maximally Ahlfors regular boundary if there exists a constant κ∂Ω such that inf k∈∂Ω # � ∂Ω ∩ k + [−n/2, n/2)d� ≥ κ∂Ω · nd−1, 1 ≤ n ≤ (#∂Ω)1/(d−1), k ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (Note the slight notational abuse: though Ω ⊆ Zd ⊆ Rd, the notions of boundary and boundary regularity are to be understood in the discrete sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=') Our last result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let d ≥ 2, Ω ⊆ Zd a finite set with maximally Ahlfors regular boundary and constant κ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let F ⊆ (−1/2, 1/2)d be compact with maximally Ahlfors regular boundary and constant κ∂F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Assume that #∂Ω · |∂F| ≥ 1, and consider the concentration operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='8) and its eigenvalues {λn : n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then for every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): # � n ∈ N : λn ∈ (ε, 1 − ε) � ≤ Aα,d · #∂Ω κ∂Ω |∂F| κ∂F log �#∂Ω · |∂F| κ∂Ω ε �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' One sided estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Finally, we remark that bounds on the number of intermediate eigenvalues, as in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3, can be equivalently formulated in terms of the distribution function Nε := {n ∈ N : λn > ε}, ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For example, for ε ∈ (0, 1) under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 we have ��#Nε(S) − |E| · |F| �� ≤ Cα,d · |∂E| κ∂E |∂F| κ∂F log � |∂E||∂F| κ∂E min{ε, 1 − ε} �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10) See Section 8 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Methods and related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We work for the most part with the discrete Fourier transform and then obtain consequences for the continuous one by a limiting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 is proved in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We first revisit Israel’s argument [8] and adapt it to prove eigenvalue estimates when one of the domains is a rectangle and the other is a general multi-dimensional domain (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' These estimates are slightly stronger than those in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2, and EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 5 the extra precision is exploited in the subsequent step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We follow the method of almost diagonalization with wavepackets, which we achieve, unlike [8], through a redundant system (frame) instead of an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The second step is a decomposition, rescaling, and dyadic approximation ar- gument, implemented by means of p-Schatten norm estimates for certain Hankel operators, and especially by quantifying those estimates as a function of p, as p → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Our intermediate result, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1, is close in spirit to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 in [9] (which appeared as we were finishing this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The estimates in [9], formulated in the context of the continuous Fourier transform and concerning dilated convex domains, are stronger than what follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 in that regime, as [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2] involves smaller powers of a certain logarithmic factor (see also Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' On the other hand, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 concerns sufficiently regular, non-dilated and possibly non-convex domains, and covers the discrete Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We also mention our recent work on concentration operators for the short-time Fourier transform [16], that also makes use of Ahlfors regularity and Schatten norm estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Though the goals and results are philosophically similar to those in the present article, the settings are rather different from the technical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Indeed, the arguments used in [16] rely on the rapid off-diagonal decay of the reproducing kernel of the range of the short-time Fourier transform, and do not seem to be applicable to Fourier concentration operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The remainder of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Section 2 sets up the notation and provides background on boundary regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Section 3 revisits the technique from [8] and implements certain adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' These are used in Section 4 to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 is proved in Section 5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 is proved in Section 6, and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3 is proved in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4 is proved in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We shall focus on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 and set up the notation accord- ingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3 will be obtained afterwards as an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We denote cubes by Qa = [−a/2, a/2)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The Euclidean norm on Rd is denoted | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For two non-negative functions f, g we write f ≲ g if there exist a constant C such that f(x) ≤ Cg(x), and write f ≍ g is f ≲ g and g ≲ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The implied constant is allowed to depend on the dimension d and the parameter α from Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3, but not on other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We enumerate the eigenvalues of a compact self adjoint operator L : H → H acting on a Hilbert space H as follows: λk = inf{∥L − S∥ : S ∈ L(L2(Rd)), dim(Range(S)) < k}, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) Then {λk : k ≥ 1} ∖ {0} = σ(L) ∖ {0} as sets with multiplicities — see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 6 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER For a set E ⊆ Rd, we write Ec L = L−1Zd ∖ EL and ∂EL for the points in EL which are at distance L−1 of Ec L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For L = 1 this is consistent with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We will work with the discrete Fourier transform FL : L2((−L/2, L/2)d) → ℓ2(L−1Zd) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='6) and reserve the notation Ff or �f for the continuous Fourier transform (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that if supp(f) ⊆ (−L/2, L/2)d, then FLf(k/L) = Ff(k/L) for every k ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We also write PE,L = F −1 L χELFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For F ⊆ (−L/2, L/2)d we define the operator T = TE,F,L : L2(F) → L2(F) by T = TE,F,L = χFPE,L and let λn = λn(T) denote its eigenvalues as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' An easy computation shows that Tt−1E,tF,tL = Mt−1TE,F,LMt, t > 0, where Mt denotes the dilation operator Mtf(x) = f(tx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In particular, λn(Tt−1E,tF,tL) = λn(TE,F,L), n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Boundary regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let us introduce regularity of sets in more generality and discuss a few properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' An Hd−1-measurable set X ⊆ Rd is said to be lower Ahlfors (d − 1)-regular (regular for short) at scale ηX > 0 if there exists a constant κX > 0 such that Hd−1� X ∩ Br(x) � ≥ κX · rd−1, 0 < r ≤ ηX, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that if X ⊆ Rd is regular at scale ηX > 0 with constant κX > 0 and t > 0, then tX ⊆ Rd is regular at scale ηtX = tηX with constant κtX = κX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By differentiation around a point of positive Hd−1-density, κX ≤ cd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) for any regular X of positive Hd−1-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We also mention that if X is regular with parameters ηX and κX, then choosing an arbitrary x ∈ X gives Hd−1� X � ≥ Hd−1� X ∩ BηX(x) � ≥ κX · ηd−1 X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4) We shall use the following basic result, derived from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' There exists a universal constant Cd > 0 such that for every compact set X ⊆ Rd that is regular at scale ηX > 0 with constant κX and every s > 0, |X + Bs(0)| ≤ Cd κX Hd−1(X) · s · � 1 + sd−1 ηd−1 X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From [4, Theorems 5 and 6] it follows that Hd−1� {x : d(x, X) = r} � ≤ Cd κX Hd−1(X) · � 1 + rd−1 ηd−1 X � , EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 7 for almost every r > 0, and in addition, |∇d(x, X)| = 1, for almost every x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From this and the coarea formula, it follows that |X + Bs(0)| = � Rd χ[0,s)(d(x, X))dx = � Rd χ[0,s)(d(x, X))|∇d(x, X)|dx = � s 0 Hd−1� {x : d(x, X) = r} � dr ≤ Cd κX Hd−1(X) � s 0 � 1 + rd−1 ηd−1 X � dr ≤ Cd κX Hd−1(X)s � 1 + sd−1 ηd−1 X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For E ⊆ Rd a compact domain with regular boundary at scale η∂E ≥ 1 with constant κ∂E and a discretization resolution L ≥ 1, we have L−d#EL ≲ |E| + |∂E| κ∂EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In particular, for d ≥ 2 L−d#EL ≲ max{|∂E|d/(d−1), 1} κ∂E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Recall that QL−1 = L−1[−1/2, 1/2)d and define E′ L = {m ∈ EL : m + QL−1 ⊆ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1, we get L−d#EL = ��� � m∈E′ L m + QL−1 ��� + ��� � m∈EL∖E′ L m + QL−1 ��� ≤ |E| + |∂E + BL−1√ d(0)| ≲ |E| + |∂E| κ∂EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Finally, the second inequality follows from the isoperimetric inequality |E| ≲ |∂E|d/(d−1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Israel’s argument revisited We now revisit the core argument of [8] and make a few adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Israel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We need a slight generalization of Lemma 1 in [8], phrasing it in terms of frames rather than orthonormal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We include a proof for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Recall that a frame for a Hilbert space H is a subset of vectors {φi}i∈I for which the exist constants 0 < A, B < ∞ — called lower and upper frame bounds — such that A∥f∥2 ≤ � i∈I |⟨f, φi⟩|2 ≤ B∥f∥2, f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let T : H → H be a positive, compact, self-adjoint operator on a Hilbert space H with ∥T∥ ≤ 1 and eigendecomposition T = � n≥1 λn⟨·, fn⟩fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let {φi}i∈I be a frame of unit norm vectors for H with lower frame bound A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' If I = I1 ∪ I2 ∪ I3, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) � i∈I1 ∥Tφi∥2 + � i∈I3 ∥(I − T)φi∥2 ≤ A 2 ε2, 8 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER then #Mε(T) ≤ 2 A#I2, where Mε(T) is defined as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let Sε = span{fn : λn ∈ (ε, 1 − ε)} and let Pε : H → Sε denote the orthogonal projection onto Sε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For f ∈ Sε one has ∥f∥ − ∥(I − T)f∥ ≤ ∥Tf∥ ≤ (1 − ε)∥f∥, which shows ε∥f∥ ≤ ∥Tf∥, and ε∥f∥ ≤ ∥(I − T)f∥, f ∈ Sε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that T and Pε commute since Sε is spanned by a collection of eigenvectors of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Therefore, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) we obtain � i∈I1∪I3 ε2∥Pεφi∥2 ≤ � i∈I1 ∥TPεφi∥2 + � i∈I3 ∥(I − T)Pεφi∥2 ≤ A 2 ε2, which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) � i∈I1∪I3 ∥Pεφi∥2 ≤ A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Using the frame property we get for f ∈ Sε: A∥f∥2 ≤ � i∈I |⟨f, φi⟩|2 = � i∈I |⟨f, Pεφi⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Now assume that dim(Sε) ≥ 1 (otherwise the result is trivial), take an orthonormal basis {ψk}dim(Sε) k=1 of Sε, and sum the inequality above over all basis elements to derive A · #Mε(T) = A · dim(Sε) = A dim(Sε) � k=1 ∥ψk∥2 ≤ dim(Sε) � k=1 � i∈I |⟨ψk, φi⟩|2 = � i∈I ∥Pεφi∥2 ≤ � i∈I2 ∥φi∥2 + A 2 = #I2 + A 2 ≤ #I2 + A 2 #Mε(T), where in the second line we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This shows that #Mε(T) ≤ 2 A#I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Local trigonometric frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In this section, we construct a tight frame that allows us to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let α > 0, and θ ∈ C∞(R) be such that (i) θ(x) = 1, for x ≥ 1, and θ(x) = 0, for x ≤ −1, (ii) θ(−x)2 + θ(x)2 = 1, for every x ∈ R, (iii) |Dkθ(x)| ≤ Ck αk(1+α)k, for all k ∈ N0, all x ∈ R, and a constant Cα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' See, for example, [8, Proposition 1] or [6, Chapter 1] for the existence of such a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let W > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We decompose the interval � − W 2 , W 2 � into disjoint intervals Ij = xj + W 3 · 2|j|+1[−1, 1), j ∈ Z, where xj = sign(j)W 2 � 1 − 1 2|j| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 9 Note that |Ij| = |I|j|| = 2|I|j|+1| for every j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We will also denote Dj = Ij∪Ij+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Now define θj(x) = θ �2(x − xj) |Ij| � θ � −2(x − xj+1) |Ij+1| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We have that θj(x) = 0 for x /∈ Dj, and furthermore by properties (i) and (ii) ∥θj∥2 2 = � Ij θ �2(x − xj) |Ij| �2 dx + � Ij+1 θ � −2(x − xj+1) |Ij+1| �2 dx = |Ij| 2 � 1 −1 θ(x)2dx + |Ij+1| 2 � 1 −1 θ(x)2dx = |Dj| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We define the set of vectors (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) φj,k(x) = � 2 |Dj| · θj(x) · exp � 2πi xk |Dj| � , j, k ∈ Z, and note that ∥φj,k∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The family {φj,k}j,k∈Z defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) forms a tight frame for L2(−W/2, W/2) with frame constants A = B = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We write fj := f|Ij so that f = � j∈Z fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since supp(θj) ⊆ Dj = Ij ∪ Ij+1, we observe � j,k∈Z |⟨f, φj,k⟩|2 = � j,k∈Z |⟨fj + fj+1, φj,k⟩|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' As � |Dj|−1/2 exp (2πikx/|Dj|) � k∈Z is an orthonormal basis for L2(Dj), we find that � k∈Z |⟨fj + fj+1, φj,k⟩|2 = 2∥(fj + fj+1)θj∥2 2 = 2∥fjθj∥2 2 + 2∥fj+1θj∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Combining both identities and using property (ii), we conclude � j,k∈Z |⟨f, φj,k⟩|2 = 2 � j∈Z � ∥fjθj∥2 2 + ∥fj+1θj∥2 2 � = 2 � j∈Z � ∥fjθj∥2 2 + ∥fjθj−1∥2 2 � = 2 � j∈Z � Ij |f(x)|2 � θ �2(x − xj) |Ij| �2 + θ � −2(x − xj) |Ij| �2� dx = 2 � j∈Z ∥fj∥2 2 = 2∥f∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ Let 0 < Wi ≤ L, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', d, and consider �d i=1(−Wi/2, Wi/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Set also Wmax := max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=',dWi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We define a frame for L2� �d i=1(−Wi/2, Wi/2) � via the tensor product Φj,k(x) = Φj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=',jd,k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='kd(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', xd) = φj1,k1(x1) · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' · φjd,kd(xd), where each family {φji,ki(xi)}ji,kk∈Z is the frame for L2(−Wi/2, Wi/2) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This construction also yields a tight frame with frame bounds equal to 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 10 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Consider ψj(x) = θj � |Dj|x + xj − W 3 · 2|j|+1 � , x ∈ R, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' A straightforward computation shows that ψj is supported on [0, 1] and satisfies |Dkψj(x)| ≤ � Cα kk(1+α)k by property (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' As shown in [8, Lemma 1] it thus follows that |� ψj(ξ)| ≤ Aα · exp � −aα|ξ|(1+α)−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since 1 − α ≤ (1 + α)−1, we derive that t(1+α)−1 ≥ t1−α − 1 for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Adjusting the constant Aα, we therefore get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4) |�θj(ξ)| ≤ Aα · |Dj| · exp � −aα(|Dj| · |ξ|)1−α� , ξ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' With this at hand, we estimate the decay of F(Φj,k)(ξ) = 2d/2 d � i=1 |Dji|−1/2 · � θji � ξi − ki |Dji| � , ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Define Mj = diag(|Dj1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', |Djd|) ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4) (possibly enlarging Aα) and |ξ|1−α ≤ �d i=1 |ξi|1−α, it follows |F(Φj,k)(ξ)| ≤ Ad α d � i=1 |Dji|1/2 · exp � −aα ��|Dji|ξi − ki ��1−α� ≤ Ad α · det(Mj)1/2 · exp � −aα ��Mj(ξ − ξj,k) ��1−α� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5) where (ξj,k)i = ki|Dji|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Consider now a compact domain E ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let s ≥ 1 be a parameter that will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For j ∈ Zd fixed, we cover the index set Zd with three subsets as follows Llow j := � k ∈ Zd : dist(k, MjEc L) ≥ s � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lmed j := � k ∈ Zd : dist(k, MjEL) < s, and dist(k, MjEc L) < s};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lhigh j := � k ∈ Zd : dist(k, MjEL) ≥ s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='6) (Here, dist is associated with the usual Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=') We claim that Lmed j ⊆ � k ∈ Zd : dist(k, Mj∂EL) < s � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='7) let us briefly sketch an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Fix k ∈ Lmed j and let k0 ∈ MjL−1Zd minimize the distance to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For any point x ∈ MjL−1Zd with |k − x| < s we can build a path of adjacent points in MjL−1Zd from x to k0 such that the distance to k is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In particular, choosing x1 ∈ MjEL and x2 ∈ MjEc L at distance less than s from k, we can connect x1 and x2 through a path of adjacent points in MjL−1Zd that stays at distance less than s from k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Necessarily, one of the points in the path must belong to Mj∂EL, which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The indices (j, k) with k ∈ Lmed j , and j satisfying a condition specified below (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10)) will play the role of I2 in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1, so we need to estimate #(Lmed j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 11 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let E ⊆ Rd be a compact domain with regular boundary at scale η∂E ≥ 1 and constant κ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let L ≥ Wmax and s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then for all j ∈ Zd we have # � k ∈ Zd : dist(k, Mj∂EL) < s � ≲ max{Wmax, 1/η∂E}d−1 · |∂E| κ∂E sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since for every x ∈ Rd the cube x + Q1 contains one point in Zd, # � k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · # � k ∈ Zd : k ∈ Mj∂EL + Q1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='8) For x ∈ R, we denote the unique integer in x + [−1/2, 1/2) by x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For a set X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', xN} ⊆ R and 0 < a ≤ 1, we write yi = axi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' It is easy to check that if x∗ i = x∗ i′, then |y∗ i − y∗ i′| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Using this, a straightforward argument shows that # � k ∈ Z : k ∈ aX + [−1/2, 1/2)} ≤ 2# � k ∈ Z : k ∈ X + [−1/2, 1/2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='8), if we apply the inequality above componentwise (noting that (Mj)i,i ≤ Wmax), we obtain for s ≥ 1 # � k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · # � k ∈ Zd : k ∈ Wmax∂EL + Q1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since for every x ∈ ∂EL there exists x′ ∈ ∂E such that |x − x′| ≤ L−1, and Wmax/L ≤ 1, it follows that # � k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · # � k ∈ Zd : k ∈ Wmax∂E + Q3} =: sd · #(KWmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='9) Now let k ∈ KWmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' There exists at least one point xk ∈ ∂E such that k ∈ Wmaxxk + Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In particular, we have that xk ∈ W −1 maxk + Q4/Wmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Therefore, for every k ∈ KWmax we get by regularity of ∂E κ∂E · min{W −1 max, η∂E}d−1 ≤ Hd−1� ∂E ∩ B1/Wmax(xk) � ≤ Hd−1� ∂E ∩ xk + Q2/Wmax � ≤ Hd−1� ∂E ∩ W −1 maxk + Q6/Wmax � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' So, κ∂E · min � W −1 max, η∂E �d−1 · #(KWmax) ≤ � k∈KWmax Hd−1� ∂E ∩ W −1 maxk + Q6/Wmax � ≲ � k∈Zd Hd−1� ∂E ∩ W −1 maxk + Q1/Wmax � = Hd−1(∂E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Plugging this estimate into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='9) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ Next, for a compact domain E ⊆ Rd and a parameter s ≥ 1 we recall the sets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='6), introduce a second auxiliary parameter 0 < δ < 1, and define the following covering of Z2d: Γlow := � (j, k) : min i |Dji| ≥ δ, k ∈ Llow j � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Γmed := � (j, k) : min i |Dji| ≥ δ, k ∈ Lmed j � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Γhigh := � (j, k) : min i |Dji| ≥ δ, k ∈ Lhigh j � ∪ � (j, k) : min i |Dji| < δ, k ∈ Zd� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10) 12 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Under the conditions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3, let 0 < δ < 1 and consider the set Γmed from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then #(Γmed) ≲ max{Wmax, 1/η∂E}d−1 · |∂E| κ∂E max{log(Wmax/δ), 1}d · sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='7) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3 it follows #(Γmed) = � j∈Zd min |Dji|≥δ #(Lmed j ) ≲ � j∈Zd min |Dji|≥δ max{Wmax, 1/η∂E}d−1|∂E| κ∂E sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In each coordinate, we have that the number of intervals Dji for which |Dji| ≥ δ is bounded by C max{log(Wmax/δ), 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Hence, we arrive at #(Γmed) ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E max{log(Wmax/δ), 1}dsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let d ≥ 2, L ≥ max{Wmax, 1}, and E ⊆ Rd be a compact domain with regular boundary at scale η∂E ≥ 1 with constant κ∂E and such that |∂E| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let s ≥ 1 and δ ∈ (0, 1) be parameters and consider the sets from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then there exists a constant c = cα > 0 such that � (j,k)∈Γlow L−d � m∈Ec L |F(Φj,k)(m)|2 ≲ max{Wmax, 1/η∂E}d−1 · |∂E| κ∂E exp � − cs1−α� max{log(Wmax/δ), 1}d, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='11) and � (j,k)∈Γhigh L−d � m∈EL |F(Φj,k)(m)|2 ≲max{Wmax, 1/η∂E}d−1 κ∂E � |∂E| d d−1 · δ + |∂E| · exp � − cs1−α� max{log(Wmax/δ), 1}d� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For j ∈ Zd and l ∈ N0 we set Llow j,l = � k ∈ Zd : dist(k, MjEc L) ∈ [s2l, s2l+1) � , and Lhigh j,l = � k ∈ Zd : dist(k, MjEL) ∈ [s2l, s2l+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Notice that Llow j,l ∪ Lhigh j,l ⊆ � k ∈ Zd : dist(k, MjEL) < s2l+1, and dist(k, MjEc L) < s2l+1} ⊆ � k ∈ Zd : dist(k, Mj∂EL) < s2l+1}, where the last step follows as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3 we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='13) #(Llow j,l ), #(Lhigh j,l ) ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E sd2dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5) it follows that if k ∈ Llow j,l L−d � m∈Ec L |F(Φj,k)(m)|2 ≤ L−d � m∈Ec L A2d α det(Mj) exp � − 2aα|Mj(m − ξj,k)|1−α� EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 13 ≤ A2d α L−d det(Mj) � m′∈MjEc L exp � − 2aα|m′ − k|1−α� ≲ � {|x|≥s2l} exp � − 2aα|x|1−α� dx ≲ exp � − c(s2l)1−α� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='14) where c can for example be chosen as aα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' A similar argument also shows that for k ∈ Lhigh j,l , L−d � m∈EL |F(Φj,k)(m)|2 ≲ exp � − c(s2l)1−α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' As Llow j = � l∈N0 Llow j,l , it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='14) that � (j,k)∈Γlow L−d � m∈Ec L |F(Φj,k)(m)|2 = � j∈Zd min |Dji|≥δ � l∈N0 � k∈Llow j,l L−d � m∈Ec L |F(Φj,k)(m)|2 ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E � j∈Zd min |Dji|≥δ � l∈N0 (s2l)d exp � − c(s2l)1−α� ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E � j∈Zd min |Dji|≥δ exp � − c′s1−α� ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E exp � − c′s1−α� max{log(Wmax/δ), 1}d, which completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Again, we can use an analogous reasoning to show � j∈Zd min |Dji|≥δ � k∈Lhigh j L−d � m∈EL |F(Φj,k)(m)|2 ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E exp � − c′s1−α� max{log(Wmax/δ), 1}d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Now suppose that j ∈ Zd is such that min1≤i≤d |Dji| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For every m ∈ Zd we can uniformly bound the subsequent series � k∈Zd exp � − 2aα|Mj(m − ξj,k)|1−α� = � k∈Zd exp � − 2aα|Mjm − k|1−α� ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since det(Mj) = |Dj|, where Dj = Dj1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' × Djd, we thus get by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5) � j∈Zd min |Dji|<δ � k∈Zd L−d � m∈EL |F(Φj,k)(m)|2 ≤ C � j∈Zd min |Dji|<δ L−d � m∈EL det(Mj) ≤ CL−d#EL � j∈Zd min |Dji|<δ |Dj| 14 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER ≲ |∂E|d/(d−1) κ∂E � j∈Zd min |Dji|<δ |Dj|, where in the last inequality we used Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Finally, � j∈Zd min |Dji|<δ |Dj| ≤ d � i=1 � j∈Zd |Dji|<δ |Dj| ≤ d � i=1 W d−1 max4δ ≲ max{Wmax, 1/η∂E}d−1δ, where we used that each interval Dji is at most at |Dji| < δ distance from the boundary of (−Wi/2, Wi/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This concludes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' General domain vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' rectangle In this section, we prove the following variant of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 for F a rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let L ≥ 1 be a discretization resolution, d ≥ 2, and E ⊆ Rd be a compact domain with regular boundary at scale η∂E ≥ 1 with constant κ∂E and such that |∂E| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For 0 < Wi ≤ L, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', d, take F = �d i=1(−Wi/2, Wi/2) and denote Wmax = maxi Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2): # � n ∈ N : λn ∈ (ε, 1 − ε) � ≤ Aα,d · max{Wmax, 1/η∂E}d−1 · |∂E| κ∂E log �max{Wmax, 1/η∂E}d−1|∂E| κ∂E ε �2d(1+α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We adopt all the notation of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Fix parameters s ≥ 1, δ ∈ (0, 1) and consider the sets from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Observe that for f ∈ L2(F) one has ∥Tf∥2 2 = ∥χFPE,Lf∥2 2 ≤ ∥PE,Lf∥2 2 = L−d � m∈EL �� �f(m) ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' and ∥f − Tf∥2 2 = ∥χFf − χFPE,Lf∥2 2 ≤ ∥(I − PE,L)f∥2 2 = L−d � m∈Ec L �� �f(m) ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5 it thus follows � (j,k)∈Γlow ∥(I − T)Φj,k∥2 2 + � (j,k)∈Γhigh ∥TΦj,k∥2 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) ≤ C max{Wmax, 1/η∂E}d−1 κ∂E � |∂E| exp � − cs1−α� max{log(Wmax/δ), 1}d + |∂E|d/(d−1)δ � , where the constants depend only on α and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 15 At last, we can now specify the parameters δ and s in order for the sets Γlow, Γmed and Γhigh to play the role of I1, I2 and I3 in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We take δ = κ∂E ε2 C max{Wmax, 1/η∂E}d−1|∂E|d/(d−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This ensures that C max{Wmax, 1/η∂E}d−1|∂E|d/(d−1) κ∂E δ ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) Also, we select s such that C max{Wmax, 1/η∂E}d−1|∂E| κ∂E exp � − cs1−α� max{log(Wmax/δ), 1}d ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) This condition on s is equivalent to s ≥ �1 c log �C max{Wmax, 1/η∂E}d−1|∂E| max{log(Wmax/δ), 1}d κ∂E ε2 ��1/(1−α) , and is satisfied if s = Aα,d log �max{Wmax, 1/η∂E}d−1|∂E| κ∂E ε �1/(1−α) , for an adequate constant Aα,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Moreover, we can guarantee that s ≥ 1, since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4), the term inside the logarithm is ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4, #Mε(T) ≤ 21−d#(Γmed) ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E max{log(Wmax/δ), 1}dsd ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E log �max{Wmax, 1/η∂E}d−1|∂E| κ∂E ε �d/(1−α)+d ≲ max{Wmax, 1/η∂E}d−1|∂E| κ∂E log �max{Wmax, 1/η∂E}d−1|∂E| κ∂E ε �2d(1+α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Eigenvalue estimates for two domains 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Schatten quasi-norm estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For 0 < p ≤ 1, and ε > 0, define the auxiliary function g = gp,ε : [0, 1] → R given by g(t) = � t − t2 ε − ε2 �p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that since χ(ε,1−ε) ≤ g, for a positive operator 0 ≤ S ≤ 1, #Mε(S) = tr(χ(ε,1−ε)S) ≤ tr(g(S)) = ∥S − S2∥p p (ε − ε2)p , where ∥ · ∥p, 0 < p ≤ 1, denotes the Schatten quasi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The next lemma shows that upper bounds for the left-hand side of the last inequality can be transferred to the right-hand side without much loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 16 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Suppose that for a positive operator 0 ≤ S ≤ 1 there are constants C, D, a > 0 such that for every ε ∈ (0, 1/2), #Mε(S) ≤ C � D + log(ε−1) �a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then, for every 0 < p ≤ 1 there is a constant Ca > 0 such that ∥S − S2∥p p ≤ CaC � D + p−1�a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By the symmetry of the function h(x) = x − x2 around 1/2, for 0 ≤ x ≤ 1, h(x)p = � min{x,1−x} 0 (hp)′(t)dt = � 1/2 0 χ(t,1−t)(x)php−1(t)h′(t)dt ≤ � 1/2 0 χ(t,1−t)(x)ptp−1dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By a monotone convergence argument we get ∥S − S2∥p p ≤ � 1/2 0 Mt(S)ptp−1dt ≤ C � 1 0 (D + log(t−1))aptp−1dt = C � ∞ 0 (D + u/p)ae−udu ≤ C(D + 1/p)a + C � ∞ 1 (D + u/p)ae−udu ≤ C(D + 1/p)a + C(D + 1/p)a � ∞ 1 uae−udu ≤ (1 + Γ(a + 1))C(D + 1/p)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Decomposition of the domain and Hankel operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In what follows, we let F ⊆ (−L/2, L/2)d be a compact domain with regular boundary at scale η∂F = |∂F|1/(d−1) ≥ 1 with constant κ∂F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We construct two auxiliary sets F − ⊆ F ⊆ F + which will be dyadic approximations of F from above and below by cubes of length at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' More precisely, let F = � k∈Z � j∈Jk Qk,j be a dyadic decomposition of F in piecewise disjoint cubes of the form Qk,j = Q2k + 2kj with k ∈ Z and j ∈ Jk ⊆ Zd, that are maximal (they are not contained in a larger dyadic cube included in F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We define F − = � k≥0 � j∈Jk Qk,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For F + we add cubes of length 1 to fully cover F and intersect them with (−L/2, L/2)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The result is a covering of F that combines the cubes from F − with rectangles of maximal side-length ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' More precisely, define V = {v ∈ Zd : (F ∖ F −) ∩ (Q1 + v) ̸= ∅}, and F + = F − ∪ � v∈V � (Q1 + v) ∩ (−L/2, L/2)d� =: F − ∪ � v∈V Rv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 17 Note that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 can be applied to each rectangle in the decomposition of F − and F +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This follows from a translation argument and the fact that the boundaries of the rectangles have null measure, so we can replace them by their interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We write T ± for TE,F ±,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For a set K ⊆ (−L/2, L/2)d define the Hankel operator on L2((−L/2, L/2)d) by HK = (I − PE,L)χKPE,L and write H± = HF ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that (HK)∗HK = PE,LχKPE,L − PE,LχKPE,LχKPE,L = PE,LχKPE,L − (PE,LχKPE,L)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since PE,LχKPE,L and TK share the same non-zero eigenvalues, for p > 0, ∥TK − (TK)2∥p p = ∥HK∥2p 2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Recall that for two operators S1, S2 in the p-Schatten class, 0 < p ≤ 1, one has ∥S1 + S2∥p p ≤ ∥S1∥p p + ∥S2∥p p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let L ≥ 1, d ≥ 2, and E, F ⊆ Rd be compact domains with regular boundaries at scales η∂E ≥ 1, η∂F = |∂F|1/(d−1) ≥ 1, with constants κ∂E, κ∂F respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Assume also that |∂E| ≥ 1 and F ⊆ (−L/2, L/2)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For ε ∈ (0, 1/2), we have #Mε(T ±) ≲ |∂E| κ∂E |∂F| κ∂F log �|∂E| max{|∂F|, 1} κ∂E ε �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' If k ∈ Z is such that Jk ̸= ∅, then there is a cube of length 2k included in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In particular, the projection of ∂F onto the hyperplane {x1 = 0} contains a (d−1)-dimensional cube of length 2k and therefore 2k(d−1) ≤ |∂F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The maximality of the dyadic decomposition of F implies that Qj,k ⊆ ∂F + B√ d2k+1(0) for j ∈ Jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 and the fact that η∂F = |∂F|1/(d−1), we thus derive 2dk#Jk ≤ |∂F + B√ d2k+1(0)| ≲ 2k |∂F| κ∂F � 1 + 2k(d−1) |∂F| � ≲ 2k |∂F| κ∂F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) Similarly, #V ≤ |∂F + B√ d(0)| ≲ |∂F| κ∂F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) For 0 < 2p ≤ 1, and ε ∈ (0, 1/2), we thus get #Mε(T +) ≤ ∥T + − (T +)2∥p p (ε − ε2)p = ∥H+∥2p 2p (ε − ε2)p ≤ (2/ε)p � k≥0 � j∈Jk ∥HQk,j∥2p 2p + (2/ε)p � v∈V ∥HRv∥2p 2p ≲ ε−p � k≥0 � j∈Jk ∥TQk,j − T 2 Qk,j∥p p + ε−p � v∈V ∥TRv − T 2 Rv∥p p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 shows that when applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 to TQk,j one can take C ≲ 2k(d−1)|∂E| κ∂E , and D = log � 2k(d−1)|∂E| κ∂E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 18 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER Similarly, the same holds for TRv with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Choosing p = log(2) � 2 log(ε−1) �−1 (which ensures that 2p ≤ 1 for every ε ∈ (0, 1/2)) thus yields #Mε(T +) ≲ |∂E| κ∂E �� k≥0 � j∈Jk 2k(d−1)log �|∂E|2k(d−1) κ∂E ε �2d(1+α) +#V ·log � |∂E| κ∂E ε �2d(1+α)� ≲ |∂E| κ∂E |∂F| κ∂F � k≥0 2k(d−1)≤max{|∂F |,1} log �|∂E|2k(d−1) κ∂E ε �2d(1+α) = |∂E| κ∂E |∂F| κ∂F � 0≤k≤⌊ log(max{|∂F |,1}) log(2)(d−1) ⌋ � log � |∂E| κ∂E ε � + (d − 1) log(2)k �2d(1+α) , where in the second to last step we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2), and the fact that 2k(d−1) ≤ |∂F| whenever Jk ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Finally, noting that for C, D, a > 0, N � k=0 (C + Dk)a ≤ � N+1 0 (C + Dx)adx ≤ (C + D(N + 1))a+1 D(a + 1) , we get, #Mε(T +) ≲ |∂E| κ∂E |∂F| κ∂F log �|∂E| max{|∂F|, 1} κ∂E ε �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The same argument works for #Mε(T −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The transition index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The following estimate is part of the proof of [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='5] (see also [16, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3]) and allows us to find the index where eigenvalues cross the 1/2 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We include a proof for the sake of complete- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For any trace class operator 0 ≤ S ≤ 1, (i) λn ≤ 1 2, for every n ≥ ⌈tr(S)⌉ + max{2 tr(S − S2), 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (ii) λn ≥ 1 2, for every 1 ≤ n ≤ ⌈tr(S)⌉ − max{2 tr(S − S2), 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' First notice that if S is an orthogonal projection, then the result holds trivially, so we can assume otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' In particular, we have that tr(S − S2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Set K = ⌈tr(S)⌉ and write tr(S) − tr(S2) = ∞ � n=1 λn(1 − λn) = K � n=1 λn(1 − λn) + ∞ � n=K+1 λn(1 − λn) ≥ λK K � n=1 (1 − λn) + (1 − λK) ∞ � n=K+1 λn EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 19 = λKK − λK K � n=1 λn + (1 − λK) � tr(S) − K � n=1 λn � = λKK + (1 − λK) tr(S) − K � n=1 λn = tr(S) − K � n=1 λn + λK(K − tr(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) ∞ � n=K+1 λn = tr(S) − K � n=1 λn ≤ tr(S) − tr(S2), and K−1 � n=1 (1 − λn) = tr(S) − K � n=1 λn + λK(K − tr(S)) − (1 − λK)(1 + tr(S) − K) ≤ tr(S) − K � n=1 λn + λK(K − tr(S)) ≤ tr(S) − tr(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4) Now let j ∈ N such that j ≥ 2(tr(S) − tr(S2)) and consider k = K + j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3) that 2(tr(S) − tr(S2)) · λk ≤ j · λK+j ≤ ∞ � n=K+1 λn ≤ tr(S) − tr(S2), which shows λk ≤ 1/2 as 0 < tr(S) − tr(S2) < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' this proves part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For part (ii), if 1 ≤ k = K − j ≤ K − 2(tr(S) − tr(S2)) for j ∈ N, then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4) implies 2(tr(S) − tr(S2)) · (1 − λk) ≤ j · (1 − λK−j) ≤ K−1 � n=1 (1 − λn) ≤ tr(S) − tr(S2), yielding λk ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' With all the preparatory work at hand, we are ready to prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) that the eigenvalues of the concentration operator remain the same if we replace E, F and L with t−1E, tF and tL respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We choose t = |∂E|1/(d−1) and notice that t−1E satisfies |∂t−1E| = 1, η∂t−1E = t−1η∂E = 1, and κ∂t−1E = κ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Furthermore, we also have that tF has regular boundary at scale η∂tF = tη∂F = (|∂E||∂F|)1/(d−1) ≥ 1 with constant κ∂tF = κ∂F, and tL ≥ 1 by assumption on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that for F ′ ⊆ (−tL/2, tL/2)d, the operator T has integral kernel K(x, y) = χF ′(x)χF ′(y) 1 (tL)d � k∈(t−1E)tL e−2πik(x−y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 20 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER Thus, tr(T) = � K(x, x)dx = � F ′ 1 (tL)d � k∈(t−1E)tL 1dx = #(t−1E)tL (tL)d |F ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' On the other hand, from Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 we have that tr � T ± − (T ±)2� ≲ |∂t−1E| κ∂t−1E |∂tF| κ∂tF log �e|∂t−1E||∂tF| κ∂t−1E �2d(1+α)+1 = |∂E| κ∂E |∂F| κ∂F log �e|∂E||∂F| κ∂E �2d(1+α)+1 =: CE,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' So from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3, λn(T +) ≤ 1 2, n ≥ �#(t−1E)tL (tL)d |(tF)+| � + 2CE,F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' λn(T −) ≥ 1 2, n ≤ �#(t−1E)tL (tL)d |(tF)−| � − 2CE,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 and |∂t−1E| = 1, #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} ≲ 1 κ∂E |(tF)+ ∖ (tF)−| + CE,F ≤ 1 κ∂E |∂tF + B√ d(0)| + CE,F ≲ 1 κ∂E td−1|∂F| κ∂F + CE,F ≲ CE,F, where in the second to last step we used Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since λn(T −) ≤ λn(T) ≤ λn(T +) for every n ∈ N, again by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2, #Mε(T) ≤#{n ∈ N : 1/2 ≤ λn(T −) < 1 − ε} + #{n ∈ N : ε < λn(T +) ≤ 1/2} + #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} ≲#Mε(T −) + #Mε(T +) + #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} ≲|∂E| κ∂E |∂F| κ∂F log �|∂E||∂F| κ∂E ε �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The continuous Fourier transform In this section we deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 by taking L → ∞ in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Fix E and F as in the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We consider a sufficiently large resolution such that L ≥ |∂E|−1/(d−1) and F ⊆ (−L/2, L/2)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let SL : L2(Rd) → L2(Rd) be the operator given by SLf = TE,F,L(χFf) = χFF −1 L χELFLχFf, f ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' An easy computation shows that SL and TE,F,L share the same non-zero eigenval- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Also, recall the operator S from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 21 Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We show that lim L→∞ ∥SL − S∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) Recall that QL−1 = L−1[−1/2, 1/2)d and define the auxiliary set ΓL = � m∈EL m + QL−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that the symmetric difference E∆ΓL is included in ∂E + BL−1√ d(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1, |E∆ΓL| ≤ |∂E + BL−1√ d(0)| ≲ |∂E| κL � 1 + (Lη∂E)−(d−1)� L→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Using this and setting RL = χFF −1χΓLFχF, for f ∈ L2(Rd) we have ∥(RL − S)f∥2 2 ≤ ∥(χΓL − χE)F(χFf)∥2 2 ≤ |E∆ΓL|∥F(χFf)∥2 ∞ ≤ |E∆ΓL|∥χFf∥2 1 ≤ |E∆ΓL||F|∥f∥2 2 L→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' To prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1), it only remains to show that ∥RL − SL∥ L→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' let f ∈ L2(Rd) and estimate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='∥SLf − RLf∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='ΓL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F(χFf)(w)e2πiwxdw − L−d � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m∈EL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F(χFf)(m)e2πimx��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m∈EL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m+QL−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F(χFf)(w)e2πiwx − F(χFf)(m)e2πimxdw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m∈EL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m+QL−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='f(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='e2πiw(x−t) − e2πim(x−t)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='dtdw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='≲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m∈EL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m+QL−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='|f(t)||w − m||x − t|dtdw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='≲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='m∈EL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='L−(d+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='|f(t)||x − t|dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='≲ (#EL)2L−2(d+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='∥f∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='|x − t|2dtdx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='≲ L−2max{|∂E|2d/(d−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' 1} κ2 ∂E |F|2 diam(F)2∥f∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' where in the inequality step we used Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Since SL and TE,F,L share the same non-zero eigenvalues, the estimates in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 apply also to SL for all sufficiently large L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By the Fischer-Courant formula, operator convergence of positive compact operators implies convergence of their eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Hence, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1), the estimate satisfied by the spectrum of SL extends to the spectrum of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 22 FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' SPECKBACHER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The discrete Fourier transform Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let us define E := Ω + Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Then Ω = EL for L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Let us apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 with L = 1 to E, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' We check the relevant hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For each point k ∈ ∂Ω, there exist at least one face and at most 2d faces of the cube k + Q1 that are contained in ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Therefore, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) #∂Ω ≤ ��∂E �� ≤ 2d · #∂Ω, and consequently |∂E||∂F| ≥ #∂Ω · |∂F| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Moreover, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) shows that the choice L = 1 satisfies L ≥ |∂E ��−1/(d−1) as ∂Ω contains at least one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Now fix 0 < r ≤ √ d·(#∂Ω)1/(d−1) and let us show that ∂E is regular at maximal scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' If r < 2 √ d, and x ∈ ∂E we clearly have Hd−1� ∂E ∩ Br(x) � ≳ rd−1 as E is a union of cubes of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' If r ≥ 2 √ d, set n = ⌊r/ √ d⌋ and let x ∈ ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' There exist kx ∈ ∂Ω such that |kx − x| ≤ √ d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that for y ∈ kx + Qn, |y − x| ≤ √ dn 2 + √ d 2 ≤ r 2 + √ d 2 < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Hence, kx + Qn ⊆ Br(x) and therefore, Hd−1� ∂E ∩ Br(x) � ≥ Hd−1� ∂E ∩ kx + Qn � ≥ # � ∂Ω ∩ kx + Qn � ≥ κ∂Ωnd−1 ≳ κ∂Ωrd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' This shows that ∂E is regular at scale √ d · (#∂Ω)1/(d−1) with constant Cd · κ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Note that if a set X is regular at scale ηX and constant κX, then it is also regular at scale αηX and constant min{1, α1−d}κX, for every α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1) we therefore see that ∂E is regular at scale η∂E = ��∂E ��1/(d−1) and constant κ∂E ≍ κ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' The desired estimates now follow by applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='2 to E and F, with L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Proof of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='4 First we combine Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 (for p = 1) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 to conclude that there exist a constant C = Cα,d > 0 such that if n ≥ ⌈|E| · |F|⌉ + C |∂E| κ∂E |∂F| κ∂F log �e|∂E||∂F| κ∂E �2d(1+α)+1 =: C1, then λn ≤ 1/2, and if n ≤ ⌈|E| · |F|⌉ − C |∂E| κ∂E |∂F| κ∂F log �e|∂E||∂F| κ∂E �2d(1+α)+1 =: C2, then λn ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' For ε ∈ (0, 1), define ε0 := min{ε, 1 − ε} ≤ 1/2 and let 0 < τ < ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Observe that {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', ⌊C2⌋} ∖ Mτ(S) ⊆ N1−ε0(S) ⊆ Nε(S) ⊆ Nε0(S) ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', ⌈C1⌉} ∪ Mτ(S), EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS 23 where we understand {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', ⌊C2⌋} to be ∅ if C2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Consequently, C2 − 1 − #Mτ(S) ≤ #Nε(S) ≤ C1 + 1 + #Mτ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Rearranging the last expression and using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='1 for τ gives ��Nε(S) − |E| · |F| �� ≲ |∂E| κ∂E |∂F| κ∂F log �|∂E||∂F| κ∂E τ �2d(1+α)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Letting τ ր ε0 yields (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Abreu, J.' metadata={'source': 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+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=', 66(22):5887–5901, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content=' Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria Email address: felipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='marceca@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='at Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria, and Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, Vienna, 1040, Austria Email address: jose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='luis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='romero@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='at Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria Email address: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='speckbacher@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} +page_content='at' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFJT4oBgHgl3EQf8y1j/content/2301.11685v1.pdf'} diff --git a/ctFKT4oBgHgl3EQfqC4I/content/tmp_files/2301.11872v1.pdf.txt b/ctFKT4oBgHgl3EQfqC4I/content/tmp_files/2301.11872v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..80886b2013eaf736612ea7bff8f345d5db3497bb --- /dev/null +++ b/ctFKT4oBgHgl3EQfqC4I/content/tmp_files/2301.11872v1.pdf.txt @@ -0,0 +1,311 @@ +1 + +Fracture properties of La(Fe,Mn,Si)13 magnetocaloric +materials +Siyang Wanga,*, Paul Burdettb, Edmund Lovellb, Rachel Bettlesb, Neil Wilsonb, Mary P. Ryana, +Finn Giuliania +aDepartment of Materials, Imperial College London, London, SW7 2AZ, UK +bCamfridge Ltd., Cambridge, CB22 3GN, UK +*Email: siyang.wang15@imperial.ac.uk +Keywords: La(Fe,Mn,Si)13; Strength; Crack; Fracture; Four-point bending + +Abstract +La(Fe,Mn,Si)13 alloys are a promising material family for magnetic refrigeration. Challenges +associated with their structural integrity during device assembly and operation requires deep +understanding of the mechanical properties. Here we developed a workflow to quantitatively +study the fracture properties of La(Fe,Mn,Si)13 plates used in magnetic cooling devices. We +employed microstructural characterisation, optical examination of defects, and four-point +bending tests of samples with known defect sizes to evaluate their mechanical performance. +We established the residual strength curve which directly links observed defects to +mechanical strength. The estimated fracture toughness KC of hydrogenated La(Fe,Mn,Si)13 is +approximately 4 MPa·m1/2 for the geometry employed. The established relationship between +strength and crack length enables the prediction of mechanical performance through +examination of defects via optical microscopy, therefore can be used industrially for directing +plate selection to guarantee the mechanical stability of refrigeration devices. + + + + +2 + +Magnetic refrigeration could drastically reduce CO2 emission by the cooling sector through +reduced electricity consumption, due to the potential high energy efficiency of this +technology [1,2]. A key challenge in its commercialisation arises from the poor mechanical +stability of the La(Fe,Mn,Si)13 magnetocaloric materials when subjected to simultaneous +action of several generalised forces during processing and service, leading to limited lifetime +[3,4]. Understanding the mechanical properties of these materials is therefore of high +demand, yet studies are limited [5,6]. We found previously that although the intrinsic strength +of La(Fe,Mn,Si)13 is up to ~6 GPa, strength of the actual materials in use is reduced by a factor +of ~20 likely due to cracks/pores in the microstructure [6,7] and potentially (pre-existing) +dislocations and grain boundaries. While complete removal of the cracks is difficult because +of technical limitations and service environment requirements, it is important to understand +the tolerance of cracks in the material for prolonged stable performance, thereby establishing +a guideline for quality control before device production. +The strength of La(Fe,Si)13 during compression varies with sample dimension [5], in line with +other defect-controlled materials. This highlights the importance of carrying out mechanical +testing on samples with dimensions identical to those employed in cooling devices, thereby +providing relevant properties that can be used to accurately predict in-service performance. +This is however difficult, as materials in use are often small and thin wafers (akin to human +nails in dimensions) with high surface area to volume ratio for high energy efficiency, making +them impossible to fit onto conventional mechanical testing platforms. Bespoke testing +fixture is thus necessary, and bending is a realistic geometry for testing such plate specimens. +Three-point bending is not ideal for exploring the effect of cracks on mechanical properties, +as the strength extracted is only sensitive to cracks within a small volume at the middle of the +specimens due to concentrated strain energy. Four-point bending is advantageous as it +subjects a wider area on the specimen to a more uniform stress field. We therefore employ +four-point bending for this study, to understand the variation of strength with (pre-existing) +crack length, and to estimate toughness. +La(Fe,Mn,Si)13 plates (hydrogenated, with the temperature where the magnetocaloric effect +is maximum, Tpeak – as measured by adiabatic temperature change for 0 to 1.5 T field change +– of 20.9 °C) with identical dimensions to those employed in magnetic cooling devices were +received from Camfridge Ltd. The plate thickness is in the sub-mm scale, and the thickness to + +3 + +width ratio is ~1:30. Microstructure of the plates was characterised using scanning electron +microscopy (SEM) imaging and electron backscatter diffraction (EBSD), on a FEI Quanta 650 +SEM with a Bruker eFlashHR (v2) EBSD camera using a beam voltage of 20 kV [6]. As shown in +Figure 1, the material consists of two main phases: the La(Fe,Mn,Si)13 (also termed main/1:13) +phase and the α-Fe phase. The nominal volume fraction of the α-Fe phase is 15%. The pole +figures indicate an absence of texture in the main phase. The composition of the 1:13 phase, +as measured by SEM-energy dispersive X-ray spectroscopy (SEM-EDX), is LaFe10.7Mn0.4Si1.2. + +Figure 1 (a) Secondary electron image, (b) phase map, (c) inverse pole figure (IPF)-out-of-page map, and (d) +pole figures (for the 1:13 phase) of an area on the as-received specimen. In (b) and (c), grain boundaries of the +1:13 phase, and phase boundaries between La(Fe,Mn,Si)13 and α-Fe are highlighted with black and white lines, +respectively. MUD denotes multiples of the uniform density [6]. Note that the image in (a) shows the cross +sections of the pores, of which the 3D shapes could be revealed by X-ray or focussed ion beam tomography. +A bespoke four-point bending fixture was designed and fabricated (Figure 2(a)), which +enabled testing of the plates on a Gatan/Deben Microtest 300 mechanical testing stage. The +loading span was designed to be ½ of the support span. Before mechanical testing, optical +microscopy examination of each individual plate (both sides) was conducted, in order to pick +out plates with (typically) one pre-existing through-thickness crack present within the region +that will be subjected to the stress field between the two loading/inner pins during the tests. +The cracks were likely generated upon cutting and/or handling of the plates, and their growth + +a +b) +La-richphase +pore +a-Fephase +a-Fe +30 μm +1:13 +c) +d) +(001) +(111) +(011) +3.2 +MUD +[001] +0 +[111] +110114 + +may be retarded by the α-Fe phase [4]. The length of each crack was measured. Five plates +without observed pre-existing cracks were tested for comparison. Tests were performed with +a displacement speed of 0.1 mm/min. Figure 2(b) shows a typical load-displacement response +of the plates during the tests, where fracture occurred immediately after linear elastic loading +without plastic flow, as per prior three-point bending tests [6]. + +Figure 2 (a) The bespoke four-point bending fixture for testing the plate specimens. (b) A typical load- +displacement curve for the plates recorded during the tests. (c) Optical micrograph showing a pre-existing +crack in a plate, and pictures of (d) a plate with a pre-existing crack and (e) a plate without observable pre- +existing cracks after four-point bending tests. (f) SEM image of the fracture surface for a sample post-test. +Figure 2(c) shows an optical micrograph of a plate captured before the tests revealing a pre- +existing crack. After the tests, plates with pre-existing cracks all fracture into two pieces. The +cracks were found to develop along the pre-existing cracks, and an example is shown in Figure +2(d). In contrast, plates without observable pre-existing cracks broke into three nearly even +pieces, where the two cracks were both within the region between the loading/inner pins for +all the plates (Figure 2(e)). The fracture mode is mainly intragranular as evidenced by Figure +2(f), in agreement with our prior work on hydrogenated La(Fe,Mn,Si)13 [6]. We found +previously that hydrogenation changes the fracture mode of this material upon bending tests +[6], but we note that in-service fracture mode may also be affected by factors such as +corrosive media [7] and/or external stress state [8]. + +2.5 +Inner/loadingpin5 +2 +1.5 +peo +O +1 +Outer/su +0.5 +0 +0.05 +0.1 +Displacement (mm) +re-existing crac +50 μm +50 μm5 + + +Figure 3 Variation of fracture strength of the plates as a function of total crack length (black) and x component +of crack length (red) which is the component of crack length along the direction perpendicular to principal +stress direction during four-point bending tests (x and y axes are illustrated in the insert). The dashed line is +the fit of the data for the fracture strength vs. x component of the crack length to Equation 1. +The variation of plate strength with pre-existing crack length, or residual strength curve, is +shown in Figure 3. Plates without observable pre-existing cracks exhibit fracture strengths of +130-170 MPa, in agreement with prior work [6]. Broadly, as the crack length increases, the +fracture strength decreases. This graph serves as a guideline for quality control of plates used +to assemble regenerators. The established relationship between strength and crack length +enables the prediction of mechanical performance through examination of defects via optical +microscopy, therefore can be used for directing plate selection to guarantee the mechanical +stability of refrigeration devices. Note that residual strength curves for brittle materials are +often accessed via analytical prediction based on the theory of fracture mechanics and +material properties, for example [9]. Instead, rarely have they been established directly +through experimental measurements to our best knowledge, and only few data sources such +as [10] are available. This likely arises from the difficulty in obtaining well-defined samples +with a broad distribution of defect sizes. However, we have shown here that through careful +examination and testing of the plates, direct establishment of the residual strength curve can +be achieved for this particular material and geometry, and is vastly useful for controlling in- +service performance. +If strength is controlled by brittle fracture under (far-field) tensile stress, then the variation of +strength with crack size should generally follow Equation 1 [11]. + +160 +Total crack +x component +Fracture strength (MPa) +120 +Fit (x component) +80 +40 +0 +0 +0.5 +1 +1.5 +2 +2.5 +3 +Cracklength（mm）6 + +𝜎 = +𝐾𝐶 +√𝜋𝑎 1 +where σ is the fracture strength, KC the fracture toughness, and a the crack length. +Fitting the data for the fracture strength vs. x component of the crack length to Equation 1 +(dashed line in Figure 3) gives an estimated KC of 4 MPa·m1/2, which falls into the characteristic +range of brittle materials [12]. Note that this is only a rough estimation of KC, given the stress +state being not uniform tensile and the crack length being not negligible compared to plate +width. However, this value extracted aligns with the load-displacement behaviour (Figure 2(b)) +which indicates the brittle nature of the material, and prior tests where even μm-scale test +pieces failed in a quasi-brittle manner [6]. + +Figure 4 Summary of the workflow established in this work to quantify the correlation between fracture +strength and pre-existing crack length, which can be used for quality control and mechanical stability +prediction of regenerators in magnetic refrigeration systems. +In summary, we established a workflow (Figure 4) which enabled quantification of the +correlation between fracture strength and pre-existing crack length of La(Fe,Mn,Si)13 +magnetocaloric material. The results, obtained through testing plate specimens with identical +geometry to those used in magnetic cooling devices, can be directly used for quality control +and mechanical stability prediction of regenerators in magnetic refrigeration systems. For the +material studied, evaluation of the fracture toughness was achieved for the first time. An + +Crack examination +Four-point bending fixture +Bending tests to +& measurement +design & manufacture +measure strength +0.5 +50 μm +0.05 +0.1 +Displacement (mm) +Assembly of device +Establish strength-crack +Automated crack examination +with longevity +& weak plate elimination +length relationship & work +out critical crack size +160 +Total cracl +0.57 + +estimated KC of 4 MPa·m1/2 for the testing geometry employed indicates the brittle nature of +the material. + +Acknowledgements +We acknowledge funding from Innovate UK (UKRI 32645). + +References +[1] +M. Balli, S. Jandl, P. Fournier, A. Kedous-Lebouc, Advanced materials for magnetic +cooling: Fundamentals and practical aspects, Appl. Phys. Rev. 4 (2017). +https://doi.org/10.1063/1.4983612. +[2] +V. Franco, J.S. Blázquez, J.J. Ipus, J.Y. Law, L.M. Moreno-Ramírez, A. Conde, +Magnetocaloric effect: From materials research to refrigeration devices, Prog. Mater. +Sci. 93 (2018) 112–232. https://doi.org/10.1016/j.pmatsci.2017.10.005. +[3] +S. Lionte, A. Barcza, M. Risser, C. Muller, M. Katter, LaFeSi-based magnetocaloric +material analysis: Cyclic endurance and thermal performance results, Int. J. Refrig. 124 +(2021) 43–51. https://doi.org/10.1016/j.ijrefrig.2020.12.004. +[4] +S. Wang, J.O. Douglas, E. Lovell, N. Wilson, L. Guo, B. Gault, M.P. Ryan, F. Giuliani, Near- +atomic scale chemical analysis of interfaces in a La(Fe,Mn,Si)13-based magnetocaloric +material, +Scr. +Mater. +224 +(2023) +115143. +https://doi.org/10.1016/j.scriptamat.2022.115143. +[5] +O. Glushko, A. Funk, V. Maier-Kiener, P. Kraker, M. Krautz, J. Eckert, A. Waske, +Mechanical properties of the magnetocaloric intermetallic LaFe11.2Si1.8 alloy at +different +length +scales, +Acta +Mater. +165 +(2019) +40–50. +https://doi.org/10.1016/j.actamat.2018.11.038. +[6] +S. Wang, O. Gavalda-Diaz, T. Luo, L. Guo, E. Lovell, N. Wilson, B. Gault, M.P. Ryan, F. +Giuliani, The effect of hydrogen on the multiscale mechanical behaviour of a +La(Fe,Mn,Si)13-based magnetocaloric material, J. Alloys Compd. 906 (2022) 164274. + +8 + +https://doi.org/10.1016/j.jallcom.2022.164274. +[7] +S. Pan, J. Yuan, C. Linsley, J. Liu, X. Li, Corrosion behavior of nano-treated AA7075 alloy +with +TiC +and +TiB2 +nanoparticles, +Corros. +Sci. +206 +(2022) +110479. +https://doi.org/10.1016/j.corsci.2022.110479. +[8] +X. Yang, S. Gao, Analysis of the crack propagation mechanism of multiple scratched +glass-ceramics by an interference stress field prediction model and experiment, Ceram. +Int. 48 (2022) 2449–2458. https://doi.org/10.1016/j.ceramint.2021.10.026. +[9] +Z.H. Jin, R.C. Batra, Some basic fracture mechanics concepts in functionally graded +materials, J. Mech. Phys. Solids. 44 (1996) 1221–1235. https://doi.org/10.1016/0022- +5096(96)00041-5. +[10] +T.W. Orange, Fracture Toughness of Wide 2014-T6 Aluminum Sheet at -320 °F, +Washington, D. C., 1967. https://ntrs.nasa.gov/search.jsp?R=19670018968. +[11] +N.E. Dowling, K.S. Prasad, R. Narayanasamy, Mechanical Behavior of Materials : +Engineering Methods for Deformation, Fracture, and Fatigue, 4th & intl ed., Pearson, +Boston, Mass., 2013. +[12] +G.A. Gogotsi, Fracture toughness of ceramics and ceramic composites, Ceram. Int. 29 +(2003) 777–784. https://doi.org/10.1016/S0272-8842(02)00230-4. + + diff --git a/ctFKT4oBgHgl3EQfqC4I/content/tmp_files/load_file.txt b/ctFKT4oBgHgl3EQfqC4I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d7f91f886ef197cfc42033bd9c4f0a03f487bcc --- /dev/null +++ b/ctFKT4oBgHgl3EQfqC4I/content/tmp_files/load_file.txt @@ -0,0 +1,270 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf,len=269 +page_content='1 Fracture properties of La(Fe,Mn,Si)13 magnetocaloric materials Siyang Wanga,*, Paul Burdettb, Edmund Lovellb, Rachel Bettlesb, Neil Wilsonb, Mary P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Ryana, Finn Giuliania aDepartment of Materials, Imperial College London, London, SW7 2AZ, UK bCamfridge Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=', Cambridge, CB22 3GN, UK Email: siyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='wang15@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='uk Keywords: La(Fe,Mn,Si)13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Strength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Crack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Fracture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Four-point bending Abstract La(Fe,Mn,Si)13 alloys are a promising material family for magnetic refrigeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Challenges associated with their structural integrity during device assembly and operation requires deep understanding of the mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Here we developed a workflow to quantitatively study the fracture properties of La(Fe,Mn,Si)13 plates used in magnetic cooling devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' We employed microstructural characterisation, optical examination of defects, and four-point bending tests of samples with known defect sizes to evaluate their mechanical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' We established the residual strength curve which directly links observed defects to mechanical strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The estimated fracture toughness KC of hydrogenated La(Fe,Mn,Si)13 is approximately 4 MPa·m1/2 for the geometry employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The established relationship between strength and crack length enables the prediction of mechanical performance through examination of defects via optical microscopy, therefore can be used industrially for directing plate selection to guarantee the mechanical stability of refrigeration devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' 2 Magnetic refrigeration could drastically reduce CO2 emission by the cooling sector through reduced electricity consumption, due to the potential high energy efficiency of this technology [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' A key challenge in its commercialisation arises from the poor mechanical stability of the La(Fe,Mn,Si)13 magnetocaloric materials when subjected to simultaneous action of several generalised forces during processing and service, leading to limited lifetime [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Understanding the mechanical properties of these materials is therefore of high demand, yet studies are limited [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' We found previously that although the intrinsic strength of La(Fe,Mn,Si)13 is up to ~6 GPa, strength of the actual materials in use is reduced by a factor of ~20 likely due to cracks/pores in the microstructure [6,7] and potentially (pre-existing) dislocations and grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' While complete removal of the cracks is difficult because of technical limitations and service environment requirements, it is important to understand the tolerance of cracks in the material for prolonged stable performance, thereby establishing a guideline for quality control before device production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The strength of La(Fe,Si)13 during compression varies with sample dimension [5], in line with other defect-controlled materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' This highlights the importance of carrying out mechanical testing on samples with dimensions identical to those employed in cooling devices, thereby providing relevant properties that can be used to accurately predict in-service performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' This is however difficult, as materials in use are often small and thin wafers (akin to human nails in dimensions) with high surface area to volume ratio for high energy efficiency, making them impossible to fit onto conventional mechanical testing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Bespoke testing fixture is thus necessary, and bending is a realistic geometry for testing such plate specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Three-point bending is not ideal for exploring the effect of cracks on mechanical properties, as the strength extracted is only sensitive to cracks within a small volume at the middle of the specimens due to concentrated strain energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Four-point bending is advantageous as it subjects a wider area on the specimen to a more uniform stress field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' We therefore employ four-point bending for this study, to understand the variation of strength with (pre-existing) crack length, and to estimate toughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' La(Fe,Mn,Si)13 plates (hydrogenated, with the temperature where the magnetocaloric effect is maximum, Tpeak – as measured by adiabatic temperature change for 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 T field change – of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='9 °C) with identical dimensions to those employed in magnetic cooling devices were received from Camfridge Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The plate thickness is in the sub-mm scale, and the thickness to 3 width ratio is ~1:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Microstructure of the plates was characterised using scanning electron microscopy (SEM) imaging and electron backscatter diffraction (EBSD), on a FEI Quanta 650 SEM with a Bruker eFlashHR (v2) EBSD camera using a beam voltage of 20 kV [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' As shown in Figure 1, the material consists of two main phases: the La(Fe,Mn,Si)13 (also termed main/1:13) phase and the α-Fe phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The nominal volume fraction of the α-Fe phase is 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The pole figures indicate an absence of texture in the main phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The composition of the 1:13 phase, as measured by SEM-energy dispersive X-ray spectroscopy (SEM-EDX), is LaFe10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='7Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='4Si1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Figure 1 (a) Secondary electron image, (b) phase map, (c) inverse pole figure (IPF)-out-of-page map, and (d) pole figures (for the 1:13 phase) of an area on the as-received specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' In (b) and (c), grain boundaries of the 1:13 phase, and phase boundaries between La(Fe,Mn,Si)13 and α-Fe are highlighted with black and white lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' MUD denotes multiples of the uniform density [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Note that the image in (a) shows the cross sections of the pores, of which the 3D shapes could be revealed by X-ray or focussed ion beam tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' A bespoke four-point bending fixture was designed and fabricated (Figure 2(a)), which enabled testing of the plates on a Gatan/Deben Microtest 300 mechanical testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The loading span was designed to be ½ of the support span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Before mechanical testing, optical microscopy examination of each individual plate (both sides) was conducted, in order to pick out plates with (typically) one pre-existing through-thickness crack present within the region that will be subjected to the stress field between the two loading/inner pins during the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The cracks were likely generated upon cutting and/or handling of the plates, and their growth a b) La-richphase pore a-Fephase a-Fe 30 μm 1:13 c) d) (001) (111) (011) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='2 MUD [001] 0 [111] 110114 may be retarded by the α-Fe phase [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The length of each crack was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Five plates without observed pre-existing cracks were tested for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Tests were performed with a displacement speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='1 mm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Figure 2(b) shows a typical load-displacement response of the plates during the tests, where fracture occurred immediately after linear elastic loading without plastic flow, as per prior three-point bending tests [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Figure 2 (a) The bespoke four-point bending fixture for testing the plate specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' (b) A typical load- displacement curve for the plates recorded during the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' (c) Optical micrograph showing a pre-existing crack in a plate, and pictures of (d) a plate with a pre-existing crack and (e) a plate without observable pre- existing cracks after four-point bending tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' (f) SEM image of the fracture surface for a sample post-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Figure 2(c) shows an optical micrograph of a plate captured before the tests revealing a pre- existing crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' After the tests, plates with pre-existing cracks all fracture into two pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The cracks were found to develop along the pre-existing cracks, and an example is shown in Figure 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' In contrast, plates without observable pre-existing cracks broke into three nearly even pieces, where the two cracks were both within the region between the loading/inner pins for all the plates (Figure 2(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The fracture mode is mainly intragranular as evidenced by Figure 2(f), in agreement with our prior work on hydrogenated La(Fe,Mn,Si)13 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' We found previously that hydrogenation changes the fracture mode of this material upon bending tests [6], but we note that in-service fracture mode may also be affected by factors such as corrosive media [7] and/or external stress state [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 Inner/loadingpin5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 peo O 1 Outer/su 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='1 Displacement (mm) re-existing crac 50 μm 50 μm5 Figure 3 Variation of fracture strength of the plates as a function of total crack length (black) and x component of crack length (red) which is the component of crack length along the direction perpendicular to principal stress direction during four-point bending tests (x and y axes are illustrated in the insert).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The dashed line is the fit of the data for the fracture strength vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' x component of the crack length to Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The variation of plate strength with pre-existing crack length, or residual strength curve, is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Plates without observable pre-existing cracks exhibit fracture strengths of 130-170 MPa, in agreement with prior work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Broadly, as the crack length increases, the fracture strength decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' This graph serves as a guideline for quality control of plates used to assemble regenerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The established relationship between strength and crack length enables the prediction of mechanical performance through examination of defects via optical microscopy, therefore can be used for directing plate selection to guarantee the mechanical stability of refrigeration devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Note that residual strength curves for brittle materials are often accessed via analytical prediction based on the theory of fracture mechanics and material properties, for example [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Instead, rarely have they been established directly through experimental measurements to our best knowledge, and only few data sources such as [10] are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' This likely arises from the difficulty in obtaining well-defined samples with a broad distribution of defect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' However, we have shown here that through careful examination and testing of the plates, direct establishment of the residual strength curve can be achieved for this particular material and geometry, and is vastly useful for controlling in- service performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' If strength is controlled by brittle fracture under (far-field) tensile stress, then the variation of strength with crack size should generally follow Equation 1 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' 160 Total crack x component Fracture strength (MPa) 120 Fit (x component) 80 40 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 3 Cracklength（mm）6 𝜎 = 𝐾𝐶 √𝜋𝑎 1 where σ is the fracture strength, KC the fracture toughness, and a the crack length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Fitting the data for the fracture strength vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' x component of the crack length to Equation 1 (dashed line in Figure 3) gives an estimated KC of 4 MPa·m1/2, which falls into the characteristic range of brittle materials [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Note that this is only a rough estimation of KC, given the stress state being not uniform tensile and the crack length being not negligible compared to plate width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' However, this value extracted aligns with the load-displacement behaviour (Figure 2(b)) which indicates the brittle nature of the material, and prior tests where even μm-scale test pieces failed in a quasi-brittle manner [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Figure 4 Summary of the workflow established in this work to quantify the correlation between fracture strength and pre-existing crack length, which can be used for quality control and mechanical stability prediction of regenerators in magnetic refrigeration systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' In summary, we established a workflow (Figure 4) which enabled quantification of the correlation between fracture strength and pre-existing crack length of La(Fe,Mn,Si)13 magnetocaloric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' The results, obtained through testing plate specimens with identical geometry to those used in magnetic cooling devices, can be directly used for quality control and mechanical stability prediction of regenerators in magnetic refrigeration systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' For the material studied, evaluation of the fracture toughness was achieved for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' An Crack examination Four-point bending fixture Bending tests to & measurement design & manufacture measure strength 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='5 50 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='1 Displacement (mm) Assembly of device Establish strength-crack Automated crack examination with longevity & weak plate elimination length relationship & work out critical crack size 160 Total cracl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content='57 estimated KC of 4 MPa·m1/2 for the testing geometry employed indicates the brittle nature of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Acknowledgements We acknowledge funding from Innovate UK (UKRI 32645).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFKT4oBgHgl3EQfqC4I/content/2301.11872v1.pdf'} +page_content=' Balli, S.' metadata={'source': 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Planetary nebula luminosity function +Azlizan A. Soemitro1, 2, Martin M. Roth1, 2, Peter M. Weilbacher1, Robin Ciardullo3, 4, George H. Jacoby5, Ana +Monreal-Ibero6, Norberto Castro1, and Genoveva Micheva1 +1 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany +e-mail: asoemitro@aip.de +2 Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany +3 Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA +4 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA +5 NSF’s NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA +6 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands +Received ; accepted +ABSTRACT +Aims. We perform a deep survey of planetary nebulae (PNe) in the spiral galaxy NGC 300 to construct its planetary nebula luminosity +function (PNLF). We aim to derive the distance using the PNLF and to probe the characteristics of the most luminous PNe. +Methods. We analyse 44 fields observed with MUSE at the VLT, covering a total area of ∼ 11 kpc2. We find [O iii]λ5007 sources +using the differential emission line filter (DELF) technique. We identify PNe through spectral classification using the aid of the BPT- +diagram. The PNLF distance is derived using the maximum likelihood estimation technique. For the more luminous PNe, we also +measure their extinction using the Balmer decrement. We estimate the luminosity and effective temperature of the central stars of the +luminous PNe, based on estimates of the excitation class and the assumption of optically thick nebulae. +Results. We identify 107 PNe and derive a most-likely distance modulus (m − M)0 = 26.48+0.11 +−0.26 (d = 1.98+0.10 +−0.23 Mpc). We find that +the PNe at the PNLF cut-off exhibit relatively low extinction, with some high extinction cases caused by local dust lanes. We present +the lower limit luminosities and effective temperatures of the central stars for some of the brighter PNe. We also identify a few Type I +PNe that come from a young population with progenitor masses > 2.5 M⊙, however do not populate the PNLF cut-off. +Conclusions. The spatial resolution and spectral information of MUSE allow precise PN classification and photometry. These ca- +pabilities also enable us to resolve possible contamination by diffuse gas and dust, improving the accuracy of the PNLF distance to +NGC 300. +Key words. galaxies: stellar content – planetary nebulae: general – galaxies: luminosity function, mass function – distance scale – +stars: AGB and post-AGB +1. Introduction +The planetary nebula luminosity function (PNLF) is a distance +determination method with a precision and accuracy that is com- +parable to those of the tip of the red giant branch (TRGB) and +Cepheid methods (Jacoby 1989; Ciardullo et al. 1989; Ciardullo +2010, 2012; Roth et al. 2021). Using evidence from narrow- +band photometric surveys in [O iii]λ5007, Ciardullo et al. (1989) +have shown that the magnitude distribution of planetary nebulae +(PNe) for a given galaxy follows an empirical power law defined +as +N(M) ∝ e0.307M{1 − e3(M∗−M)} +(1) +where the brightest PN at the cut-off has an absolute magni- +tude of M∗ = −4.53 ± 0.06 with a possible minor dependency +on metallicity (Jacoby 1989; Dopita et al. 1992; Ciardullo et al. +2002; Ciardullo 2012). While a number of different formulations +have been developed to model the various shapes of the PNLF +at fainter magnitudes (Rodríguez-González et al. 2015; Longob- +ardi et al. 2013; Bhattacharya et al. 2019, 2021), such faint-end +variation do not affect the definition of the PNLF’s bright end +cut-off, which is the critical feature for distance determinations +(Spriggs et al. 2021; Ciardullo 2022). +Until the early 2010s, most PNLF distance measurements +were obtained using 4-meter class telescopes and narrow-band +interference filters, and as a result, the method has been tradi- +tionally limited to distances of ∼ 20 Mpc (Jacoby et al. 1990; +Ciardullo 2010, 2012, 2022). Although 8-meter class telescopes +were available and even observed PNe at the Coma cluster (∼ +100 Mpc, Gerhard et al. 2005), most of the instruments had +wider bandpass filters, which increased the inclusion of sky +background signal. This limited the PN detection sensitivity, that +was necessary to significantly improve the distance range of the +PNLF (Ciardullo 2022). This situation has now changed due to +the use of the Multi Unit Spectroscopic Explorer (MUSE, Ba- +con et al. 2010) integral-field spectrograph on the 8.2-meter Very +Large Telescope to survey PNe in distant systems (Spriggs et al. +2020, 2021; Roth et al. 2021; Scheuermann et al. 2022). In fact, +Roth et al. (2021) have shown that by using a differential emis- +sion line filter technique on MUSE data, PNLF measurements +are now possible out to distances of ∼ 40 Mpc under excellent +seeing condition and with the aid of the adaptive optics system. +This is mainly due to the narrow effective bandpass of MUSE, +Article number, page 1 of 20 +arXiv:2301.03437v1 [astro-ph.GA] 9 Jan 2023 + +A&A proofs: manuscript no. pnlf_ngc300 +that is five times narrower than the typical narrow-band filters, +which can substantially suppressed the background sky noise +(Roth et al. 2021). +Previous +PNLF +studies +of +late +type +galaxies +using +[O iii]λ5007 narrow-band filters were also hampered by the pos- +sible confusion with supernova remnants (SNRs) or H ii regions +(Herrmann et al. 2008; Herrmann & Ciardullo 2009; Frew & +Parker 2010). While Hα narrow-band image allows the exclu- +sion of H ii regions, it cannot be used to exclude the SNRs, +whose classification typically rely on the [S ii]λ6716, 6731 lines. +In M31 and M33, Davis et al. (2018) found that the SNR con- +tamination does not change the shape nor the position of the +PNLF cutoff. In contrary, Kreckel et al. (2017) have shown with +MUSE observations of NGC 628 that the presence of SNR con- +taminants can affect the bright cut-off. However, Scheuermann +et al. (2022) demonstrated that the latter study had an issue with +the background subtraction in Hα, which affected the classifica- +tion. Their reanalysis concluded that the PNLF cutoff was indeed +unaffected by the contaminants. Nevertheless, the possible con- +tamination by SNRs is relevant for the investigation of the faint +end of the PNLF. The impressive spatial resolution and spec- +troscopic capability of the MUSE instrument allows the instant +identification of interlopers, even in star forming disk galaxies +(Kreckel et al. 2017; Roth et al. 2018, 2021; Scheuermann et al. +2022). +NGC 300 is a spiral galaxy in the foreground of the Sculp- +tor group. Being fairly isolated from its neighbouring galaxies +(Karachentsev et al. 2003) and close in distance (Gieren et al. +2005; Rizzi et al. 2006, 2007) makes it interesting for studying +star formation histories and galactic evolution (Muñoz-Mateos +et al. 2007; Kudritzki et al. 2008; Bernard-Salas et al. 2009; +Gogarten et al. 2010; Jang et al. 2020). A previous PNLF study, +Soffner et al. (1996) identified 34 PNe, a small number that was +not ideal to make a proper PNLF and therefore opted the cu- +mulative PNLF to derive the distance by using the LMC as a +yardstick. More recently, Peña et al. (2012) observed 104 PN +candidates using narrow-band imaging from the central and the +eastern outskirt region to construct the PNLF, with a follow-up +spectroscopy for the brighter candidates (Stasi´nska et al. 2013). +In Paper I (Roth et al. 2018), seven 1′ × 1′ MUSE fields in the +central region of NGC 300 were observed with the goal of re- +solving stellar populations in crowded fields of nearby galaxies, +from which they discovered 45 PN candidates. Again, this num- +ber was too small to create a useful PNLF, since the sample spans +a very wide magnitude range of 22 ≲ m5007 ≲ 29. In the present +work, using publicly available archival data from McLeod et al. +(2020, 2021) – or ML20 – and 2 additional MUSE-GTO fields, +we expand the observed area from 7 to 44 MUSE fields in or- +der to detect more PNe and obtain a PNLF distance to NGC 300 +using integral field spectroscopy. +Our lack of a complete understanding of the underlying +physics behind the invariance of the PNLF cut-off has prevented +the PNLF technique to become a primary standard candle (Ciar- +dullo 2010, 2012). Although simulations have provided an im- +pression of the physical properties of the most luminous PNe +(Jacoby 1989; Dopita & Meatheringham 1990, 1991; Mendez +& Soffner 1997; Méndez et al. 2008b; Schönberner et al. 2007, +2010; Valenzuela et al. 2019), an observational characterisation +is still limited to the LMC (Dopita & Meatheringham 1991; Do- +pita et al. 1992; Reid & Parker 2010a,b), and M31 (Kwitter et al. +2012; Davis et al. 2018; Galera-Rosillo et al. 2022). If the most +luminous PNe at the PNLF cut-off have indeed originated from +a single-star stellar evolution, then placing the central stars in +the HR-diagram will provide insights into the underlying stellar +population, and also the nature of the cut-off itself. Using the +data quality that MUSE offers, we aim to constrain the lumi- +nosity and effective temperature of the central stars for some of +the bright PNe to understand their origin and expand our under- +standing of PNe beyond the Local Group. +The structure of this paper is as follows: details on obser- +vations and data reduction are described in Section 2. The data +analysis regarding the PN detection and classification, the liter- +ature comparison of the PN number, the [O iii]λ5007 photome- +try, and the measurement of the Balmer decrement is explained +in Section 3. The resulting luminosity function and the distance +measurement are presented in Section 4. The discussion and the +implications of this work follow in Section 5. Lastly, the conclu- +sions are given in Section 6. +2. Observations and data reduction +The data for this project were acquired using the MUSE spec- +trograph on the 8.2-meter Very Large Telescope (Bacon et al. +2010). 9 fields were obtained as part of the MUSE guaranteed +time observation (GTO) program1, while 35 fields were taken +from the ESO Archive2. The area covered by these observations +is shown in Figure 1. +The initial MUSE-GTO data (field A, B, C, D, E, I, J) were +obtained between the years 2014 – 2016 using the extended wide +field mode with a spatial coverage of 1′×1′ and spectral coverage +of 4650 − 9350 Å with 1.25 Å sampling. First results from the 7 +fields in the centre area were reported in Paper I, covering the +nucleus, part of the spiral arm that extends from the nucleus to +the north-west, and inter-arm regions of the galaxy. In late 2018, +additional fields P and Q were observed to cover the part of the +outer spiral arm. Moreover, fields A, B, and C were re-observed +with adaptive optics support to obtain better image quality. In the +adaptive optics mode, a notch filter at 5750 − 6100 Å blocks the +laser light, which is, however, not affecting the emission lines of +interest. Most of the fields were obtained with an exposure time +of 6 × 900 s, with the exception of field J (4 × 900 s), field C +(8 × 900 s), and field B, D (11 × 900 s). +The 9 MUSE-GTO fields were reduced using the MUSE +pipeline (Weilbacher et al. 2020) within the MUSE-WISE envi- +ronment (Vriend 2015), as explained in more detail in Paper III +(Micheva et al. 2022). In addition to the field distortion correc- +tion produced by the pipeline, the astrometry is also calibrated +using the Gaia DR2 catalogue (Gaia Collaboration 2018), pro- +viding absolute astrometry within 0′′.1. Sky subtraction was per- +formed using an offset field outside the galaxy. Since we will per- +form photometry in [O iii]λ5007, we measure the seeing quality +at this wavelength based on the FWHM of 3 to 4 stars for each +field. These stars, which are typically giants or supergiants in the +disk of NGC 300, have apparent magnitudes of F606W ≳ 21 in +the HST ACS magnitude system (Roth et al. 2018). Unlike the +situation in more distant systems, such as Fornax cluster ellip- +ticals (Sextl et al. 2021), confusion with globular clusters is not +a concern. For the MUSE GTO data, the image quality ranges +from 0′′.6 − 0′′.8 FWHM, as presented in Appendix C. +Another 35 fields, the ML20 data, publicly available at the +ESO Archive, were originally observed to study stellar feedback +in NGC 300 (McLeod et al. 2020, 2021). The data were obtained +using the nominal wide field mode, which has the same spatial +1 Program IDs 094.D-0116, 095.D-0173, 097.D-0348, and 0102.B- +0317 – PI: Roth +2 Program ID 098.B-0193(A) – PI: McLeod +Article number, page 2 of 20 + +Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +Fig. 1. MUSE fields of NGC 300 are marked with red, green, and cyan. The red fields are MUSE-GTO data from the Paper I pilot study. The green +fields labelled P and Q are additional MUSE-GTO observation for the outer spiral arm. The cyan fields indicate ML20 fields. The magenta fields +are the previous PNe survey area of Peña et al. (2012) using the FORS2 instrument. Image: NGC 300 in Hα taken with the Wide Field Imager +(ESO) – Program ID 065.N-0076 +. +coverage of 1′ × 1′, but with a slightly shorter wavelength cov- +erage of 4750 − 9350 Å. The observations were conducted in the +period of 2016 – 2018 without the support of adaptive optics, and +each field was observed with an exposure time of 3×900s. These +data were reduced using the fully automated MUSE pipeline +(Weilbacher et al. 2020) with default parameters, as provided +in the ESO Archive. The astrometry of the ML20 data only re- +lied on the distortion correction within each field, which limited +the absolute positional accuracy of the object catalogue to ∼ 3′′. +Moreover, the sky subtraction was performed using a reference +region within each field instead of an offset field; this resulted +in sky oversubtraction, especially in areas where diffuse gas is +prominent. However, since we perform local sky subtraction for +flux measurements of individual objects (see Section 3), the ef- +fect cancels out. Based on our measurements, the seeing qual- +ity of the ML20 data in [O iii]λ5007 ranges between 0′′.8 − 1′′.5 +FWHM. These [O iii]λ5007 seeing measurements are presented +in Appendix C. +3. Data analysis +3.1. PN detection and classification +To find PN candidates, we employed the differential emission +line filter (DELF) method described by Roth et al. (2021). +This is performed by extracting 15 datacube layers around the +wavelength of redshifted [O iii]λ5007 (the systematic velocity of +NGC 300 is vsys = 144 km/s; Lauberts & Valentijn 1989) and +treating each layer as an on-band image; this 18.75 Å range ac- +counts for the different line-of-sight velocities (LOSV) within +the galaxy. Then, an intermediate broadband continuum image is +constructed from the wavelength range between λ5063−5188 Å, +which is free from strong absorption line features; this is used +as the off-band image. By subtracting the scaled off-band im- +age (see scaling factor in Equation 8, Roth et al. 2021) from +the on-band images, we obtain a series of continuum-free dif- +ferential images. Using the DS9 software (Joye & Mandel +2003), the differential images are visually inspected to find the +[O iii]λ5007 sources. After experimenting unsuccessfully with +DAOPHOT FIND (Stetson 1987) to identify PN candidates, +which turned out to be unable to cope with the spatially variable +emission line background in [O iii]λ5007, we resorted to the dat- +acube layer blinking technique, that is described in Roth et al. +(2021). +The typical physical size of planetary nebulae is of the of +order ∼ 0.3 pc (Osterbrock & Ferland 2006). If we assume a dis- +tance of 1.88 Mpc (Gieren et al. 2005) and a scale of ∼ 9 pc/′′, +we expect the PNe in NGC 300 to appear as point sources. After +marking the coordinates of the point sources in [O iii]λ5007, we +apply aperture photometry (Stetson 1987) at each wavelength +layer along the datacube for these objects to obtain their spec- +tra. We employ an aperture diameter of 3 spaxels (0′′.6), an in- +ner sky annulus of 12 spaxels, and an outer sky annulus of 15 +spaxels. Although the seeing conditions of the datacubes vary +Article number, page 3 of 20 + +PE12centel +PE12outskil +H13 +H15A&A proofs: manuscript no. pnlf_ngc300 +between ∼ 0′′.6 − 1′′.5, we extract the spectra using the same pa- +rameters. The small aperture of 3 spaxels is chosen to minimise +contamination of background gas or nearby H ii regions. Then, +the line fluxes are extracted using Gaussian fitting with the LM- +FIT routine in Python (Newville et al. 2016), keeping in mind +that the MUSE data has a wavelength sampling of 1.25 Å. Since +the MUSE-GTO and the ML20 data have overlapping areas, the +classifications were done independently for each data set. Cross- +matching was performed after the PNe were identified. +To classify the sources into PNe, H ii regions, and super- +nova remnants (SNR), we employed the BPT-diagram (Baldwin +et al. 1981) that is based on the line ratio of [O iii]λ5007/Hβ and +[S ii]λλ6716, 6731/Hα. Besides the application of classifying ac- +tive galaxies (Kewley et al. 2001, 2006), the BPT-diagram has +been demonstrated to effectively discriminate the PNe from their +mimics (Kniazev et al. 2008; Frew & Parker 2010; Sabin et al. +2013; Roth et al. 2021). As the classification rely on emission +line ratios with very similar wavelengths, we can assume that +the relative line fluxes have negligible extinction and seeing de- +pendency on wavelength. +Our BPT-diagrams for the MUSE-GTO and ML20 data are +presented in Figure 2. To separate the SNRs, we adopt the value +log [S ii]λλ6716, 6731/Hα ≥ −0.5 from Roth et al. (2021). To +discriminate PNe from H ii regions, we employ the theoretical +line by Kewley et al. (2001), which was originally intended to +differentiate starburst galaxies. We also consider the line ratio of +[S ii]λ6731/6716 as a proxy for density, since bright PNe are ex- +pected to be denser than both H ii regions and SNRs (Osterbrock +& Ferland 2006). While the brightest [O iii]λ5007 sources have +sufficient line fluxes for the BPT-diagram classification, fainter +sources may lack the weaker emission lines, i.e. the Hβ line +or the [S ii]λ6716, 6731 lines. In such cases, we assume lower +limits for the line fluxes and classify the sources as PNe if the +[O iii]λ5007 line is stronger than the Hα line, which may in- +troduce deviation from the separation lines in the diagram. We +also found some faint objects, which only have the detection of +the [O iii]λ5007 and the [N ii]λ6584 line without Hα detection, +which are possibly Type I PNe (Frew & Parker 2010). Moreover, +we also put remarks for PNe, which are only classified solely +based on [O iii]λ5007 detection. This is the case for few of our +faintest PN candidates. Nevertheless, such cases will not affect +the distance determination because the PNLF cut-off is only de- +fined by the brightest PNe. +In the MUSE-GTO data, we classified 37 PNe, 62 H ii re- +gions, and 59 SNRs. In ML20 data, we classified 85 PNe, 176 +H ii regions, and 105 SNRs. To cross-match the PNe candidates +in the overlap area between the two dataset, we attempted an +automated algorithm by comparing the sky coordinates. How- +ever, since the astrometric accuracy of both data differs, our +attempt was not successful. Therefore, the cross-matching was +performed visually using the DS9 software (Joye & Mandel +2003). The final PN number from the MUSE-GTO and ML20 +fields: 105 PNe in the central region, and 2 in the P and Q fields +at higher galactocentric distance. The PN catalogue is presented +in Appendix D. +3.2. PN number comparison +Previous PN surveys of NGC 300 were conducted by Soffner +et al. (1996) – SO96, Peña et al. (2012) – PE12, and Roth et al. +(2018) – Paper I, who identified 34, 104, and 45 PN candidates, +respectively. To demonstrate the accuracy of our classification, +we employed the sample by PE12 as comparison, since it covers +Fig. 2. BPT-diagram of MUSE-GTO data (upper) and ML20 data +(lower). The orange dot-dashed line is taken from Kewley et al. (2001) +and the purple dashed line is defined by Roth et al. (2021). Open cir- +cles indicate PN candidates, which only have [O iii]λ5007 as the diag- +nostic line for the diagram. The deviation from the separation lines is +explained in the text. +more area and contains more PNe than the other studies. PE12 +observed NGC 300 with the FORS2 imager (Appenzeller et al. +1998) in two 6.8′ × 6.8′ fields, one in the centre, and another +in the eastern outskirts of the galaxy. The study employed the +on/off-band technique to detect PN candidates and classified ob- +jects based on the criterion of whether or not a central star was +present in their 5105 Å image. The expectation was that cen- +tral stars of PNe would be too faint to be detected in the vi- +sual, while the ionising O stars in H ii regions could be seen. +For brighter candidates with m5007 < 25, they also performed +additional spectroscopy using the MXU-mode with the same in- +strument (Stasi´nska et al. 2013). Since our observations cover a +smaller area on the sky, we only made the comparison for the +Article number, page 4 of 20 + +PNe +2.0 +SNR +Hll Region +1.5 +(dH / L00S[III ])6o +1.0 +I +0.5 ++ +0.0 +-0.5 +-1.0 +一1.5 +3.0 -2.5 -2.0 -1.5 -1.0 -0.5 +0.0 +0.5 +1.0 +log([S II]入6716+31 / Hα)PNe +2.0 +SNR +Hll Region +1.5 +( / 0[III )l +1.0 +0.5 +个国国 +0.0 +-0.5 +-1.0 +-1.5 +-3.0 -2.5-2.0 -1.5 -1.0 -0.5 +0.0 +0.5 +1.0 +log([S II]6716+31 /Hα)Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +intersecting 5′ × 7′ region in the centre of the galaxy, which is +also indicated in Figure 1. +In the overlapping region at the centre of the galaxy, we iden- +tify 105 PNe, compared to 58 in the PE12 sample. Moreover, +although we recover all 58 sources found by PE12, our classi- +fication indicates several discrepancies. While 43 of the PE12 +sources are confirmed as PNe, we classify 9 objects as compact +H ii regions and 3 as SNRs. These misclassifications could have +happened due to the fact that PE12 only have the spectral clas- +sification for candidates with m5007 < 25, while the fainter ob- +jects completely relied on the detection of a central star. This +approach also lacked the ability to identify SNRs amongst the +fainter candidates, as such objects can be discriminated through +the detection of the [S ii]λ6716, 6731 lines (Frew & Parker 2010; +Sabin et al. 2013). Since all of our candidates are classified on +the basis of their spectral properties, we believe that our classi- +fication is more reliable. Moreover, in terms of the number of +detection, we also demonstrate that the MUSE observations are +more sensitive and able to reach fainter magnitudes. +3.3. [O III]λ5007 photometry +The [O iii]λ5007 fluxes were obtained using DAOPHOT aperture +photometry (Stetson 1987), applied to the PNe candidates in the +15 differential layers for each datacube. Then, the magnitudes +were computed using the V-band equivalent conversion (Jacoby +1989) defined as +m5007 = −2.5 log F5007 − 13.74 +(2) +where the flux is in erg cm−2 s−1. Here, the aperture radius was +adjusted to a value of approximately the FWHM of the PSF in a +given exposure to accommodate the respective seeing condition. +The inner and outer sky annulus were fixed to 12 and 15 spaxels, +respectively. Most of the flux of the PSF was obtained by adding +the 5 bins closest to the Gaussian peak and the remaining flux +is recovered through the use of an aperture correction based on +the information of a PSF reference. This correction is crucial to +obtain accurate fluxes, which however can be a challenge when +there is no reference available especially with the small field of +view of MUSE. The aperture correction method is explained in +Appendix A. +The photometric uncertainty was calculated from the Gaus- +sian fit errors, convoluted with an assumed flux calibration er- +ror of 5% (Weilbacher et al. 2020). In high surface brightness +regions of distant galaxies, double-peaked profiles are occasion- +ally found and indicate the presence of two superposed PNe with +different radial velocities (Roth et al. 2021). Unsurprisingly, we +do not find such cases in our sample. Since NGC 300 is a quite +nearby galaxy, spatial coincidences are less likely. Additionally, +the five datacube layers containing the total flux for [O iii]λ5007 +were also inspected to insure the PN candidate was not extended +or contaminated by surrounding gas emission. +As an internal test of our photometry, we used the regions +of field overlap to compare our PN measurements made in the +MUSE-GTO fields to those from the ML20 data. This com- +parison is presented in the upper panels of Figure 3. We find +that the ML20 observations obtained under poor seeing condi- +tion tend to be systematically fainter than the MUSE-GTO data, +while the photometry of the same object from different datacubes +with similar image quality gives identical results (exception for +the faintest PN in the comparison). Thus, the difference in see- +ing conditions can introduce a magnitude error; this is most +likely due to the choice of a too small aperture for the asymp- +totic assumption for the aperture correction. Because the disk of +Fig. 3. Comparison between the MUSE-GTO and ML20 [O iii]λ5007 +magnitudes before (upper) and after (lower) photometric correction for +PNe in the overlapping area. The markers are linearly scaled with the +seeing FWHM of the field and sorted from the brightest to the faintest. +NGC 300 contains a large amount of diffuse emission-line gas, +we chose to not to increase this radius. However, in order to ob- +tain the same photometric quality between the two data sets, we +applied and additional corrections of 0.2 mag for objects with +seeing FWHM ∼ 1′′.2 (∼ 6 spaxels), and 0.3 mag for seeing +FWHM ∼ 1′′.4 (∼ 7 spaxels). We found these values based on +empirical trial and error to achieve the minimum average dis- +crepancy between the two sets of magnitudes. +The comparison after applying the correction is presented in +the lower panels of Figure 3. The average discrepancy is now +0.05 mag, which is still within the typical measurement error of +0.06 mag. For PNe with m5007 ∼ 27 or fainter, the correction +has no meaningful implication, because the candidates are close +Article number, page 5 of 20 + +28 +MUSE-GTO +27 +Archival +26 +m5007 +25 +24 +23 +22 +0.6 +0.4 +0.2 +△m5007 +0.0 +-0.2 +-0.4 +-0.6 +4 +5 +6 +8 +9 +8869899860 +PN ID28 +MUSE-GTO +27 +Archival +26 +m5007 +25 +24 +23 +22 +0.6 +0.4 +0.2 +△m5007 +0.0 +-0.2 +-0.4 +-0.6 +4 +5 +6 +8 +9 +20 21 25 27 29 37 45 48 51 +55 62 68 69 73 78 85 +PN IDA&A proofs: manuscript no. pnlf_ngc300 +to the detection limit. In the pilot study, Roth et al. (2018) per- +formed a completeness simulation for the MUSE-GTO data of +given exposure time, with seeing quality ranging from 0′′.6−1′′.2. +This was conducted by embedding artificial PNe with different +magnitudes into the real datacubes. It was found that for a see- +ing of 1′′.2, the expected completeness is 90% at m5007 = 27. +Although we are able to detect a PN as faint as m5007 = 28.91, +the seeing quality on average for all our data is 1′′.0, with 23% +of the fields exhibiting larger than 1′′.2 FWHM. This shows that +completeness of 90% at m5007 = 27 is only achieved for 77% of +our fields. However, since the emphasis of this work is on the +bright candidates that define the PNLF cut-off for the distance +determination, our results are not suffering from sample incom- +pleteness at the faint end. For the final [O iii]λ5007 magnitudes, +we preferred the MUSE-GTO data, if available, and otherwise +we employed the ML20 data. We also applied the correction for +fields outside the overlapping area with seeing FWHM > 1′′.2. In +total, 11 PNe from 4 fields were corrected in this manner. +To test the accuracy of our photometry, we compared our +magnitudes to the results from the literature. Figure 4 shows a +comparison with SO96 and PE12. While our data is in reason- +able agreement with SO96 within 0.01 mag on average, there +is a systematic offset with regard to PE12. We find that our +magnitudes are systematically brighter by an average of 0.71 +mag. PE12 obtained instrumental [O iii]λ5007 magnitudes for +the FORS2 on-band image using aperture photometry with the +aperture diameter of 5 pixels (1′′.25), based on the average PSF +FWHM of 2.9 pixels. To obtain the apparent m5007 magnitudes, +they calibrated the instrumental measurements using an empiri- +cal relation derived from the objects’ spectroscopic fluxes, which +are only available for the brightest PNe in their sample. We can +try to understand what may be the reason for the discrepancy. +First of all, we note that flux calibration is an established MUSE +procedure in operation at the VLT and part of the data reduction +pipeline. According to Weilbacher et al. (2020), flux calibration +has been measured to be accurate to within 3-5%. If a signifi- +cant number of our MUSE exposures would have been affected +by non-photometric observing conditions – for which we have +no evidence, we would expect a scattered, but not the tightly +constrained linear correlation, that we see in Figure 4, in partic- +ular for the brightest 3 magnitudes. Secondly, Roth et al. (2018) +have tested synthetic MUSE datacube broadband photometry of +stars against published HST photometry for the same GTO dat- +acube subset that has been used in our work, showing no hint +of an offset to within a magnitude of F606W=22.7. Thirdly, the +agreement with SO96, who obtained their data with narrow-band +imaging at the ESO NTT, i.e. a different instrument at a different +telescope, gives us reasonable confidence that our flux calibra- +tion cannot be off by as much as a factor of almost 2. Finally, +we can follow the argument put forward by Roth et al. (2004), +that by definition, integral field spectroscopy is an ideal tool for +spectrophotometry, as is does not suffer from any kind of slit +effects. +We can speculate, though, that the spectrophotometry from +PE12 might have been affected in various ways to give rise to the +observed systematic offset. In case of PE12, the calibration relies +on spectroscopic fluxes, which were obtained using a slit spec- +trograph, and thus may suffered from slit-losses that were esti- +mated to be 10 − 15%. However, from our comparison, we infer +that the loss might be underestimated since 0.71 mag difference +is equivalent to a loss of ∼ 48%. To test this, we performed a +slit-loss simulation based on a model of the PSF with the quoted +seeing conditions of 0′′.7 − 0′′.9, and a slit size of 1′′. The simula- +tion is done in the R-band, as the seeing measurement is typically +Fig. 4. Comparison of m5007 between this work, SO96, and PE12. The +photometry of SO96 agrees within 0.01 mag. The photometry of PE12 +is systematically fainter by 0.71 mag. For m5007 > 25, the relation with +PE12 becomes scattered. +done in this band. Based on these parameters, our simulation +predicts that the slit-losses should be between 8 − 20%. How- +ever, since the PSF FWHM is expected to be larger in the blue +wavelength region, the slit-loss in [O iii]λ5007 will be larger than +that in the R-band. Moreover, additional losses can be introduced +by positioning and guiding errors, as investigated by Jacoby & +Kaler (1993). Spectrophotometry with a slit spectrograph also +requires the slit to be oriented at the parallactic angle to min- +imise the effect of atmospheric dispersion (Filippenko 1982; Ja- +coby & Kaler 1993). Since our observations are performed with +an IFU, we are not affected by any of these problems. While it +is possible that our use of small apertures has caused some flux +to be lost, we are able to compensate for this loss using aperture +corrections as described above. Such a procedure is not easily +performed for data observed with a slit spectrograph. +Besides the issue of slit-losses, the follow-up spectroscopy +by PE12 is limited to PNe with m5007 < 25, which is less +than 40% of their whole sample. This implies that most of their +PNe are dependant on the measurement accuracy of the brighter +PNe, which are likely to be affected by systematic errors. More- +over, the use a 5 pixel aperture to measure the flux for a 2.9 +pixel FWHM PSF incurs a risk of including light from back- +ground contamination. In such crowded fields with a variable +background and ubiquitous diffuse emission-line gas, the aper- +ture might unexpectedly collect [O iii]λ5007 flux of the ambient +interstellar medium. Without proper background inspection and +subtraction, this might lead to an overestimation of brightness. +The inclusion of background emission likely explains the scat- +ter for m5007 > 25 in Figure 4. The spatial resolution of MUSE +allows us to carefully check and analyse the condition of the +background on a case-by-case basis and provide more accurate +photometry. The variable background is also the main considera- +tion to opt for a smaller aperture size for our flux measurements, +and to rely on the aperture correction to deliver the final values. +Article number, page 6 of 20 + +SO96 +29 +PE12 +28 +27 +(SO96,PE12) +26 +25 +m5007 ( +24 +23 +22 +21 +21 +22 +23 +24 +25 +26 +27 +28 +29 +m5007 (This work)Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +3.4. Balmer decrement +Measurement of extinction using the Balmer decrement with +Hα/Hβ ratio have been demonstrated on MUSE data for differ- +ent objects, i.e. Pillars of Creation in M16 (McLeod et al. 2015), +core of R136 in the LMC (Castro et al. 2018), faint H ii regions +in NGC 300 (Micheva et al. 2022). To obtain this, we employed +the spectra extracted for the classification, as explained in Sec- +tion 3.1, using the aperture of 3 spaxels, with the inner and outer +sky annulus of 12 and 15 spaxels, respectively. We also apply +aperture correction for the Balmer lines, which can be referred +to in Appendix A. +However, not all of our PNe candidates are detected at these +two wavelengths. In order to filter out the candidates, we put +a threshold of F(Hα) = 2 × 10−17 erg cm−2 s−1. For typical +PNe with electron temperature Te = 10.000 K, the expected +Balmer ratio is Hα/Hβ = 2.86 (Osterbrock & Ferland 2006), +which corresponds to F(Hβ) ∼ 8.75 × 10−18 erg cm−2 s−1 for the +Hα threshold. Any Hβ flux lower than the threshold is too close +to the detection limit. For such cases, we assume the upper limit +of Hβ flux derived from the Hα line, which consequently also as- +sumes no extinction. To avoid possible biased exclusion of high +extinction PNe, we flag the objects with upper limit Hβ flux. +If the Hα flux is below the threshold, then the extinction mea- +surement is not performed. Based on the Hα threshold criterion, +we have a complete sample for PNe down to m5007 = 24.5. +If we extend the sample to fainter magnitudes, we reach 87% +completeness until m5007 = 26.0 and 64% completeness un- +til m5007 = 27.0. To calculate the extinction, we then used the +Balmer decrement defined as +Aλ = k(λ) c(Hβ) = +k(λ) 2.5 +k(Hβ) − k(Hα) +� +log +�Hα +Hβ +� +− log (2.86) +� +(3) +where k(λ) is the wavelength dependant extinction constant. For +the foreground extinction, we employed the extinction curve of +Cardelli et al. (1989) with RV = 3.1 and E(B − V) = 0.011 +(Schlafly & Finkbeiner 2011). For NGC 300, Bresolin et al. +(2009) measured the present day metallicity of 12 + log(O/H) ∼ +8.1 − 8.5 using the H ii regions. Recent measurement using the +same MUSE-GTO data based on faint H ii regions and dif- +fuse interstellar gas (DIG) also agrees with the latter value as +12 + log(O/H) ∼ 8.5 (Micheva et al. 2022). Since the chemical +abundance of NGC 300 in the observation area are similar to the +LMC, with 12+log(O/H) ∼ 8.4−8.5 (Toribio San Cipriano et al. +2017), we employed the average LMC extinction curve to obtain +the extinction of our PNe. The uncertainty of our extinction mea- +surement is highly dependent to the aperture correction method. +Therefore, we quote an estimated error based on the comparison +of extinction calculated from different aperture correction meth- +ods, as explained in Appendix B. +In spiral galaxies, Herrmann & Ciardullo (2009) found that +the typical extinction for the PNe in [O iii]λ5007 is A5007 ∼ 0.7. +However, we discovered three high extinction cases with A5007 > +1.5, including one with an extreme value of A5007 ∼ 3.3. While +it is possible that some PNe exhibit high intrinsic extinction, as +high metallicity populations and massive progenitors tend to pro- +duce dustier PNe (Stanghellini et al. 2012), we suspect that the +Balmer decrement might not always be accurate due to the local +contamination. To investigate this further, and to highlight pos- +sible pitfalls that may play a role in studies based on slit spec- +troscopy, we examined spatial maps of the high-extinction ob- +jects in the wavelengths of Hβ, [O iii]λ5007, and Hα, using the +p3d software (Sandin et al. 2010). We found that these PNe can- +didates are co-spatial with nearby H ii regions. In Figure 5, a PN +Fig. 5. False-colour spatial map in [O iii]λ5007 (left) and Hα (right). +The flux scaling is identical and logarithmic. The images are 20′′ × 20′′ +(∼ 180 × 180 pc) each. The green marker illustrate the main and sky +aperture. The candidate is isolated in [O iii]λ5007, but overlapped with +nearby H ii region in Hα. +is shown to be an isolated point source in [O iii]λ5007. However, +in the spatial map of Hα, the point source is entirely embedded +inside the extended emission surface brightness distribution of +an unrelated nebula. This clearly shows that the Balmer line flux +of the PN candidate is contaminated, and an accurate extinction +measurement of the PN itself cannot be obtained. All three of the +objects in question show similar patterns of contamination. We +therefore excluded them from the sample. +We compared our PN extinction measurements with the re- +sults from Stasi´nska et al. (2013) – also referred as ST13 – who +observed PNe in NGC 300 with the FORS2-MXU instrument at +the VLT (Appenzeller et al. 1998). They used 3 grisms 600B, +600RI, and 300T to cover spectral ranges of 3600 − 5100Å, +5000 − 7500 Å, and 6500 − 9500 Å, respectively. To avoid un- +certainties from the flux calibration of different bands, they em- +ployed the Hγ and Hβ lines from the 600B grism spectra to mea- +sure the Balmer decrement. The extinction values for 18 PNe in +common are presented in Table 1. +We have contemplated several reasons to explain the dis- +crepancy. Firstly, slit losses that occur for the measurement of +[O iii]λ5007 most likely also occur for the Hγ and Hβ lines. The +short baseline between the lines is also very sensitive to sys- +tematic errors, making it difficult to derive accurate extinction +values. Moreover, higher order Balmer lines are typically weak. +The accuracy of measuring their flux depends on the precision +to which the background of stellar absorption line spectra can be +subtracted (Jacoby & Kaler 1993; Roth et al. 2004). Using the +PMAS instrument, Roth et al. (2004) compared the accuracy of +the Hβ line flux obtained with the IFU and simulated slits. They +demonstrated that the orientation of the slit can introduce a dif- +ferent sampling of the background, leading to systematic differ- +ences of the derived flux measurements. Since ST13 performed +their measurements with slit spectroscopy, they were susceptible +to this type of error. +For cases where the internal extinction of ST13 is reported +higher than our values, we find that 3 of their PNe are discarded +from our sample (L6-5, H1-1, and H12-1 in Table 1), either be- +cause of severe contamination as shown in Figure 5, or by their +low excitation, which we consider typical for compact H ii re- +gions (Frew & Parker 2010). We also found that in some of these +cases, the PN is embedded in diffuse gas. In our sample, if the +diffuse gas is assumed to be uniformly distributed, the flux ex- +cess can be corrected using the background sky annulus. It re- +mains an open question whether the background correction of +diffuse gas was accurately accounted for in the slit spectroscopy +of ST13, but we conclude that a careful consideration of back- +Article number, page 7 of 20 + +O +0A&A proofs: manuscript no. pnlf_ngc300 +Table 1. Extinction comparison between this work and ST13. +IDMUSE +IDST13 +c(Hβ)MUSE +c(Hβ)ST13 +E-11 +12 +0.12±0.07 +0.00 +E-2 +14 +0.10±0.07 +0.00 +L6-5a +20 +– +1.17 +L6-7a +22 +0.20±0.09 +0.44 +I-2b +24 +0.02±0.04 +0.12 +C-7b +25 +0.03±0.10 +0.07 +H9-1 +35 +0.04±0.08 +0.00 +L9-8 +40 +0.13±0.07 +0.00 +H2-6 +45 +0.17±0.11 +0.00 +A-23 +48 +0.42±0.05 +0.21 +A-11 +51 +0.10±0.05 +0.00 +H1-8 +54 +0.19±0.06 +0.00 +H7-2 +58 +0.11±0.08 +0.00 +H1-1c +63 +– +0.41 +H6-5 +65 +0.10±0.07 +0.00 +L2-3 +66 +0.05±0.08 +0.00 +H6-3 +69 +0.26±0.07 +0.00 +H12-1a +74 +– +0.64 +Notes. +(a) severe Balmer contamination +(b) uniform diffuse Hα background +(c) low excitation – possibly compact H ii region +ground subtraction is critical for the extinction measurements +based on the Balmer decrement. +4. The PNLF +The PN luminosity function of this work is presented in Figure +6. It exhibits the dip between 1 and 3 magnitudes below the cut- +off. Such dip is typically observed in star-forming galaxies (Ja- +coby & De Marco 2002; Ciardullo 2010; Reid & Parker 2010a), +which is possibly caused by multiple episodes of star formation +(Rodríguez-González et al. 2015; Bhattacharya et al. 2021) or +difference in opacity and mass range of the PN formation (Valen- +zuela et al. 2019). +To determine the distance, we employed the maximum like- +lihood technique, where the empirical PNLF is treated as a prob- +ability function (Ciardullo et al. 1989), assuming M∗ = −4.53 ± +0.06 and a fixed slope parameter of 0.307. When the number of +PNe at the bright end cut-off is less than ∼ 50, distance deter- +minations based on χ2 minimisation depend significantly on the +details of how the PNLF is binned. Such methods are not rec- +ommended (Ciardullo et al. 1989; Roth et al. 2021). Although +our observation extends to m5007 ∼ 29, the PNLF fit is only per- +formed for the sample brighter than the dip until m5007 = 23.6, +since equation (1) does not consider the dip feature, which nev- +ertheless is insignificant for the distance determination (Spriggs +et al. 2021; Ciardullo 2022). By taking the foreground extinction +of E(B − V) = 0.011 (Schlafly & Finkbeiner 2011) into account, +the most-likely distance modulus is (m − M)0 = 26.48+0.11 +−0.26 with +the uncertainties representing the statistical error of the fit and +the M∗ uncertainty. +We also calculated the distance using PE12 photometry to +make a comparison. Assuming a completeness limit of m5007 = +1 +10 +2 +5 +20 +This Work +Number of Objects +21 +22 +23 +24 +25 +26 +27 +28 +29 +1 +10 +2 +5 +20 +Pena et al. (2012) +Number of Objects +Fig. 6. PNLF of NGC 300 using MUSE (top) and PE12 photometry +(bottom). Completeness limit for distance measurement is assumed at +m5007 = 23.6 and m5007 = 24.0 for ours and PE12, respectively. Open +symbols indicate incompleteness. The PNLF dip is visible for both. +24.0, our maximum likelihood approach yields (m − M)0 = +27.30+0.09 +−0.20, a value that is significantly larger than our MUSE +distance. We argue that this is due to the systematically fainter +magnitudes of PE12 photometry, as discussed in Section 3.3. It +is important to mention that PE12 measured a modulus distance +of (m − M)0 = 26.29+0.12 +−0.22, a value much smaller than our maxi- +mum likelihood values, including our distance measurement us- +ing the PE12 data. This difference is possibly due to their use +of the Levenberg-Marquardt χ2 fitting technique, which is de- +pendant on the binning method to construct the luminosity func- +tion. Since the sample size is limited, they employed rather wide +magnitude bins of 1.16 mag, which did result in a luminosity +function shape that closely resembles the empirical law. How- +ever, in the luminosity function of PE12, their first magnitude +bin is located at m5007 ∼ 22, despite the fact that their brightest +PN has a magnitude of m5007 = 22.99. Thus, when the fit is per- +formed, this systematic shift to brighter magnitudes results in a +smaller distance modulus. This demonstrates that the choice of +bin size can produce unintended systematical shifts of the lumi- +nosity function when the number of PNe in the top ∼ 0.5 mag of +the luminosity function is small. Similarly, PE12’s choice of bin +size also smeared out detail in the PNLF’s shape, as they did not +report the observation of the PNLF dip. In Figure 6, we present +the PNLF that we plot with the original data from PE12 using +higher binning resolution than the original work. In fact, the dip +is present in the PE12 data, confirming that it is not an artefact +in our measurements. +Finally, PE12 employed a larger extinction correction for +the photometry with A5007 = 0.2 compared to our value of +A5007 = 0.05. PE12 assumed this as the intermediate value be- +Article number, page 8 of 20 + +Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +tween found by Gieren et al. (2005) with E(B − V) = 0.096 +(A5007 = 0.3) and Schlegel et al. (1998) with E(B − V) = 0.013 +(A5007 = 0.05). In the case of Gieren et al. (2005), the extinction +value is the sum of both the foreground extinction of Schlegel +et al. (1998) and internal extinction derived from the Cepheids. +For Cepheid distances, the internal extinction correction is nec- +essary since they are originated from Population I stars, that +are typically surrounded by galactic dust. However, this is less +true for the PNe, so the foreground extinction correction for the +PNLF distance is sufficient (Ciardullo 2010). This implies that +the extinction correction A5007 = 0.2 by PE12 is overestimated +and also contributes to the smaller distance modulus. Therefore, +the discrepancy between our calculation of the PE12 data and +the original calculation is traced back to the issue of binning a +limited sample and also the extinction correction. +5. Discussion +5.1. PNLF Distance +To demonstrate the accuracy of our distance measurement, we +compare our result with previous distances in the literature de- +rived using Cepheids and tip of the red giant branch (TRGB), +taken from NED, in Figure 7. We can see that most of the +distances are within the uncertainties of our PNLF result. One +aspect that may introduce the discrepancy is the correction of +extinction. For instance, the Cepheid distance of Gieren et al. +(2005) and the TRGB distance of Rizzi et al. (2006), both part +of the Auracaria project, are corrected with foreground and inter- +nal extinction of E(B − V) = 0.096. However, Rizzi et al. (2007) +argue that the extinction derived from dusty young Cepheids by +Gieren et al. (2005) are not representative for the whole galaxy; +their result only applies the foreground component of extinction. +This shows the importance of having the same zero-point when +comparing different distances derived from different methods. +Moreover, we also show that the MUSE observation, com- +bined with the differential emission line filter (DELF, Roth +et al. 2021) and maximum likelihood technique (Ciardullo et al. +1989), has improved the accuracy of PNLF method, as shown +in Figure 7. The early result by Soffner et al. (1996) is based +on the limited sample of only 34 PNe, from which they only +construct a cumulative PNLF and employed the distance mod- +ulus of the LMC as a yardstick. Peña et al. (2012) identified a +significantly larger sample of 104 PNe, but as shown in Sec- +tion 3.3, their data may suffer from slit-losses and contamina- +tion. The systematically fainter PN magnitudes then led to larger +distance modulus, as described in Section 4. Since the cut-off +of the PNLF of NGC 300 is defined by a very small number of +PNe, minimisation fitting methods become too dependant on the +binning (Ciardullo et al. 1989). In a study by Jacoby (1997), a +correction for PNLF distance based on the number of PN sam- +ple is suggested. For a PNLF cut-off sample < 20 PNe, they +estimated a distance correction of ∼ 0.1 mag (see Figure 5 in +Jacoby 1997). However, since the Cepheid and TRGB distance +also varies with standard deviation of 0.1 mag, there are no solid +distance reference to test if the correction is appropriate. Never- +theless, we have shown that the PNLF distance derived with the +maximum likelihood technique is more robust. We take this as a +motivation to improve PNLF distance measurements for nearby +galaxies with our method. +5.2. Local dust effect on PN extinction +Dust formation plays important role in the early stages of PN +evolution since it occurs at the surface of the progenitor AGB +star and presumably plays an important role in the envelope ejec- +tion (Herwig 2005; Stanghellini et al. 2012). Infrared studies in +the Milky Way and the LMC have revealed that the dust produc- +tion is dependant on metallicity, with dustier systems found in +higher metallicity environments (Stanghellini et al. 2007, 2012; +Bernard-Salas et al. 2009). Although it cannot tell the proper- +ties of the dust, Balmer decrement extinction measurements can +also probe the presence of dust in PNe. In the study of Davis +et al. (2018), a comparison was made between the PN extinction +distribution in the bulge of M31 and several other galaxies: the +LMC (Reid & Parker 2010a), NGC 4697 (Méndez et al. 2008a), +and NGC 5128 (Walsh et al. 2012). Despite the limited samples +involved, the authors found that the average extinction of PNe in +each galaxy roughly follows the metallicity of the system. +To investigate such trends in NGC 300, we plot the extinc- +tion distribution in [O iii]λ5007 for our PNe until m5007 = 23.6 +(15 PNe) in Figure 8. These PNe are the ones employed for +the maximum likelihood distance measurement. We find that in +general these bright PNe have low extinction in [O iii]λ5007. +The average extinction value for this sample is A5007 = 0.31 +(c(Hβ) ∼ 0.09), which is lower than the average of the bright +PNe sample in the LMC with A5007 = 0.57 (Reid & Parker +2010a; Davis et al. 2018). However, we refrain from further in- +terpreting the extinction distribution with the PN dust produc- +tion, since the distribution is likely to be affected by local dust +clouds, which can vary from one object to another. Such a prob- +lem has been reported in NGC 5128, where the high extinction +of some PNe was attributed to local dust clouds rather than the +PNe themselves (Walsh et al. 2012). +At the distance of NGC 300, our MUSE observations offer a +spatial resolution of between 6 and 14 pc. This resolution should +be sufficient to visually resolve the spatial variation of dust ex- +tinction (Kreckel et al. 2013; Tomiˇci´c et al. 2017). To test this, +we inspected several objects with high extinction. As an exam- +ple, we present the spatial map in [O iii]λ5007 and RGB colours, +which is constructed from Johnson-VRI filters, for the PN with +the highest extinction value (PN A-23, A5007 = 1.18) in Figure +9. Although it is not obvious in the [O iii]λ5007 image, the RGB +image shows a dust lane patch, extending from the lower left +corner to the centre. Since PN A-23 is in proximity to the dust +lane, we suggest that the measured high extinction of this object +is composed of both local dust within the galaxy and circumneb- +ular extinction associated with the PN itself. +Based on the comparison study between Balmer decrement +extinction and infrared dust distribution in M31, Tomiˇci´c et al. +(2017) concluded that vertical distribution of diffuse interstellar +gas (DIG) and dust can vary in different locations of the galaxy +and thus cause differing amounts of extinction. For NGC 300, +variation of extinction also has been reported by Roussel et al. +(2005). Therefore, there is currently no guarantee that the mea- +sured extinction of individual PNe is free from local effects, +which is confirmed with our images. We must therefore refrain +from making conclusions based on the extinction values alone, +until the different components of the extinction can be quantita- +tively resolved. +Although the extinction of individual PNe might be affected +by local dust lanes, such effects are less significant for the lumi- +nosity function as a whole. The effect of dust scale height in the +PNLF distances of late-type disk galaxies has been discussed by +Feldmeier et al. (1997). They modelled PNLF with varying ex- +Article number, page 9 of 20 + +A&A proofs: manuscript no. pnlf_ngc300 +Fig. 7. Distance modulus difference between our PNLF result with Cepheids (blue triangles) and TRGB (red squares), obtained from NED and +sorted based on publication date. The Cepheid distances are from Willick & Batra (2001); Paturel et al. (2002); Gieren et al. (2004, 2005); Saha +et al. (2006); Bono et al. (2010); Bhardwaj et al. (2016). The TRGB distances are from Butler et al. (2004); Sakai et al. (2004); Tikhonov et al. +(2005); Tully et al. (2006); Rizzi et al. (2006, 2007); Jacobs et al. (2009); Dalcanton et al. (2009). Previous PNLF distances of Soffner et al. (1996) +and Peña et al. (2012) is also presented (green circles). The green shadow indicated the uncertainty of our PNLF distance. +Fig. 8. Distribution of extinction measurement for the PNe in +[O iii]λ5007 until m5007 = 23.6. The average extinction is A5007 = 0.31 +(c(Hβ) ∼ 0.09). +Fig. 9. False colour (left) and RGB colour (right) map of the region +surrounding PN A-23. The flux scaling is logarithmic. The images are +20′′ × 20′′ (∼ 180 × 180 pc) each. The green marker illustrate the main +and sky aperture. The dust patch is clearly visible in the RGB image, +overlapped with PN A-23. +tinction in [O iii]λ5007 and concluded that the inferred distance +modulus should always be within 0.1 mag of the derived distance +without extinction. A similar result also obtained by Rekola et al. +(2005), who modelled the PNLF with different scale heights of +dust in the starburst galaxy NGC 253. They found that even when +the disk was optically thick with 1 mag of extinction, the PNLF +distance is robust to within 0.1 mag. Both studies suggest that +the brighter PNe tend to be located above the dust layer from the +point of view of the observer, or for other reasons suffer little +extinction from within the galaxy. With these arguments, we do +not expect the occurrence of dust lane extinction to significantly +affect our distance result and a correction for internal extinction +is at this point not necessary. +5.3. PN parent populations +To gain a better understanding of the parent population of the +PNe, we estimate the luminosity and the effective temperature of +the central stars of the planetary nebula (CSPNs). These param- +eters are calculated for PNe until m5007 = 26 with measurable +extinction, which corresponds to 87% of the objects within this +magnitude limit. +Simulation studies suggest that the maximum conver- +sion efficiency of a central star luminosity into nebular +[O iii]λ5007 emission is ∼ 11% (Jacoby 1989; Dopita et al. 1992; +Schönberner et al. 2007, 2010; Gesicki et al. 2018). This oc- +curs under the ideal assumption of optically thick nebula and +assumes that [O iii]λ5007 acts as the sole coolant. If the PNe is +optically thin, then the efficiency of [O iii]λ5007 production is +less, and the luminosity inferred for a PN’s central star will be +underestimated (Mendez et al. 1992). A high abundance of ni- +trogen, such as that typically found in Type I PNe (Peimbert & +Torres-Peimbert 1983; Phillips 2005), can also increase cooling, +and lead to an underestimation of central star luminosity. More- +over, the assumption of lower limit extinction for some cases can +also underestimate the luminosity. Therefore, we only consider +our luminosity estimates as the lower limits. +To estimate the central stars’ effective temperatures, we em- +ployed the excitation class method based on the PNe in the +LMC (Dopita & Meatheringham 1990; Reid & Parker 2010a). +For optically thick PNe, the excitation class temperatures are +found to have an empirical correlation with temperature as de- +rived from photo-ionisation modelling (Dopita & Meathering- +ham 1991; Dopita et al. 1992; Reid & Parker 2010b). To em- +ploy this method, we also assume that the metallicity difference +between the LMC and NGC 300 is negligible (Bresolin et al. +2009). The revised excitation classes by Reid & Parker (2010b) +are defined as +Elow = 0.45 +�F(λ5007) +F(Hβ) +� +(4) +Ehigh = 5.54 +�F(λ4686) +F(Hβ) ++ log10 +F(λ4959) + F(λ5007) +F(Hβ) +� +(5) +with Elow employed for low excitation PNe (0 < E < 5) and Ehigh +for medium- to high excitation PNe (5 ≤ E < 12). The empirical +relation between the excitation class and effective temperature +for optically thick PNe is then defined by Reid & Parker (2010b) +as +log Teff = 4.439 + [0.1174(E)] − [0.00172(E2)] +(6) +Since only the extended mode used in the MUSE-GTO dataset +has the wavelength coverage to include the He ii λ4686 line, +Article number, page 10 of 20 + +5 +Number of PNe +4 +3 +2 +1 +0 +0.0 +0.5 +1.0 +A500700.8 +0.4 +4 不 +回国 +0.0 +本不 +国 +中中 +中国 +国国 +国 +d +-0.4 +-0.8Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +Fig. 10. HR-diagram of CSPNs in NGC 300. The evolutionary tracks +are H-rich post-AGB models by Miller Bertolami (2016). The luminos- +ity are lower limits, assuming maximum [O iii]λ5007 conversion effi- +ciency of 11%. For measurements within error of log Teff > 4.98, lower +limit effective temperatures are assumed. The ratio [N ii]λ6584/Hα is +the indicator of optical thickness, with the value less than 0.3 for more +likely optically thin. Type I PNe are classified with [N ii]λ6584/Hα > +1.0. +for uniformity, we determine the excitation class using just the +[O iii]λ5007 line and Hβ line. This implies that the effective tem- +peratures for medium- and high excitation PNe with E ≥ 5 (or +logTeff ≥ 4.98) are only lower limits. This includes the measure- +ments, which have uncertainties beyond the condition for low- +excitation PNe. Based on 6 PNe in our sample that have He ii +λ4686 detection, we estimate that the effective temperatures can +be underestimated by 1.5 − 3 times if we only rely on the Elow. +On the other hand, if the nebula is optically thin, based on the +study in the LMC, the excitation class temperatures can be over- +estimated by at least 50% compared to the Zanstra temperatures +(Villaver et al. 2007; Reid & Parker 2010b). +Both luminosity and effective temperature estimates rely on +the optical thickness of the PNe. To obtain this, we adopt the +criterion of [N ii]λ6584/Hα ≤ 0.3 as the condition for optically +thin PNe (Kaler & Jacoby 1989; Jacoby & Kaler 1989; Reid +& Parker 2010b). Since this criterion is not based on nebular +modelling in our sample, we use the indication as a more likely +condition rather than a definite indicator to explain the possible +limitation in our estimations. The estimated stellar parameters +are presented in Figure 10. We include the post-AGB tracks from +Miller Bertolami (2016) with a stellar metallicity of Z⊙ = 0.01, +which is the closest to the observed value at the central area of +NGC 300 with Z⊙ = 0.007 (Kudritzki et al. 2008; Gogarten et al. +2010). +A stellar population study by Jang et al. (2020) using the +Hubble Space Telescope found young stars of ∼ 300 Myr, AGB +stars with an age between 1 − 3 Gyr and significant number of +RGB stars older than 3 Gyr. From a single stellar evolution per- +spective, the stellar population of NGC 300 can produce a PN +central star mass of ∼ 0.7 M⊙ from a progenitor mass of 3.0 M⊙, +which would have a main sequence lifetime of τMS > 320 Myr +(Miller Bertolami 2016). This implies that, theoretically, central +stars within any of the stellar tracks in Figure 10 can be expected. +Unfortunately, since most of our luminosities and effective tem- +peratures are lower limits, we are unable to put more constraints +on the central star masses at this point. +Within our sample, we also identified several objects as Type +I PNe (Peimbert & Torres-Peimbert 1983; Phillips 2005), highly +enriched in nitrogen, and classified using [N ii]λ6584/Hα > 1. +These objects likely to arise from younger and more massive +stars. For a progenitor mass above ∼ 2.5 M⊙, the convective en- +velope in the thermal pulsing AGB phase is likely to extend to +the hydrogen-shell burning layer and produce “hot bottom burn- +ing” (HBB). This can dredge up the products of the CNO cycle to +the surface, to be later expelled by the stellar wind, therefore in- +creasing the nitrogen-to-oxygen ratio in the nebula (Henry et al. +2018). Observations of PNe in M31 by Fang et al. (2018) put +a lower limit of ∼ 2.0 M⊙ for HBB. A more thorough analysis, +performed for a Type I PN in the M31 young open cluster B477- +D075, yields a HBB lower mass limit of ∼ 3.4 M⊙ (Davis et al. +2019). This suggests that our approximation for the central star +luminosities of Type I PNe is greatly underestimated. For this +particular case, we argue that the assumption of [O iii]λ5007 as +the only coolant is not true. Since the nitrogen-to-oxygen ratio +is high, the nitrogen contribution as additional coolant cannot +be neglected (Jacoby 1989), causing the underestimation. This +might also explain why we did not see the Type I PNe at our +PNLF cut-off, although they are expected to have more massive +cores than the typical PNe. +More implications regarding the underlying stellar popula- +tion can also be inferred from the faint end of the PNLF (Ciar- +dullo 2010). However, the current observational study is still +limited to a relatively small sample. Recently, based on a very +deep survey in M31, Bhattacharya et al. (2021) found that the +steep rise in the number of PN fainter M∗ + 5 mag is caused +by the increased mass fraction of a population older than 5 Gyr. +For NGC 300, this implies that the photometry should be com- +plete for m5007 > 27. Since our PNLF completeness breaks after +m5007 = 27.5, we are unable to provide any insights on this mat- +ter at the moment. +5.4. Insights on the most luminous PNe +Numerous simulation studies have been conducted to investigate +the nature of the PNe at the cut-off of the luminosity function (Ja- +coby 1989; Schönberner et al. 2007, 2010; Méndez et al. 2008b; +Gesicki et al. 2018; Valenzuela et al. 2019). We review some of +them and compare it to our estimated properties to investigate the +nature of the most luminous PNe in NGC 300. Using the most re- +cent post-AGB models by Miller Bertolami (2016), simulations +of the [O iii]λ5007 fluxes for different progenitor mass have been +performed by Gesicki et al. (2018). They found that progenitors +with the mass range between 1.5−3.0 M⊙ are able reach the cut- +off absolute magnitude M∗ = −4.5, assuming that the fluxes at +the stellar evolution stages are maximised – also known as maxi- +mum nebula hypothesis. It is important to note that the timescale +of the 3.0 M⊙ track is too short and less likely to be observed. +Additionally, they also performed a simulation with an inter- +mediate nebula hypothesis, where the PNe are predominately +opaque; this model suggests that the brightest PNe in the lumi- +nosity function will have the luminosity log L/L⊙ = 3.75 ± 0.13. +For comparison, the intrinsically most luminous PN in our sam- +ple, PN H9-1, has a lower limit luminosity of log L/L⊙ > 3.53. +It is also indicated as more likely optically thick, which is in +agreement with the simulation. Again, we note that this assumes +Article number, page 11 of 20 + +4.0 +3.5 +L/L o +3.0 +3.0 Mo --> 0.706 Mo +2.5 Mo --> 0.616 Mo +2.0 Mo --> 0.583 M o +2.5 +1.5 Mo --> 0.583 Mo +1.25 Mo --> 0.566 M +1.0 Mo --> 0.532 Mo +likely thick PN +2.0 +likely thin PN +Type I PN +5.5 +5.0 +4.5 +4.0 +log TeffA&A proofs: manuscript no. pnlf_ngc300 +the ideal 11% maximum efficiency. For example, based on the +chemical abundance analysis, the bright PNe in M31 exhibit less +conversion efficiency (Jacoby & Ciardullo 1999; Kwitter et al. +2012). Therefore, the actual central star luminosity is likely to +be brighter. +Simulation of [O iii]λ5007 flux evolution has also been +conducted by Schönberner et al. (2007), who employed 1- +dimensional radiative-hydrodynamical simulations for the neb- +ulae. They calculated that the most luminous PNe that popu- +late the PNLF cut-off will achieve their maximum luminosity +at log Teff = 5.00 K and spend ∼ 500 years in this phase. For PN +H9-1, the lower limit temperature is log Teff > 4.97. They also +suggest that UV- to [O iii]λ5007 flux conversion process hap- +pens most efficiently for central star mass of ∼ 0.62 M⊙, if the +nebular shell remains optically thick during the evolution. Re- +ferring to the post-AGB models by Miller Bertolami (2016), the +initial mass of the progenitor star would be ∼ 2.5 M⊙, and the ob- +ject would spend less than 1000 years before entering the white +dwarf cooling sequence. +Similarly, hydrodynamical models have been used to inves- +tigate PNe in nearby galaxies by Schönberner et al. (2010). In +these simulations, it was found that central star masses greater +than 0.65 M⊙ do not exist at the PNLF cut-off. This also sup- +ports the result from Gesicki et al. (2018), in that a progenitor +mass of 3.0 M⊙ for PNe is not expected. This also agrees with +the progenitor masses between 2.0 − 2.5 M⊙ for the bright PNe +in NGC 300 predicted by Stasi´nska et al. (2013), despite the con- +cerns we mentioned regarding their spectroscopic fluxes in Sec- +tion 3.3. They derived the progenitor masses using the stellar +tracks of Bloecker (1995), which evolve slower than the recent +models of Miller Bertolami (2016). This implies the possibility +of less massive progenitors if the new stellar tracks are adopted, +which is however beyond the scope of our current study. +Recently, the properties of luminous PNe near the PNLF cut- +off of M31 have been studied by Davis et al. (2018) for the bulge, +and by Galera-Rosillo et al. (2022) for the disk. For the disk, it +was found that the four brightest PNe have an average progeni- +tor mass of 1.5 M⊙, which is lower than the values predicted by +Schönberner et al. (2007), but still in agreement with Gesicki +et al. (2018). Galera-Rosillo et al. (2022) also measure a rela- +tively low average extinction of the PNe with c(Hβ) ∼ 0.1. This +means that the PNe originated from an older stellar population, +although the disk of M31 also exhibits star forming regions. +In contrast, in the older population of the bulge of M31, +Davis et al. (2018) found that the brightest PN have a central star +mass > 0.66 M⊙, which means progenitor masses of > 2.5 M⊙. +This is found for cases with high extinction, one even reaching +c(Hβ) ∼ 0.6, with the average of c(Hβ) ∼ 0.3 for 23 PNe. Cur- +rent simulations do not predict such massive central stars to be +observable, if they exist at all (Schönberner et al. 2010; Gesicki +et al. 2018). In old systems, the most luminous PNe are sug- +gested to be products from of blue stragglers – stars that re- +sult from a merger during the main sequence (Ciardullo et al. +2005), or symbiotic nebula (Soker 2006). However, both scenar- +ios still do not predict such massive central stars to exist. While +the bright Hα background might overestimate the measured ex- +tinction, which can lead to the overestimation of luminosity and +subsequently the progenitor mass, Davis et al. (2018) in fact did +their measurement with an IFU. Their sky subtraction was based +on a PSF model, which was claimed to be accurate within 10%. +Lately, Ueta & Otsuka (2021) suggested that the extinction +measurement should be solved iteratively, considering the de- +pendency of Hα/Hβ ratio on the electron temperature (Te) and +electron density (ne). Assuming those two parameter as con- +stants would increase the uncertainty of the extinction, and sub- +sequently the stellar parameters. They demonstrate this approach +by reanalysing the M31 disk PNe, worked by Galera-Rosillo +et al. (2022), and found that the iterative approach yields an +average progenitor mass of 2.2 M⊙, instead of 1.5 M⊙ for the +four brightest PNe (Ueta & Otsuka 2022). While the extinction +does not necessarily affect the Te and ne, it may compromise the +ionic and elemental abundance analysis (Ueta & Otsuka 2021, +2022). Since we also assume constant Te and ne for our parame- +ters, we are not excluded from this problem. However, as we are +missing the diagnostic lines in the blue spectral region and the +ones within MUSE wavelength coverage are below the detection +limit, we are unable to put constraints on the Te and ne. +It would be interesting to repeat the exercise of modelling +PN spectra on the basis of improved IFU observations that we +believe are superior to slit-based spectroscopy in controlling sys- +tematic errors, with the more recent stellar evolution tracks and +more careful plasma diagnostics. The future BlueMUSE instru- +ment for the VLT (Richard et al. 2019) will offer the capability +with a wavelength coverage down to the atmospheric limit in the +UV, which includes the necessary nebular lines for such study. +6. Conclusions +We analyse 44 fields, obtained with the MUSE instrument to +find PNe and construct the PNLF. Using the differential emis- +sion line filter (DELF, Roth et al. 2021), we identified more than +500 point sources in [O iii]λ5007, 107 of which were designated +as PNe based on spectral classification with the aid of the BPT- +diagram (Baldwin et al. 1981). The [O iii]λ5007 magnitudes for +the PNe were obtained using DAOPHOT aperture photometry +(Stetson 1987) with aperture corrections. With the sample com- +pleteness at m5007 = 27 for most fields, we constructed the PNLF, +which exhibits the dip that has been observed in other star form- +ing galaxies (Jacoby & De Marco 2002; Ciardullo 2010; Reid +& Parker 2010a). To derive the distance, we employed the max- +imum likelihood estimation method (Ciardullo et al. 1989) to +yield a most likely distance modulus (m − M)0 = 26.48+0.11 +−0.26 +(d = 1.98+0.10 +−0.23 Mpc). For PNe, that are isolated from surround- +ing emission line sources, and that exhibit bright enough Balmer +lines, we measured their extinction. We estimated parameters of +the central stars using the extinction corrected fluxes in an at- +tempt to track their origin from the underlying stellar population. +We discuss the accuracy of our distance measurement, the effect +of local dust for our PNe extinction measurements, and the prop- +erties of the most luminous PNe in our sample. The conclusions +are as follows: +1. The PNLF distance measurement to NGC 300 is improved +with our method and is in excellent agreement with Cepheids +and TRGB distances. This is due to the spectral information +and spatial resolution of MUSE, that provides a higher PN +detection per area, better classification, and accurate photom- +etry. +2. With a limited sample, distance determination based on the +minimisation technique is very dependent on the binning. Al- +though coarse binning might provide a better apparent shape +of the luminosity function for fitting, it can introduce an un- +intended systematic shift. Moreover, the details of the PNLF +shape, which can provide insights on the stellar population, +are also smeared out. +3. The extinction derived for the PNe cannot be disentangled +completely from the local dust lane extinction within the +galaxy. However, with the spatial resolution of MUSE, we +Article number, page 12 of 20 + +Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +were able to resolve several PNe that are likely obstructed +by dust lanes. Any attempt to link the internal extinction and +the underlying stellar population requires a quantitative tech- +nique to separate the local and internal PNe extinction. +4. We found a few Type I PNe, that evolved from main se- +quence mass > 2.5 M⊙. Their luminosities are likely under- +estimated due to the high abundance of nitrogen that serves +as a competing coolant with oxygen. They do not populate +our PNLF cut-off. +With these results, and other works reported in the litera- +ture, we feel encouraged to further develop the IFU observing +technique with MUSE to study extragalactic PNe. One of the in- +herent parameters that we have as yet not utilised is the radial +velocity of individual PNe that comes for free as a by-product +of the analysis. It will be interesting to find out whether the +kinematics can provide hints as to the membership in different +populations in NGC 300. Such study was recently done for other +disc galaxies: NGC 628 (Aniyan et al. 2018), NGC 6946 (Aniyan +et al. 2021), and M31 (Bhattacharya et al. 2019). In the interest +of understanding the physical parameters of the PNe, we are cur- +rently dependant on the ideal assumption of [O iii]λ5007 maxi- +mum conversion and excitation classes to derive the central star +parameters, which is not ideal, especially when most cases have +no He ii λ4686 coverage. Better constraints on the luminosities +and effective temperatures are obtainable through nebular abun- +dance modelling. However, our current wavelength coverage of +the MUSE instrument limit us to explore this possibility. Future +IFUs, that are optimised in the blue wavelength, such as Blue- +MUSE (Richard et al. 2019), will play an important role and +allow us to gain more understanding about PNe in the nearby +galaxies beyond the Local Group, getting us closer to compre- +hend the underlying physics behind the constancy of PNLF cut- +off across galaxies. +Acknowledgements. We thank the anonymous referee for a critical reading of +the manuscript and helpful suggestions to improve the quality of this paper. Part +of this work was supported by the German BMBF program Unternehmen Re- +gion, grant 03Z22AN11. PMW gratefully acknowledges support by the BMBF +from the ErUM program (project VLT-BlueMUSE, grant 05A20BAB). N. 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Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +Appendix A: Aperture correction +The radial profile of a PSF is best modelled with a Moffat func- +tion, as a Gaussian often does not accurately match the wings of +the PSF (Peng et al. 2002; Kamann et al. 2013). Moreover, flux +measurements using a discrete aperture are not able to collect all +of the flux from the PSF wings. To recover the lost flux and ob- +tain accurate photometry, we therefore need to apply an aperture +correction to our measurements. In order to do this, we need at +least one star in a given field as a reference for the observation’s +PSF. We examined 3-4 objects to infer the average PSF FWHM +of the frame and chose the best star for the aperture correction. +Moreover, we also examined the behaviour of the PSF across +wavelengths, as the PSF is expected to be more extended in the +blue, and to show a monotonic decrease of the FWHM toward +the red (Fried 1966; Boyd 1978; Kamann et al. 2013). +To obtain the aperture correction value, we collected the flux +of the reference star using a large aperture radius of 2′′.4 (or 12 +spaxels), assuming that almost all of the flux will be recorded +(Howell 1989). Then, by taking the flux of the same star with +the aperture size employed for the PNe, we were able to obtain +the correction value by taking the ratio of the two fluxes. We then +applied this constant to all PNe measurements within the field. +For the Balmer lines, we have to make sure that both lines are +corrected in a consistent manner, especially with respect to the +wavelength dependence of the PSF. Since the seeing at the tele- +scope is decreasing monotonically with wavelength, the FWHM +for Hβ is expected to be larger than the one for Hα, thus changing +the aperture correction. The reference stars in each field there- +fore have to be well behaved across this wavelength range which +was found to not always be the case. We found several appar- +ent point sources that unexpectedly exhibit an increasing PSF +FWHM trend to the red. Closer inspection revealed that stel- +lar crowding with luminous red stars, e.g., M giants and carbon +stars, were responsible for this problem. For Balmer line correc- +tions, we decided to discard the problematic stars as useful PSF +references. +As an alternative, we used the brightest PNe that happen to +be sufficiently isolated from nearby diffuse gas and H ii regions. +We then used the PNe’s image profile at the wavelengths of the +strong lines of [O iii]λ5007 and Hα, while assuming a negli- +gible difference between the PSF at 5007 Å and Hβ. Unfortu- +nately, we found that some of our fields have neither a well be- +haved star, nor bright isolated PNe, so another alternative was +needed. Using the best reference stars from different fields and +seeing conditions, we derived a simple polynomial relation be- +tween the seeing FWHM and the aperture correction value for +Hβ, [O iii]λ5007, and Hα; these curves are presented in Figure +A.1. +We also confirmed that the difference between PSF at Hβ and +[O iii]λ5007 is not significant, as the polynomial fit is almost +identical for the two wavelengths. However, it is important to +mention that the relation is only derived using a limited sample +of 23 stars. It is not possible to determine the true distribution of +this relation and identify the variables that affect it. While this is +worthwhile for further investigation, we will not explore it in the +current study. We employed the empirical relation as the final al- +ternative, after the bright PN method and the main reference star +method. From the 50 PNe that are within the Hα threshold, we +corrected 16 PNe with the reference star method, 18 PNe with +the bright PN method, and 16 PNe with the empirical relation. +In future studies, the aperture correction can be improved. +Firstly, the uncertainties can be minimised under excellent see- +ing conditions, ideally 0′′.6 PSF FWHM at the wavelength of +Fig. A.1. Empirical polynomial relation between the seeing FWHM and +aperture correction for Hβ, [O iii]λ5007, and Hα. The relation is derived +using 23 stars from different fields. +[O iii]λ5007, that can be achieved using the adaptive optics mode +of MUSE. In cases where no field star is available to serve as a +PSF reference, modelling the wavelength and seeing dependant +PSF on the basis of instrumental data from the adaptive optics +control software may provide a way out (Fusco et al. 2020). +Appendix B: Extinction uncertainties +The main uncertainty of our extinction measurement is the aper- +ture correction. Since we have only a limited number of objects +observed with each correction method, every PN has its own un- +certainty, making it difficult to we derive a proper statistical er- +ror. As an alternative, we estimate the error based on the com- +parison of extinction calculated from different aperture correc- +tion methods. To perform this, we considered PN candidates that +were measured with a well behaved reference star in the field, +and a bright PN in the same field. The comparison is presented +in Table B.1. +Since we expect the aperture correction at Hβ to be larger +than the one for Hα, the application of this factor will reduce the +inferred extinction, as Hβ appears in the denominator of equa- +tion (3). In field E, where the initially selected reference star +shows the unusual trend of an increasing PSF width to the red, +we computed the extinction to be larger after the aperture cor- +rection, prompting us to restrict the reference star method only to +cases where a well behaved star is available in the field. More- +over, we also see that the use of PNe that are not completely +isolated from the ambient gas tends to underestimate the extinc- +tion, if compared to other aperture correction methods. Based +on our choice of priority of the methods, marked as bold in Ta- +ble B.1, the difference between the extinction with and without +aperture correction (∆A) is always larger than the difference be- +tween extinction values derived using various aperture correc- +tion methods. Therefore, we decided to select the ∆A as our error +Article number, page 15 of 20 + +Seeing FWHM ["] +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.014x2 + 0.044x - 0.019 +1.2 +0.014x2 + 0.051x - 0.036 +0.013x2 + 0.043x - 0.046 +Hβ +[O III] +1.0 +Ha +O +Aperture correction +D +0.8 +0.6 +0.4 +0.2 +2 +3 +4 +5 +6 +7 +8 +Seeing FWHM [spaxel]A&A proofs: manuscript no. pnlf_ngc300 +Table B.1. Comparison for extinction values in [O iii]λ5007 derived us- +ing different aperture correction methods. The preferred extinction val- +ues are marked in bold. +ID +A0 +A1 +A2 +A3 +∆A +E-2 +0.542 +0.589a +0.353 +0.457 +0.189 +E-11 +0.615 +0.663a +0.445 +0.531 +0.170 +P-2 +1.342 +1.229 +1.093b +1.252 +0.113 +H9-1 +0.358 +0.147 +0.257 +0.223 +0.211 +H9-6 +0.490 +0.279 +-0.027b +0.354 +0.211 +H10-2 +0.694 +0.420 +0.344 +0.506 +0.274 +H11-1 +0.570 +0.494 +0.412 +0.413 +0.076 +H11-2 +1.440 +1.364 +1.276 +1.283 +0.076 +L9-8 +0.633 +0.466 +0.463 +0.410 +0.167 +Notes. A0 – no aperture correction; A1 – reference star method; A2 – +bright PN method; A3 – empirical relation method. +(a) bad reference star +(b) uniform diffuse Hα background +estimates. In cases where the error estimates exceed nonphysical +negative extinction, the lower limit of the uncertainty is assumed +until zero extinction. +We should note, as discussed in Section 5.4, that assuming +a constant electron temperature Te and ne also introduce uncer- +tainties (Ueta & Otsuka 2021, 2022). Since we did not have the +capability to measure Te and ne, we did not include this aspect +in our measurement error. +Appendix C: MUSE observation fields +The details of the MUSE fields, both the MUSE-GTO and ML20 +(McLeod et al. 2020, 2021), can be referred in Table C.1. We also +include the seeing in [O iii]λ5007, which obtained based on the +average FWHM of 3-4 point sources, preferably stars, in each +field. +Appendix D: MUSE-PN catalogue +The MUSE-PN catalogue of NGC 300 is presented in Table +D.1. MUSE-GTO coordinates (accuracy of ∼ 0′′.1) are pre- +sented if available, otherwise ML20 coordinates (accuracy of +∼ 3′′) are provided. The table will be available through the CDS +Archive and will also include the columns for aperture corrected +line fluxes of Hβ, [O iii]λ5007, Hα, [N ii]λ6584, [S ii]λ6716, +[S ii]λ6731, and the classification remarks. +Article number, page 16 of 20 + +Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +Table C.1. The MUSE observation fields for this work. The upper part represent the MUSE-GTO data and the lower part the ML20 data. +Field +RA(2000) +DEC(2000) +Observation date +FWHM5007 ["] +A +00:54:53.62 +-37:41:05.1 +2018-10-15 +0′′.67 +B +00:54:48.54 +-37:41:05.3 +2018-10-15 +0′′.69 +C +00:54:43.49 +-37:41:05.1 +2018-11-13 +0′′.82 +D +00:54:42.32 +-37:42:05.1 +2015-08-24 +0′′.79 +E +00:54:48.17 +-37:42:13.7 +2015-09-13 +0′′.60 +I +00:54:37.08 +-37:40:52.6 +2014-10-30 +0′′.66 +J +00:54:39.49 +-37:39:50.4 +2014-11-26 +0′′.82 +P +00:54:24.00 +-37:36:29.0 +2016-09-03 +0′′.59 +Q +00:54:22.00 +-37:37:47.0 +2016-09-03 +0′′.55 +H1 +00:54:59.83 +–37:39:42.0 +2016-10-01 +1′′.00 +H2 +00:54:55.40 +-37:39:17.0 +2016-10-01 +1′′.10 +H3 +00:54:50.99 +–37:38:51.8 +2016-10-01 +0′′.87 +H4 +00:54:46.55 +–37:38:26.7 +2016-10-04 +1′′.19 +H5 +00:55:06.51 +–37:41:25.5 +2016-10-05 +1′′.49 +H6 +00:55:02.08 +–37:41:00.9 +2016-10-05 +1′′.18 +H7 +00:54:57.65 +–37:40:35.7 +2016-10-05 +1′′.30 +H8 +00:54:53.22 +–37:40:10.3 +2016-10-05 +1′′.06 +H9 +00:54:48.81 +–37:39:45.4 +2016-11-07 +0′′.82 +H10 +00:54:44.37 +–37:39:20.3 +2016-11-08 +0′′.96 +H11 +00:54:39.95 +–37:38:55.1 +2016-11-08 +0′′.86 +H12 +00:55:04.35 +–37:42:19.4 +2016-11-08 +0′′.88 +H13 +00:54:59.90 +–37:41:54.7 +2016-11-08 +0′′.94 +H14 +00:54:55.47 +–37:41:29.3 +2016-11-08 +1′′.14 +H15 +00:54:57.70 +–37:42:48.2 +2016-11-08 +1′′.23 +H16 +00:54:55.52 +–37:43:42.0 +2016-12-19 +1′′.02 +H17 +00:54:51.08 +–37:43:16.8 +2016-12-23 +1′′.02 +L1 +00:55:08.69 +–37:40:32.3 +2016-12-23 +1′′.13 +L2 +00:55:04.26 +–37:40:06.8 +2016-12-23 +1′′.01 +L3 +00:54:42.13 +–37:38:01.3 +2016-12-24 +1′′.08 +L4 +00:54:51.04 +–37:41:04.2 +2016-12-26 +1′′.11 +L5 +00:54:46.63 +–37:40:38.9 +2017-01-02 +1′′.42 +L6 +00:54:42.19 +–37:40:13.7 +2017-01-02 +1′′.50 +L7 +00:54:37.77 +–37:39:48.5 +2017-01-04 +1′′.49 +L8 +00:55:02.15 +–37:43:13.1 +2018-07-03 +0′′.77 +L9 +00:54:53.27 +–37:42:22.6 +2017-01-05 +1′′.16 +L10 +00:54:48.84 +–37:41:57.9 +2017-01-06 +0′′.84 +L11 +00:54:44.43 +–37:41:32.7 +2017-01-06 +0′′.84 +L12 +00:54:39.99 +–37:41:07.4 +2017-01-06 +0′′.96 +L13 +00:54:35.57 +–37:40:42.2 +2017-01-07 +0′′.84 +L14 +00:54:59.95 +–37:44:06.8 +2017-01-07 +1′′.01 +L15 +00:54:46.64 +–37:42:51.6 +2017-01-16 +1′′.00 +L16 +00:54:42.21 +–37:42:26.4 +2017-01-27 +1′′.16 +L17 +00:54:37.79 +–37:42:00.9 +2018-07-04 +1′′.01 +L18 +00:54:33.37 +–37:41:36.2 +2018-07-04 +0′′.95 +Article number, page 17 of 20 + +A&A proofs: manuscript no. pnlf_ngc300 +Table D.1. MUSE-PN catalogue of NGC 300 +No +IDGTO +IDMcLeod +IDPE12 +RA(2000) +DEC(2000) +m5007 +c(Hβ) +log L [L⊙]a +log Teff [K] +1 +- +H9-1 +PN35 +0:54:48.44 +-37:39:48.39 +22.10±0.06 +0.04±0.08 +3.53 +4.97* +2 +- +H7-3 +PN53 +0:54:56.54 +-37:40:28.99 +22.48±0.06 +0.00±0.07 +3.37 +5.02* +3 +- +H7-2 +PN58 +0:54:58.41 +-37:40:44.29 +22.68±0.06 +0.11±0.08 +3.36 +5.05* +4 +C-7 +L11-2 +PN25 +0:54:44.45 +-37:41:29.36 +22.85±0.06 +0.03±0.10 +3.17 +4.90* +5 +E-2 +L7-8 +PN14 +0:54:38.95 +-37:39:43.26 +22.86±0.06 +0.10±0.07 +3.29 +4.74* +6 +A-11 +H14-7 +PN51 +0:54:55.34 +-37:41:28.34 +22.98±0.06 +0.10±0.05 +3.27 +4.80* +7 +- +L6-7 +PN22 +0:54:42.24 +-37:40:04.82 +22.99±0.06 +0.20±0.09 +3.29 +5.16* +8 +A-23 +H14-8 +PN48 +0:54:54.95 +-37:41:32.88 +23.01±0.06 +0.42±0.05 +3.54 +5.06* +9 +I-2 +L11-11 +PN24 +0:54:43.75 +-37:41:51.52 +23.18±0.06 +0.02±0.04 +3.13 +5.07* +10 +- +L9-8 +PN40 +0:54:52.16 +-37:42:43.27 +23.20±0.06 +0.13±0.07 +3.18 +4.82* +11 +P-2 +- +- +0:54:25.37 +-37:36:29.93 +23.25±0.06 +0.38±0.05 +3.42 +4.92* +12 +- +H6-3 +PN69 +0:55:04.25 +-37:40:52.20 +23.29±0.06 +0.26±0.07 +3.27 +4.77* +13 +- +H2-6 +PN45 +0:54:54.03 +-37:39:28.14 +23.36±0.06 +0.17±0.11 +3.08 +4.75* +14 +- +H1-8 +PN54 +0:54:57.03 +-37:39:44.09 +23.39±0.06 +0.19±0.06 +3.17 +4.81* +15 +- +L2-3 +PN66 +0:55:02.42 +-37:39:54.64 +23.40±0.06 +0.05±0.08 +3.00 +4.91* +16 +- +H10-2 +PN23 +0:54:43.57 +-37:39:35.80 +23.65±0.06 +0.13±0.09 +2.96 +4.96* +17 +- +H7-7 +- +0:54:58.44 +-37:41:09.84 +23.82±0.06 +0.00* +2.89 +5.06* +18 +- +H14-19 +PN60 +0:54:58.34 +-37:41:16.14 +23.88±0.06 +0.05±0.11 +2.77 +4.68* +19 +- +H6-5 +PN65 +0:55:02.00 +-37:40:28.21 +23.93±0.06 +0.10±0.07 +2.87 +4.86* +20 +E-11 +L7-2 +PN12 +0:54:37.93 +-37:40:14.29 +24.21±0.06 +0.12±0.07 +2.77 +5.12* +21 +A-31 +L4-6 +PN42 +0:54:53.28 +-37:40:54.37 +24.45±0.06 +0.24±0.05 +2.81 +5.05* +22 +- +H14-18 +PN61 +0:54:58.49 +-37:41:13.81 +24.49±0.06 +0.00±0.07 +2.55 +4.82±0.11 +23 +- +L17-6 +- +0:54:38.46 +-37:42:29.27 +24.53±0.06 +- +- +- +24 +Q-1 +- +- +0:54:22.61 +-37:37:52.67 +24.64±0.06 +0.11±0.03 +2.64 +4.85* +25 +I-5 +L15-2 +- +0:54:44.81 +-37:42:27.13 +24.96±0.06 +0.04±0.04 +2.43 +4.95* +26 +- +H12-14 +- +0:55:04.02 +-37:41:51.29 +25.01±0.06 +0.00* +2.41 +5.00* +27 +C-6 +L6-6 +PN21 +0:54:41.95 +-37:40:43.32 +25.07±0.06 +0.29±0.10 +2.51 +4.77* +28 +- +H11-2 +PN10 +0:54:37.65 +-37:39:03.73 +25.07±0.06 +0.42±0.03 +2.74 +4.65±0.04 +29 +B-1 +L4-8 +PN37 +0:54:49.46 +-37:40:42.51 +25.13±0.06 +0.13±0.04 +2.45 +4.96* +30 +- +H12-13 +- +0:55:04.02 +-37:41:52.26 +25.21±0.06 +0.00* +2.33 +4.86* +31 +- +H1-3 +- +0:55:00.90 +-37:39:39.74 +25.23±0.06 +0.00* +2.33 +5.08* +32 +- +L5-8 +PN27 +0:54:45.05 +-37:40:28.67 +25.32±0.06 +0.00* +2.29 +4.91* +33 +- +H6-9 +PN62 +0:54:59.83 +-37:41:00.24 +25.41±0.06 +0.00* +2.25 +4.70±0.02 +34 +- +H9-6 +PN29 +0:54:45.85 +-37:39:58.04 +25.52±0.06 +0.08±0.08 +2.20 +4.70* +35 +- +H6-2 +PN72 +0:55:05.13 +-37:40:45.33 +25.57±0.06 +- +- +- +36 +A-25 +H14-4 +- +0:54:55.04 +-37:41:17.31 +25.59±0.06 +- +- +- +37 +C-2 +L11-1 +- +0:54:45.79 +-37:41:30.80 +25.60±0.06 +0.21±0.10 +2.23 +4.76* +38 +- +H9-11 +PN29 +0:54:49.31 +-37:40:19.15 +25.65±0.06 +0.00* +2.16 +4.63±0.02 +39 +- +H16-1 +PN52 +0:54:56.02 +-37:43:14.52 +25.66±0.07 +0.00* +2.15 +4.89* +40 +- +H12-2 +PN67 +0:55:03.18 +-37:42:08.97 +25.69±0.06 +0.00* +2.14 +4.75±0.04 +41 +- +L14-2 +- +0:55:01.17 +-37:44:36.29 +25.72±0.07 +0.00* +2.13 +4.81* +42 +- +H9-2 +PN31 +0:54:48.11 +-37:39:41.62 +25.78±0.06 +0.00* +2.11 +4.86* +43 +- +L9-2 +PN43 +0:54:53.45 +-37:41:56.10 +25.80±0.07 +- +- +- +44 +- +H9-4 +PN38 +0:54:49.77 +-37:39:48.14 +25.84±0.06 +- +- +- +45 +E-4 +L7-7 +PN13 +0:54:38.17 +-37:39:41.42 +25.92±0.06 +0.02±0.07 +2.00 +4.81* +46 +- +H8-8 +- +0:54:51.02 +-37:40:03.35 +25.97±0.07 +- +- +- +Article number, page 18 of 20 + +Azlizan A. Soemitro et al.: MUSE crowded field 3D spectroscopy in NGC 300 +Table D.1. continued. +No +IDGTO +IDMcLeod +IDPE12 +RA(2000) +DEC(2000) +m5007 +c(Hβ) +log L [L⊙]a +log Teff [K] +47 +- +H6-1 +PN68 +0:55:04.00 +-37:40:41.44 +26.00±0.06 +0.00* +2.02 +4.69±0.02 +48 +C-5 +L12-1 +- +0:54:42.12 +-37:41:00.04 +26.04±0.06 +0.00* +- +- +49 +- +H13-3 +- +0:54:59.92 +-37:41:50.68 +26.08±0.07 +- +- +- +50 +- +H14-14 +PN56 +0:54:57.22 +-37:41:22.89 +26.09±0.07 +0.00* +- +- +51 +A-29 +H14-6 +- +0:54:55.72 +-37:41:26.39 +26.12±0.06 +0.00* +- +- +52 +- +H17-5 +- +0:54:49.75 +-37:42:59.50 +26.12±0.06 +- +- +- +53 +- +H2-1 +- +0:54:53.10 +-37:39:34.10 +26.17±0.07 +- +- +- +54 +- +H16-5 +- +0:54:52.90 +-37:44:00.13 +26.18±0.08 +0.00* +- +- +55 +J-1 +L10-3 +- +0:54:50.62 +-37:41:46.48 +26.19±0.07 +- +- +- +56 +- +L9-6 +PN47 +0:54:54.36 +-37:42:32.71 +26.24±0.07 +- +- +- +57 +- +H16-2 +PN50 +0:54:55.00 +-37:43:11.53 +26.28±0.08 +- +- +- +58 +- +H10-5 +- +0:54:44.42 +-37:38:49.86 +26.35±0.07 +0.00* +- +- +59 +- +H11-1 +PN15 +0:54:39.03 +-37:38:43.34 +26.36±0.07 +0.00* +- +- +60 +- +H14-9 +- +0:54:54.27 +-37:41:35.21 +26.38±0.07 +- +- +- +61 +- +L12-4 +PN18 +0:54:39.81 +-37:41:34.87 +26.48±0.09 +- +- +- +62 +D-7 +L12-5 +PN11 +0:54:37.74 +-37:41:18.90 +26.50±0.06 +- +- +- +63 +- +H2-4 +- +0:54:54.52 +-37:39:12.97 +26.53±0.07 +- +- +- +64 +- +H2-3 +- +0:54:54.58 +-37:39:06.09 +26.56±0.07 +- +- +- +65 +- +L6-11 +- +0:54:43.66 +-37:40:10.98 +26.60±0.10 +- +- +- +66 +- +H10-6 +- +0:54:45.69 +-37:38:58.76 +26.61±0.09 +- +- +- +67 +- +H13-7 +- +0:54:57.08 +-37:42:03.65 +26.64±0.08 +- +- +- +68 +A-24 +L4-17 +- +0:54:52.16 +-37:41:32.42 +26.73±0.07 +- +- +- +69 +E-5 +L6-17 +PN19 +0:54:40.02 +-37:40:02.33 +26.74±0.06 +- +- +- +70 +- +L2-9 +- +0:55:04.57 +-37:40:30.39 +26.77±0.07 +- +- +- +71 +- +H8-13 +- +0:54:51.58 +-37:39:54.83 +26.85±0.08 +- +- +- +72 +- +L2-12 +- +0:55:04.30 +-37:40:24.76 +26.88±0.08 +- +- +- +73 +A-7 +H14-3 +- +0:54:54.56 +-37:41:10.83 +26.90±0.07 +- +- +- +74 +- +H14-12 +- +0:54:54.22 +-37:41:50.64 +26.94±0.08 +- +- +- +75 +- +H12-12 +- +0:55:03.12 +-37:42:26.65 +26.94±0.09 +0.00* +- +- +76 +- +H1-5 +- +0:54:58.93 +-37:39:46.00 +27.04±0.08 +- +- +- +77 +- +H9-7 +- +0:54:47.73 +-37:39:31.07 +27.06±0.08 +- +- +- +78 +J-4 +L10-8 +- +0:54:48.98 +-37:42:09.55 +27.10±0.07 +- +- +- +79 +- +H13-8 +- +0:55:02.07 +-37:42:06.89 +27.13±0.10 +- +- +- +80 +- +H13-4 +- +0:55:01.30 +-37:42:19.86 +27.15±0.10 +- +- +- +81 +- +H8-7 +- +0:54:53.07 +-37:39:45.15 +27.26±0.09 +- +- +- +82 +- +H13-1 +- +0:54:59.75 +-37:41:35.90 +27.29±0.11 +- +- +- +83 +C-11 +- +- +-0:54:45.86 +-37:40:46.33 +27.34±0.07 +- +- +- +84 +- +H17-3 +- +0:54:51.91 +-37:43:52.93 +27.34±0.10 +- +- +- +85 +B-19 +L11-4 +- +0:54:47.24 +-37:41:19.98 +27.50±0.07 +- +- +- +86 +D-9 +- +- +0:54:38.99 +-37:41:10.65 +27.56±0.07 +- +- +- +87 +- +H12-17 +- +0:55:06.27 +-37:42:07.55 +27.58±0.11 +- +- +- +88 +- +H17-8 +- +0:54:50.96 +-37:42:50.68 +27.59±0.11 +- +- +- +89 +- +H15-5 +- +0:54:56.36 +-37:42:39.91 +27.62±0.13 +- +- +- +90 +J-14 +- +- +0:54:46.56 +-37:42:29.36 +27.67±0.09 +- +- +- +91 +- +L1-7 +- +0:55:07.22 +-37:40:44.58 +27.67±0.12 +- +- +- +92 +E-1 +- +- +0:54:41.25 +-37:39:40.11 +27.69±0.08 +- +- +- +93 +E-9 +- +- +0:54:38.85 +-37:39:58.80 +27.73±0.08 +- +- +- +94 +J-8 +- +- +0:54:46.53 +-37:41:55.03 +27.75±0.10 +- +- +- +Article number, page 19 of 20 + +A&A proofs: manuscript no. pnlf_ngc300 +Table D.1. continued. +No +IDGTO +IDMcLeod +IDPE12 +RA(2000) +DEC(2000) +m5007 +c(Hβ) +log L [L⊙]a +log Teff [K] +95 +D-19 +- +- +0:54:38.89 +-37:40:38.47 +27.76±0.08 +- +- +- +96 +- +H11-3 +- +0:54:40.57 +-37:38:28.75 +27.80±0.13 +- +- +- +97 +- +H2-5 +- +0:54:57.74 +-37:39:12.84 +27.86±0.13 +- +- +- +98 +C-14 +- +- +0:54:42.26 +-37:41:21.15 +27.88±0.10 +- +- +- +99 +- +L2-10 +- +0:55:03.98 +-37:40:06.30 +27.90±0.13 +- +- +- +100 +A-6 +- +- +0:54:52.56 +-37:40:45.39 +28.02±0.10 +- +- +- +101 +- +H12-15 +- +0:55:04.64 +-37:41:59.55 +28.06±0.16 +- +- +- +102 +I-7 +- +- +0:54:42.12 +-37:42:14.90 +28.32±0.12 +- +- +- +103 +A-15 +- +- +0:54:53.80 +-37:41:30.34 +28.40±0.13 +- +- +- +104 +A-52 +- +- +0:54:51.93 +-37:41:26.37 +28.62±0.15 +- +- +- +105 +E-7 +- +- +0:54:40.27 +-37:39:56.77 +28.62±0.12 +- +- +- +106 +E-13 +- +- +0:54:40.56 +-37:40:01.23 +28.84±0.13 +- +- +- +107 +B-39 +- +- +0:54:46.25 +-37:40:49.75 +28.91±0.15 +- +- +- +Notes. +(a) Lower limits assuming maximum [O iii]λ5007 conversion efficiency of 11% +(*) Lower limit value +Article number, page 20 of 20 + diff --git a/f9E1T4oBgHgl3EQfywWE/content/tmp_files/load_file.txt b/f9E1T4oBgHgl3EQfywWE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f012aae0a08076f0384bd586530733ba9c5a47af --- /dev/null +++ b/f9E1T4oBgHgl3EQfywWE/content/tmp_files/load_file.txt @@ -0,0 +1,2623 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf,len=2622 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 ©ESO 2023 January 10, 2023 MUSE crowded field 3D spectroscopy in NGC 300 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Planetary nebula luminosity function Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro1, 2, Martin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth1, 2, Peter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Weilbacher1, Robin Ciardullo3, 4, George H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Jacoby5, Ana Monreal-Ibero6, Norberto Castro1, and Genoveva Micheva1 1 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany e-mail: asoemitro@aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='de 2 Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 24/25, 14476 Potsdam, Germany 3 Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 4 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA 5 NSF’s NOIRLab, 950 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Cherry Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Tucson, AZ 85719, USA 6 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands Received ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' accepted ABSTRACT Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We perform a deep survey of planetary nebulae (PNe) in the spiral galaxy NGC 300 to construct its planetary nebula luminosity function (PNLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We aim to derive the distance using the PNLF and to probe the characteristics of the most luminous PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We analyse 44 fields observed with MUSE at the VLT, covering a total area of ∼ 11 kpc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We find [O iii]λ5007 sources using the differential emission line filter (DELF) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We identify PNe through spectral classification using the aid of the BPT- diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The PNLF distance is derived using the maximum likelihood estimation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For the more luminous PNe, we also measure their extinction using the Balmer decrement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We estimate the luminosity and effective temperature of the central stars of the luminous PNe, based on estimates of the excitation class and the assumption of optically thick nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We identify 107 PNe and derive a most-likely distance modulus (m − M)0 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='26 (d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='23 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We find that the PNe at the PNLF cut-off exhibit relatively low extinction, with some high extinction cases caused by local dust lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We present the lower limit luminosities and effective temperatures of the central stars for some of the brighter PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also identify a few Type I PNe that come from a young population with progenitor masses > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙, however do not populate the PNLF cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The spatial resolution and spectral information of MUSE allow precise PN classification and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These ca- pabilities also enable us to resolve possible contamination by diffuse gas and dust, improving the accuracy of the PNLF distance to NGC 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' galaxies: stellar content – planetary nebulae: general – galaxies: luminosity function, mass function – distance scale – stars: AGB and post-AGB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Introduction The planetary nebula luminosity function (PNLF) is a distance determination method with a precision and accuracy that is com- parable to those of the tip of the red giant branch (TRGB) and Cepheid methods (Jacoby 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2010, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using evidence from narrow- band photometric surveys in [O iii]λ5007, Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1989) have shown that the magnitude distribution of planetary nebulae (PNe) for a given galaxy follows an empirical power law defined as N(M) ∝ e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='307M{1 − e3(M∗−M)} (1) where the brightest PN at the cut-off has an absolute magni- tude of M∗ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='06 with a possible minor dependency on metallicity (Jacoby 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While a number of different formulations have been developed to model the various shapes of the PNLF at fainter magnitudes (Rodríguez-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Longob- ardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019, 2021), such faint-end variation do not affect the definition of the PNLF’s bright end cut-off, which is the critical feature for distance determinations (Spriggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Until the early 2010s, most PNLF distance measurements were obtained using 4-meter class telescopes and narrow-band interference filters, and as a result, the method has been tradi- tionally limited to distances of ∼ 20 Mpc (Jacoby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2010, 2012, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although 8-meter class telescopes were available and even observed PNe at the Coma cluster (∼ 100 Mpc, Gerhard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2005), most of the instruments had wider bandpass filters, which increased the inclusion of sky background signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This limited the PN detection sensitivity, that was necessary to significantly improve the distance range of the PNLF (Ciardullo 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This situation has now changed due to the use of the Multi Unit Spectroscopic Explorer (MUSE, Ba- con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010) integral-field spectrograph on the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2-meter Very Large Telescope to survey PNe in distant systems (Spriggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In fact, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2021) have shown that by using a differential emis- sion line filter technique on MUSE data, PNLF measurements are now possible out to distances of ∼ 40 Mpc under excellent seeing condition and with the aid of the adaptive optics system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This is mainly due to the narrow effective bandpass of MUSE, Article number, page 1 of 20 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='03437v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='GA] 9 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 that is five times narrower than the typical narrow-band filters, which can substantially suppressed the background sky noise (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Previous PNLF studies of late type galaxies using [O iii]λ5007 narrow-band filters were also hampered by the pos- sible confusion with supernova remnants (SNRs) or H ii regions (Herrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Herrmann & Ciardullo 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Frew & Parker 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While Hα narrow-band image allows the exclu- sion of H ii regions, it cannot be used to exclude the SNRs, whose classification typically rely on the [S ii]λ6716, 6731 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In M31 and M33, Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) found that the SNR con- tamination does not change the shape nor the position of the PNLF cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In contrary, Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2017) have shown with MUSE observations of NGC 628 that the presence of SNR con- taminants can affect the bright cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2022) demonstrated that the latter study had an issue with the background subtraction in Hα, which affected the classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Their reanalysis concluded that the PNLF cutoff was indeed unaffected by the contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Nevertheless, the possible con- tamination by SNRs is relevant for the investigation of the faint end of the PNLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The impressive spatial resolution and spec- troscopic capability of the MUSE instrument allows the instant identification of interlopers, even in star forming disk galaxies (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' NGC 300 is a spiral galaxy in the foreground of the Sculp- tor group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Being fairly isolated from its neighbouring galaxies (Karachentsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2003) and close in distance (Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Rizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2006, 2007) makes it interesting for studying star formation histories and galactic evolution (Muñoz-Mateos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Kudritzki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Gogarten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A previous PNLF study, Soffner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1996) identified 34 PNe, a small number that was not ideal to make a proper PNLF and therefore opted the cu- mulative PNLF to derive the distance by using the LMC as a yardstick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' More recently, Peña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2012) observed 104 PN candidates using narrow-band imaging from the central and the eastern outskirt region to construct the PNLF, with a follow-up spectroscopy for the brighter candidates (Stasi´nska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In Paper I (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018), seven 1′ × 1′ MUSE fields in the central region of NGC 300 were observed with the goal of re- solving stellar populations in crowded fields of nearby galaxies, from which they discovered 45 PN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Again, this num- ber was too small to create a useful PNLF, since the sample spans a very wide magnitude range of 22 ≲ m5007 ≲ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the present work, using publicly available archival data from McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2020, 2021) – or ML20 – and 2 additional MUSE-GTO fields, we expand the observed area from 7 to 44 MUSE fields in or- der to detect more PNe and obtain a PNLF distance to NGC 300 using integral field spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Our lack of a complete understanding of the underlying physics behind the invariance of the PNLF cut-off has prevented the PNLF technique to become a primary standard candle (Ciar- dullo 2010, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although simulations have provided an im- pression of the physical properties of the most luminous PNe (Jacoby 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Dopita & Meatheringham 1990, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Mendez & Soffner 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Méndez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2007, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Valenzuela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019), an observational characterisation is still limited to the LMC (Dopita & Meatheringham 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Do- pita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010a,b), and M31 (Kwitter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Galera-Rosillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' If the most luminous PNe at the PNLF cut-off have indeed originated from a single-star stellar evolution, then placing the central stars in the HR-diagram will provide insights into the underlying stellar population, and also the nature of the cut-off itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using the data quality that MUSE offers, we aim to constrain the lumi- nosity and effective temperature of the central stars for some of the bright PNe to understand their origin and expand our under- standing of PNe beyond the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The structure of this paper is as follows: details on obser- vations and data reduction are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The data analysis regarding the PN detection and classification, the liter- ature comparison of the PN number, the [O iii]λ5007 photome- try, and the measurement of the Balmer decrement is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The resulting luminosity function and the distance measurement are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The discussion and the implications of this work follow in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Lastly, the conclu- sions are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Observations and data reduction The data for this project were acquired using the MUSE spec- trograph on the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2-meter Very Large Telescope (Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 9 fields were obtained as part of the MUSE guaranteed time observation (GTO) program1, while 35 fields were taken from the ESO Archive2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The area covered by these observations is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The initial MUSE-GTO data (field A, B, C, D, E, I, J) were obtained between the years 2014 – 2016 using the extended wide field mode with a spatial coverage of 1′×1′ and spectral coverage of 4650 − 9350 Å with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='25 Å sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' First results from the 7 fields in the centre area were reported in Paper I, covering the nucleus, part of the spiral arm that extends from the nucleus to the north-west, and inter-arm regions of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In late 2018, additional fields P and Q were observed to cover the part of the outer spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, fields A, B, and C were re-observed with adaptive optics support to obtain better image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the adaptive optics mode, a notch filter at 5750 − 6100 Å blocks the laser light, which is, however, not affecting the emission lines of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Most of the fields were obtained with an exposure time of 6 × 900 s, with the exception of field J (4 × 900 s), field C (8 × 900 s), and field B, D (11 × 900 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The 9 MUSE-GTO fields were reduced using the MUSE pipeline (Weilbacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020) within the MUSE-WISE envi- ronment (Vriend 2015), as explained in more detail in Paper III (Micheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In addition to the field distortion correc- tion produced by the pipeline, the astrometry is also calibrated using the Gaia DR2 catalogue (Gaia Collaboration 2018), pro- viding absolute astrometry within 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Sky subtraction was per- formed using an offset field outside the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since we will per- form photometry in [O iii]λ5007, we measure the seeing quality at this wavelength based on the FWHM of 3 to 4 stars for each field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These stars, which are typically giants or supergiants in the disk of NGC 300, have apparent magnitudes of F606W ≳ 21 in the HST ACS magnitude system (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Unlike the situation in more distant systems, such as Fornax cluster ellip- ticals (Sextl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021), confusion with globular clusters is not a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For the MUSE GTO data, the image quality ranges from 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 − 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 FWHM, as presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Another 35 fields, the ML20 data, publicly available at the ESO Archive, were originally observed to study stellar feedback in NGC 300 (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The data were obtained using the nominal wide field mode, which has the same spatial 1 Program IDs 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='D-0116, 095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='D-0173, 097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='D-0348, and 0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='B- 0317 – PI: Roth 2 Program ID 098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='B-0193(A) – PI: McLeod Article number, page 2 of 20 Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' MUSE fields of NGC 300 are marked with red, green, and cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The red fields are MUSE-GTO data from the Paper I pilot study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The green fields labelled P and Q are additional MUSE-GTO observation for the outer spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The cyan fields indicate ML20 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The magenta fields are the previous PNe survey area of Peña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2012) using the FORS2 instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Image: NGC 300 in Hα taken with the Wide Field Imager (ESO) – Program ID 065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='N-0076 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' coverage of 1′ × 1′, but with a slightly shorter wavelength cov- erage of 4750 − 9350 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The observations were conducted in the period of 2016 – 2018 without the support of adaptive optics, and each field was observed with an exposure time of 3×900s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These data were reduced using the fully automated MUSE pipeline (Weilbacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020) with default parameters, as provided in the ESO Archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The astrometry of the ML20 data only re- lied on the distortion correction within each field, which limited the absolute positional accuracy of the object catalogue to ∼ 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, the sky subtraction was performed using a reference region within each field instead of an offset field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' this resulted in sky oversubtraction, especially in areas where diffuse gas is prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, since we perform local sky subtraction for flux measurements of individual objects (see Section 3), the ef- fect cancels out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Based on our measurements, the seeing qual- ity of the ML20 data in [O iii]λ5007 ranges between 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 − 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These [O iii]λ5007 seeing measurements are presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Data analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PN detection and classification To find PN candidates, we employed the differential emission line filter (DELF) method described by Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This is performed by extracting 15 datacube layers around the wavelength of redshifted [O iii]λ5007 (the systematic velocity of NGC 300 is vsys = 144 km/s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Lauberts & Valentijn 1989) and treating each layer as an on-band image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' this 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='75 Å range ac- counts for the different line-of-sight velocities (LOSV) within the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Then, an intermediate broadband continuum image is constructed from the wavelength range between λ5063−5188 Å, which is free from strong absorption line features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' this is used as the off-band image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' By subtracting the scaled off-band im- age (see scaling factor in Equation 8, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021) from the on-band images, we obtain a series of continuum-free dif- ferential images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using the DS9 software (Joye & Mandel 2003), the differential images are visually inspected to find the [O iii]λ5007 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' After experimenting unsuccessfully with DAOPHOT FIND (Stetson 1987) to identify PN candidates, which turned out to be unable to cope with the spatially variable emission line background in [O iii]λ5007, we resorted to the dat- acube layer blinking technique, that is described in Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The typical physical size of planetary nebulae is of the of order ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 pc (Osterbrock & Ferland 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' If we assume a dis- tance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='88 Mpc (Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2005) and a scale of ∼ 9 pc/′′, we expect the PNe in NGC 300 to appear as point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' After marking the coordinates of the point sources in [O iii]λ5007, we apply aperture photometry (Stetson 1987) at each wavelength layer along the datacube for these objects to obtain their spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We employ an aperture diameter of 3 spaxels (0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6), an in- ner sky annulus of 12 spaxels, and an outer sky annulus of 15 spaxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although the seeing conditions of the datacubes vary Article number, page 3 of 20 PE12centel PE12outskil H13 H15A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 between ∼ 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 − 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5, we extract the spectra using the same pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The small aperture of 3 spaxels is chosen to minimise contamination of background gas or nearby H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Then, the line fluxes are extracted using Gaussian fitting with the LM- FIT routine in Python (Newville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2016), keeping in mind that the MUSE data has a wavelength sampling of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='25 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since the MUSE-GTO and the ML20 data have overlapping areas, the classifications were done independently for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Cross- matching was performed after the PNe were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To classify the sources into PNe, H ii regions, and super- nova remnants (SNR), we employed the BPT-diagram (Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1981) that is based on the line ratio of [O iii]λ5007/Hβ and [S ii]λλ6716, 6731/Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Besides the application of classifying ac- tive galaxies (Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2001, 2006), the BPT-diagram has been demonstrated to effectively discriminate the PNe from their mimics (Kniazev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Frew & Parker 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Sabin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' As the classification rely on emission line ratios with very similar wavelengths, we can assume that the relative line fluxes have negligible extinction and seeing de- pendency on wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Our BPT-diagrams for the MUSE-GTO and ML20 data are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To separate the SNRs, we adopt the value log [S ii]λλ6716, 6731/Hα ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 from Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To discriminate PNe from H ii regions, we employ the theoretical line by Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2001), which was originally intended to differentiate starburst galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also consider the line ratio of [S ii]λ6731/6716 as a proxy for density, since bright PNe are ex- pected to be denser than both H ii regions and SNRs (Osterbrock & Ferland 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While the brightest [O iii]λ5007 sources have sufficient line fluxes for the BPT-diagram classification, fainter sources may lack the weaker emission lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' the Hβ line or the [S ii]λ6716, 6731 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In such cases, we assume lower limits for the line fluxes and classify the sources as PNe if the [O iii]λ5007 line is stronger than the Hα line, which may in- troduce deviation from the separation lines in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also found some faint objects, which only have the detection of the [O iii]λ5007 and the [N ii]λ6584 line without Hα detection, which are possibly Type I PNe (Frew & Parker 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, we also put remarks for PNe, which are only classified solely based on [O iii]λ5007 detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This is the case for few of our faintest PN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Nevertheless, such cases will not affect the distance determination because the PNLF cut-off is only de- fined by the brightest PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the MUSE-GTO data, we classified 37 PNe, 62 H ii re- gions, and 59 SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In ML20 data, we classified 85 PNe, 176 H ii regions, and 105 SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To cross-match the PNe candidates in the overlap area between the two dataset, we attempted an automated algorithm by comparing the sky coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' How- ever, since the astrometric accuracy of both data differs, our attempt was not successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, the cross-matching was performed visually using the DS9 software (Joye & Mandel 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The final PN number from the MUSE-GTO and ML20 fields: 105 PNe in the central region, and 2 in the P and Q fields at higher galactocentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The PN catalogue is presented in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PN number comparison Previous PN surveys of NGC 300 were conducted by Soffner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1996) – SO96, Peña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2012) – PE12, and Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) – Paper I, who identified 34, 104, and 45 PN candidates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To demonstrate the accuracy of our classification, we employed the sample by PE12 as comparison, since it covers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' BPT-diagram of MUSE-GTO data (upper) and ML20 data (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The orange dot-dashed line is taken from Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2001) and the purple dashed line is defined by Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Open cir- cles indicate PN candidates, which only have [O iii]λ5007 as the diag- nostic line for the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The deviation from the separation lines is explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' more area and contains more PNe than the other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PE12 observed NGC 300 with the FORS2 imager (Appenzeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1998) in two 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8′ × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8′ fields, one in the centre, and another in the eastern outskirts of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The study employed the on/off-band technique to detect PN candidates and classified ob- jects based on the criterion of whether or not a central star was present in their 5105 Å image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The expectation was that cen- tral stars of PNe would be too faint to be detected in the vi- sual, while the ionising O stars in H ii regions could be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For brighter candidates with m5007 < 25, they also performed additional spectroscopy using the MXU-mode with the same in- strument (Stasi´nska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since our observations cover a smaller area on the sky, we only made the comparison for the Article number, page 4 of 20 PNe 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 SNR Hll Region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 (dH / L00S[III ])6o 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 一1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 log([S II]入6716+31 / Hα)PNe 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 SNR Hll Region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 ( / 0[III )l 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 个国国 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 log([S II]6716+31 /Hα)Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 intersecting 5′ × 7′ region in the centre of the galaxy, which is also indicated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the overlapping region at the centre of the galaxy, we iden- tify 105 PNe, compared to 58 in the PE12 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, although we recover all 58 sources found by PE12, our classi- fication indicates several discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While 43 of the PE12 sources are confirmed as PNe, we classify 9 objects as compact H ii regions and 3 as SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These misclassifications could have happened due to the fact that PE12 only have the spectral clas- sification for candidates with m5007 < 25, while the fainter ob- jects completely relied on the detection of a central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This approach also lacked the ability to identify SNRs amongst the fainter candidates, as such objects can be discriminated through the detection of the [S ii]λ6716, 6731 lines (Frew & Parker 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Sabin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since all of our candidates are classified on the basis of their spectral properties, we believe that our classi- fication is more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, in terms of the number of detection, we also demonstrate that the MUSE observations are more sensitive and able to reach fainter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' [O III]λ5007 photometry The [O iii]λ5007 fluxes were obtained using DAOPHOT aperture photometry (Stetson 1987), applied to the PNe candidates in the 15 differential layers for each datacube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Then, the magnitudes were computed using the V-band equivalent conversion (Jacoby 1989) defined as m5007 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 log F5007 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='74 (2) where the flux is in erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Here, the aperture radius was adjusted to a value of approximately the FWHM of the PSF in a given exposure to accommodate the respective seeing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The inner and outer sky annulus were fixed to 12 and 15 spaxels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Most of the flux of the PSF was obtained by adding the 5 bins closest to the Gaussian peak and the remaining flux is recovered through the use of an aperture correction based on the information of a PSF reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This correction is crucial to obtain accurate fluxes, which however can be a challenge when there is no reference available especially with the small field of view of MUSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The aperture correction method is explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The photometric uncertainty was calculated from the Gaus- sian fit errors, convoluted with an assumed flux calibration er- ror of 5% (Weilbacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In high surface brightness regions of distant galaxies, double-peaked profiles are occasion- ally found and indicate the presence of two superposed PNe with different radial velocities (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Unsurprisingly, we do not find such cases in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since NGC 300 is a quite nearby galaxy, spatial coincidences are less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Additionally, the five datacube layers containing the total flux for [O iii]λ5007 were also inspected to insure the PN candidate was not extended or contaminated by surrounding gas emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' As an internal test of our photometry, we used the regions of field overlap to compare our PN measurements made in the MUSE-GTO fields to those from the ML20 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This com- parison is presented in the upper panels of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We find that the ML20 observations obtained under poor seeing condi- tion tend to be systematically fainter than the MUSE-GTO data, while the photometry of the same object from different datacubes with similar image quality gives identical results (exception for the faintest PN in the comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Thus, the difference in see- ing conditions can introduce a magnitude error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' this is most likely due to the choice of a too small aperture for the asymp- totic assumption for the aperture correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Because the disk of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Comparison between the MUSE-GTO and ML20 [O iii]λ5007 magnitudes before (upper) and after (lower) photometric correction for PNe in the overlapping area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The markers are linearly scaled with the seeing FWHM of the field and sorted from the brightest to the faintest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' NGC 300 contains a large amount of diffuse emission-line gas, we chose to not to increase this radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, in order to ob- tain the same photometric quality between the two data sets, we applied and additional corrections of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 mag for objects with seeing FWHM ∼ 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 (∼ 6 spaxels), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 mag for seeing FWHM ∼ 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 (∼ 7 spaxels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We found these values based on empirical trial and error to achieve the minimum average dis- crepancy between the two sets of magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The comparison after applying the correction is presented in the lower panels of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The average discrepancy is now 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='05 mag, which is still within the typical measurement error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='06 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For PNe with m5007 ∼ 27 or fainter, the correction has no meaningful implication, because the candidates are close Article number, page 5 of 20 28 MUSE-GTO 27 Archival 26 m5007 25 24 23 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 △m5007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 4 5 6 8 9 8869899860 PN ID28 MUSE-GTO 27 Archival 26 m5007 25 24 23 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 △m5007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 4 5 6 8 9 20 21 25 27 29 37 45 48 51 55 62 68 69 73 78 85 PN IDA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 to the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the pilot study, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) per- formed a completeness simulation for the MUSE-GTO data of given exposure time, with seeing quality ranging from 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6−1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This was conducted by embedding artificial PNe with different magnitudes into the real datacubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It was found that for a see- ing of 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2, the expected completeness is 90% at m5007 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although we are able to detect a PN as faint as m5007 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='91, the seeing quality on average for all our data is 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0, with 23% of the fields exhibiting larger than 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This shows that completeness of 90% at m5007 = 27 is only achieved for 77% of our fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, since the emphasis of this work is on the bright candidates that define the PNLF cut-off for the distance determination, our results are not suffering from sample incom- pleteness at the faint end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For the final [O iii]λ5007 magnitudes, we preferred the MUSE-GTO data, if available, and otherwise we employed the ML20 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also applied the correction for fields outside the overlapping area with seeing FWHM > 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In total, 11 PNe from 4 fields were corrected in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To test the accuracy of our photometry, we compared our magnitudes to the results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Figure 4 shows a comparison with SO96 and PE12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While our data is in reason- able agreement with SO96 within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='01 mag on average, there is a systematic offset with regard to PE12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We find that our magnitudes are systematically brighter by an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='71 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PE12 obtained instrumental [O iii]λ5007 magnitudes for the FORS2 on-band image using aperture photometry with the aperture diameter of 5 pixels (1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='25), based on the average PSF FWHM of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='9 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To obtain the apparent m5007 magnitudes, they calibrated the instrumental measurements using an empiri- cal relation derived from the objects’ spectroscopic fluxes, which are only available for the brightest PNe in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We can try to understand what may be the reason for the discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' First of all, we note that flux calibration is an established MUSE procedure in operation at the VLT and part of the data reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' According to Weilbacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2020), flux calibration has been measured to be accurate to within 3-5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' If a signifi- cant number of our MUSE exposures would have been affected by non-photometric observing conditions – for which we have no evidence, we would expect a scattered, but not the tightly constrained linear correlation, that we see in Figure 4, in partic- ular for the brightest 3 magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Secondly, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) have tested synthetic MUSE datacube broadband photometry of stars against published HST photometry for the same GTO dat- acube subset that has been used in our work, showing no hint of an offset to within a magnitude of F606W=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Thirdly, the agreement with SO96, who obtained their data with narrow-band imaging at the ESO NTT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' a different instrument at a different telescope, gives us reasonable confidence that our flux calibra- tion cannot be off by as much as a factor of almost 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Finally, we can follow the argument put forward by Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2004), that by definition, integral field spectroscopy is an ideal tool for spectrophotometry, as is does not suffer from any kind of slit effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We can speculate, though, that the spectrophotometry from PE12 might have been affected in various ways to give rise to the observed systematic offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In case of PE12, the calibration relies on spectroscopic fluxes, which were obtained using a slit spec- trograph, and thus may suffered from slit-losses that were esti- mated to be 10 − 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, from our comparison, we infer that the loss might be underestimated since 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='71 mag difference is equivalent to a loss of ∼ 48%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To test this, we performed a slit-loss simulation based on a model of the PSF with the quoted seeing conditions of 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7 − 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='9, and a slit size of 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The simula- tion is done in the R-band, as the seeing measurement is typically Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Comparison of m5007 between this work, SO96, and PE12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The photometry of SO96 agrees within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='01 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The photometry of PE12 is systematically fainter by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='71 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For m5007 > 25, the relation with PE12 becomes scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' done in this band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Based on these parameters, our simulation predicts that the slit-losses should be between 8 − 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' How- ever, since the PSF FWHM is expected to be larger in the blue wavelength region, the slit-loss in [O iii]λ5007 will be larger than that in the R-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, additional losses can be introduced by positioning and guiding errors, as investigated by Jacoby & Kaler (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Spectrophotometry with a slit spectrograph also requires the slit to be oriented at the parallactic angle to min- imise the effect of atmospheric dispersion (Filippenko 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ja- coby & Kaler 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since our observations are performed with an IFU, we are not affected by any of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While it is possible that our use of small apertures has caused some flux to be lost, we are able to compensate for this loss using aperture corrections as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Such a procedure is not easily performed for data observed with a slit spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Besides the issue of slit-losses, the follow-up spectroscopy by PE12 is limited to PNe with m5007 < 25, which is less than 40% of their whole sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This implies that most of their PNe are dependant on the measurement accuracy of the brighter PNe, which are likely to be affected by systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' More- over, the use a 5 pixel aperture to measure the flux for a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='9 pixel FWHM PSF incurs a risk of including light from back- ground contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In such crowded fields with a variable background and ubiquitous diffuse emission-line gas, the aper- ture might unexpectedly collect [O iii]λ5007 flux of the ambient interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Without proper background inspection and subtraction, this might lead to an overestimation of brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The inclusion of background emission likely explains the scat- ter for m5007 > 25 in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The spatial resolution of MUSE allows us to carefully check and analyse the condition of the background on a case-by-case basis and provide more accurate photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The variable background is also the main considera- tion to opt for a smaller aperture size for our flux measurements, and to rely on the aperture correction to deliver the final values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Article number, page 6 of 20 SO96 29 PE12 28 27 (SO96,PE12) 26 25 m5007 ( 24 23 22 21 21 22 23 24 25 26 27 28 29 m5007 (This work)Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Balmer decrement Measurement of extinction using the Balmer decrement with Hα/Hβ ratio have been demonstrated on MUSE data for differ- ent objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Pillars of Creation in M16 (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2015), core of R136 in the LMC (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018), faint H ii regions in NGC 300 (Micheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To obtain this, we employed the spectra extracted for the classification, as explained in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1, using the aperture of 3 spaxels, with the inner and outer sky annulus of 12 and 15 spaxels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also apply aperture correction for the Balmer lines, which can be referred to in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, not all of our PNe candidates are detected at these two wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In order to filter out the candidates, we put a threshold of F(Hα) = 2 × 10−17 erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For typical PNe with electron temperature Te = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='000 K, the expected Balmer ratio is Hα/Hβ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='86 (Osterbrock & Ferland 2006), which corresponds to F(Hβ) ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='75 × 10−18 erg cm−2 s−1 for the Hα threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Any Hβ flux lower than the threshold is too close to the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For such cases, we assume the upper limit of Hβ flux derived from the Hα line, which consequently also as- sumes no extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To avoid possible biased exclusion of high extinction PNe, we flag the objects with upper limit Hβ flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' If the Hα flux is below the threshold, then the extinction mea- surement is not performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Based on the Hα threshold criterion, we have a complete sample for PNe down to m5007 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' If we extend the sample to fainter magnitudes, we reach 87% completeness until m5007 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 and 64% completeness un- til m5007 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To calculate the extinction, we then used the Balmer decrement defined as Aλ = k(λ) c(Hβ) = k(λ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 k(Hβ) − k(Hα) � log �Hα Hβ � − log (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='86) � (3) where k(λ) is the wavelength dependant extinction constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For the foreground extinction, we employed the extinction curve of Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1989) with RV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 and E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='011 (Schlafly & Finkbeiner 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For NGC 300, Bresolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2009) measured the present day metallicity of 12 + log(O/H) ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 using the H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Recent measurement using the same MUSE-GTO data based on faint H ii regions and dif- fuse interstellar gas (DIG) also agrees with the latter value as 12 + log(O/H) ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 (Micheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since the chemical abundance of NGC 300 in the observation area are similar to the LMC, with 12+log(O/H) ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 (Toribio San Cipriano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2017), we employed the average LMC extinction curve to obtain the extinction of our PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The uncertainty of our extinction mea- surement is highly dependent to the aperture correction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, we quote an estimated error based on the comparison of extinction calculated from different aperture correction meth- ods, as explained in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In spiral galaxies, Herrmann & Ciardullo (2009) found that the typical extinction for the PNe in [O iii]λ5007 is A5007 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, we discovered three high extinction cases with A5007 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5, including one with an extreme value of A5007 ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While it is possible that some PNe exhibit high intrinsic extinction, as high metallicity populations and massive progenitors tend to pro- duce dustier PNe (Stanghellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012), we suspect that the Balmer decrement might not always be accurate due to the local contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To investigate this further, and to highlight pos- sible pitfalls that may play a role in studies based on slit spec- troscopy, we examined spatial maps of the high-extinction ob- jects in the wavelengths of Hβ, [O iii]λ5007, and Hα, using the p3d software (Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We found that these PNe can- didates are co-spatial with nearby H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In Figure 5, a PN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' False-colour spatial map in [O iii]λ5007 (left) and Hα (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The flux scaling is identical and logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The images are 20′′ × 20′′ (∼ 180 × 180 pc) each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The green marker illustrate the main and sky aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The candidate is isolated in [O iii]λ5007, but overlapped with nearby H ii region in Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' is shown to be an isolated point source in [O iii]λ5007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, in the spatial map of Hα, the point source is entirely embedded inside the extended emission surface brightness distribution of an unrelated nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This clearly shows that the Balmer line flux of the PN candidate is contaminated, and an accurate extinction measurement of the PN itself cannot be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' All three of the objects in question show similar patterns of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We therefore excluded them from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We compared our PN extinction measurements with the re- sults from Stasi´nska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2013) – also referred as ST13 – who observed PNe in NGC 300 with the FORS2-MXU instrument at the VLT (Appenzeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They used 3 grisms 600B, 600RI, and 300T to cover spectral ranges of 3600 − 5100Å, 5000 − 7500 Å, and 6500 − 9500 Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To avoid un- certainties from the flux calibration of different bands, they em- ployed the Hγ and Hβ lines from the 600B grism spectra to mea- sure the Balmer decrement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The extinction values for 18 PNe in common are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We have contemplated several reasons to explain the dis- crepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Firstly, slit losses that occur for the measurement of [O iii]λ5007 most likely also occur for the Hγ and Hβ lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The short baseline between the lines is also very sensitive to sys- tematic errors, making it difficult to derive accurate extinction values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, higher order Balmer lines are typically weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The accuracy of measuring their flux depends on the precision to which the background of stellar absorption line spectra can be subtracted (Jacoby & Kaler 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using the PMAS instrument, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2004) compared the accuracy of the Hβ line flux obtained with the IFU and simulated slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They demonstrated that the orientation of the slit can introduce a dif- ferent sampling of the background, leading to systematic differ- ences of the derived flux measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since ST13 performed their measurements with slit spectroscopy, they were susceptible to this type of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For cases where the internal extinction of ST13 is reported higher than our values, we find that 3 of their PNe are discarded from our sample (L6-5, H1-1, and H12-1 in Table 1), either be- cause of severe contamination as shown in Figure 5, or by their low excitation, which we consider typical for compact H ii re- gions (Frew & Parker 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also found that in some of these cases, the PN is embedded in diffuse gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In our sample, if the diffuse gas is assumed to be uniformly distributed, the flux ex- cess can be corrected using the background sky annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It re- mains an open question whether the background correction of diffuse gas was accurately accounted for in the slit spectroscopy of ST13, but we conclude that a careful consideration of back- Article number, page 7 of 20 O 0A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Extinction comparison between this work and ST13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' IDMUSE IDST13 c(Hβ)MUSE c(Hβ)ST13 E-11 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 E-2 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 L6-5a 20 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='17 L6-7a 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='44 I-2b 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='12 C-7b 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='07 H9-1 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 L9-8 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H2-6 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 A-23 48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='21 A-11 51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H1-8 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H7-2 58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H1-1c 63 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='41 H6-5 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 L2-3 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H6-3 69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H12-1a 74 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='64 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (a) severe Balmer contamination (b) uniform diffuse Hα background (c) low excitation – possibly compact H ii region ground subtraction is critical for the extinction measurements based on the Balmer decrement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The PNLF The PN luminosity function of this work is presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It exhibits the dip between 1 and 3 magnitudes below the cut- off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Such dip is typically observed in star-forming galaxies (Ja- coby & De Marco 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010a), which is possibly caused by multiple episodes of star formation (Rodríguez-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021) or difference in opacity and mass range of the PN formation (Valen- zuela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To determine the distance, we employed the maximum like- lihood technique, where the empirical PNLF is treated as a prob- ability function (Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1989), assuming M∗ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='06 and a fixed slope parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' When the number of PNe at the bright end cut-off is less than ∼ 50, distance deter- minations based on χ2 minimisation depend significantly on the details of how the PNLF is binned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Such methods are not rec- ommended (Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although our observation extends to m5007 ∼ 29, the PNLF fit is only per- formed for the sample brighter than the dip until m5007 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6, since equation (1) does not consider the dip feature, which nev- ertheless is insignificant for the distance determination (Spriggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' By taking the foreground extinction of E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='011 (Schlafly & Finkbeiner 2011) into account, the most-likely distance modulus is (m − M)0 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='26 with the uncertainties representing the statistical error of the fit and the M∗ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also calculated the distance using PE12 photometry to make a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Assuming a completeness limit of m5007 = 1 10 2 5 20 This Work Number of Objects 21 22 23 24 25 26 27 28 29 1 10 2 5 20 Pena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2012) Number of Objects Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PNLF of NGC 300 using MUSE (top) and PE12 photometry (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Completeness limit for distance measurement is assumed at m5007 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 and m5007 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 for ours and PE12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Open symbols indicate incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The PNLF dip is visible for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0, our maximum likelihood approach yields (m − M)0 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='20, a value that is significantly larger than our MUSE distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We argue that this is due to the systematically fainter magnitudes of PE12 photometry, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It is important to mention that PE12 measured a modulus distance of (m − M)0 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='22, a value much smaller than our maxi- mum likelihood values, including our distance measurement us- ing the PE12 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This difference is possibly due to their use of the Levenberg-Marquardt χ2 fitting technique, which is de- pendant on the binning method to construct the luminosity func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since the sample size is limited, they employed rather wide magnitude bins of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='16 mag, which did result in a luminosity function shape that closely resembles the empirical law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' How- ever, in the luminosity function of PE12, their first magnitude bin is located at m5007 ∼ 22, despite the fact that their brightest PN has a magnitude of m5007 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Thus, when the fit is per- formed, this systematic shift to brighter magnitudes results in a smaller distance modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This demonstrates that the choice of bin size can produce unintended systematical shifts of the lumi- nosity function when the number of PNe in the top ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 mag of the luminosity function is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Similarly, PE12’s choice of bin size also smeared out detail in the PNLF’s shape, as they did not report the observation of the PNLF dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In Figure 6, we present the PNLF that we plot with the original data from PE12 using higher binning resolution than the original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In fact, the dip is present in the PE12 data, confirming that it is not an artefact in our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Finally, PE12 employed a larger extinction correction for the photometry with A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 compared to our value of A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PE12 assumed this as the intermediate value be- Article number, page 8 of 20 Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 tween found by Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005) with E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='096 (A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3) and Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1998) with E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='013 (A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the case of Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005), the extinction value is the sum of both the foreground extinction of Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1998) and internal extinction derived from the Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For Cepheid distances, the internal extinction correction is nec- essary since they are originated from Population I stars, that are typically surrounded by galactic dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, this is less true for the PNe, so the foreground extinction correction for the PNLF distance is sufficient (Ciardullo 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This implies that the extinction correction A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 by PE12 is overestimated and also contributes to the smaller distance modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, the discrepancy between our calculation of the PE12 data and the original calculation is traced back to the issue of binning a limited sample and also the extinction correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PNLF Distance To demonstrate the accuracy of our distance measurement, we compare our result with previous distances in the literature de- rived using Cepheids and tip of the red giant branch (TRGB), taken from NED, in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We can see that most of the distances are within the uncertainties of our PNLF result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' One aspect that may introduce the discrepancy is the correction of extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For instance, the Cepheid distance of Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005) and the TRGB distance of Rizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2006), both part of the Auracaria project, are corrected with foreground and inter- nal extinction of E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, Rizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2007) argue that the extinction derived from dusty young Cepheids by Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005) are not representative for the whole galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' their result only applies the foreground component of extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This shows the importance of having the same zero-point when comparing different distances derived from different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, we also show that the MUSE observation, com- bined with the differential emission line filter (DELF, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021) and maximum likelihood technique (Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1989), has improved the accuracy of PNLF method, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The early result by Soffner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1996) is based on the limited sample of only 34 PNe, from which they only construct a cumulative PNLF and employed the distance mod- ulus of the LMC as a yardstick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Peña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2012) identified a significantly larger sample of 104 PNe, but as shown in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3, their data may suffer from slit-losses and contamina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The systematically fainter PN magnitudes then led to larger distance modulus, as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since the cut-off of the PNLF of NGC 300 is defined by a very small number of PNe, minimisation fitting methods become too dependant on the binning (Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In a study by Jacoby (1997), a correction for PNLF distance based on the number of PN sam- ple is suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For a PNLF cut-off sample < 20 PNe, they estimated a distance correction of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 mag (see Figure 5 in Jacoby 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, since the Cepheid and TRGB distance also varies with standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 mag, there are no solid distance reference to test if the correction is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Never- theless, we have shown that the PNLF distance derived with the maximum likelihood technique is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We take this as a motivation to improve PNLF distance measurements for nearby galaxies with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Local dust effect on PN extinction Dust formation plays important role in the early stages of PN evolution since it occurs at the surface of the progenitor AGB star and presumably plays an important role in the envelope ejec- tion (Herwig 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Stanghellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Infrared studies in the Milky Way and the LMC have revealed that the dust produc- tion is dependant on metallicity, with dustier systems found in higher metallicity environments (Stanghellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2007, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Bernard-Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although it cannot tell the proper- ties of the dust, Balmer decrement extinction measurements can also probe the presence of dust in PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the study of Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018), a comparison was made between the PN extinction distribution in the bulge of M31 and several other galaxies: the LMC (Reid & Parker 2010a), NGC 4697 (Méndez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008a), and NGC 5128 (Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Despite the limited samples involved, the authors found that the average extinction of PNe in each galaxy roughly follows the metallicity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To investigate such trends in NGC 300, we plot the extinc- tion distribution in [O iii]λ5007 for our PNe until m5007 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 (15 PNe) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These PNe are the ones employed for the maximum likelihood distance measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We find that in general these bright PNe have low extinction in [O iii]λ5007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The average extinction value for this sample is A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='31 (c(Hβ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='09), which is lower than the average of the bright PNe sample in the LMC with A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='57 (Reid & Parker 2010a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, we refrain from further in- terpreting the extinction distribution with the PN dust produc- tion, since the distribution is likely to be affected by local dust clouds, which can vary from one object to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Such a prob- lem has been reported in NGC 5128, where the high extinction of some PNe was attributed to local dust clouds rather than the PNe themselves (Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' At the distance of NGC 300, our MUSE observations offer a spatial resolution of between 6 and 14 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This resolution should be sufficient to visually resolve the spatial variation of dust ex- tinction (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Tomiˇci´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To test this, we inspected several objects with high extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' As an exam- ple, we present the spatial map in [O iii]λ5007 and RGB colours, which is constructed from Johnson-VRI filters, for the PN with the highest extinction value (PN A-23, A5007 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='18) in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although it is not obvious in the [O iii]λ5007 image, the RGB image shows a dust lane patch, extending from the lower left corner to the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since PN A-23 is in proximity to the dust lane, we suggest that the measured high extinction of this object is composed of both local dust within the galaxy and circumneb- ular extinction associated with the PN itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Based on the comparison study between Balmer decrement extinction and infrared dust distribution in M31, Tomiˇci´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2017) concluded that vertical distribution of diffuse interstellar gas (DIG) and dust can vary in different locations of the galaxy and thus cause differing amounts of extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For NGC 300, variation of extinction also has been reported by Roussel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, there is currently no guarantee that the mea- sured extinction of individual PNe is free from local effects, which is confirmed with our images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We must therefore refrain from making conclusions based on the extinction values alone, until the different components of the extinction can be quantita- tively resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Although the extinction of individual PNe might be affected by local dust lanes, such effects are less significant for the lumi- nosity function as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The effect of dust scale height in the PNLF distances of late-type disk galaxies has been discussed by Feldmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They modelled PNLF with varying ex- Article number, page 9 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Distance modulus difference between our PNLF result with Cepheids (blue triangles) and TRGB (red squares), obtained from NED and sorted based on publication date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The Cepheid distances are from Willick & Batra (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Paturel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Gieren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2004, 2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Bono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Bhardwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The TRGB distances are from Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Sakai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Tikhonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Rizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2006, 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Previous PNLF distances of Soffner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (1996) and Peña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2012) is also presented (green circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The green shadow indicated the uncertainty of our PNLF distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Distribution of extinction measurement for the PNe in [O iii]λ5007 until m5007 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The average extinction is A5007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='31 (c(Hβ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' False colour (left) and RGB colour (right) map of the region surrounding PN A-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The flux scaling is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The images are 20′′ × 20′′ (∼ 180 × 180 pc) each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The green marker illustrate the main and sky aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The dust patch is clearly visible in the RGB image, overlapped with PN A-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' tinction in [O iii]λ5007 and concluded that the inferred distance modulus should always be within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 mag of the derived distance without extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A similar result also obtained by Rekola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2005), who modelled the PNLF with different scale heights of dust in the starburst galaxy NGC 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They found that even when the disk was optically thick with 1 mag of extinction, the PNLF distance is robust to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Both studies suggest that the brighter PNe tend to be located above the dust layer from the point of view of the observer, or for other reasons suffer little extinction from within the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' With these arguments, we do not expect the occurrence of dust lane extinction to significantly affect our distance result and a correction for internal extinction is at this point not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PN parent populations To gain a better understanding of the parent population of the PNe, we estimate the luminosity and the effective temperature of the central stars of the planetary nebula (CSPNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These param- eters are calculated for PNe until m5007 = 26 with measurable extinction, which corresponds to 87% of the objects within this magnitude limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Simulation studies suggest that the maximum conver- sion efficiency of a central star luminosity into nebular [O iii]λ5007 emission is ∼ 11% (Jacoby 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2007, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Gesicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This oc- curs under the ideal assumption of optically thick nebula and assumes that [O iii]λ5007 acts as the sole coolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' If the PNe is optically thin, then the efficiency of [O iii]λ5007 production is less, and the luminosity inferred for a PN’s central star will be underestimated (Mendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A high abundance of ni- trogen, such as that typically found in Type I PNe (Peimbert & Torres-Peimbert 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Phillips 2005), can also increase cooling, and lead to an underestimation of central star luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' More- over, the assumption of lower limit extinction for some cases can also underestimate the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, we only consider our luminosity estimates as the lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To estimate the central stars’ effective temperatures, we em- ployed the excitation class method based on the PNe in the LMC (Dopita & Meatheringham 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For optically thick PNe, the excitation class temperatures are found to have an empirical correlation with temperature as de- rived from photo-ionisation modelling (Dopita & Meathering- ham 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To em- ploy this method, we also assume that the metallicity difference between the LMC and NGC 300 is negligible (Bresolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The revised excitation classes by Reid & Parker (2010b) are defined as Elow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='45 �F(λ5007) F(Hβ) � (4) Ehigh = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='54 �F(λ4686) F(Hβ) + log10 F(λ4959) + F(λ5007) F(Hβ) � (5) with Elow employed for low excitation PNe (0 < E < 5) and Ehigh for medium- to high excitation PNe (5 ≤ E < 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The empirical relation between the excitation class and effective temperature for optically thick PNe is then defined by Reid & Parker (2010b) as log Teff = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='439 + [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1174(E)] − [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00172(E2)] (6) Since only the extended mode used in the MUSE-GTO dataset has the wavelength coverage to include the He ii λ4686 line, Article number, page 10 of 20 5 Number of PNe 4 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 A500700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 4 不 回国 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 本不 国 中中 中国 国国 国 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' HR-diagram of CSPNs in NGC 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The evolutionary tracks are H-rich post-AGB models by Miller Bertolami (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The luminos- ity are lower limits, assuming maximum [O iii]λ5007 conversion effi- ciency of 11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For measurements within error of log Teff > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='98, lower limit effective temperatures are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The ratio [N ii]λ6584/Hα is the indicator of optical thickness, with the value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 for more likely optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Type I PNe are classified with [N ii]λ6584/Hα > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' for uniformity, we determine the excitation class using just the [O iii]λ5007 line and Hβ line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This implies that the effective tem- peratures for medium- and high excitation PNe with E ≥ 5 (or logTeff ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='98) are only lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This includes the measure- ments, which have uncertainties beyond the condition for low- excitation PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Based on 6 PNe in our sample that have He ii λ4686 detection, we estimate that the effective temperatures can be underestimated by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 − 3 times if we only rely on the Elow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' On the other hand, if the nebula is optically thin, based on the study in the LMC, the excitation class temperatures can be over- estimated by at least 50% compared to the Zanstra temperatures (Villaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Both luminosity and effective temperature estimates rely on the optical thickness of the PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To obtain this, we adopt the criterion of [N ii]λ6584/Hα ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 as the condition for optically thin PNe (Kaler & Jacoby 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Jacoby & Kaler 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since this criterion is not based on nebular modelling in our sample, we use the indication as a more likely condition rather than a definite indicator to explain the possible limitation in our estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The estimated stellar parameters are presented in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We include the post-AGB tracks from Miller Bertolami (2016) with a stellar metallicity of Z⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='01, which is the closest to the observed value at the central area of NGC 300 with Z⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='007 (Kudritzki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Gogarten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A stellar population study by Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2020) using the Hubble Space Telescope found young stars of ∼ 300 Myr, AGB stars with an age between 1 − 3 Gyr and significant number of RGB stars older than 3 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' From a single stellar evolution per- spective, the stellar population of NGC 300 can produce a PN central star mass of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7 M⊙ from a progenitor mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 M⊙, which would have a main sequence lifetime of τMS > 320 Myr (Miller Bertolami 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This implies that, theoretically, central stars within any of the stellar tracks in Figure 10 can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Unfortunately, since most of our luminosities and effective tem- peratures are lower limits, we are unable to put more constraints on the central star masses at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Within our sample, we also identified several objects as Type I PNe (Peimbert & Torres-Peimbert 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Phillips 2005), highly enriched in nitrogen, and classified using [N ii]λ6584/Hα > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' These objects likely to arise from younger and more massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For a progenitor mass above ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙, the convective en- velope in the thermal pulsing AGB phase is likely to extend to the hydrogen-shell burning layer and produce “hot bottom burn- ing” (HBB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This can dredge up the products of the CNO cycle to the surface, to be later expelled by the stellar wind, therefore in- creasing the nitrogen-to-oxygen ratio in the nebula (Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Observations of PNe in M31 by Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) put a lower limit of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 M⊙ for HBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A more thorough analysis, performed for a Type I PN in the M31 young open cluster B477- D075, yields a HBB lower mass limit of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 M⊙ (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This suggests that our approximation for the central star luminosities of Type I PNe is greatly underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For this particular case, we argue that the assumption of [O iii]λ5007 as the only coolant is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since the nitrogen-to-oxygen ratio is high, the nitrogen contribution as additional coolant cannot be neglected (Jacoby 1989), causing the underestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This might also explain why we did not see the Type I PNe at our PNLF cut-off, although they are expected to have more massive cores than the typical PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' More implications regarding the underlying stellar popula- tion can also be inferred from the faint end of the PNLF (Ciar- dullo 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, the current observational study is still limited to a relatively small sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Recently, based on a very deep survey in M31, Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2021) found that the steep rise in the number of PN fainter M∗ + 5 mag is caused by the increased mass fraction of a population older than 5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For NGC 300, this implies that the photometry should be com- plete for m5007 > 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since our PNLF completeness breaks after m5007 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5, we are unable to provide any insights on this mat- ter at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Insights on the most luminous PNe Numerous simulation studies have been conducted to investigate the nature of the PNe at the cut-off of the luminosity function (Ja- coby 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2007, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Méndez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2008b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Gesicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Valenzuela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We review some of them and compare it to our estimated properties to investigate the nature of the most luminous PNe in NGC 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using the most re- cent post-AGB models by Miller Bertolami (2016), simulations of the [O iii]λ5007 fluxes for different progenitor mass have been performed by Gesicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They found that progenitors with the mass range between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 M⊙ are able reach the cut- off absolute magnitude M∗ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5, assuming that the fluxes at the stellar evolution stages are maximised – also known as maxi- mum nebula hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It is important to note that the timescale of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 M⊙ track is too short and less likely to be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Additionally, they also performed a simulation with an inter- mediate nebula hypothesis, where the PNe are predominately opaque;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' this model suggests that the brightest PNe in the lumi- nosity function will have the luminosity log L/L⊙ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For comparison, the intrinsically most luminous PN in our sam- ple, PN H9-1, has a lower limit luminosity of log L/L⊙ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It is also indicated as more likely optically thick, which is in agreement with the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Again, we note that this assumes Article number, page 11 of 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 L/L o 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 Mo --> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='706 Mo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 Mo --> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='616 Mo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 Mo --> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='583 M o 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 Mo --> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='583 Mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='25 Mo --> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='566 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 Mo --> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='532 Mo likely thick PN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 likely thin PN Type I PN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 log TeffA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 the ideal 11% maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For example, based on the chemical abundance analysis, the bright PNe in M31 exhibit less conversion efficiency (Jacoby & Ciardullo 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Kwitter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, the actual central star luminosity is likely to be brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Simulation of [O iii]λ5007 flux evolution has also been conducted by Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2007), who employed 1- dimensional radiative-hydrodynamical simulations for the neb- ulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They calculated that the most luminous PNe that popu- late the PNLF cut-off will achieve their maximum luminosity at log Teff = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 K and spend ∼ 500 years in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For PN H9-1, the lower limit temperature is log Teff > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They also suggest that UV- to [O iii]λ5007 flux conversion process hap- pens most efficiently for central star mass of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='62 M⊙, if the nebular shell remains optically thick during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Re- ferring to the post-AGB models by Miller Bertolami (2016), the initial mass of the progenitor star would be ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙, and the ob- ject would spend less than 1000 years before entering the white dwarf cooling sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Similarly, hydrodynamical models have been used to inves- tigate PNe in nearby galaxies by Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In these simulations, it was found that central star masses greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='65 M⊙ do not exist at the PNLF cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This also sup- ports the result from Gesicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018), in that a progenitor mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 M⊙ for PNe is not expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This also agrees with the progenitor masses between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙ for the bright PNe in NGC 300 predicted by Stasi´nska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2013), despite the con- cerns we mentioned regarding their spectroscopic fluxes in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They derived the progenitor masses using the stellar tracks of Bloecker (1995), which evolve slower than the recent models of Miller Bertolami (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This implies the possibility of less massive progenitors if the new stellar tracks are adopted, which is however beyond the scope of our current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Recently, the properties of luminous PNe near the PNLF cut- off of M31 have been studied by Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) for the bulge, and by Galera-Rosillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2022) for the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For the disk, it was found that the four brightest PNe have an average progeni- tor mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙, which is lower than the values predicted by Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2007), but still in agreement with Gesicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Galera-Rosillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2022) also measure a rela- tively low average extinction of the PNe with c(Hβ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This means that the PNe originated from an older stellar population, although the disk of M31 also exhibits star forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In contrast, in the older population of the bulge of M31, Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) found that the brightest PN have a central star mass > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='66 M⊙, which means progenitor masses of > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This is found for cases with high extinction, one even reaching c(Hβ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6, with the average of c(Hβ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 for 23 PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Cur- rent simulations do not predict such massive central stars to be observable, if they exist at all (Schönberner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Gesicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In old systems, the most luminous PNe are sug- gested to be products from of blue stragglers – stars that re- sult from a merger during the main sequence (Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2005), or symbiotic nebula (Soker 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, both scenar- ios still do not predict such massive central stars to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While the bright Hα background might overestimate the measured ex- tinction, which can lead to the overestimation of luminosity and subsequently the progenitor mass, Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2018) in fact did their measurement with an IFU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Their sky subtraction was based on a PSF model, which was claimed to be accurate within 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Lately, Ueta & Otsuka (2021) suggested that the extinction measurement should be solved iteratively, considering the de- pendency of Hα/Hβ ratio on the electron temperature (Te) and electron density (ne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Assuming those two parameter as con- stants would increase the uncertainty of the extinction, and sub- sequently the stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They demonstrate this approach by reanalysing the M31 disk PNe, worked by Galera-Rosillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (2022), and found that the iterative approach yields an average progenitor mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 M⊙, instead of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙ for the four brightest PNe (Ueta & Otsuka 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While the extinction does not necessarily affect the Te and ne, it may compromise the ionic and elemental abundance analysis (Ueta & Otsuka 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since we also assume constant Te and ne for our parame- ters, we are not excluded from this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, as we are missing the diagnostic lines in the blue spectral region and the ones within MUSE wavelength coverage are below the detection limit, we are unable to put constraints on the Te and ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It would be interesting to repeat the exercise of modelling PN spectra on the basis of improved IFU observations that we believe are superior to slit-based spectroscopy in controlling sys- tematic errors, with the more recent stellar evolution tracks and more careful plasma diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The future BlueMUSE instru- ment for the VLT (Richard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019) will offer the capability with a wavelength coverage down to the atmospheric limit in the UV, which includes the necessary nebular lines for such study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Conclusions We analyse 44 fields, obtained with the MUSE instrument to find PNe and construct the PNLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using the differential emis- sion line filter (DELF, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021), we identified more than 500 point sources in [O iii]λ5007, 107 of which were designated as PNe based on spectral classification with the aid of the BPT- diagram (Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The [O iii]λ5007 magnitudes for the PNe were obtained using DAOPHOT aperture photometry (Stetson 1987) with aperture corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' With the sample com- pleteness at m5007 = 27 for most fields, we constructed the PNLF, which exhibits the dip that has been observed in other star form- ing galaxies (Jacoby & De Marco 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Ciardullo 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Reid & Parker 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To derive the distance, we employed the max- imum likelihood estimation method (Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1989) to yield a most likely distance modulus (m − M)0 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='26 (d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='23 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For PNe, that are isolated from surround- ing emission line sources, and that exhibit bright enough Balmer lines, we measured their extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We estimated parameters of the central stars using the extinction corrected fluxes in an at- tempt to track their origin from the underlying stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We discuss the accuracy of our distance measurement, the effect of local dust for our PNe extinction measurements, and the prop- erties of the most luminous PNe in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The conclusions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The PNLF distance measurement to NGC 300 is improved with our method and is in excellent agreement with Cepheids and TRGB distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' This is due to the spectral information and spatial resolution of MUSE, that provides a higher PN detection per area, better classification, and accurate photom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' With a limited sample, distance determination based on the minimisation technique is very dependent on the binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Al- though coarse binning might provide a better apparent shape of the luminosity function for fitting, it can introduce an un- intended systematic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, the details of the PNLF shape, which can provide insights on the stellar population, are also smeared out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The extinction derived for the PNe cannot be disentangled completely from the local dust lane extinction within the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, with the spatial resolution of MUSE, we Article number, page 12 of 20 Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 were able to resolve several PNe that are likely obstructed by dust lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Any attempt to link the internal extinction and the underlying stellar population requires a quantitative tech- nique to separate the local and internal PNe extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We found a few Type I PNe, that evolved from main se- quence mass > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Their luminosities are likely under- estimated due to the high abundance of nitrogen that serves as a competing coolant with oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' They do not populate our PNLF cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' With these results, and other works reported in the litera- ture, we feel encouraged to further develop the IFU observing technique with MUSE to study extragalactic PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' One of the in- herent parameters that we have as yet not utilised is the radial velocity of individual PNe that comes for free as a by-product of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It will be interesting to find out whether the kinematics can provide hints as to the membership in different populations in NGC 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Such study was recently done for other disc galaxies: NGC 628 (Aniyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2018), NGC 6946 (Aniyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2021), and M31 (Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In the interest of understanding the physical parameters of the PNe, we are cur- rently dependant on the ideal assumption of [O iii]λ5007 maxi- mum conversion and excitation classes to derive the central star parameters, which is not ideal, especially when most cases have no He ii λ4686 coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Better constraints on the luminosities and effective temperatures are obtainable through nebular abun- dance modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, our current wavelength coverage of the MUSE instrument limit us to explore this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Future IFUs, that are optimised in the blue wavelength, such as Blue- MUSE (Richard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2019), will play an important role and allow us to gain more understanding about PNe in the nearby galaxies beyond the Local Group, getting us closer to compre- hend the underlying physics behind the constancy of PNLF cut- off across galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We thank the anonymous referee for a critical reading of the manuscript and helpful suggestions to improve the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Part of this work was supported by the German BMBF program Unternehmen Re- gion, grant 03Z22AN11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' PMW gratefully acknowledges support by the BMBF from the ErUM program (project VLT-BlueMUSE, grant 05A20BAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Cas- tro gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) – CA 2551/1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' References Aniyan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Freeman, K.' metadata={'source': 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614, A147 Ciardullo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2010, PASA, 27, 149 Ciardullo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012, Ap&SS, 341, 151 Ciardullo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2022, Frontiers in Astronomy and Space Sciences, 9, 896326 Ciardullo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Feldmeier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Jacoby, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2002, ApJ, 577, 31 Ciardullo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Jacoby, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Ford, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2009, ApJS, 183, 67 Davis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Bond, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2005, ApJ, 621, L93 Gesicki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Zijlstra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', & Miller Bertolami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' M.' metadata={'source': 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39 Jacoby, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 1997, in The Extragalactic Distance Scale, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Livio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Don- ahue, & N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Panagia, 197 Jacoby, G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', de Jong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Minchev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020, A&A, 640, L19 Joye, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', & Walton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2012, A&A, 544, A70 Weilbacher, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Palsa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', Streicher, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020, A&A, 641, A28 Willick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' & Batra, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2001, ApJ, 548, 564 Article number, page 14 of 20 Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 Appendix A: Aperture correction The radial profile of a PSF is best modelled with a Moffat func- tion, as a Gaussian often does not accurately match the wings of the PSF (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, flux measurements using a discrete aperture are not able to collect all of the flux from the PSF wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To recover the lost flux and ob- tain accurate photometry, we therefore need to apply an aperture correction to our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In order to do this, we need at least one star in a given field as a reference for the observation’s PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We examined 3-4 objects to infer the average PSF FWHM of the frame and chose the best star for the aperture correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Moreover, we also examined the behaviour of the PSF across wavelengths, as the PSF is expected to be more extended in the blue, and to show a monotonic decrease of the FWHM toward the red (Fried 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Boyd 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To obtain the aperture correction value, we collected the flux of the reference star using a large aperture radius of 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 (or 12 spaxels), assuming that almost all of the flux will be recorded (Howell 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Then, by taking the flux of the same star with the aperture size employed for the PNe, we were able to obtain the correction value by taking the ratio of the two fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We then applied this constant to all PNe measurements within the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For the Balmer lines, we have to make sure that both lines are corrected in a consistent manner, especially with respect to the wavelength dependence of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since the seeing at the tele- scope is decreasing monotonically with wavelength, the FWHM for Hβ is expected to be larger than the one for Hα, thus changing the aperture correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The reference stars in each field there- fore have to be well behaved across this wavelength range which was found to not always be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We found several appar- ent point sources that unexpectedly exhibit an increasing PSF FWHM trend to the red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Closer inspection revealed that stel- lar crowding with luminous red stars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=', M giants and carbon stars, were responsible for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' For Balmer line correc- tions, we decided to discard the problematic stars as useful PSF references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' As an alternative, we used the brightest PNe that happen to be sufficiently isolated from nearby diffuse gas and H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We then used the PNe’s image profile at the wavelengths of the strong lines of [O iii]λ5007 and Hα, while assuming a negli- gible difference between the PSF at 5007 Å and Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Unfortu- nately, we found that some of our fields have neither a well be- haved star, nor bright isolated PNe, so another alternative was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Using the best reference stars from different fields and seeing conditions, we derived a simple polynomial relation be- tween the seeing FWHM and the aperture correction value for Hβ, [O iii]λ5007, and Hα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' these curves are presented in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also confirmed that the difference between PSF at Hβ and [O iii]λ5007 is not significant, as the polynomial fit is almost identical for the two wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' However, it is important to mention that the relation is only derived using a limited sample of 23 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' It is not possible to determine the true distribution of this relation and identify the variables that affect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' While this is worthwhile for further investigation, we will not explore it in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We employed the empirical relation as the final al- ternative, after the bright PN method and the main reference star method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' From the 50 PNe that are within the Hα threshold, we corrected 16 PNe with the reference star method, 18 PNe with the bright PN method, and 16 PNe with the empirical relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In future studies, the aperture correction can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Firstly, the uncertainties can be minimised under excellent see- ing conditions, ideally 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 PSF FWHM at the wavelength of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Empirical polynomial relation between the seeing FWHM and aperture correction for Hβ, [O iii]λ5007, and Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The relation is derived using 23 stars from different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' [O iii]λ5007, that can be achieved using the adaptive optics mode of MUSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In cases where no field star is available to serve as a PSF reference, modelling the wavelength and seeing dependant PSF on the basis of instrumental data from the adaptive optics control software may provide a way out (Fusco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Appendix B: Extinction uncertainties The main uncertainty of our extinction measurement is the aper- ture correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since we have only a limited number of objects observed with each correction method, every PN has its own un- certainty, making it difficult to we derive a proper statistical er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' As an alternative, we estimate the error based on the com- parison of extinction calculated from different aperture correc- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' To perform this, we considered PN candidates that were measured with a well behaved reference star in the field, and a bright PN in the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The comparison is presented in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since we expect the aperture correction at Hβ to be larger than the one for Hα, the application of this factor will reduce the inferred extinction, as Hβ appears in the denominator of equa- tion (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In field E, where the initially selected reference star shows the unusual trend of an increasing PSF width to the red, we computed the extinction to be larger after the aperture cor- rection, prompting us to restrict the reference star method only to cases where a well behaved star is available in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' More- over, we also see that the use of PNe that are not completely isolated from the ambient gas tends to underestimate the extinc- tion, if compared to other aperture correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Based on our choice of priority of the methods, marked as bold in Ta- ble B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1, the difference between the extinction with and without aperture correction (∆A) is always larger than the difference be- tween extinction values derived using various aperture correc- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Therefore, we decided to select the ∆A as our error Article number, page 15 of 20 Seeing FWHM ["] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='014x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='044x - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='019 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='014x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='051x - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='013x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='043x - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='046 Hβ [O III] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 Ha O Aperture correction D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 2 3 4 5 6 7 8 Seeing FWHM [spaxel]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Comparison for extinction values in [O iii]λ5007 derived us- ing different aperture correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The preferred extinction val- ues are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' ID A0 A1 A2 A3 ∆A E-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='589a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='189 E-11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='663a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='170 P-2 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='076 L9-8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='167 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A0 – no aperture correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A1 – reference star method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A2 – bright PN method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' A3 – empirical relation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (a) bad reference star (b) uniform diffuse Hα background estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' In cases where the error estimates exceed nonphysical negative extinction, the lower limit of the uncertainty is assumed until zero extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We should note, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4, that assuming a constant electron temperature Te and ne also introduce uncer- tainties (Ueta & Otsuka 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Since we did not have the capability to measure Te and ne, we did not include this aspect in our measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Appendix C: MUSE observation fields The details of the MUSE fields, both the MUSE-GTO and ML20 (McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' 2020, 2021), can be referred in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' We also include the seeing in [O iii]λ5007, which obtained based on the average FWHM of 3-4 point sources, preferably stars, in each field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Appendix D: MUSE-PN catalogue The MUSE-PN catalogue of NGC 300 is presented in Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' MUSE-GTO coordinates (accuracy of ∼ 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1) are pre- sented if available, otherwise ML20 coordinates (accuracy of ∼ 3′′) are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The table will be available through the CDS Archive and will also include the columns for aperture corrected line fluxes of Hβ, [O iii]λ5007, Hα, [N ii]λ6584, [S ii]λ6716, [S ii]λ6731, and the classification remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Article number, page 16 of 20 Azlizan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Soemitro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' : MUSE crowded field 3D spectroscopy in NGC 300 Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The MUSE observation fields for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' The upper part represent the MUSE-GTO data and the lower part the ML20 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' Field RA(2000) DEC(2000) Observation date FWHM5007 ["] A 00:54:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='62 37:41:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 2018-10-15 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='67 B 00:54:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='54 37:41:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 2018-10-15 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='69 C 00:54:43.' metadata={'source': 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+page_content='17 37:42:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7 2015-09-13 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='60 I 00:54:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 37:40:52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='6 2014-10-30 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='66 J 00:54:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='49 37:39:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 2014-11-26 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='82 P 00:54:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 37:36:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 2016-09-03 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='59 Q 00:54:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 37:37:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 2016-09-03 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='55 H1 00:54:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='83 –37:39:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 2016-10-01 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='00 H2 00:54:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='40 37:39:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 2016-10-01 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10 H3 00:54:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='99 –37:38:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 2016-10-01 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='87 H4 00:54:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='55 –37:38:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7 2016-10-04 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='19 H5 00:55:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='51 –37:41:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 2016-10-05 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='49 H6 00:55:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 –37:41:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='9 2016-10-05 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='18 H7 00:54:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='65 –37:40:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7 2016-10-05 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='30 H8 00:54:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='22 –37:40:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 2016-10-05 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='06 H9 00:54:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='81 –37:39:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 2016-11-07 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='82 H10 00:54:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='37 –37:39:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 2016-11-08 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='96 H11 00:54:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='95 –37:38:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 2016-11-08 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='86 H12 00:55:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='35 –37:42:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 2016-11-08 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='88 H13 00:54:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='90 –37:41:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='23 H16 00:54:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='52 –37:43:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='0 2016-12-19 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='02 H17 00:54:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 –37:43:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='8 2016-12-23 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='13 –37:38:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='3 2016-12-24 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='08 L4 00:54:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='04 –37:41:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 2016-12-26 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='11 L5 00:54:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='63 –37:40:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='9 2017-01-02 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='42 L6 00:54:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='19 –37:40:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='7 2017-01-02 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='50 L7 00:54:37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='77 –37:39:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='5 2017-01-04 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='49 L8 00:55:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='15 –37:43:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1 2018-07-03 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='77 L9 00:54:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='27 –37:42:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='84 L12 00:54:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='99 –37:41:07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='4 2017-01-06 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='96 L13 00:54:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='57 –37:40:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='2 2017-01-07 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} 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+page_content='03 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='10 Article number, page 19 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' pnlf_ngc300 Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' No IDGTO IDMcLeod IDPE12 RA(2000) DEC(2000) m5007 c(Hβ) log L [L⊙]a log Teff 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+page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='13 107 B-39 0:54:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='25 37:40:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='75 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content='15 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} +page_content=' (a) Lower limits assuming maximum [O iii]λ5007 conversion efficiency of 11% (*) Lower limit value Article number, page 20 of 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E1T4oBgHgl3EQfywWE/content/2301.03437v1.pdf'} diff --git a/fdE3T4oBgHgl3EQf3Qt-/content/tmp_files/2301.04762v1.pdf.txt b/fdE3T4oBgHgl3EQf3Qt-/content/tmp_files/2301.04762v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0ac52c59c55a1e27daa15cc5394b4c5a5b640c1 --- /dev/null +++ b/fdE3T4oBgHgl3EQf3Qt-/content/tmp_files/2301.04762v1.pdf.txt @@ -0,0 +1,932 @@ +ADJOINT-BASED ESTIMATION OF SENSITIVITY OF CLINICAL +MEASURES TO BOUNDARY CONDITIONS FOR ARTERIES +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +Abstract. The use of adjoint solvers is considered in order to obtain the sensitivity of +clinical measures in aneurysms to incomplete (or unknown) boundary conditions and/or +geometry. It is shown that these techniques offer interesting theoretical insights and viable +computational tools to obtain these sensitivities. +1. Introduction +The analysis of haemodynamic phenomena and their clinical relevance via computational +mechanics (fluids, solids, . . . ) is now common in research and development. Yet a recurring +question has been the influence of boundary conditions and geometry on ‘clinically relevant +measures’. +As an example, consider flows in aneurysms. +A crucial question is how far +upstream the geometry has to be modeled accurately in order to obtain sufficiently accurate +flow predictions, as well as their associated loads on vessel walls (shear, pressures) and +clinically relevant measures (such as kinetic and vortical energy, vortex line length, etc.). In +many cases, users may not have sufficient upstream information, so this question is of high +relevance. The thesis of Castro and subsequent publications [3, 4] have shown how dramatic +the difference between well resolved upstream geometries and so-called ‘cut’ geometries can +be. In some cases, completely different types of flow were seen, which in turn could have led +to different clinical decisions. Figures 1-2 show two examples. +To complicate matters further, the flow is transient/pulsating, and the flowrate and flow +profile coming in at the upstream boundary in most cases is unknown. It is a common +practice to simply set some kind of pipe flow profile (Poiseuille, Womersley) at the inflow, +adjusting the analytical parameters to the estimated/known flux. +The central question remains: what is the influence of a change of boundary conditions (e.g. +inflow profiles) or geometry (e.g. more upstream/downstream geometry) on the clinically +relevant measures ? +A simple way to answer this question is to perform several runs, each with a different +geometry or different boundary condition. This finite difference approach can then yield +the sensitivity of a ‘measure of clinical relevance’ I to a change in geometry or boundary +condition z. Another possibility is via adjoints [16, 23, 13, 12, 2]. We also refer to a series of +works by Glowinski and collaborators on the role of adjoints in optimization [8, 11, 21, 1, 7, 9]. +See also [22, 14, 10]. We emphasize that this list is incomplete as many authors have made +fundamental contributions to this topic. +Key words and phrases. +incomplete Boundary Conditions, Adjoint Solvers, CFD, Sensitivity Analysis . +Dedicated to Prof. Roland Glowinski. +This work is partially supported by NSF grant DMS-2110263 and the AirForce Office of Scientific Research +under Award NO: FA9550-22-1-0248. +1 +arXiv:2301.04762v1 [physics.med-ph] 11 Jan 2023 + +2 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +Figure 1. Vessel 1: Difference in flow features between properly resolved and +unresolved upstream geometry. +Figure 2. Vessel 2: Difference in flow features between properly resolved and +unresolved upstream geometry +1.1. Upstream Boundary Conditions for the Flow. It is known from empirical evidence +and simple fluid mechanics that given any steady inflow velocity profile, after a given number +of diameters along the pipe the flow will revert to a simple pipe flow (Poiseuille). This so- +called hydrodynamic entry length Lh is a function of the Reynolds number Re, and for +laminar flow and uniform inflow is given by: +Lh = 0.05Re D , +Re = ρUeD +µ +, +(1) +where ρ, Ue, µ denote the density, mean entrance velocity and viscosity of the flow and D +the vessel diameter. For blood and a typical artery ρ = 1 g/cm3, Ue = 50 cm/sec, µ = +0.04 g/cm/sec, D = 0.1 cm, so Re = O(100) and Lh = 5 D. Note that this estimate is +only valid for steady flows and a uniform inflow. As far as the authors are aware, similar +estimates for vessels with high curvatures (tortuosity) as typically encountered in arteries are + +3DRA +original +truncated +Elevated WSS in the +Low wss +body, dome & neck +wss 650.0 +wss 650.0 +20.0 +520.0 +90.0 +90.0 +260.0 +260.0 +30.0 +130.0 +0.0 +Complex flow pattern +Simpler flow +pattern +Secondary flows +Laminar flow +Swirling flows3DRA +original +truncated +Short M1 +Higher WSS +Lower wSS +YSS +wSs +200.0 +200.0 +150.0 +150.0 +100.0 +100.0 +50.0 +50.0 +0.0 +0.0 +Double’ vortex +Single' vortex +pattern +pattern +Secondary flows +Laminar flowADJOINT-BASED ESTIMATION OF SENSITIVITY +3 +not available. We note in passing that for the unsteady cases analyzed by [3, 4] the number +of upstream diameters required before the flow did not change in the aneurysms was much +higher than the estimate given above. +1.2. Possible Mathematical Approaches. In order to formulate the problem mathemat- +ically, we can consider different approaches. +a) Empirical Data: for any given geometry/case, one could perform a series of studies, +changing the type of inflow (vortical flows, unsteady flows) and seeing how long the +observed hydrodynamic entry lengths are; +b) Sensitivity Analysis I: one could try to obtain a ‘topological derivative’ that measures +the sensitivity of the flow in the aneurysm with respect to movement of the upstream +boundary. +c) Sensitivity Analysis II: one could obtain a ‘flow derivative’ that measures the sensi- +tivity of the clinical measure of the flow in the aneurysm with respect to changes of +the entry flow in the upstream boundary. +Outline: The remainder of the paper is organized as follows. In Section 2, we first introduce +a generic optimization problem formulation and adjoint framework. This generic discussion +is well-known. This is followed by an example of Navier-Stokes specific to the aneurysm +problem. We study the sensitivity with respect to the inflow velocity and inflow position. +Section 3 focuses on numerical implementation. In Section 4.1, we present a specific example +corresponding to the 2-D channel flow. +For this example, we are able to derive explicit +expressions for the state variables, adjoint variables, and the sensitivities (see Appendix A). +This is followed by a realistic aneurysm example in Section 4.2, where we study the sensitivity +of the ‘measure of clinical relevance’ I. All the numerical examples confirm the proposed +approach. +2. General Adjoint Formulation +Suppose we have a ‘measure of clinical relevance’ I for a region that is in or close to an +aneurysm. This could be the kinetic or vortical energy, the shear stress or the length of +vortex lines - all of which have been proposed in the literature [19, 5, 6]. +The question then becomes: how sensitive is this measure to the (often unknown) boundary +conditions imposed or the (often approximate) geometric accuracy ? +Given that I is a +function of the unknowns u and these in turn are a function of a set of parameters z describing +the boundary conditions or the geometry, the answer to this question is given by the gradient +of I. Consider the well-known generic minimization problem +min +u,z I(u, z) +subject to +e(u, z) = 0 , +where I : U × Z → R is the cost functional and e(·, ·) : U × Z → Y is the PDE constraint. +Here U, Y and Z are function spaces. Typically, U, Y are Banach spaces and Z is a Hilbert +space. Under very generic conditions, one can establish existence of solution to the above +optimization problems, see [12, 2]. As it has been known in the literature, there are two +ways to derive the expression of the adjoint and the gradient of objective function I. The +first approach is the so-called reduced formulation, where assuming that the PDE is uniquely +solvable, one considers the well-defined control-to-state map +z �→ u(z) + +4 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +with (u(z), z) solving the PDE e(u(z), z) = 0. The reduced objective functional is then +given by I(z) = I(u(z), z). Then one obtains the derivative of I with respect to z which also +requires computing the sensitivites of u with respect to z. The second approach is the full +space formulation and it requires forming the Lagrangian. Under fairly generic conditions +(constraint qualificiations), one can establish the existence of Lagrange multipliers in this +setting, see [25, 12]. Regardless, in both cases, the same expression of gradient is obtained +[2, Pg. 14]. +We briefly sketch the Lagrangian approach and refer to [12, 2] for details. Let p denotes +the adjoint variable, then the Lagrangian functional is given by +L(u, z, p) = I(u, z) − ⟨e(u, z), p⟩Y,Y ∗ . +(2) +Then at a stationary point (u, z, p) the following conditions hold +Lp(u, z, p) = 0, +Lu(u, z, p) = 0, +Lz(u, z, p) = 0. +(3) +Our goal for the application under consideration is not to solve the above optimization +problem, but rather derive the expression of the gradient Lz(u, z, p). In view of the expression +of the Lagrangian given in (2), it is not difficult to see that conditions in (3) are equivalent +to +e(u, z) = 0, +(State equation) +eu(u, z)∗p = Iu(u, z), +(Adjoint equation) +Iz(u, z) − ez(u, z)∗p = 0. +(Gradient equation) +(4) +Namely, the gradient is given by (cf. [2, Pg. 14]) +∇I(z) = Iz(u, z) − ez(u, z)∗p. +(5) +The consequences of the above formulation are profound: +• The variation of I in (5) exhibits only derivatives with respect to z, i.e., no explicit +derivatives with respect to u appear; +• The cost of evaluation of gradients is independent of the number of design variables +(!). +In the next section, we will apply this abstract framework to the case where the PDE +e(u, z) = 0 is given by the incompressible Navier-Stokes equations. These equations are used +to model the flow in the aneurysms. +2.1. Incompressible Navier-Stokes and Sensitivity with Respect to Inflow. Let +the domain Ω ⊂ Rd be sufficiently smooth, and consisting of two subdomains Ωaneurysm +and the remainder of the domain Ω \ Ωaneurysm consisting of vascular vessels. Furthermore, +let the boundary Γ of Ω consist of three parts Γin (inflow), Γfixed (fixed / wall), and Γout +(outflow). Moreover, let (u, p) denote the velocity-pressure pair solving the incompressible + +ADJOINT-BASED ESTIMATION OF SENSITIVITY +5 +Navier-Stokes equations: +−div(µ∇u) + (u · ∇)u + ∇p = f +in Ω +div u = 0 +in Ω +u = z +on Γin +u = 0 +on Γfixed +(µ∇u − pI) · n = 0 +on Γout +(6) +where f denotes a given force (for the current set of applications f = 0), µ is viscosity, +and n is the outward unit normal. Finally, z is some given velocity profile on the inflow +boundary Γin. +Given a quantity of interest (measure of clinical relevance), I(u, p, z), the goal is to obtain +the derivative of I with respect z with the help of adjoint formulation as discussed in the +previous section. We begin by stating the following result, see [24, Appendix C] +Lemma 1. Let u, v and ˜u be smooth vector fields, then +� +Ω +[(u · ∇)v]˜u = − +� +Ω +(div u)(v · ˜u) + [(u · ∇)˜u] · v + +� +Γ +(u · n)(v · ˜u). +When v = u and div u = 0, then +� +Ω +[(u · ∇)u]˜u = − +� +Ω +[(u · ∇)˜u] · u + +� +Γ +(u · n)(u · ˜u). +Next, a derivation of sensitivity is provided using the adjoint approach. We begin by +writing the Lagrangian functional +L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − +�� +Ω +(−div(µ∇u) + (u · ∇)u + ∇p − f) · ˜u − ˜pdiv u dx ++ +� +Γin +(u − z) · ˜uΓ ds +� +. +Applying integration-by-parts, and using Lemma 1, along with u = 0 on Γfixed and (µ∇u − +pI)n = 0 on Γout, we obtain that +L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − +�� +Ω +µ∇u : ∇˜u − [(u · ∇)˜u] · u − pdiv ˜u + u · ∇˜p dx ++ +� +Γin∪Γfixed +˜u · (−µ∇u + pI) n ds − +� +Γin∪Γout +u · n˜p ds ++ +� +Γin +(u − z) · ˜uΓ ds + +� +Γin∪Γout +(u · n)(u · ˜u)ds +� +. + +6 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +Applying integration-by-parts again, we arrive at +L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − +�� +Ω +(−div(µ∇˜u) + ∇˜p) · u − [(u · ∇)˜u] · u − pdiv ˜u dx ++ +� +Γin∪Γfixed +˜u · (−µ∇u + pI) n ds + +� +Γin +u · (µ∇˜u − ˜pI)n ds ++ +� +Γout +u · (µ∇˜u − ˜pI)n ds ++ +� +Γin +(u − z) · ˜uΓ ds + +� +Γin∪Γout +(u · n)(u · ˜u)ds +� +. +(7) +In view of (3), taking a variation of L with respect to (u, p) and setting it equal to zero, +we obtain the adjoint equation +−div(µ∇˜u) − (u · ∇)˜u − (∇˜u)⊤u + ∇˜p = Iu(u, p, z) +in Ω +div ˜u = −Ip(u, p, z) +in Ω +˜u = 0 +on Γin ∪ Γfixed +(µ∇˜u − ˜pI)n = − [(u · ˜u)n + (u · n)˜u] +on Γout. +(8) +We note the compatibility condition: +˜uΓ = −(µ∇˜u − ˜pI)n − (u · ˜u)n − (u · n)˜u = −(µ∇˜u − ˜pI)n +on Γin, +where in the last equality we used the fact that ˜u = 0 on Γin. +We notice that, if I is +independent of p, then we obtain the standard incompressibility condition for ˜u in (8). +Finally, the required variation of I with respect to z is given by +DzI(u, p, z) = Iz(u, p, z) − [(µ∇˜u − ˜pI)n + (u · ˜u)n + (u · n)˜u] +on Γin += Iz(u, p, z) − [(µ∇˜u − ˜pI)n] +on Γin , +(9) +where we have again used the fact that ˜u = 0 on Γin. Note that if the clinical measure I +is not a function of the control variable (in this case the inflow velocity), for a channel with +constant flow in the normal direction n (i.e. µ∇˜u · n = 0) the sensitivity reverts to (recall +that I is the reduced objective) +DzI(z) = ˜pn +on Γin . +(10) +i.e. the sensitivity to inflow velocities is the adjoint pressure. +2.1.1. Sensitivity to Changes in Inflow Position. Consider next the variation of the La- +grangian L given in (7) with respect to the normal n. We recall that after simplifications, +we have +L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − +� +Γin +(u − z) [(µ∇˜u − ˜pI)n] . + +ADJOINT-BASED ESTIMATION OF SENSITIVITY +7 +Then +DnL(u, p, ˜u, ˜p, ˜uΓ)h = DnI(u, p, z)h − +� +Γin +Dn [(u − z) ((µ∇˜u − ˜pI)n)] h += DnI(u, p, z)h +− +� +Γin +(Dnuh) [((µ∇˜u − ˜pI)n)] + (u − z)Dn [((µ∇˜u − ˜pI)n)] h += DnI(u, p, z)h − +� +Γin +(Dnuh) [((µ∇˜u − ˜pI)n)] , +where, in the last step, we have used the fact that u = z on Γin. In case, I is independent +of n, we then obtain that +DnL(u, p, ˜u, ˜p, ˜uΓ)h = − +� +Γin +(Dnuh) [((µ∇˜u − ˜pI)n)] . +Note that if µ∇˜u · n = 0 (as is often the case) the sensitivity reverts to (recall that I is the +reduced objective) +DnI(n) = un +n˜p +on Γin +(11) +i.e. the sensitivity to changes in inflow position is the adjoint pressure multiplied by the +normal derivative of the inflow velocity. +2.2. In- and Outflow Boundary Conditions for the Adjoint. Consider the aneurysm +shown in Figure 3. +Inflow +Outflow +Figure 3. Schematic of Aneurysm +For the usual (forward) incompressible Navier-Stokes calculation, one would prescribe a +velocity profile (u = z) at the inflow boundary and the ‘do nothing’ ((η∇u − pI)n = 0) +or pressure boundary condition (p = pou) at the outflow boundary. This implies letting the +pressure ‘free’ at the inflow and the velocity ‘free’ at the outflow. At the walls the velocity +is zero, i.e. u|Γfixed = 0. Consider now the adjoint problem. The boundary conditions in +this case are described in (8), i.e., we obtain zero velocity at the inflow and ‘do nothing’ +or prescribed zero adjoint pressure at the outflow. The adjoint velocity is also zero on the +walls. +3. Numerical Implementation +In a strict mathematical sense, the adjoint solver obtained by discretizing the adjoint +partial differential equation should be as close as possible to the discrete adjoint obtained +from transposing and manipulating the discretization of the forward problem. In this way +‘optimize-then-discretize’ and ’discretize-then-optimize’ are as close as possible. This was + +8 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +not adopted in the present case. Instead, while the forward problem was solved for the +incompressible Navier-Stokes equations, the adjoint equations were derived for the quasi- +incompressible Navier-Stokes equations, which for steady flows give the same results. Fur- +thermore, while the forward problem was integrated to steady state using a fractional step +solver with implicit solution of the viscous terms and the pressure increments, and edge- +based upwinding for the velocities and 4th order pressure stabilization [17], the adjoint was +discretized in space using the following scheme, which for each point i in the mesh is given +by: +� +Ak�T +i Mi∇k(˜u)i + BT(µi + µj)Kij(˜ui − ˜uj) + MiIΩ +u + Di = 0 , +(∗∗) +where A, Mi, ∇k, Kij, Di denote the Jacobians of the advective fluxes, lumped mass-matrix, +discrete gradient in direction k, Laplacian edge-based coefficients and damping vector, and +∇k(˜u)i = Ck +ij(˜ui + ˜uj) , +where Ck +ij are the edge-based coefficients for the gradient (see [17], Chapter 20). Furthermore +Di = −λ(ij) +� +˜ui − ˜uj + β +2 lij · (∇(˜u)i + ∇(˜u)j) +� +, +where c is the speed of sound, ˜p the adjoint pressure, λ = |u| + c the maximum eigenvalue +of the system and 0 < β < 1 denotes a pressure sensor function of the form [20]. +β = 1 − +˜pi − ˜pj + 0.5lij · (∇(˜p)i + ∇(˜p)j)| +|˜pi − ˜pj| + |0.5lij · (∇(˜p)i + ∇(˜p)j)| . +(9.4) +For β = 0, 1, second and fourth order damping operators are obtained respectively. Several +other forms are possible for the sensor function β [18]. +Although this discretization of the adjoint Euler fluxes looks like a blend of second and +fourth order dissipation, it has no adjustable parameters. Defining U = (u, p), ˜U = (˜u, ˜p) +Eqn.(**) may be re-written as +R(U, ˜U) = 0 , +the system re-written as an unsteady equation of the form: +˜U,τ + R(U, ˜U) = 0 , +and integrated in pseudo-time τ via a classic explicit multistep Runge-Kutta [15]. +4. Numerical Examples +We will focus on two main examples. At first, we consider Poisuille flow through a channel +in Section 4.1. Remarkably enough, we are able to derive the explicit expressions for all +the quantities, such as solution to the state equation, adjoint equation and sensitivities, see +Appendix A. These theoretical results are also confirmed by numerical results. In Section 4.2, +we focus on a realistic aneurysm scenario, where we truly see the benefits of the proposed +sensitivity approach. + +ADJOINT-BASED ESTIMATION OF SENSITIVITY +9 +4.1. Poiseuille Flow. The 2-D channel flow provides a good test to verify the implementa- +tion of the forward and adjoint solvers. The domain considered is of dimension 0.0 ≤ x ≤ 0.5, +−0.05 ≤ y ≤ 0.05 and −0.005 ≤ z ≤ 0.005. A parabolic inflow with maximum velocity of +umax = 1.0 was prescribed. The velocity at the top and bottom walls (ymin, ymax) was pre- +scribed to zero, and the velocity in the z-direction was prescribed to zero for the back and +front walls (zmin, zmax). The other relevant parameter is µ = 0.01. Two ‘clinically relevant +measures’ (i.e. cost functions) were considered: kinetic energy I = 1 +2 +� +Ω ρu2 dx and vortical +energy I = 1 +2 +� +Ω ρ|∇×u|2 dx. We set ρ = 1.0 in our experiments. The derivation of the exact +solutions for the adjoint equations for these cost functions may be found in Appendix A. Let +u = (u, v, w)⊤, then the x-component of u is given by: +u = +� +1 − 4 +H2y2 +� +u0 , +where u0 = umax and H is the total height of the channel, i.e. ymax = −ymin = H/2. We +thus obtain +∂yu = −8u0 +H2 y, +∂yyu = −8u0 +H2 , +∂xp = −8µu0 +H2 . +The pressure, velocity magnitude, and velocity vectors are shown in Figures 4-6. +Figure 4. Poiseuille Flow: Pressure +Figure 5. Poiseuille Flow: Velocity Magnitude +4.1.1. Kinetic Energy. Consider the cost function +I = 1 +2 +� +ρ|u|2 dx , +implying +Iu = ρu. +As can be seen in Appendix 1, the adjoint pressure for this cost function is: +∂x˜p = 4 +5ρu0 , + +pressure +40 +ILI +30 +20 +10velocity +1010 +7.510 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +Figure 6. Poiseuille Flow: Velocity +i.e. the gradient of the adjoint pressure is also constant and linearly dependent of u0. The +results obtained are shown in Figures 7-9. +Figure 7. Poiseuille Flow: Adjoint Pressure +Figure 8. Poiseuille Flow: Magnitude of Adjoint Velocity. +Here the cost +function is Kinetic Energy. +Figure 9. Poiseuille Flow: Adjoint Velocity. Here the cost function is Kinetic Energy. + +10 +:10 +7.5adj_press +0.01 +0.0075 +0.005 +0.0025adjvelo +0.007 +0.0020.007 +0.006 +00ADJOINT-BASED ESTIMATION OF SENSITIVITY +11 +4.1.2. Vortical Energy. The cost function is given by +I = 1 +2 +� +ρ |∇ × u|2 dΩ . +For the 2-D channel (u = u(y), v = 0, w = z) +(∇ × u)2 = (∂yu)2 , +so that +I,u = ρu,y(u,y),u = −ρu,yy = −ρ +µp,x = 8ρu0 +H2 , +i.e. constant. As can be seen in Appendix 1, the adjoint velocities and pressure are given +by: +˜u(x, y) = 0 , ˜v(x, y) = 0 , −˜p = ρ +µp . +Figure 10. Poiseuille Flow: Adjoint Pressure. Cost Function: Vortical Energy. +4.2. Aneurysm with Simple Flow Pattern. As an example, we include an aneurysm +with simple flow pattern. +The geometry and discretization may be discerned from Fig- +ures 11a-c which show the surface triangulation, pressure and magnitude of the velocity. +The region for the source-terms of the adjoint is shown in Figure 12 a and the adjoint pres- +sure, as well as the magnitude of the adjoint velocities obtained in Figures 12 b,c. The +adjoint velocites can also be seen in Figures 13 a,b. Note the effect of the source-term that +pushes the adjoint flow and forms a double vortex. +Figure 11. a,b,c Aneurysm: Surface Triangulation, Surface Pressure and +Magnitude of Velocity in Cut Plane + +adj_press +9.5 +80420 +. +1.5pressure +200 +100velocity +9.5 +8 +612 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +Figure 12. a,b,c Aneurysm: Source, Adjoint Pressure and Magnitude of +Adjoint Velocity in Cut Plane +Figure 13. a,b Aneurysm: Adjoint Velocity in Cut Plane +5. Conclusions and Outlook +The use of adjoint solvers to assess the sensitivity of incomplete boundary (inflow, geometry) +information has been considered. The results of this investigation indicate that the sensitivity +of clinical measures or other flow features that are inside the flow domain with respect to +inflow velocity is proportional to the adjoint pressure, while the sensitivity with respect to +inflow geometry is given by the product of the adjoint pressure and the normal derivative of +the inflow velocity. Thus, the adjoint pressure may be a good indicator to see if the inflow +boundary of haemodynamic cases is far enough from the region of interest so that errors can +be avoided. The use of adjoint solvers is not unproblematic. Unlike running a series of cases, +varying inflow profiles and geometry, and seeing their influence on many clinically relevant +measures, adjoints require a different run for each of the clinical measures. +Appendix A. Appendix 1: Analytical Expressions for Poiseuille Flow +A.1. Exact Forward Solution. Let us consider a long 2-D channel of length 0 ≤ x ≤ L +and width −H/2 ≤ y ≤ H/2 with incompressible viscous flow. Let u = (u, v, w)⊤, then the +equation for the x-velocity u is given by: +u∂xu + v∂yu + ∂xp = µ∆u . +Assuming a constant velocity profile in x, i.e. u = u(y) and laminar flow with v = 0, the +solution is the Poiseuille solution, given by: +u = +� +1 − 4 +H2y2 +� +u0 , +(12) + +AbsDIDv +1.21e-05 +1e-5 +7.5e-6 +5e-6 +2.5e-6adj_press +0.000228 +0.0002 +0.0001 +0 +-0.0001 +-0.000129adj_veloc +0.0188 +0.016 +0.012 +0.008 +0.004adj_veloc +0.0188 +0:016 +0.012 +0.008 +0.004adi_veloc +0.0188 +0.016 +0.012 +三0.008 +0.004ADJOINT-BASED ESTIMATION OF SENSITIVITY +13 +where u0 is the maximum velocity at the center of the channel, and the channel extends in +height from −H/2 ≤ y ≤ H/2, implying +∂yu = −8u0 +H2 y , +and +∂yyu = −8u0 +H2 , +so that the constant pressure gradient is given by: +∂xp = −8µu0 +H2 , +where we have used the fact that ∂xu = ∂xxu = 0. The average velocity is then: +u = 1 +H +� H/2 +−H/2 +u dy = 2 +3u0 . +A.2. Adjoint Equations. The equation for the adjoint x-velocity ˜u is given by: +−u∂x˜u − v∂y˜u + ∂x˜p = µ∆˜u,xx + Iu +Here I is the cost function. For the channel u is given by (12) and v = 0. +Kinetic Energy: If the cost function is given by the kinetic energy +I = 1 +2 +� +ρ|u|2 dx , +then +Iu = ρu . +Assuming a long channel with no change in x of the variables, the equation for the adjoint +x-velocity ˜u simplifies to: +∂x˜p = µ∂yy˜u + ρu0 +� +1 − 4 +H2y2 +� +. +Assuming furthermore that ∂x˜p is constant, and applying the boundary conditions ˜u = 0 for +y = −H/2 and y = H/2 this yields +˜u = 1 +2µ [−∂x˜p + ρu0] +�H2 +4 − y2 +� +− ρu0 +3µH2 +�H4 +16 − y4 +� +. +If we consider that at the inflow boundary ˜u = 0, then as the adjoint velocity field is also +divergence-free, in any section of x we must have: +� +˜udy = 0. +This implies: +� H/2 +−H/2 +˜u dy = 1 +2µ [−∂x˜p + ρu0] +�H2 +4 y − y3 +3 +�H/2 +−H/2 +− ρu0 +3µH2 +�H4 +16 y − y5 +5 +�H/2 +−H/2 += 0. +Evaluation of all terms leads to the remarkable result: +∂x˜p = 4 +5ρu0 = −ρH2 +10µ ∂xp , + +14 +RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT +i.e. the gradient of the adjoint pressure is also constant and linearly dependent of u0. Given +that the base level of the pressure p is arbitrary, we might set it so that it vanishes at the +exit, i.e. p = 0. We finally obtain the remarkable result that: +−˜p = ρH2 +10µ p , +i.e. the pressure and adjoint pressure are related by the factor ρH2 +10µ and have a constant +gradient in the field. The adjoint velocity is given by: +˜u = ρu0 +µ +� 1 +10 +�H2 +4 − y2 +� +− +1 +3H2 +�H4 +16 − y4 +�� +. +At the center of the channel the velocity is given by: +˜u(y = 0) = ρu0H2 +240µ . +Vortical Energy: If the cost function is given by the vortical energy +I = 1 +2 +� +ρ |∇ × u|2 dx , +then, for the 2-D channel (u = u(y), v = 0, w = z) +|∇ × u|2 = (∂yu)2 , +so that +Iu = ρ∂yu(∂yu),u = −ρ∂yyu = −ρ +µ∂xp = 8ρu0 +H2 , +i.e. constant (!). Assuming a long channel with no change in x for the variables, the equation +for the adjoint x-velocity ˜u simplifies to: +∂x˜p = µ∂yy˜u − ρ +µ∂xp . +As this is a long channel and the source-term is constant, the assumption that ∂x˜p is constant +is warranted. This implies that ∂yy˜u should also be a constant. Applying the boundary +conditions ˜u = 0 for y = −H/2 and y = H/2 yields: +˜u = +� +1 − 4 +H2y2 +� +˜u0 . +However, if we again consider that at the inflow boundary ˜u = 0, and given that the adjoint +velocity field is divergence-free, then in any section of x we must have: +� +˜udy = 0 , +which implies that the only possible solution is ˜u(x, y) = 0, and therefore: +−∂x˜p = ρ +µ∂xp . +As at the exit the pressure p vanishes, i.e. p = 0, we finally obtain the remarkable result +that: +−˜p = ρ +µp , + +ADJOINT-BASED ESTIMATION OF SENSITIVITY +15 +i.e. the pressure and adjoint pressure are related by the factor ρ +µ and have a constant gradient +in the field. +A.3. Exact Derivatives of Cost Functions. Kinetic Energy: +Ike = 1 +2 +� +ρ|u|2 dx . +Given that u = u(y), v = 0 this results in: +Ike = 1 +2ρ +� +x +dx +� +y +u2dy = 1 +2ρL +� +u2 +0 +� +1 − 4 +H2y2 +�2 +dy +Ike = 1 +2 +8 +15LHρu2 +0 , +Ike +,u0 = 8 +15LHρu0 = 2 +3H ˜pin , +i.e. linear in the length L and the velocity u0, and +Ike +,x = 1 +2 +8 +15Hρu2 +0 = 1 +2 +2 +3H ˜pinu0 , +i.e. +not dependent (constant) of the length L and quadratic in the velocity u0. +In the +previous equations we assumed pout = 0, and used the analytical results that relate mass +flow, viscosity and pressure gradient for the Poiseuille flow. One should remark that if the +domain that is of interest does not change (e.g. only a certain region inside the channel is +considered), the correct value is: +Ike +,x = 0 +as the flow is constant in x and therefore the cost functional does not change if the upstream +boundary is moved. +Vortical Energy (Dissipation): +Ive = 1 +2 +� +ρ|∇ × u|2 dx . +Given that u = u(y), v = 0 this results in: +Ive = 1 +2ρ +� +x +dx +� +y +|∂yu|2dy = 8 +3 +ρu2 +0 +H2 LH +This implies: +Ive +,u0 = 16 +3 Lρu0 +H = 2LH ˜p +3 +, +i.e. linear in the length L and the velocity u0, and +Ive +,x = 8 +3 +ρu2 +0 +H = LH ˜pu0 +3 +, +i.e. not dependent (constant) of the length L and quadratic in the velocity u0. 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Applied computational fluid dynamics techniques: an introduction based on finite element +methods. John Wiley & Sons, 2008. +[18] E. Mestreau, R. L¨ohner, and S. Aita. Tgv tunnel entry simulations using a finite element code with +automatic remeshing. In 31st Aerospace Sciences Meeting, page 890, 1993. +[19] F. Mut, R. L¨ohner, A. Chien, S. Tateshima, F. Vi˜nuela, C. Putman, and J. R. Cebral. Computational +hemodynamics framework for the analysis of cerebral aneurysms. International journal for numerical +methods in biomedical engineering, 27(6):822–839, 2011. + +ADJOINT-BASED ESTIMATION OF SENSITIVITY +17 +[20] J. Peraire, J. Peir´o, and K. Morgan. A 3d finite element multigrid solver for the euler equations. In 30th +Aerospace Sciences Meeting and Exhibit, page 449, 1992. +[21] A. M. Ramos, R. Glowinski, and J. Periaux. Nash equilibria for the multiobjective control of linear +partial differential equations. J. Optim. Theory Appl., 112(3):457–498, 2002. +[22] S. S. Ravindran. Numerical solutions of optimal control for thermally convective flows. Internat. J. +Numer. Methods Fluids, 25(2):205–223, 1997. +[23] F. Tr¨oltzsch. Optimal control of partial differential equations, volume 112 of Graduate Studies in Math- +ematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, +Translated from the 2005 German original by J¨urgen Sprekels. +[24] S. W. Walker and M. J. Shelley. Shape optimization of peristaltic pumping. J. Comput. Phys., +229(4):1260–1291, 2010. +[25] J. Zowe and S. Kurcyusz. Regularity and stability for the mathematical programming problem in Banach +spaces. Appl. Math. Optim., 5(1):49–62, 1979. +R. L¨ohner and H. Antil. +Center for Computational Fluid Dynamics and Center for +Mathematics and Artificial Intelligence, 4400 University Dr., George Mason University, +Fairfax, VA 22030-4444, USA +J. Cebral and F. Mut. Dept. of Biomedical Engineering, George Mason University, 4400 +University Dr., George Mason University, Fairfax, VA 22030-4444, USA + diff --git a/fdE3T4oBgHgl3EQf3Qt-/content/tmp_files/load_file.txt b/fdE3T4oBgHgl3EQf3Qt-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fada38179fa278dabea31c59e3306e27f2bfdf7 --- /dev/null +++ b/fdE3T4oBgHgl3EQf3Qt-/content/tmp_files/load_file.txt @@ -0,0 +1,618 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf,len=617 +page_content='ADJOINT-BASED ESTIMATION OF SENSITIVITY OF CLINICAL MEASURES TO BOUNDARY CONDITIONS FOR ARTERIES RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The use of adjoint solvers is considered in order to obtain the sensitivity of clinical measures in aneurysms to incomplete (or unknown) boundary conditions and/or geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' It is shown that these techniques offer interesting theoretical insights and viable computational tools to obtain these sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Introduction The analysis of haemodynamic phenomena and their clinical relevance via computational mechanics (fluids, solids, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' ) is now common in research and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Yet a recurring question has been the influence of boundary conditions and geometry on ‘clinically relevant measures’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As an example, consider flows in aneurysms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' A crucial question is how far upstream the geometry has to be modeled accurately in order to obtain sufficiently accurate flow predictions, as well as their associated loads on vessel walls (shear, pressures) and clinically relevant measures (such as kinetic and vortical energy, vortex line length, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In many cases, users may not have sufficient upstream information, so this question is of high relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The thesis of Castro and subsequent publications [3, 4] have shown how dramatic the difference between well resolved upstream geometries and so-called ‘cut’ geometries can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In some cases, completely different types of flow were seen, which in turn could have led to different clinical decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figures 1-2 show two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' To complicate matters further, the flow is transient/pulsating, and the flowrate and flow profile coming in at the upstream boundary in most cases is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' It is a common practice to simply set some kind of pipe flow profile (Poiseuille, Womersley) at the inflow, adjusting the analytical parameters to the estimated/known flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The central question remains: what is the influence of a change of boundary conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' inflow profiles) or geometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' more upstream/downstream geometry) on the clinically relevant measures ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' A simple way to answer this question is to perform several runs, each with a different geometry or different boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This finite difference approach can then yield the sensitivity of a ‘measure of clinical relevance’ I to a change in geometry or boundary condition z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Another possibility is via adjoints [16, 23, 13, 12, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We also refer to a series of works by Glowinski and collaborators on the role of adjoints in optimization [8, 11, 21, 1, 7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' See also [22, 14, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We emphasize that this list is incomplete as many authors have made fundamental contributions to this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' incomplete Boundary Conditions, Adjoint Solvers, CFD, Sensitivity Analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Dedicated to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Roland Glowinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This work is partially supported by NSF grant DMS-2110263 and the AirForce Office of Scientific Research under Award NO: FA9550-22-1-0248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='04762v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='med-ph] 11 Jan 2023 2 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Vessel 1: Difference in flow features between properly resolved and unresolved upstream geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Vessel 2: Difference in flow features between properly resolved and unresolved upstream geometry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Upstream Boundary Conditions for the Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' It is known from empirical evidence and simple fluid mechanics that given any steady inflow velocity profile, after a given number of diameters along the pipe the flow will revert to a simple pipe flow (Poiseuille).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This so- called hydrodynamic entry length Lh is a function of the Reynolds number Re, and for laminar flow and uniform inflow is given by: Lh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='05Re D , Re = ρUeD µ , (1) where ρ, Ue, µ denote the density, mean entrance velocity and viscosity of the flow and D the vessel diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' For blood and a typical artery ρ = 1 g/cm3, Ue = 50 cm/sec, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='04 g/cm/sec, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1 cm, so Re = O(100) and Lh = 5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Note that this estimate is only valid for steady flows and a uniform inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As far as the authors are aware, similar estimates for vessels with high curvatures (tortuosity) as typically encountered in arteries are 3DRA original truncated Elevated WSS in the Low wss body, dome & neck wss 650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 wss 650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 Complex flow pattern Simpler flow pattern Secondary flows Laminar flow Swirling flows3DRA original truncated Short M1 Higher WSS Lower wSS YSS wSs 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content="0 Double’ vortex Single' vortex pattern pattern Secondary flows Laminar flowADJOINT-BASED ESTIMATION OF SENSITIVITY 3 not available." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We note in passing that for the unsteady cases analyzed by [3, 4] the number of upstream diameters required before the flow did not change in the aneurysms was much higher than the estimate given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Possible Mathematical Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In order to formulate the problem mathemat- ically, we can consider different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' a) Empirical Data: for any given geometry/case, one could perform a series of studies, changing the type of inflow (vortical flows, unsteady flows) and seeing how long the observed hydrodynamic entry lengths are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' b) Sensitivity Analysis I: one could try to obtain a ‘topological derivative’ that measures the sensitivity of the flow in the aneurysm with respect to movement of the upstream boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' c) Sensitivity Analysis II: one could obtain a ‘flow derivative’ that measures the sensi- tivity of the clinical measure of the flow in the aneurysm with respect to changes of the entry flow in the upstream boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Outline: The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In Section 2, we first introduce a generic optimization problem formulation and adjoint framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This generic discussion is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This is followed by an example of Navier-Stokes specific to the aneurysm problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We study the sensitivity with respect to the inflow velocity and inflow position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Section 3 focuses on numerical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1, we present a specific example corresponding to the 2-D channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' For this example, we are able to derive explicit expressions for the state variables, adjoint variables, and the sensitivities (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This is followed by a realistic aneurysm example in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2, where we study the sensitivity of the ‘measure of clinical relevance’ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' All the numerical examples confirm the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' General Adjoint Formulation Suppose we have a ‘measure of clinical relevance’ I for a region that is in or close to an aneurysm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This could be the kinetic or vortical energy, the shear stress or the length of vortex lines - all of which have been proposed in the literature [19, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The question then becomes: how sensitive is this measure to the (often unknown) boundary conditions imposed or the (often approximate) geometric accuracy ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Given that I is a function of the unknowns u and these in turn are a function of a set of parameters z describing the boundary conditions or the geometry, the answer to this question is given by the gradient of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Consider the well-known generic minimization problem min u,z I(u, z) subject to e(u, z) = 0 , where I : U × Z → R is the cost functional and e(·, ·) : U × Z → Y is the PDE constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Here U, Y and Z are function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Typically, U, Y are Banach spaces and Z is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Under very generic conditions, one can establish existence of solution to the above optimization problems, see [12, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As it has been known in the literature, there are two ways to derive the expression of the adjoint and the gradient of objective function I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The first approach is the so-called reduced formulation, where assuming that the PDE is uniquely solvable, one considers the well-defined control-to-state map z �→ u(z) 4 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT with (u(z), z) solving the PDE e(u(z), z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The reduced objective functional is then given by I(z) = I(u(z), z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Then one obtains the derivative of I with respect to z which also requires computing the sensitivites of u with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The second approach is the full space formulation and it requires forming the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Under fairly generic conditions (constraint qualificiations), one can establish the existence of Lagrange multipliers in this setting, see [25, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Regardless, in both cases, the same expression of gradient is obtained [2, Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We briefly sketch the Lagrangian approach and refer to [12, 2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Let p denotes the adjoint variable, then the Lagrangian functional is given by L(u, z, p) = I(u, z) − ⟨e(u, z), p⟩Y,Y ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (2) Then at a stationary point (u, z, p) the following conditions hold Lp(u, z, p) = 0, Lu(u, z, p) = 0, Lz(u, z, p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (3) Our goal for the application under consideration is not to solve the above optimization problem, but rather derive the expression of the gradient Lz(u, z, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In view of the expression of the Lagrangian given in (2), it is not difficult to see that conditions in (3) are equivalent to e(u, z) = 0, (State equation) eu(u, z)∗p = Iu(u, z), (Adjoint equation) Iz(u, z) − ez(u, z)∗p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (Gradient equation) (4) Namely, the gradient is given by (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' [2, Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 14]) ∇I(z) = Iz(u, z) − ez(u, z)∗p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (5) The consequences of the above formulation are profound: The variation of I in (5) exhibits only derivatives with respect to z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=', no explicit derivatives with respect to u appear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The cost of evaluation of gradients is independent of the number of design variables (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In the next section, we will apply this abstract framework to the case where the PDE e(u, z) = 0 is given by the incompressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' These equations are used to model the flow in the aneurysms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Incompressible Navier-Stokes and Sensitivity with Respect to Inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Let the domain Ω ⊂ Rd be sufficiently smooth, and consisting of two subdomains Ωaneurysm and the remainder of the domain Ω \\ Ωaneurysm consisting of vascular vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Furthermore, let the boundary Γ of Ω consist of three parts Γin (inflow), Γfixed (fixed / wall), and Γout (outflow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Moreover, let (u, p) denote the velocity-pressure pair solving the incompressible ADJOINT-BASED ESTIMATION OF SENSITIVITY 5 Navier-Stokes equations: −div(µ∇u) + (u · ∇)u + ∇p = f in Ω div u = 0 in Ω u = z on Γin u = 0 on Γfixed (µ∇u − pI) · n = 0 on Γout (6) where f denotes a given force (for the current set of applications f = 0), µ is viscosity, and n is the outward unit normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Finally, z is some given velocity profile on the inflow boundary Γin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Given a quantity of interest (measure of clinical relevance), I(u, p, z), the goal is to obtain the derivative of I with respect z with the help of adjoint formulation as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We begin by stating the following result, see [24, Appendix C] Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Let u, v and ˜u be smooth vector fields, then � Ω [(u · ∇)v]˜u = − � Ω (div u)(v · ˜u) + [(u · ∇)˜u] · v + � Γ (u · n)(v · ˜u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' When v = u and div u = 0, then � Ω [(u · ∇)u]˜u = − � Ω [(u · ∇)˜u] · u + � Γ (u · n)(u · ˜u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Next, a derivation of sensitivity is provided using the adjoint approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We begin by writing the Lagrangian functional L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − �� Ω (−div(µ∇u) + (u · ∇)u + ∇p − f) · ˜u − ˜pdiv u dx + � Γin (u − z) · ˜uΓ ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Applying integration-by-parts, and using Lemma 1, along with u = 0 on Γfixed and (µ∇u − pI)n = 0 on Γout, we obtain that L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − �� Ω µ∇u : ∇˜u − [(u · ∇)˜u] · u − pdiv ˜u + u · ∇˜p dx + � Γin∪Γfixed ˜u · (−µ∇u + pI) n ds − � Γin∪Γout u · n˜p ds + � Γin (u − z) · ˜uΓ ds + � Γin∪Γout (u · n)(u · ˜u)ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 6 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT Applying integration-by-parts again, we arrive at L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − �� Ω (−div(µ∇˜u) + ∇˜p) · u − [(u · ∇)˜u] · u − pdiv ˜u dx + � Γin∪Γfixed ˜u · (−µ∇u + pI) n ds + � Γin u · (µ∇˜u − ˜pI)n ds + � Γout u · (µ∇˜u − ˜pI)n ds + � Γin (u − z) · ˜uΓ ds + � Γin∪Γout (u · n)(u · ˜u)ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (7) In view of (3), taking a variation of L with respect to (u, p) and setting it equal to zero, we obtain the adjoint equation −div(µ∇˜u) − (u · ∇)˜u − (∇˜u)⊤u + ∇˜p = Iu(u, p, z) in Ω div ˜u = −Ip(u, p, z) in Ω ˜u = 0 on Γin ∪ Γfixed (µ∇˜u − ˜pI)n = − [(u · ˜u)n + (u · n)˜u] on Γout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (8) We note the compatibility condition: ˜uΓ = −(µ∇˜u − ˜pI)n − (u · ˜u)n − (u · n)˜u = −(µ∇˜u − ˜pI)n on Γin, where in the last equality we used the fact that ˜u = 0 on Γin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We notice that, if I is independent of p, then we obtain the standard incompressibility condition for ˜u in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Finally, the required variation of I with respect to z is given by DzI(u, p, z) = Iz(u, p, z) − [(µ∇˜u − ˜pI)n + (u · ˜u)n + (u · n)˜u] on Γin = Iz(u, p, z) − [(µ∇˜u − ˜pI)n] on Γin , (9) where we have again used the fact that ˜u = 0 on Γin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Note that if the clinical measure I is not a function of the control variable (in this case the inflow velocity), for a channel with constant flow in the normal direction n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' µ∇˜u · n = 0) the sensitivity reverts to (recall that I is the reduced objective) DzI(z) = ˜pn on Γin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the sensitivity to inflow velocities is the adjoint pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Sensitivity to Changes in Inflow Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Consider next the variation of the La- grangian L given in (7) with respect to the normal n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We recall that after simplifications, we have L(u, p, ˜u, ˜p, ˜uΓ) = I(u, p, z) − � Γin (u − z) [(µ∇˜u − ˜pI)n] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' ADJOINT-BASED ESTIMATION OF SENSITIVITY 7 Then DnL(u, p, ˜u, ˜p, ˜uΓ)h = DnI(u, p, z)h − � Γin Dn [(u − z) ((µ∇˜u − ˜pI)n)] h = DnI(u, p, z)h − � Γin (Dnuh) [((µ∇˜u − ˜pI)n)] + (u − z)Dn [((µ∇˜u − ˜pI)n)] h = DnI(u, p, z)h − � Γin (Dnuh) [((µ∇˜u − ˜pI)n)] , where, in the last step, we have used the fact that u = z on Γin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In case, I is independent of n, we then obtain that DnL(u, p, ˜u, ˜p, ˜uΓ)h = − � Γin (Dnuh) [((µ∇˜u − ˜pI)n)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Note that if µ∇˜u · n = 0 (as is often the case) the sensitivity reverts to (recall that I is the reduced objective) DnI(n) = un n˜p on Γin (11) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the sensitivity to changes in inflow position is the adjoint pressure multiplied by the normal derivative of the inflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In- and Outflow Boundary Conditions for the Adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Consider the aneurysm shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Inflow Outflow Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Schematic of Aneurysm For the usual (forward) incompressible Navier-Stokes calculation, one would prescribe a velocity profile (u = z) at the inflow boundary and the ‘do nothing’ ((η∇u − pI)n = 0) or pressure boundary condition (p = pou) at the outflow boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This implies letting the pressure ‘free’ at the inflow and the velocity ‘free’ at the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' At the walls the velocity is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' u|Γfixed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Consider now the adjoint problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The boundary conditions in this case are described in (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=', we obtain zero velocity at the inflow and ‘do nothing’ or prescribed zero adjoint pressure at the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The adjoint velocity is also zero on the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Numerical Implementation In a strict mathematical sense, the adjoint solver obtained by discretizing the adjoint partial differential equation should be as close as possible to the discrete adjoint obtained from transposing and manipulating the discretization of the forward problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In this way ‘optimize-then-discretize’ and ’discretize-then-optimize’ are as close as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This was 8 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT not adopted in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Instead, while the forward problem was solved for the incompressible Navier-Stokes equations, the adjoint equations were derived for the quasi- incompressible Navier-Stokes equations, which for steady flows give the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Fur- thermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' while the forward problem was integrated to steady state using a fractional step solver with implicit solution of the viscous terms and the pressure increments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' and edge- based upwinding for the velocities and 4th order pressure stabilization [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the adjoint was discretized in space using the following scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' which for each point i in the mesh is given by: � Ak�T i Mi∇k(˜u)i + BT(µi + µj)Kij(˜ui − ˜uj) + MiIΩ u + Di = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (∗∗) where A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' ∇k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Kij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Di denote the Jacobians of the advective fluxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' lumped mass-matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' discrete gradient in direction k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Laplacian edge-based coefficients and damping vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' and ∇k(˜u)i = Ck ij(˜ui + ˜uj) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' where Ck ij are the edge-based coefficients for the gradient (see [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Chapter 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Furthermore Di = −λ(ij) � ˜ui − ˜uj + β 2 lij · (∇(˜u)i + ∇(˜u)j) � , where c is the speed of sound, ˜p the adjoint pressure, λ = |u| + c the maximum eigenvalue of the system and 0 < β < 1 denotes a pressure sensor function of the form [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' β = 1 − ˜pi − ˜pj + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5lij · (∇(˜p)i + ∇(˜p)j)| |˜pi − ˜pj| + |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5lij · (∇(˜p)i + ∇(˜p)j)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='4) For β = 0, 1, second and fourth order damping operators are obtained respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Several other forms are possible for the sensor function β [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Although this discretization of the adjoint Euler fluxes looks like a blend of second and fourth order dissipation, it has no adjustable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Defining U = (u, p), ˜U = (˜u, ˜p) Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' (**) may be re-written as R(U, ˜U) = 0 , the system re-written as an unsteady equation of the form: ˜U,τ + R(U, ˜U) = 0 , and integrated in pseudo-time τ via a classic explicit multistep Runge-Kutta [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Numerical Examples We will focus on two main examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' At first, we consider Poisuille flow through a channel in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Remarkably enough, we are able to derive the explicit expressions for all the quantities, such as solution to the state equation, adjoint equation and sensitivities, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' These theoretical results are also confirmed by numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2, we focus on a realistic aneurysm scenario, where we truly see the benefits of the proposed sensitivity approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' ADJOINT-BASED ESTIMATION OF SENSITIVITY 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The 2-D channel flow provides a good test to verify the implementa- tion of the forward and adjoint solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The domain considered is of dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 ≤ x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='05 ≤ y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='05 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='005 ≤ z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' A parabolic inflow with maximum velocity of umax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 was prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The velocity at the top and bottom walls (ymin, ymax) was pre- scribed to zero, and the velocity in the z-direction was prescribed to zero for the back and front walls (zmin, zmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The other relevant parameter is µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Two ‘clinically relevant measures’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' cost functions) were considered: kinetic energy I = 1 2 � Ω ρu2 dx and vortical energy I = 1 2 � Ω ρ|∇×u|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We set ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The derivation of the exact solutions for the adjoint equations for these cost functions may be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Let u = (u, v, w)⊤, then the x-component of u is given by: u = � 1 − 4 H2y2 � u0 , where u0 = umax and H is the total height of the channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' ymax = −ymin = H/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We thus obtain ∂yu = −8u0 H2 y, ∂yyu = −8u0 H2 , ∂xp = −8µu0 H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The pressure, velocity magnitude, and velocity vectors are shown in Figures 4-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Pressure Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Velocity Magnitude 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Kinetic Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Consider the cost function I = 1 2 � ρ|u|2 dx , implying Iu = ρu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As can be seen in Appendix 1, the adjoint pressure for this cost function is: ∂x˜p = 4 5ρu0 , pressure 40 ILI 30 20 10velocity 1010 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='510 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Velocity i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the gradient of the adjoint pressure is also constant and linearly dependent of u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The results obtained are shown in Figures 7-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Adjoint Pressure Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Magnitude of Adjoint Velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Here the cost function is Kinetic Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Adjoint Velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Here the cost function is Kinetic Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 10 :10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5adj_press 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0025adjvelo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='006 00ADJOINT-BASED ESTIMATION OF SENSITIVITY 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Vortical Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The cost function is given by I = 1 2 � ρ |∇ × u|2 dΩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' For the 2-D channel (u = u(y), v = 0, w = z) (∇ × u)2 = (∂yu)2 , so that I,u = ρu,y(u,y),u = −ρu,yy = −ρ µp,x = 8ρu0 H2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As can be seen in Appendix 1, the adjoint velocities and pressure are given by: ˜u(x, y) = 0 , ˜v(x, y) = 0 , −˜p = ρ µp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Poiseuille Flow: Adjoint Pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Cost Function: Vortical Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Aneurysm with Simple Flow Pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As an example, we include an aneurysm with simple flow pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The geometry and discretization may be discerned from Fig- ures 11a-c which show the surface triangulation, pressure and magnitude of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The region for the source-terms of the adjoint is shown in Figure 12 a and the adjoint pres- sure, as well as the magnitude of the adjoint velocities obtained in Figures 12 b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The adjoint velocites can also be seen in Figures 13 a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Note the effect of the source-term that pushes the adjoint flow and forms a double vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' a,b,c Aneurysm: Surface Triangulation, Surface Pressure and Magnitude of Velocity in Cut Plane adj_press 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5 80420 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5pressure 200 100velocity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5 8 612 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' a,b,c Aneurysm: Source, Adjoint Pressure and Magnitude of Adjoint Velocity in Cut Plane Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' a,b Aneurysm: Adjoint Velocity in Cut Plane 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Conclusions and Outlook The use of adjoint solvers to assess the sensitivity of incomplete boundary (inflow, geometry) information has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The results of this investigation indicate that the sensitivity of clinical measures or other flow features that are inside the flow domain with respect to inflow velocity is proportional to the adjoint pressure, while the sensitivity with respect to inflow geometry is given by the product of the adjoint pressure and the normal derivative of the inflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Thus, the adjoint pressure may be a good indicator to see if the inflow boundary of haemodynamic cases is far enough from the region of interest so that errors can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The use of adjoint solvers is not unproblematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Unlike running a series of cases, varying inflow profiles and geometry, and seeing their influence on many clinically relevant measures, adjoints require a different run for each of the clinical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Appendix 1: Analytical Expressions for Poiseuille Flow A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Exact Forward Solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Let us consider a long 2-D channel of length 0 ≤ x ≤ L and width −H/2 ≤ y ≤ H/2 with incompressible viscous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Let u = (u, v, w)⊤, then the equation for the x-velocity u is given by: u∂xu + v∂yu + ∂xp = µ∆u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Assuming a constant velocity profile in x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' u = u(y) and laminar flow with v = 0, the solution is the Poiseuille solution, given by: u = � 1 − 4 H2y2 � u0 , (12) AbsDIDv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='21e-05 1e-5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5e-6 5e-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='5e-6adj_press 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='000228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0001 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='000129adj_veloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='004adj_veloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0188 0:016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='004adi_veloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='0188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='012 三0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='004ADJOINT-BASED ESTIMATION OF SENSITIVITY 13 where u0 is the maximum velocity at the center of the channel, and the channel extends in height from −H/2 ≤ y ≤ H/2, implying ∂yu = −8u0 H2 y , and ∂yyu = −8u0 H2 , so that the constant pressure gradient is given by: ∂xp = −8µu0 H2 , where we have used the fact that ∂xu = ∂xxu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The average velocity is then: u = 1 H � H/2 −H/2 u dy = 2 3u0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Adjoint Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The equation for the adjoint x-velocity ˜u is given by: −u∂x˜u − v∂y˜u + ∂x˜p = µ∆˜u,xx + Iu Here I is the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' For the channel u is given by (12) and v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Kinetic Energy: If the cost function is given by the kinetic energy I = 1 2 � ρ|u|2 dx , then Iu = ρu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Assuming a long channel with no change in x of the variables, the equation for the adjoint x-velocity ˜u simplifies to: ∂x˜p = µ∂yy˜u + ρu0 � 1 − 4 H2y2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Assuming furthermore that ∂x˜p is constant, and applying the boundary conditions ˜u = 0 for y = −H/2 and y = H/2 this yields ˜u = 1 2µ [−∂x˜p + ρu0] �H2 4 − y2 � − ρu0 3µH2 �H4 16 − y4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' If we consider that at the inflow boundary ˜u = 0, then as the adjoint velocity field is also divergence-free, in any section of x we must have: � ˜udy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This implies: � H/2 −H/2 ˜u dy = 1 2µ [−∂x˜p + ρu0] �H2 4 y − y3 3 �H/2 −H/2 − ρu0 3µH2 �H4 16 y − y5 5 �H/2 −H/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Evaluation of all terms leads to the remarkable result: ∂x˜p = 4 5ρu0 = −ρH2 10µ ∂xp , 14 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the gradient of the adjoint pressure is also constant and linearly dependent of u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Given that the base level of the pressure p is arbitrary, we might set it so that it vanishes at the exit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' We finally obtain the remarkable result that: −˜p = ρH2 10µ p , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the pressure and adjoint pressure are related by the factor ρH2 10µ and have a constant gradient in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' The adjoint velocity is given by: ˜u = ρu0 µ � 1 10 �H2 4 − y2 � − 1 3H2 �H4 16 − y4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' At the center of the channel the velocity is given by: ˜u(y = 0) = ρu0H2 240µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Vortical Energy: If the cost function is given by the vortical energy I = 1 2 � ρ |∇ × u|2 dx , then, for the 2-D channel (u = u(y), v = 0, w = z) |∇ × u|2 = (∂yu)2 , so that Iu = ρ∂yu(∂yu),u = −ρ∂yyu = −ρ µ∂xp = 8ρu0 H2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' constant (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Assuming a long channel with no change in x for the variables, the equation for the adjoint x-velocity ˜u simplifies to: ∂x˜p = µ∂yy˜u − ρ µ∂xp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As this is a long channel and the source-term is constant, the assumption that ∂x˜p is constant is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' This implies that ∂yy˜u should also be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Applying the boundary conditions ˜u = 0 for y = −H/2 and y = H/2 yields: ˜u = � 1 − 4 H2y2 � ˜u0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' However, if we again consider that at the inflow boundary ˜u = 0, and given that the adjoint velocity field is divergence-free, then in any section of x we must have: � ˜udy = 0 , which implies that the only possible solution is ˜u(x, y) = 0, and therefore: −∂x˜p = ρ µ∂xp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' As at the exit the pressure p vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' p = 0, we finally obtain the remarkable result that: −˜p = ρ µp , ADJOINT-BASED ESTIMATION OF SENSITIVITY 15 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' the pressure and adjoint pressure are related by the factor ρ µ and have a constant gradient in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Exact Derivatives of Cost Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Kinetic Energy: Ike = 1 2 � ρ|u|2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Given that u = u(y), v = 0 this results in: Ike = 1 2ρ � x dx � y u2dy = 1 2ρL � u2 0 � 1 − 4 H2y2 �2 dy Ike = 1 2 8 15LHρu2 0 , Ike ,u0 = 8 15LHρu0 = 2 3H ˜pin , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' linear in the length L and the velocity u0, and Ike ,x = 1 2 8 15Hρu2 0 = 1 2 2 3H ˜pinu0 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' not dependent (constant) of the length L and quadratic in the velocity u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' In the previous equations we assumed pout = 0, and used the analytical results that relate mass flow, viscosity and pressure gradient for the Poiseuille flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' One should remark that if the domain that is of interest does not change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' only a certain region inside the channel is considered), the correct value is: Ike ,x = 0 as the flow is constant in x and therefore the cost functional does not change if the upstream boundary is moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Vortical Energy (Dissipation): Ive = 1 2 � ρ|∇ × u|2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Given that u = u(y), v = 0 this results in: Ive = 1 2ρ � x dx � y |∂yu|2dy = 8 3 ρu2 0 H2 LH This implies: Ive ,u0 = 16 3 Lρu0 H = 2LH ˜p 3 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' linear in the length L and the velocity u0, and Ive ,x = 8 3 ρu2 0 H = LH ˜pu0 3 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' not dependent (constant) of the length L and quadratic in the velocity u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Notice, though, that as before if the domain that is of interest does not change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' only a certain region inside the channel is considered), the correct value is: Ive ,x = 0 16 RAINALD L¨OHNER, HARBIR ANTIL, JUAN CEBRAL, FERNANDO MUT as the flow is constant in x and the cost functional will not change if the upstream boundary is moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Antil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Kouri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Lacasse, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Ridzal, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Frontiers in PDE-constrained optimization, volume 163 of The IMA Volumes in Mathematics and its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Springer, New York, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Papers based on the workshop held at the Institute for Mathematics and its Applications, Minneapolis, MN, June 6–10, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Mut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=' of Biomedical Engineering, George Mason University, 4400 University Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} +page_content=', George Mason University, Fairfax, VA 22030-4444, USA' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE3T4oBgHgl3EQf3Qt-/content/2301.04762v1.pdf'} diff --git a/iNE2T4oBgHgl3EQfcwcu/content/tmp_files/2301.03898v1.pdf.txt b/iNE2T4oBgHgl3EQfcwcu/content/tmp_files/2301.03898v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4a13a5b9480a0dfa71012aec19e2c545de23038f --- /dev/null +++ b/iNE2T4oBgHgl3EQfcwcu/content/tmp_files/2301.03898v1.pdf.txt @@ -0,0 +1,1992 @@ +1 + +Synergetic Effect of Wall-Slip and Compressibility During Startup Flow of Complex +Fluids +Aniruddha Sanyal, Sachin Balasaheb Shinde, Lalit Kumar* +Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Mumbai 400076, +Maharashtra, India +*Corresponding Author: lalit.kumar@ese.iitb.ac.in +ORCID ID: orcid.org/0000-0002-1946-8231 +The present letter explains the synergetic effect of wall-slip, compressibility, and thixotropy in a pressurized +flow startup operation of various structured fluids. Opposite to the intuition, experimental and numerical +simulations suggest that the wall-slip (adhesive failure) is facilitating gel degradation (cohesive failure), +revealing a new flow-startup mechanism. The thixotropic rheological model includes structural degradation +kinetics at the bulk. Whereas, a static slip-based model addresses the near-wall phenomenon. The near-wall +transient variations in axial velocity or strain evolution, and the initial pressure propagation mechanism +along the axis of the circular pipe explain the essence of the aforementioned synergy. + +Shear-induced forces during flow startup operation in a pipeline carrying complex fluids cause +structural disintegration through compression, creep, shear-stress-localization, shear-banding, and +hammering. Wall-slip occasionally instigates flow startup when the shearing strength is low [1-3]. +Transportation of complex fluids like waxy crude oil gel, polymeric melts, paints, toothpaste, +sewage waste, foodstuffs, and several suspensions or emulsions show such wall-slip effects during +flow startup [4-6]. +Some high molecular-weight organic compounds characteristically disobey hydrodynamic +no-slip at the fluid-wall interface (FWI) beyond a certain stress 𝜏𝑐 (often termed as “sliding yield +stress” [7] or “critical stress” for wall-slip [8]) during flow initiation. The wall-slip in a pipe is a +shear-dependent phenomenon wherein velocity discontinuities at the wall accounts for highly + +2 + +sheared thin region adjacent to the wall having very low viscosity compared to the bulk [9, 10]. +Wall-slip reduces the yield stress requirement at the FWI (as seen for colloidal silica gels [11]) +without changing the rheological properties of the fluid [11, 12]. The rheology at the FWI is +governed by its surface properties during flow startup operation [13]. In the case of polymer melts, +Brochard & De Gennes [14] interpreted the interface as a region grafted with few chains identical +to polymer melt flowing in bulk. Above 𝜏𝑐, the grafted chains undergo a coil stretch leading to +disentanglement and subsequent slippage. Wall-slip may also happen when gaseous films are at +the FWI or where water flows through hydrophobic capillaries [15]. Consequently, the velocity +discontinuity at the wall is a common feature of the wall-slip phenomenon in all these scenarios +(Mooney [16] was the first person to report this). +The wall-slip causes flow instability, resulting in non-linear dynamics (quasi-periodic and +chaotic flow) at the FWI. According to Graham & Coworkers [5, 8], the stress history at the wall- +boundary influences this instability, and one should incorporate it in the slip-based rheological +model. Spikes et al. [10] broadened this shear stress-based criteria using critical wall-shear stress +at which the slip begins, thereafter, confined within a constant slip-length. The overall deformation +in the fluid’s structure at the wall is quantitatively explained by the apparent shear rate 𝛾̇𝑎𝑝𝑝 which +is the combined effect of the nominal shear rate due to bulk flow 𝛾̇𝑛 and the slip at the wall (𝑢𝑠 𝑏 +⁄ ). +At low shear rates, 𝛾̇𝑎𝑝𝑝 is only due to the surface effects [6, 8, 11, 17-19]. The deformation due +to slip is shown to be a power-law function of the shear stress at the wall (discussed more in detail +in the Methods section). +In startup flow, the complex fluid initially ruptures due to an adhesive failure at the wall +resulting from the shearing confinement. At a later stage, continuous shear deformation results in +cohesive failure (or disengagement) [20-21]. One can expect the wall-slip, through adhesive + +3 + +failure, initiates complex gel movement from inlet to outlet of a pipe at the smallest time scale of +flow, i.e., during initial pressure propagation (IPP). The initial gel rupture mechanism can have a +lasting effect on the flow startup operation, as it dictates the pressure gradient in the subsequent +section of the pipeline [22-25]. For example, the wall-slip phenomenon prevails during the flow +assurance of waxy crude oil pipelines at subsea conditions. Literature indicates that waxy crude +oil, similar to other complex fluids, can exhibit different phenomenological or indirect- +microstructure-based complex rheology [21-23, 26-31]. The investigations on flow startup using +weakly compressible waxy crude oil gel can create a benchmark analysis for operations involving +a larger group of complex fluids. +Flow startup operation is theoretically best understood through initial compressive pressure +wave propagation for complex fluids and subsequent shear-layer development, leading to +destructing of the complex fluids structure [22, 24, 32-35]. During IPP the pressure gradient is +generally high at the compressive pressure front (CPF) in most parts of the pipeline. The high local +pressure gradient at the front may cause an adhesive gel failure, resulting in slip flow. In theory, +the wall-slip effects during adhesive breakage remains unexplained for transient compressive +pressure wave movement. The slip can result in the un-attenuated propagation of pressure signals +along the pipeline axis. It intuitively indicates that a high pressure gradient may not result in shear +deformation and subsequent de-structuring of the fluid. However, the intuition of low overall +structural degradation compared to the no-slip scenario is far from true for most complex fluids. +The analysis involving IPP phenomenon must address the contribution of wall-slip in rheological +formulations for correct assessment of the flow behavior. +Results + +4 + +The mechanism for elasto-hydrodynamic slip at the interface of a soft fluid or glassy +material and wall-surface has been comprehensively studied in the literature [7, 36]. However, the +wall-slip effects during compressive pressure propagation and gel degradation for flow startup +remains unknown. Hence, we carefully examine the combined role of wall-slip and compressibility +in various complex gel degradation processes during startup flow. Initially, experiments are +performed to decipher the wall-slip effects on the gel degradation mechanism at the bulk. A model +oil with 10% wax concentration is cooled from 45℃ to 4℃ with a cooling rate of 1 ℃/min. +Following a 10 min hold, the sample is subjected to a constant stress of 100 Pa until the material +breaks (or 1 hour whichever is earlier). The results are compared for the cases with smooth and +rough inner surfaces (the exact parameters for smoothness and roughness are discussed in +“Methods” section). Counter-intuitively, one may see that the smooth surfaces show increasing +gel deformation quantified through strain parameter compared to negligible deformation for the +rough surfaces. Our preliminary numerical analysis, as discussed hereafter, gratifies the +experimental outcome (Figure 1b). + + +(a) +(b) +Figure 1. Comparison of strain evolution showing flow and no-flow scenario for smooth surfaces +(signifying wall-slip) and rough surfaces (signifying no-slip) using (a) experimental (for parallel plate +configuration in Anton Paar MCR 301 rheometer) and (b) numerical simulations (at a location near the inner +wall of a pipeline). +t' +' +0 +50 +100 +150 +200 +0 +20000 +40000 +60000 +Rough Surface +Smooth Surface + +00000 +300000 +Rough Surface +200000 +Smooth Surface +100000 +0 +0 +5 +10 +15 +20 +t(in s)5 + +Model Development: As a schematic, we consider a horizontally aligned cylindrical pipeline +clogged homogeneously with elasto-viscoplastic or shear-thinning-based thixotropic fluids (e.g., +waxy crude oil gel). An isothermal startup operation of the pipeline is initiated by applying +pressure P at the inlet using a Newtonian fluid having a property equivalent to that of the complex +fluids at completely destructed state. The compressibility of the gel 𝜅Θ vary between 10-10 Pa-1 to +10-7 Pa-1, signifying nearly-incompressible and moderately-weak compressibility limits [22, 24]. +The present numerical study assumes an axisymmetric domain Ω in the range [0, L] × [0, R] in +polar coordinate system (r, θ, z). +The following constitutive functional form is used to represent tangential slip velocity at +the wall: +𝑢𝑠 = ∅(𝜏) = { +0, |𝜏| < 𝜏𝑐 +𝐵(|𝜏| − 𝜏𝑐)𝑚, |𝜏| ≥ 𝜏𝑐 … … … … … (1), +where m is a power-law parameter governing slip. The variable B depends on kinetic parameters, +and for isothermal study it is a constant [37, 38]. When shear-thinning-based slip is possible, 𝜏𝑐 +becomes 0 [32]. However, for the yielding fluids, we have considered partial slip where 𝜏𝑐 +becomes some fraction of the yield stress 𝜏𝑦 (e.g., 𝜏𝑐 = 2 3 +⁄ 𝜏𝑦, estimated from the experimental +probes for various complex fluids, as shown in the “Methods” section). The best-suited numerical +values for B and m are finalized after analyzing various fluids through rheometric experiments +(details are provided in “Methods” section). +The problem is defined through the conservation principles for mass and momentum [25] +along with strain evolution equations to assert the coupling of structural degradation-based kinetics +with the constitutive model for extra stress tensor 𝝉̈. The extra stress tensor 𝝉̈ is represented in +terms of viscous and elastic components of stress as follows: + +6 + +𝝉̈ = 𝜇((𝛁𝑼) + (𝛁𝑼)𝑇) − +2 +3 𝜇𝜵𝑼 𝑰̈ + 𝐺𝜸̈ ………(2), +with G, 𝜸̈ , and μ being the effective elastic modulus, nominal strain tensor and effective gel +viscosity, respectively. Finally, the evolution of 𝜸̈ for infinitely high relaxation time is represented +in frame-invariant upper convection derivative form as follows [24, 39]: +𝜸̈̌ = 𝜕𝜸̈ +𝜕𝑡 + 𝑼 ∙ (𝛁𝜸̈ ) − (𝛁𝑼)𝑇 ∙ 𝜸̈ − 𝜸̈ ∙ (𝛁𝑼) = ((𝛁𝑼) + (𝛁𝑼)𝑇) … … … (3), +where 𝜸̈̌ is the upper convection derivative. The magnitude of strain tensor 𝜸̈ can be determined +after solving for each component of the strain tensor (γrr, γθθ, γzz, γrz) in Eq. (3) using the following +form: +𝛾 = ‖𝜸̈ ‖ = √𝛾𝑟𝑧 +2 + 1 +2 (𝛾𝑟𝑟 +2 + 𝛾𝜃𝜃 +2 + 𝛾𝑧𝑧 +2 ) … … … (4). +The structural degradation-based constitutive model for stress is written as: +𝜏 = 𝜇∞ (1 + +𝜇𝑟 +(1 + 2𝑘𝛾)1/2) 𝛾̇ + +𝐺0 +(1 + 2𝑘𝛾)3/2 𝛾 … … … (5), +where µr is the viscosity ratio (= 𝜇𝑔0/𝜇∞). Eq. (5) follows the flow curve at varying values of +constant strain rate 𝛾̇ < 10 s-1 in the experimental results of Zhao et al. [40, 41]. In the creep flow +regime, the major contribution for stress comes from the elastic response. The value of strain at +maximum stress in such condition is considered as “yield strain”, which depends on the +degradation rate constant (i.e. 𝛾𝑚𝑎𝑥 = 1/𝑘). The static yield stress 𝜏𝑦 can be determined at 𝛾 = +𝛾𝑚𝑎𝑥 in Eq. (5), which turns up to be: 𝜏𝑦 = 𝐺0 +33/2 +⁄ +. The present constitutive model consistently +replicates the patterns from the literature [40-42] e.g., overshoot in a shear-rate-controlled stress- +strain flow curve, shear hysteresis and shear banding. For shear-thinning fluids, the assumption of + +7 + +a high shear rate makes the second term on the right-hand side of Eq. (5) negligible. However, for +a material showing Kelvin-Voigt type response to loading, the viscosity and elastic-modulus are +time and strain-independent. The present letter purposefully avoids mechanical analogs which +capture yielding behavior using extreme limiting parameters (e.g. infinite viscosity [43]). One may +note that the structural degradation kinetics considers that the timescale for gel-breakdown or IPP +during startup is much smaller than the gel-buildup timescale (also realistic for waxy crude oil gel +[44, 45]). Eq. (5) is derived following the mathematical sequences stated by Cheng & Evans [46]. +The equations are scaled in terms of the aspect-ratio of the pipeline (1 𝜀 +⁄ ), the timescale capturing +compressive pressure movement 𝑡′and the ratio between the actual pipe length to the critical length +𝛼 signifying the pressure force to wall-shear related force balance (discussed in detail in +“Methods” section). In this problem, FVM-based methodology is applied in a numerically +adequate staggered-type orthogonal grid setup to solve for primary variables like U, p and 𝜸̈ using +the point-by-point iterative method (verification of the algorithm is provided in point 1 of the +supplementary material). The symbol ‘′’ throughout indicates the dimensionless forms. +Analysis: The gel deformation and subsequent gel degradation mechanisms during wall-slip can +be best understood by determining the physical phenomena occurring near the wall at various time +instants. The time evolution of local axial velocity 𝑤′near the wall (𝑟′ = 0.975) at 𝛼𝑧′ = +0.1 (Figure 2a) indicates the scenario in which the pressurized fluid has just entered the pipe from +the inlet. At compressibility number 𝛿 = 4 × 10−4 (= 𝑃𝜅𝛩), the acoustic speed 𝑤𝑓 (≈ 1 √𝜌0𝜅𝛩 +⁄ +) +for initial “compressive” pressure propagation is 333 m/s. Based on this, the CPF will travel a 7.5 +m long pipeline in 𝑡′ = 2.252. The minimum time for the pressure signal to reach 𝛼𝑧′ = 0.1 is +𝑡′ = 0.15. Furthermore, the gel starts degrading substantially at 𝑡′ ≈ 9.16 causing a sharp rise in +velocity. At this juncture, the applied pressure overcomes the net viscoelastic force across the wall + +8 + +(this concept was explicitly explained for a no-flow startup [31]). One can subsequently note the +earliest hint of flow from the outlet at 𝑡′ ≈ 9.16 due to inertial puncture [23] (Figure 4 in the +supplementary material). However, the density barrier at the outlet of the pipe causes reflection of +pressure waves through the impedance phenomenon [47]. Additionally, an increase in 𝑤′is noted +accompanied by continuous reduction in local elastic forces for 𝑡′ > 9.16. This understanding is +consistent with the strain evolution pattern at 𝛼𝑧′ = 0.1 for 𝑡′ > 9.16 (Figure 3a). The magnitude +of local strain increases sharply at 𝑡′ ≈ 15 (Figure 3a) along with a sharp rise in 𝑤′, as observed +in Figure 2a. The local near-wall elastic forces at 𝛼𝑧′ = 0.1, and net elastic forces along the entire +axial stretch of the wall decline to insignificance at 𝑡′ ≈ 50. Once the gel is uniformly compressed, +one may see the inception of flow with a constant axial pressure gradient indicating uniform +resistance to flow at all cross-sections. Thereafter, 𝑤′monotonically increases with time to a +steady-state at 𝑡′ > 300. As per the established sequence for gel degradation where the +accumulated strain diffuses radially inward with time [24], the sheared region spreads from the +near-wall region to the interior bulk. The initial transient feature of 𝑤′is caused by the viscous +decaying of shear layers along with multiple flow reflections from the outlet to the inlet due to +impedance [47]. Other than the fact that a sizeable portion in the upstream is already compressed +and partially deformed, the flow characteristics (in terms of 𝑤′) along the wall at 𝛼𝑧′ = 0.75 +(Figure 2b) remains similar to that at 𝛼𝑧′ = 0.1. The initial CPF reaches 𝛼𝑧′ = 0.75 at 𝑡′ ≈ 1.126. +The inlet-based influence on flow rearrangement at 𝛼𝑧′ = 0.1 diminishes at 𝛼𝑧′ = 0.75. Hence, +𝑤′ for wall-slip at 𝛼𝑧′ = 0.75 remains higher during initial time (also verifiable from the time +evolution of inlet and outlet flowrates (Figure 4 in the supplementary material)). + +9 + + +Figure 2. Comparison for time-dependent variations in local axial velocity 𝒘′near the fluid-wall interface at +𝝁𝒈𝟎 = 𝟏𝟎𝟎 Pa s, α = 1.5, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 at (a) 𝜶𝒛′ = 𝟎. 𝟏, and (b) 𝜶𝒛′ = 𝟎. 𝟕𝟓; with the +cases for no-slip, conditional slip (τc = +𝟐 +𝟑 𝝉𝒚) and shear-thinning slip (τc = 0). (c) Variations in slip velocity with +time at τc = 0 for different axial locations. (d) Comparison for 𝒘′at same gel condition with 𝜹 = 𝟒 × 𝟏𝟎−𝟔, at +τc = 0. + +For wall-slip scenario, an altered compressional deformation and wall-stress causes +increase in 𝑤′ with respect to no-slip. Here, the local flow commences when the CPF reaches +𝛼𝑧′ = 0.1 and 𝛼𝑧′ = 0.75, irrespective of wall conditions. The initial increment in 𝑤′for wall-slip +at 𝛼𝑧′ = 0.1 is greater than that at 𝛼𝑧′ = 0.75, indicating prominence of initial inertial +compression near the inlet. Interestingly, the sudden rise in 𝑤′comprises the cumulative effect of +slip and no-slip velocities at the nearby wall, which initially support enhanced compressional +deformation at CPF during IPP. Furthermore, if we refer to the inset of Figure 2a, we see a drop +in 𝑤′ till 𝑡′ ≈ 4.729. From an apparent viewpoint, one may expect the initial CPF to propagate to +the end of the pipeline by 𝑡′ ≈ 2.252 and reflect to the inlet by 𝑡′ ≈ 4.504. Once the information +of “no-resistance” reaches to the inlet, the inlet flow increases as a part of recurrent process during +t' +w' +50 +100 +150 +200 +250 +300 +0 +0.01 +0.02 +0.03 +0.04 +no slip +c=2/3 y +c=0 +(a) +A +Local Velocity at z' = 0.1,  = 4E-4 +t' +w' +50 +100 +150 +200 +250 +300 +0 +0.01 +0.02 +0.03 +0.04 +no slip +c=2/3 y +c=0 +(b) Local Velocity at z' = 0.75,  = 4E-4 +0 +5 +10 +15 +0 +0.004 +0.008 +0.012 +0 +5 +10 +15 +0 +0.004 +0.008 +0.012 +t' +us' +50 +100 +150 +200 +10 +-9 +10 +-8 +10 +-7 +10 +-6 +10 +-5 +10 +-4 +10 +-3 +at z' = 0.1 +at z' = 0.75 +(c) +O(-5) - O(-6) +Slip velocity,  = 4E-4 +t +us +0 +3 +6 +9 +10 +-9 +10 +-8 +10 +-7 +10 +-6 +10 +-5 +10 +-4 +t' +w' +50 +100 +150 +200 +0 +0.01 +0.02 +0.03 +at z' = 0.1 +at z' = 0.75 +(d) Local Velocity at  = 4E-6, c=0 +2 +4 +6 +8 +10 +0 +0.002 +0.004 +0.006 + +10 + +pressure wave propagation. The additional flow resistances due to wall-yield critical stress at 𝜏𝑐 = +2 3 +⁄ 𝜏𝑦 leads to greater initial drop in 𝑤′compared to 𝜏𝑐 = 0. At very low compressibility (𝛿 = +4 × 10−6) with wall-slip, 𝑤′increases depending upon axial position during IPP (Figure 2d). One +may also note that the rate of increase in 𝑤′in case of 𝛿 = 4 × 10−6 is almost same irrespective +of the axial location. Compressive deformation during IPP is negligible, and the flow is governed +by the slip alone. In such case, a linear pressure profile develops before shear deformation becomes +significant to counter applied pressure (as shown in Figure 5 of the supplementary material). This +allows the gel to deform uniformly throughout the pipeline. With increasing time, as the overall +gel starts deforming, the magnitude of successive hikes in the 𝑤′ subsides. However, once the +entire upstream undergoes deformation, 𝑤′ at 𝛼𝑧′ = 0.1 shows a hike, indicating accelerated flow. +This is consistent with the plots for the time evolution of the strain (in Figure 3c). The time- +evolution of strain in Figures 3(a, c) shows an instantaneous hike concomitant with the +understanding from the local flow variations. One may note from Figure 3c that the nominal (or +actual) strain 𝛾′ for the no-slip scenario eventually becomes more than that for the wall-slip +induced scenario. This suggests that while the wall-slip influences the initial strain, but once IPP +is done with, the local strain due to overall deformation in a no-slip condition surpasses that of the +wall-slip scenario. To verify strain characteristics, we have checked for time-evolution of 𝛾′ at a +nearby radially inward location (Figure 3b). One can clearly see that in all cases, strain away from +the wall remains substantially lower than the nominal strain at 𝑟′ = 0.975 for initial time instants. +One expects that at the earliest instant when there is hardly any bulk deformation in the gel, +including the locations near the wall at the inlet, the strain for no-slip condition should be more +than the wall-slip induced scenario. This is precisely recovered at 𝛼𝑧′ = 0.1 at 𝑡′ < 0.1, when the +pressure force induced at the inlet is limited to 20 kPa (Figure 3d). In addition, it may be shown + +11 + +that these counter-intuitive outcomes on wall-slip-effects reverses when the fluid is almost +incompressible. +The non-periodic variations in 𝑤′in Figures 2(a, b) symbolize the slip-stick mechanism at +𝜏𝑐 = 2 3 +⁄ 𝜏𝑦. This nonlinearity can be qualitatively explained by the wall-slip model [8]. The +present rheological model accurately predicts decreasing strain rate during the initial stage of flow +startup (a decrease in strain rate at earlier times during flow startup is due to “shear localization” +[45]). + +Figure 3. Comparison for the time evolution of strain 𝜸′ between (a) no-slip, conditional-slip and shear- +thinning slip at 𝒓′ = 𝟎. 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎. 𝟏 and P = 40 kPa, (b) at a radial location 𝒓′ = 𝟎. 𝟗, 𝜶𝒛′ = 𝟎. 𝟏, and P = +40 kPa, (c) no-slip and shear-thinning slip at 𝒓′ = 𝟎. 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎. 𝟕𝟓 and P = 40 kPa, and (d) no-slip and +shear-thinning slip at 𝒓′ = 𝟎. 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎. 𝟏 and P = 20 kPa; while the other gel conditions remaining the +same as in Figure 2. +Furthermore, we extend these understandings for other structured materials (like shear- +thinning-fluids, Kelvin-Voigt (KV) type viscoelastic solid) to establish a benchmark overview to +the complex flow physics associated with the synergetic effect of wall-slip and compressibility +t' +' +5 +10 +15 +0 +0.5 +1 +1.5 +2 +(d) z' = 0.1, r' = 0.975, P = 20 kPa +t' +' +0 +5 +10 +15 +20 +25 +30 +0 +0.5 +1 +1.5 +2 +(b)z' = 0.1, r' = 0.9, P = 40 kPa +0.07 +0.08 +0.09 +0.1 +0 +0.0002 +0.0004 +t' +' +4 +8 +12 +16 +20 +0 +0.4 +0.8 +1.2 +1.6 +2 +(c) z' = 0.75, r' = 0.975, P = 40 kPa +0.06 +0.08 +0.1 +0.12 +0.0002 +0.0004 +t' +' +0 +4 +8 +12 +16 +0 +0.5 +1 +1.5 +2 +no slip +c=2/3 y +c=0 +(a) z' = 0.1, r' = 0.975, P = 40 kPa + +12 + +during flow startup. In the process, we also exhibit robustness of our model in the presence of wall- +slip through predicting continuous flow. +For weakly compressible shear-thinning (ST) gel ( 𝛿 = 4 × 10−4) with no-slip, 𝑤′ (at 𝑟′ = +0.975 and 𝛼𝑧′ = 0.1) is initially higher than that of the thixotropic elasto-viscoplastic fluids +(TEVP) (Figure 4a). Unlike TEVP, the applied pressure force for ST does not require to overcome +elastic forces at the wall. For TEVP, 𝑤′ increases drastically after gel degradation (when 𝛾′ > +𝛾𝑚𝑎𝑥′). The deformation in the present problem is directly associated with microstructural +rearrangement of the gel’s network guided through Eq. (5). For ST, the deformation in the bulk +segment of the gel (except at the outset) during IPP is high compared to TEVP. Hence, at 𝑡′ > 21, +𝑤′for ST at the vicinity of the wall is affected by shearing actions between the adjacent shear +layers along the bulk radial direction, which is evident from parabolic-type flow axial profiles. +Whereas in TEVP, the flow occurs in the form of a plug due to strong shear bands in the bulk [24, +31]. During IPP with or without wall-slip, 𝛾′ in the bulk portion of the gel remains less than the +yield strain in TEVP. Besides, the wall-slip causes additional flow due to extra compression at the +CPF by un-attenuated pressure force. However, the ST is only subjected to viscous shearing action, +which causes continuous deformation in the bulk region of the gel. Unlike TEVP, the redistribution +of energy for gel degradation is not just localized close to the wall but is widespread for ST. +Consequently, the accumulation of shearing stress in the vicinity of the wall is lesser than that of +the TEVP. Hence, a higher magnitude of 𝑤′ occurs for TEVP in Figures 4(b, c) (i.e. irrespective +of the compressive resistances in the gel) during IPP. + +13 + + +Figure 4. Comparison for time-variant local axial velocity for shear-thinning and elasto-viscoplastic fluids at +𝒓′ = 𝟎. 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎. 𝟏, P = 40 kPa for the case of (a) no-slip scenario at 𝜹 = 𝟒 × 𝟏𝟎−𝟒, (b) 𝝉𝒄 = 𝟎 at 𝜹 = +𝟒 × 𝟏𝟎−𝟒, and (c) 𝝉𝒄= 0 at 𝜹 = 𝟒 × 𝟏𝟎−𝟔. +For a weakly compressible KV-type material, the synergy between compressibility and +wall-slip explains nature of pressure propagation and its continuous movement. The pressure +profiles in Figure 5a show that the compressive resistances in the flow delay IPP. Despite the no- +slip scenario, the CPF advances with initial inertial compression causing deformation near the wall +at the inlet. This actuates small movement in the axial direction without sustainable flow. For wall- +slip at 𝜏𝑐 = 0, one may note higher 𝑤′ compared to no-slip cases after the subsidence of the initial +flow transients triggered from inertia-based compression at the outset (Figure 5b). After a certain +time, 𝑤′ attains a velocity having contributions from the bulk flow overcoming compressive +resistances at the upstream (in addition to slip-velocity). In KV-type material with no-slip, the +pressure will eventually balance by the elastic force and flow stops. However, unlike the no-slip +scenario, the wall-slip allows continuous movement of the KV-type material. In the slip-flow, a +higher pressure drop in the upstream portion of the gel setup during IPP is observed (Figure 5c). +For the wall-slip scenario, overcoming the reduced wall-stress requirement is enough to push the +KV-type material to the outlet in a stable plug-like format. Accordingly, a scenario occurs at some +𝛿 (within 4 × 10−5 − 4 × 10−6) where the KV-type material may flow to the outlet due to wall- +slip, and the flow does not occur in a no-slip scenario. Thus, the wall-slip governs the flow of a +t' +w' +5 +10 +15 +20 +25 +0 +0.005 +0.01 +0.015 +0.02 +shear-thinning +elasto-viscoplastic +(a) No slip +t' +w' +5 +10 +15 +20 +25 +0 +0.005 +0.01 +0.015 +0.02 +(b) Slip at z' = 0.1,  = 4E-4 +t' +w' +5 +10 +15 +20 +25 +0 +0.005 +0.01 +0.015 +0.02 +(c) Slip at z' = 0.1,  = 4E-6 + +14 + +viscoelastic-type solid material (like KV-type) at a later time during startup operation, opposite to +what is seen for TEVP or shear-thinning fluids. + +Figure 5. (a) Effect of compressibility on flow startup during pressure propagation through a Kelvin-Voigt +material at a later time 𝒕′ = 𝟐𝟎, 𝜶 = 𝟏. 𝟏, and 𝝉𝒄 = 𝟎. (b) Transients in local axial velocity at 𝒓′ = 𝟎. 𝟗𝟕𝟓, +𝜶𝒛′ = 𝟎. 𝟏, 𝜶 = 𝟏. 𝟏, and 𝝉𝒄 = 𝟎. Comparison for the time evolution of pressure propagation between no-slip +and shear-thinning slip scenario (𝝉𝒄 = 𝟎) for Kelvin-Voigt material (at 𝜹 = 𝟒 × 𝟏𝟎−𝟔) (c) 𝒕′ = +𝟎. 𝟎𝟎𝟔𝟓 & 𝟎. 𝟎𝟏𝟏, (d) 𝒕′ = 𝟎. 𝟎𝟏𝟐𝟓, 𝟎. 𝟎𝟏𝟓 & 𝟎. 𝟎𝟐, and (e) 𝒕′ = 𝟎. 𝟎𝟒, 𝟎. 𝟎𝟒𝟐𝟓 & 𝟎. 𝟎𝟓. +Finally, a comparison of pressure propagation at various time instants is shown in Figures +6(a-d) for cases involving slip and no-slip at the FWI for weakly compressible TEVP fluids. At an +earlier time 𝑡′ = 0.1, pressure builds up near the entrance due to inertial compression, as discussed +earlier. The CPF diffuses further into the downstream at 𝑡′ = 1 (Figure 6a). For the cases involving +wall-slip, the viscous attenuation is less, and hence, the oscillations travel downstream. A steady +decreasing pressure slope in upstream for a no-slip scenario indicates substantial bulk gel +deformation. For wall-slip cases, this deformation is less, and the majority of the gel in the bulk +region away from the wall remains intact at 𝑡′ = 1. One may note that at 𝑡′ = 2.3, the pressure +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +c= 0,  =4E - 3 +c= 0,  =4E - 6 +(a) +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +c = 0, 0.0065 +no slip, 0.0065 +no slip, 0.011 +c = 0, 0.011 +(c) +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +no slip, 0.025 +c = 0, 0.025 +no slip, 0.035 +c = 0, 0.035 +(e) +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +no slip, 0.04 +c = 0, 0.04 +no slip, 0.0425 +c = 0, 0.0425 +no slip, 0.05 +c = 0, 0.05 +(e) +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +no slip, 0.0125 +c = 0, 0.0125 +no slip, 0.015 +c = 0, 0.015 +no slip, 0.02 +c = 0, 0.02 +(d) +t' +w' +0.5 +1 +1.5 +10 +-4 +10 +-3 +no slip,  =4E - 6, z' = 0.1 +c= 0,  =4E - 6, z' = 0.1 +(b) + +15 + +propagates downstream at a higher speed for the wall-slip cases. In conclusion, the slip tends to +dominate initial CPF movement during startup. + +Figure 6. Comparison for the time evolution of axial pressure profile at 𝝁𝒈𝟎 = 100 Pa s, 𝜶 = 𝟏. 𝟓, P = 40 kPa, +k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 with the cases for no-slip, shear-thinning slip (τc = 0) and conditional slip scenario (τc = +2/3 τy) at 𝒕′ = (a) 1, (b) 2.3, (c) 5, and (d) 75. +The pressure propagation mechanism at 𝛿 = 4 × 10−6 sees multiple reflections of pressure +waves from the outlet with slow gel deformation for a wall-slip scenario (the mechanism is +explained in detail in point 3 of the supplementary material). For an energy efficient flow startup, +pressure requirement estimations in the longer pipeline become intriguing. It is important to realize +the importance of wall-slip in such scenarios. The present rheological model improves startup +estimations for a longer pipeline (𝛼 = 4) and low gel degradation rate constant (k = 50) (shown in +point 4 of the supplementary material). +Concluding Remarks: This analysis creates a benchmark for any flow showing synergy between +compressibility and wall-slip. The study can be extended to various structural degradation kinetics +involving the effects of structural buildup (often realized for wormlike micelles). The analysis may +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(c) 5 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(c) 10 +z' +p' +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +no slip +c=0 +c=2/3 y +(a) 0.1 +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +no slip +c=0 +c=2/3 y +(a) 1 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(b) 2.3 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(d) 75 + +16 + +be useful for demarcating the effect of shear banding in complex fluids where wall-slip inherently +occurs [2]. To date, the concept of critical stress calls for bigger clarity on what yields and what +might not yield. This letter hints at the importance for such clarity. +METHODS +Experimental determination of parameters of the slip-model (Eq. (1)) +Materials +Examinations are carried out based on complex fluids like model oil with 5-7% wax concentration (TEVP fluid), +toothpaste (yield-stress fluid) and 1.5% Carbopol solution (Herschel-Bulkley fluid). The sample of model waxy oil is +prepared by adding different macro-crystalline wax (Sasolwax 5054) concentrations ranging from 6 to 10 wt.% in +Dodecane solvent. The sample of model oil was heated 10-20 ℃ above WAT to assure complete solubility of the wax +in Dodecane. The Carbopol solution is prepared by adding 1.5 wt.% of Carbopol powder-940 in distilled water. This +mixture is then rotated at 1100-1300 rpm for 30 min to ensure a homogeneous gel formation. In addition, commercially +available toothpaste is used. +Experimentation +A series of rheological experiments are performed with the Anton Paar MCR 301 rheometer to investigate the wall- +slip for complex fluids with yielding behavior. The fluid sample is kept on the fixed bottom plate of the rheometer. +Two parallel-plate geometries (smooth and rough types) with 50 mm diameter are used for all rheological experiments +(Figure 7). The surface roughness of the plates is measured using a Surface Profilometer: Alicona. The surface +roughness of the smooth and rough plates is in the range of 1.5-2.2 𝜇𝑚 and 69.2-70.9 𝜇𝑚, respectively A constant gap +of 1 mm prevails between parallel plates during measurements. A Peltier plate controller from the bottom plate +maintains the temperature of the sample. For the case of waxy oil samples, the initial temperature is kept above WAT, +and further, it is cooled to below the gelation temperature with a cooling rate of 1 ℃/min. However, in the case of the +toothpaste and Carbopol solution, an isothermal temperature of 25 ℃ is maintained throughout. After holding the +sample for sufficient time, the yielding behavior of soft gelled fluid is investigated with the stress-ramp test. The stress +ramp of 20 Pa/min for the case of toothpaste and waxy oil, and 6 Pa/min for Carbopol solution is applied during +measurement. The shear stress corresponding to the sudden change in shear rate during the test is regarded as the yield + +17 + +stress of the sample. Additionally, the yielding behavior of the samples is investigated through the constant shear-rate +method. A constant shear rate varying from 0.001 s-1 to 10 s-1 is applied to the gelled sample till complete degradation. +Finally, the shear resistance offered by the material against deformation is recorded. The maximum shear stress in the +constant shear-rate method is considered to be the yield stress of the material where it starts to flow. + + + + +(a) +(b) +Figure 7. Parallel plate with overall and microscopic views for (a) smooth and (b) cross-hatched rough +surfaces. +The stress-ramp experiment on rough surfaces shows initial elastic deformation followed by a sudden increase in shear +rate i.e. a stress plateau (Figures 8(a-d)). This is the classical solid-liquid transition stress referred to as static yield +stress 𝜏𝑦 [48]. However, at a later segment the sudden rise in shear stress can be attributed to the fragmentation of the + +18 + +gel network [49]. For smooth surfaces, early yielding can be located. This is referred to as the critical shear stress for +wall-slip 𝜏𝑐. However, the stress plateau for rough or smooth surfaces indicates the presence of shear banding in +complex fluids [1]. Deducing data from Figures 8(a-d), we plotted 𝜏 versus the difference between the apparent shear +rate 𝛾̇𝑎𝑝𝑝 (calculated for the cases of rough surfaces) and the nominal shear rate 𝛾̇ (calculated for the cases of smooth +surfaces) to calculate slip-velocity 𝑢𝑠, and parameters B and m in Eq. (1) of the letter. Figures 8(a-d) suggest that m +varies from 1.5 to 3, depending upon the type of fluids. For a model waxy oil, m varies from 2.2 to 2.8 with increasing +wax concentration. The variable B depends on kinetic parameters, and for isothermal study it is a constant with an +order varying from 10-5 m Pa-1s-1 at m =1 to 10-17 m Pa-3s-1 at m = 3. The variation of m and B is consistent with the +literature [7-9, 37]. Figures 8(a-d) indicates that 𝜏𝑐/𝜏𝑦 evolves in a manner that it satisfies the range of parameters +assumed for the simulations in our letter. Furthermore, the flow curves at shear-rate controlled experiments in Figures +9(a, b) can be qualitatively tallied with the results for stress-ramp experiments for the determination of 𝜏𝑦 and 𝜏𝑐. + + +Figure 8. Stress ramp experiments showing variations in stress 𝝉 with strain rate 𝜸̇ for different types of +surfaces (rough and smooth) for (a) commercial toothpaste, (b) 1.5% Carbopol solution, (c) 6% model waxy +oil, and (d) 7.5% model waxy oil + +(a) +(b) +250 +50 +200 +40 +2 +Rough + 100 +Smooth +T +20 +50 +10 +Tc +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +25 +30 + (in s-l) +Y (in s'l) +(c) +(d) +30 +200 +25 +150 +20 +Pa) +Pa) +P +10 +50 +5 +Tc +0 +0 +200 +400 +600 +800 +1000 +0 +200 +400 +600 +800 +1000 + (in s"l) +Y (in s"l)19 + + +Figure 9. Results for flow curve from shear-rate 𝜸̇ controlled experiments showing variation of stress 𝝉 with +strain 𝜸 for different types of surfaces (rough and smooth) for (a) 1.5% Carbopol solution and (b) 7.5% +model waxy oil. + +Governing equations, scaling and solution methodology +The set of equations (1) to (5) are scaled to accommodate parameters like aspect ratio with less low simulation time +while solving for series of non-linear partial differential equations. The axial coordinate and radial coordinates are +scaled based on the length L and radius R of the pipeline as follows: 𝑧′ = 𝑧 𝐿 , +⁄ +𝑟′ = 𝑟 𝑅 +⁄ . The standardized velocity +Ws used for scaling of axial and radial velocities is calculated based on the magnitude of static yield stress 𝜏𝑦 as 𝑊𝑠 = +𝑅𝜏𝑦 2𝜇0 +⁄ +. Therefore the axial and radial velocity components are written as 𝑢′ = 𝑢 𝜀𝑊𝑠 +⁄ + and 𝑤′ = 𝑤 𝑊𝑠 +⁄ +, respectively. +The critical length Lc till which the flow always restart for a yield-stress fluid is defined as 𝐿𝑐 = 𝑃𝑅 2𝜏𝑦 +⁄ +. The +dimensionless variable related to the aspect ratio of the pipeline 𝜀 = 𝑅 𝐿𝑐 +⁄ + and a factor α which defined the ratio +between the actual pipe length to the critical length, denoted as 𝛼 = 𝐿 𝐿𝑐 +⁄ +. The scaling of time 𝑡′ is done by resolving +the smallest time scale phenomena, i.e., the compressive pressure wave propagation during initial stage of the flow +restart [25, 50]. The scaled pressure and time are represented as 𝑝′ = 𝑝 𝑃 +⁄ and 𝑡′ = 𝑡 +(𝐿𝑐√𝛿 𝑊𝑠 +⁄ +) +⁄ + respectively. The +viscosity is scaled based on the viscosity 𝜇′ = +𝜇 +𝜇∞(𝑃 𝜏𝑦 +⁄ +). Finally, dimensionless numbers like modified Reynolds +number Re* and compressibility number δ are used in the present study which helps in rewriting the governing +equations in non-trivial form which can be used for larger parametric analysis. These dimensionless numbers are + +20 + +written as follows: 𝑅𝑒∗ = +𝜌0𝑅𝑊𝑠 +𝜇∞(𝑃 𝜏𝑦 +⁄ +) , 𝛿 = 𝑃𝜅𝛩. Finally, the shear or elastic modulus G is scaled by a factor of P/2 and +the dimensionless form of strain 𝛾′ remains the same as the dimensional form. + +The governing equations are written in dimensionless forms as follows: +Mass conservation equation: +√𝛿 (𝜕𝑝′ +𝜕𝑡′ + √𝛿 [𝑢′ 𝜕𝑝′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝑝′ +𝜕𝑧′]) + 1 +𝑟′ +𝜕(𝑟′𝑢′) +𝜕𝑟′ ++ 1 +𝛼 +𝜕𝑤′ +𝜕𝑧′ = 0 … … … … … (1𝑆). +Axial momentum conservation equation: +𝜕𝑤′ +𝜕𝑡′ + √𝛿 (𝑢′ 𝜕𝑤′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝑤′ +𝜕𝑧′) = − +1 +𝛼𝜀𝑅𝑒∗ +𝜕𝑝′ +𝜕𝑧′ + +𝜀 +2𝑅𝑒∗ (1 +𝑟′ +𝜕(𝑟′𝜏𝑟𝑧′) +𝜕𝑟′ ++ 1 +𝛼 +𝜕(𝜏𝑧𝑧′) +𝜕𝑧′ +) … … … … … (2𝑆). +Radial momentum conservation equation: +𝜕𝑢′ +𝜕𝑡′ + √𝛿 (𝑢′ 𝜕𝑢′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝑢′ +𝜕𝑧′) = − +1 +𝜀2𝑅𝑒∗ +𝜕𝑝′ +𝜕𝑟′ + +1 +2𝜀𝑅𝑒∗ (1 +𝑟′ +𝜕(𝑟′𝜏𝑟𝑟′) +𝜕𝑟′ ++ 1 +𝛼 +𝜕(𝜏𝑟𝑧′) +𝜕𝑧′ +− 𝜏𝜃𝜃′ +𝑟′ ) … … … … (3𝑆). +Strain evolution equation for each component of strain tensor (𝛾𝑟𝑧′, 𝛾𝑧𝑧′, 𝛾𝑟𝑟′, 𝛾𝜃𝜃′): +1 +√𝛿 +𝜕𝛾𝑟𝑧′ +𝜕𝑡′ + (𝑢′ 𝜕𝛾𝑟𝑧′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝛾𝑟𝑧′ +𝜕𝑧′ ) − (𝛾𝑟𝑟′ +𝜀 +𝜕𝑤′ +𝜕𝑟′ + 𝛾𝑟𝑧′ +𝛼 +𝜕𝑤′ +𝜕𝑧′ + 𝛾𝑟𝑧′ 𝜕𝑢′ +𝜕𝑟′ + 𝜀𝛾𝑧𝑧′ +𝛼 +𝜕𝑢′ +𝜕𝑧′) = 𝛾̇𝑟𝑧′ … … … … (21), +1 +√𝛿 +𝜕𝛾𝑧𝑧′ +𝜕𝑡′ + (𝑢′ 𝜕𝛾𝑧𝑧′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝛾𝑧𝑧′ +𝜕𝑧′ ) − 2 (𝛾𝑟𝑧′ +𝜀 +𝜕𝑤′ +𝜕𝑟′ + 𝛾𝑧𝑧′ +𝛼 +𝜕𝑤′ +𝜕𝑧′) = 𝛾̇𝑧𝑧′ … … … … … (4𝑆), +1 +√𝛿 +𝜕𝛾𝑟𝑟′ +𝜕𝑡′ + (𝑢′ 𝜕𝛾𝑟𝑟′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝛾𝑟𝑟′ +𝜕𝑧′ ) − 2 (𝛾𝑟𝑟′ 𝜕𝑢′ +𝜕𝑟′ + 𝜀𝛾𝑟𝑧′ +𝛼 +𝜕𝑢′ +𝜕𝑧′) = 𝛾̇𝑟𝑟′ … … … … … (5𝑆), +1 +√𝛿 +𝜕𝛾𝜃𝜃′ +𝜕𝑡′ + (𝑢′ 𝜕𝛾𝜃𝜃′ +𝜕𝑟′ + 𝑤′ +𝛼 +𝜕𝛾𝜃𝜃′ +𝜕𝑧′ ) − 2 (𝛾𝜃𝜃 +𝑢′ +𝑟′) = 𝛾̇𝜃𝜃′ … … … … (6𝑆). +where 𝛾̇𝑟𝑧′, 𝛾̇𝑧𝑧′, 𝛾̇𝑟𝑟′, 𝛾̇𝜃𝜃′ are the components of strain rate tensor. Furthermore, the equation for dimensionless extra +stress tensor is written as: +𝝉̈′ = 2𝜇′𝒅̈ ′ − 2 +3 𝜇′ (𝛁′ ∙ 𝑼′)𝑰̈ + 𝐺′𝜸̈ ′ … … … … … (7𝑆). + +21 + +Boundary Conditions +In the present problem, we consider gel degradation only after a pressured fluid is inserted at the inlet. The boundary +conditions based on Dirichlet’s and Neumann’s convention are imposed for dimensionless pressure 𝑝′, strain, radial +velocity 𝑢′, axial velocity 𝑤′ and extra shear stress 𝜏′ as follows: +At inlet: 𝑝′ = 1, 𝑢′ = 0, 𝜏𝑧𝑧′ = 0 and the pressurized fluid at the inlet has a strain 𝛾′ = 𝛾𝑖𝑛′. +At outlet: 𝑝′ = 0, 𝑢′ = 0, 𝜏𝑧𝑧′ = 0 and constant flux condition is applied for strain; +𝜕𝛾′ +𝜕𝑧′ = 0. +Symmetric conditions prevail at the axis of the pipeline with no flow across the axis causing 𝑢′ = 0, 𝜏𝑟𝑧′ = 0 and +similar to the outlet boundary, Neumann’s condition is applied for pressure and strain; +𝜕𝑝′ +𝜕𝑟′ = +𝜕𝛾′ +𝜕𝑟′ = 0. +At the upper wall, slip based boundary conditions are set: 𝑤′ = 𝑢𝑠′, 𝑢′ = 0. The wall-slip phenomenon in the present +problem is isotropic in nature due to smooth walls [51]. + +Figure 10. Schematic diagram for flow representation +Solution methodology +In the present problem, we considered a uniform and orthogonal staggered grid arrangement to represent the numerical +domain Ω. The governing equations are subjected to boundary conditions to solve for the primary variables (𝑢′, 𝑤′, 𝑝′) +and components of strain (𝛾𝑟𝑧′, 𝛾𝑧𝑧′, 𝛾𝑟𝑟′, 𝛾𝜃𝜃′) using finite volume methodology. Central difference scheme is applied +for spatial discretization of velocity, pressure, stress, viscosity and strain-based components. The transient +formulations are done using second-order implicit method. The staggered grid arrangement comprises of flux-related + +Equilibrium Conditionsatwall:w'=ug,u'=o +Slip Region +Fully Developed +Shear-thinned + Plug Region +Flow Layers +inlet +Outlet +p'=1,u'= 0, +p'= 0, u' = 0,t'zz = 0 +ar' +Axisymmetric Boundary. u'= 0, trz = 0 +ap'_ +e22 + +variables like velocity at the face centers of the cell; whereas, the properties like pressure is calculated at the volumetric +center of the cell. The partial difference equations, thus formed after discretization, are solved using point-by-point +iterative technique. The convergence criteria for velocity and pressure based variables are maintained at a +dimensionless value of 10-12. A higher value for convergence criteria can lead to the divergence of the solution and a +lower value of convergence criteria incurs high simulation time in addition to possibility of round-off errors. It is to +be noted that the degradation rate constant k in the present study is varied from 10 to 200. 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Phenomenological characterization of the rheological +behaviour of inelastic reversible thixotropic and antithixotropic fluids. Br. J. Appl. Phys. +16, 1599–1617 (1965). +47. +Hirose, A and K. Lonngren, K.E. Introduction to Wave Phenomena. (Wiley & Sons, New +York, 1985). +48. +Chang, C., Boger, D. V. & Nguyen, Q. D. The Yielding of Waxy Crude Oils. Ind. Eng. +Chem. Res. 37, 1551–1559 (1998). +49. +Fakroun, A. & Benkreira, H. Rheology of waxy crude oils in relation to restart of gelled +pipelines. Chem. Eng. Sci. 211, 115212 (2020). +50. +Vinay, G., Wachs, A. & Frigaard, I. Start-up transients and efficient computation of +isothermal waxy crude oil flows. J. Nonnewton. Fluid Mech. 143, 141–156 (2007). +51. +Asmolov, E. S., Schmieschek, S., Harting, J. & Vinogradova, O. I. Flow past +superhydrophobic surfaces with cosine variation in local slip length. Phys. Rev. E - Stat. +Nonlinear, Soft Matter Phys. 87, 1–8 (2013). + + + + + + + + + + + + +26 + +Supplementary to the article “Synergetic Effect of Wall-Slip and Compressibility During +Startup Flow of Complex Fluids” by A. Sanyal, S.B. Shinde, L. Kumar +1. Grid independence and model verification +The benchmark for grid arrangement for the present problem is obtained from one of our previous +studies [1]. In the present study, we compared several grid arrangements with number of grid +points varying from 100×10 to 400×100 along axial (Nz) × radial (Nr) direction. The solutions at +200×20 is seen to be numerically adequate with the finest grid arrangement of 400×100. In the +present problem, the wall-slip velocity near the wall is compared for numerical adequacy. +Furthermore, the time-step of 10-4 following CFL criteria is taken as a benchmark for time-step +independence tests. After comparisons of results for wall-slip velocity at two time instants (𝑡′) for +several values of time-step, ∆𝑡′ =10-5 is finalized as numerically adequate. +The formulations for strain evolution based on upper-convection-derivative terms poses +some intriguing issues based on the applicability of such formulation for a problem like the present +one. Tikariha & Kumar [2] have shown that the upper-convection derivative based strain evolution +methodology when compared to the ones adopted by [3] shows dissimilar results after the initial +pressure propagation front has reached the outlet. However, in the present study, a similar +verification is carried out at a longer pipe length. One can see that the results for pressure +propagation at various instants of time remains same, irrespective of the type of formulation +(Figure 1). The variations in results are marginal (< 1% relative deviation) in the regime of low +compressibility number. Furthermore, the velocity profile (Figure 2) at a 𝛼𝑧′ = 0, 0.75 and 1.5 for +a combined effect of wall-slip and elasto-viscoplastic rheology shows variations from plug-like +profile during gel degradation (at 𝑡′ < 100 in Figures 2b and 2c) to a parabolic profile at a steady- +state indicating Newtonian characteristics (𝑡′ = 400). This is qualitatively consistent with the + +27 + +numerical results for the velocity profiles of Damianou et al. [4]. In addition, the velocity +magnitude at steady-state condition quantitatively complies with the analytical value obtained +from Hagen–Poiseuille equation (𝑤𝑚𝑎𝑥 = +∆𝑃 +4𝜇𝐿 𝑅2). Finally, the code is subjected to the verification +of the idea inspired from the experiments of El-Gendy et al. [5] which shows that the flow need +not necessarily restart when the initial pressure propagation front reaches the outlet. One such +scenario is shown in Figure 3 which occurs at 𝛼 = 2, 𝑘 = 50 and 𝜏𝑐 = +2 +3 𝜏𝑦 for an elasto- +viscoplastic based rheology involving slip at 𝛿 = 4 × 10−4. + +Figure 1. Verification of pressure profiles from upper convection derivatives based strain evolution with +strain evolution equation having only material derivative based strain evolution terms as used by Kumar et +al. (2014) for different values of compressibility number (a) 𝜹 = 𝟒 × 𝟏𝟎−𝟒, and (b) 𝜹 = 𝟒 × 𝟏𝟎−𝟔 (at initial gel +viscosity of 100 Pa s, P = 40 kPa and α = 1.5). +z' +p' +0 +0.5 +1 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +(b)  = 4E-6 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0.5 +1 +4 +25 +0.5 +1 +4 +25 +(a)  = 4E-4 + +28 + + +Figure 2. Time evolution of axial velocity variations along radial direction at different fixed axial locations +(for initial gel viscosity of 100 Pa s, P = 40 kPa, 𝜹 = 𝟒 × 𝟏𝟎−𝟒, and α = 1.5) at 𝝉𝒄=0; (a) 𝜶𝒛′ = 𝟎, (b) 𝜶𝒛′ = +𝟎. 𝟕𝟓 and (c) 𝜶𝒛′ = 𝟏. 𝟓. + +Figure 3. (a) Pressure propagation, and (b) time evolution of inlet and outlet flowrates; at initial gel +viscosity of 100 Pa s, 𝜶 = 𝟐, P = 40 kPa, k = 50, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 and τc = 2/3 τy. +2. Time-dependent inlet and outlet flowrate variation at 𝜹 = 𝟒 × 𝟏𝟎−𝟒 +r' +w' +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +400 +100 +40 +10 +3 +1.5 +0.8 +(a) z' = 0 +r' +w' +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +400 +100 +40 +10 +3 +1.5 +0.8 +(b) z' = 0.75 +r' +w' +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +400 +100 +40 +10 +3 +1.5 +0.8 +(c) z' = 1.5 +z' +p' +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1 +3 +15 +(a) +t' +Flowrate +20 +40 +60 +80 +100 +120 +0 +0.003 +0.006 +0.009 +0.012 +0.015 +inlet +outlet +(b) + +29 + + +Figure 4. Comparison for (a) inlet and (b) outlet flowrates with time at initial gel viscosity of 100 Pa s, 𝜶 = +𝟏. 𝟓, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 with the cases for no-slip, shear-thinning slip (τc = 0) and conditional +slip scenario (τc = 2/3 τy). + +3. Pressure propagation at very low gel compressibility at 𝜹 = 𝟒 × 𝟏𝟎−𝟔 +Figures 5(a-c) show pressure propagation from inlet to the outlet with marginal pressure drop along +the axial direction in the downstream for a scenario involving wall-slip. For a wall-slip scenario, +the overall gel deformation (combining net deformation at the bulk portion of the gel and the fluid- +wall interface) during initial pressure propagation stage is less. This can be understood through +negligible pressure gradient along the downstream and a sudden fall in pressure to zero at the +outlet. Following a decay of transients, the no-slip scenario shows a linear profile at 𝑡′ = 40 +(Figure 5f). +t' +inlet flowrate +50 +100 +150 +200 +250 +300 +0.1 +0.2 +0.3 +0.4 +no slip +c= 0 +c=2/3 y +(a) +10 +20 +30 +0 +0.01 +0.02 +0.03 +0.04 +t' +outlet flowrate +50 +100 +150 +200 +250 +300 +0.1 +0.2 +0.3 +0.4 +(b) +10 +20 +30 +0 +0.01 +0.02 +0.03 +0.04 + +30 + + +Figure 5. Comparison for the time evolution of axial pressure profile at initial gel viscosity of 100 Pa s, 𝜶 = +𝟏. 𝟓, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟔 with the cases for no-slip, shear-thinning slip (τc = 0) and conditional +slip scenario (τc = 2/3 τy) at 𝒕′ = (a) 0.1, (b) 1, (c) 2.3, (d) 4.5 (e) 5, (f) 10, (g) 40, (h) 75 and (i) 150. + +The reflected pressure wave causes a decrease in flow velocity compared to the flow +induced from the approaching pressure front. It is to be noted that the reflected flow traverses back +easily due to the slippage region without any significant viscosity attenuation at 𝜏𝑐 = 0 (Figure +5d). At earlier time instants like 𝑡′ = 5 (Figure 5e) for the cases at 𝜏𝑐 = 0, the elongated trailing +part in pressure propagation signal (𝛼𝑧′ > 0.32) denotes region of almost no gel deformation, and +pressure signal moves back and forth via slip. At 𝛼𝑧′ < 0.3, the inertial compressive pressure wave +moves forward with an amplitude lower than the ones in Figure 5b. This indicates the gel +movement with true velocity is causing viscous attenuation of pressure signal. A sudden drop in +pressure signal at 0.3 < 𝛼𝑧′ < 0.32 specifies the region where the net resistive force from the gel’s +z' +p' +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +c=2/3 y +c=0 +no slip +(a) 0.1 +z' +p' +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(b) 1 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(c) 2.3 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(e) 5 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(f) 10 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(h) 75 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(i) 150 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(g) 40 +z' +p' +0 +0.3 +0.6 +0.9 +1.2 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(d) 4.5 + +31 + +elastic strength is counterbalanced by the force associated with the approaching pressure wave +(this region is analogous to a region of pressure front). However, for a subsequent span of time +during the flow restart operation, the pressure wave continues to traverse through the slip region +causing slower gel deformation and multiple reflection of pressure waves from the outlet to the +inlet. The enhanced slope in the trailing part of the pressure profiles at subsequent time instants +(𝑡′ = 10 (Figure 5f) and 40 (Figure 5g)) indicates gel degradation through viscous shearing forces +from the interference between the backwardly reflected and forward approaching flow. +4. Wall-slip effects for pressure estimation during flow restart in longer pipelines and in +scenarios for low gel degradation constants +In this case, we consider the overall aspect ratio (1/ε) of the pipeline as 400, which has a length +equivalent to 4 times the critical length of the pipeline (Lc). Figure 6 shows that the initial pressure +wave front reaches the outlet at 𝑡′ = 10 at 𝜏𝑐 = 0. Whereas, the pressure does not propagate beyond +𝛼𝑧′ = 3.4 at 𝜏𝑐 = 2 3 +⁄ 𝜏𝑦 and 𝛼𝑧′ = 2.3 at no-slip scenarios, respectively. + +z' +p' +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +t' = 1, no slip +t' = 1, c = 2/3 y +t' = 1, c = 0 +t' = 10, c = 2/3 y +t' = 10, no slip +t' = 10, c = 0 +t' = 50, c = 2/3 y +t' = 50, no slip +t' = 50, c = 0 +t' = 200, c = 2/3 y +t' = 200, no slip +t' = 200, c = 0 + +32 + +Figure 6. Comparison for time evolution of axial pressure propagation in longer pipeline for the cases of no- +slip, 𝝉𝒄 = 𝟎, and 𝝉𝒄 = 𝟐 𝟑 +⁄ 𝝉𝒚 at the gel-wall interface for an initial gel viscosity 100 Pa s, 𝜶 =4, P = 40 kPa and +𝜹 = 𝟒 × 𝟏𝟎−𝟒. +At low gel degradation rate, i.e. k = 50, the maximum yield strain required for beginning of the gel +network’s disengagement increases to 0.2 from 0.1 in the earlier-mentioned results. Unlike the +case of no-slip (Figure 7a), the wall-slip allows pressure signals to propagate to the outlet for a +pipeline having an overall aspect ratio (1/ε) of 200 or α = 2 (Figures 11(b, c)). At 𝜏𝑐 = 2 3 +⁄ 𝜏𝑦, the +initial pressure wave diffuses by compressing the gel network with minimum flow resistance near +the wall. But, this does not guarantee a successful flow restart. One may see only inertial puncture +at the outlet at 𝑡′ = 3 in comparison to the fully-developed flow at the outlet at 𝑡′ = 10. The non- +linear pressure signals at late time instants like 𝑡′ = 500 in Figure 7b suggests that the compressive +resistances and viscous resistances will not be overcome in the long run, thereby leading to a halt +in the flow. However, in Figure 7c, one may note the development of a linear profile at 𝑡′ = 50, +indicating the occurrence of complete gel compression and a tendency for steady-state flow at a +later time. + +z' +p' +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0.4 +2 +3 +10 +50 +500 +(c) c = 0 +z' +p' +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0.4 +10 +50 +400 +(a) no-slip +z' +p' +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0.4 +2 +3 +10 +30 +500 +(b) c = 2/3 y + +33 + +Figure 7. Comparison for time evolution of axial pressure propagation at low gel degradation scenario k = 50, +for the cases of (a) no-slip, (b) 𝝉𝒄 = 𝟐 𝟑 +⁄ 𝝉𝒚, and (c) 𝝉𝒄 = 𝟎 at the gel-wall interface; at initial gel viscosity 100 +Pa s, 𝜶 =2, P = 40 kPa and 𝜹 = 𝟒 × 𝟏𝟎−𝟒. +5. Local shear stress variations + +Figure 8. Comparison for the time evolution of true local stress 𝝉𝒆 between no-slip, conditional slip (τc = +𝟐 +𝟑 𝝉𝒚) +and shear-thinning slip (τc =0) at 𝒓′ = 𝟎. 𝟗𝟕𝟓 at the axial location (a) 𝜶𝒛′ = 𝟎. 𝟏 and (b) 𝜶𝒛′ = 𝟎. 𝟕𝟓; while the +other gel conditions remaining the same as the ones mentioned in Figure 1 in the letter. + + +References +1. Sanyal, A., Tikariha, L. & Kumar, L. The effects of partial preheating on pressure propagation +and Flow-Restart phenomena in a clogged pipeline with a weakly compressible gel. Phys. Fluids +33, (2021). +2. Tikariha, L. & Kumar, L. Pressure propagation and flow restart in the multi-plug gelled +pipeline. J. Fluid Mech. 911, 1–26 (2021). +3. Kumar, L., Lawrence, C. & Sjöblom, J. Mechanism of pressure propagation and weakly +compressible homogeneous and heterogeneous thixotropic gel breakage to study flow restart. +RSC Adv. 4, 27493–27501 (2014). +4. Damianou, Y., Georgiou, G. C. & Moulitsas, I. Combined effects of compressibility and slip +in flows of a Herschel-Bulkley fluid. J. Nonnewton. Fluid Mech. 193, 89–102 (2013). +5. El-Gendy, H. et al. The propagation of pressure in a gelled waxy oil pipeline as studied by +particle imaging velocimetry. AIChE J. 58, 302-312 (2012). + + + +t' +e +0 +5 +10 +15 +20 +0 +50 +100 +150 +200 +no slip +c=2/3 y +c=0 +(a) z' = 0.1,  = 4E-4 +t' +e +0 +5 +10 +15 +20 +0 +50 +100 +150 +200 +(b) z = 0.75,  = 4E-6 + diff --git a/iNE2T4oBgHgl3EQfcwcu/content/tmp_files/load_file.txt b/iNE2T4oBgHgl3EQfcwcu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fd9b724c2d14ec186f8c7757d8ec155a71c5805 --- /dev/null +++ b/iNE2T4oBgHgl3EQfcwcu/content/tmp_files/load_file.txt @@ -0,0 +1,1417 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf,len=1416 +page_content='1 Synergetic Effect of Wall-Slip and Compressibility During Startup Flow of Complex Fluids Aniruddha Sanyal, Sachin Balasaheb Shinde, Lalit Kumar* Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Mumbai 400076, Maharashtra, India Corresponding Author: lalit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='kumar@ese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='in ORCID ID: orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='org/0000-0002-1946-8231 The present letter explains the synergetic effect of wall-slip, compressibility, and thixotropy in a pressurized flow startup operation of various structured fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Opposite to the intuition, experimental and numerical simulations suggest that the wall-slip (adhesive failure) is facilitating gel degradation (cohesive failure), revealing a new flow-startup mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The thixotropic rheological model includes structural degradation kinetics at the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Whereas, a static slip-based model addresses the near-wall phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The near-wall transient variations in axial velocity or strain evolution, and the initial pressure propagation mechanism along the axis of the circular pipe explain the essence of the aforementioned synergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Shear-induced forces during flow startup operation in a pipeline carrying complex fluids cause structural disintegration through compression, creep, shear-stress-localization, shear-banding, and hammering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall-slip occasionally instigates flow startup when the shearing strength is low [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Transportation of complex fluids like waxy crude oil gel, polymeric melts, paints, toothpaste, sewage waste, foodstuffs, and several suspensions or emulsions show such wall-slip effects during flow startup [4-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Some high molecular-weight organic compounds characteristically disobey hydrodynamic no-slip at the fluid-wall interface (FWI) beyond a certain stress 𝜏𝑐 (often termed as “sliding yield stress” [7] or “critical stress” for wall-slip [8]) during flow initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The wall-slip in a pipe is a shear-dependent phenomenon wherein velocity discontinuities at the wall accounts for highly 2 sheared thin region adjacent to the wall having very low viscosity compared to the bulk [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall-slip reduces the yield stress requirement at the FWI (as seen for colloidal silica gels [11]) without changing the rheological properties of the fluid [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The rheology at the FWI is governed by its surface properties during flow startup operation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In the case of polymer melts, Brochard & De Gennes [14] interpreted the interface as a region grafted with few chains identical to polymer melt flowing in bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Above 𝜏𝑐, the grafted chains undergo a coil stretch leading to disentanglement and subsequent slippage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall-slip may also happen when gaseous films are at the FWI or where water flows through hydrophobic capillaries [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Consequently, the velocity discontinuity at the wall is a common feature of the wall-slip phenomenon in all these scenarios (Mooney [16] was the first person to report this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The wall-slip causes flow instability, resulting in non-linear dynamics (quasi-periodic and chaotic flow) at the FWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' According to Graham & Coworkers [5, 8], the stress history at the wall- boundary influences this instability, and one should incorporate it in the slip-based rheological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Spikes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' [10] broadened this shear stress-based criteria using critical wall-shear stress at which the slip begins, thereafter, confined within a constant slip-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The overall deformation in the fluid’s structure at the wall is quantitatively explained by the apparent shear rate 𝛾̇𝑎𝑝𝑝 which is the combined effect of the nominal shear rate due to bulk flow 𝛾̇𝑛 and the slip at the wall (𝑢𝑠 𝑏 ⁄ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At low shear rates, 𝛾̇𝑎𝑝𝑝 is only due to the surface effects [6, 8, 11, 17-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The deformation due to slip is shown to be a power-law function of the shear stress at the wall (discussed more in detail in the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In startup flow, the complex fluid initially ruptures due to an adhesive failure at the wall resulting from the shearing confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At a later stage, continuous shear deformation results in cohesive failure (or disengagement) [20-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One can expect the wall-slip, through adhesive 3 failure, initiates complex gel movement from inlet to outlet of a pipe at the smallest time scale of flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', during initial pressure propagation (IPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The initial gel rupture mechanism can have a lasting effect on the flow startup operation, as it dictates the pressure gradient in the subsequent section of the pipeline [22-25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For example, the wall-slip phenomenon prevails during the flow assurance of waxy crude oil pipelines at subsea conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Literature indicates that waxy crude oil, similar to other complex fluids, can exhibit different phenomenological or indirect- microstructure-based complex rheology [21-23, 26-31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The investigations on flow startup using weakly compressible waxy crude oil gel can create a benchmark analysis for operations involving a larger group of complex fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Flow startup operation is theoretically best understood through initial compressive pressure wave propagation for complex fluids and subsequent shear-layer development, leading to destructing of the complex fluids structure [22, 24, 32-35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' During IPP the pressure gradient is generally high at the compressive pressure front (CPF) in most parts of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The high local pressure gradient at the front may cause an adhesive gel failure, resulting in slip flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In theory, the wall-slip effects during adhesive breakage remains unexplained for transient compressive pressure wave movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The slip can result in the un-attenuated propagation of pressure signals along the pipeline axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' It intuitively indicates that a high pressure gradient may not result in shear deformation and subsequent de-structuring of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, the intuition of low overall structural degradation compared to the no-slip scenario is far from true for most complex fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The analysis involving IPP phenomenon must address the contribution of wall-slip in rheological formulations for correct assessment of the flow behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Results 4 The mechanism for elasto-hydrodynamic slip at the interface of a soft fluid or glassy material and wall-surface has been comprehensively studied in the literature [7, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, the wall-slip effects during compressive pressure propagation and gel degradation for flow startup remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Hence, we carefully examine the combined role of wall-slip and compressibility in various complex gel degradation processes during startup flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Initially, experiments are performed to decipher the wall-slip effects on the gel degradation mechanism at the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A model oil with 10% wax concentration is cooled from 45℃ to 4℃ with a cooling rate of 1 ℃/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Following a 10 min hold, the sample is subjected to a constant stress of 100 Pa until the material breaks (or 1 hour whichever is earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The results are compared for the cases with smooth and rough inner surfaces (the exact parameters for smoothness and roughness are discussed in “Methods” section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Counter-intuitively, one may see that the smooth surfaces show increasing gel deformation quantified through strain parameter compared to negligible deformation for the rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Our preliminary numerical analysis, as discussed hereafter, gratifies the experimental outcome (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (a) (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison of strain evolution showing flow and no-flow scenario for smooth surfaces (signifying wall-slip) and rough surfaces (signifying no-slip) using (a) experimental (for parallel plate configuration in Anton Paar MCR 301 rheometer) and (b) numerical simulations (at a location near the inner wall of a pipeline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" t' \uf067' 0 50 100 150 200 0 20000 40000 60000 Rough Surface Smooth Surface 00000 300000 Rough Surface 200000 Smooth Surface 100000 0 0 5 10 15 20 t(in s)5 Model Development: As a schematic, we consider a horizontally aligned cylindrical pipeline clogged homogeneously with elasto-viscoplastic or shear-thinning-based thixotropic fluids (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', waxy crude oil gel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' An isothermal startup operation of the pipeline is initiated by applying pressure P at the inlet using a Newtonian fluid having a property equivalent to that of the complex fluids at completely destructed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The compressibility of the gel 𝜅Θ vary between 10-10 Pa-1 to 10-7 Pa-1, signifying nearly-incompressible and moderately-weak compressibility limits [22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The present numerical study assumes an axisymmetric domain Ω in the range [0, L] × [0, R] in polar coordinate system (r, θ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The following constitutive functional form is used to represent tangential slip velocity at the wall: 𝑢𝑠 = ∅(𝜏) = { 0, |𝜏| < 𝜏𝑐 𝐵(|𝜏| − 𝜏𝑐)𝑚, |𝜏| ≥ 𝜏𝑐 … … … … … (1), where m is a power-law parameter governing slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The variable B depends on kinetic parameters, and for isothermal study it is a constant [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' When shear-thinning-based slip is possible, 𝜏𝑐 becomes 0 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, for the yielding fluids, we have considered partial slip where 𝜏𝑐 becomes some fraction of the yield stress 𝜏𝑦 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', 𝜏𝑐 = 2 3 ⁄ 𝜏𝑦, estimated from the experimental probes for various complex fluids, as shown in the “Methods” section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The best-suited numerical values for B and m are finalized after analyzing various fluids through rheometric experiments (details are provided in “Methods” section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The problem is defined through the conservation principles for mass and momentum [25] along with strain evolution equations to assert the coupling of structural degradation-based kinetics with the constitutive model for extra stress tensor 𝝉̈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The extra stress tensor 𝝉̈ is represented in terms of viscous and elastic components of stress as follows: 6 𝝉̈ = 𝜇((𝛁𝑼) + (𝛁𝑼)𝑇) − 2 3 𝜇𝜵𝑼 𝑰̈ + 𝐺𝜸̈ ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='…(2), with G, 𝜸̈ , and μ being the effective elastic modulus, nominal strain tensor and effective gel viscosity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Finally, the evolution of 𝜸̈ for infinitely high relaxation time is represented in frame-invariant upper convection derivative form as follows [24, 39]: 𝜸̈̌ = 𝜕𝜸̈ 𝜕𝑡 + 𝑼 ∙ (𝛁𝜸̈ ) − (𝛁𝑼)𝑇 ∙ 𝜸̈ − 𝜸̈ ∙ (𝛁𝑼) = ((𝛁𝑼) + (𝛁𝑼)𝑇) … … … (3), where 𝜸̈̌ is the upper convection derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The magnitude of strain tensor 𝜸̈ can be determined after solving for each component of the strain tensor (γrr, γθθ, γzz, γrz) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (3) using the following form: 𝛾 = ‖𝜸̈ ‖ = √𝛾𝑟𝑧 2 + 1 2 (𝛾𝑟𝑟 2 + 𝛾𝜃𝜃 2 + 𝛾𝑧𝑧 2 ) … … … (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The structural degradation-based constitutive model for stress is written as: 𝜏 = 𝜇∞ (1 + 𝜇𝑟 (1 + 2𝑘𝛾)1/2) 𝛾̇ + 𝐺0 (1 + 2𝑘𝛾)3/2 𝛾 … … … (5), where µr is the viscosity ratio (= 𝜇𝑔0/𝜇∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (5) follows the flow curve at varying values of constant strain rate 𝛾̇ < 10 s-1 in the experimental results of Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In the creep flow regime, the major contribution for stress comes from the elastic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The value of strain at maximum stress in such condition is considered as “yield strain”, which depends on the degradation rate constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝛾𝑚𝑎𝑥 = 1/𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The static yield stress 𝜏𝑦 can be determined at 𝛾 = 𝛾𝑚𝑎𝑥 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (5), which turns up to be: 𝜏𝑦 = 𝐺0 33/2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The present constitutive model consistently replicates the patterns from the literature [40-42] e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', overshoot in a shear-rate-controlled stress- strain flow curve, shear hysteresis and shear banding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For shear-thinning fluids, the assumption of 7 a high shear rate makes the second term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (5) negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, for a material showing Kelvin-Voigt type response to loading, the viscosity and elastic-modulus are time and strain-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The present letter purposefully avoids mechanical analogs which capture yielding behavior using extreme limiting parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' infinite viscosity [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One may note that the structural degradation kinetics considers that the timescale for gel-breakdown or IPP during startup is much smaller than the gel-buildup timescale (also realistic for waxy crude oil gel [44, 45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (5) is derived following the mathematical sequences stated by Cheng & Evans [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The equations are scaled in terms of the aspect-ratio of the pipeline (1 𝜀 ⁄ ), the timescale capturing compressive pressure movement 𝑡′and the ratio between the actual pipe length to the critical length 𝛼 signifying the pressure force to wall-shear related force balance (discussed in detail in “Methods” section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In this problem, FVM-based methodology is applied in a numerically adequate staggered-type orthogonal grid setup to solve for primary variables like U, p and 𝜸̈ using the point-by-point iterative method (verification of the algorithm is provided in point 1 of the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The symbol ‘′’ throughout indicates the dimensionless forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Analysis: The gel deformation and subsequent gel degradation mechanisms during wall-slip can be best understood by determining the physical phenomena occurring near the wall at various time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The time evolution of local axial velocity 𝑤′near the wall (𝑟′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='975) at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 (Figure 2a) indicates the scenario in which the pressurized fluid has just entered the pipe from the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At compressibility number 𝛿 = 4 × 10−4 (= 𝑃𝜅𝛩), the acoustic speed 𝑤𝑓 (≈ 1 √𝜌0𝜅𝛩 ⁄ ) for initial “compressive” pressure propagation is 333 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Based on this, the CPF will travel a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 m long pipeline in 𝑡′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The minimum time for the pressure signal to reach 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 is 𝑡′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Furthermore, the gel starts degrading substantially at 𝑡′ ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='16 causing a sharp rise in velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At this juncture, the applied pressure overcomes the net viscoelastic force across the wall 8 (this concept was explicitly explained for a no-flow startup [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One can subsequently note the earliest hint of flow from the outlet at 𝑡′ ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='16 due to inertial puncture [23] (Figure 4 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, the density barrier at the outlet of the pipe causes reflection of pressure waves through the impedance phenomenon [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Additionally, an increase in 𝑤′is noted accompanied by continuous reduction in local elastic forces for 𝑡′ > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This understanding is consistent with the strain evolution pattern at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 for 𝑡′ > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='16 (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The magnitude of local strain increases sharply at 𝑡′ ≈ 15 (Figure 3a) along with a sharp rise in 𝑤′, as observed in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The local near-wall elastic forces at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1, and net elastic forces along the entire axial stretch of the wall decline to insignificance at 𝑡′ ≈ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Once the gel is uniformly compressed, one may see the inception of flow with a constant axial pressure gradient indicating uniform resistance to flow at all cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Thereafter, 𝑤′monotonically increases with time to a steady-state at 𝑡′ > 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' As per the established sequence for gel degradation where the accumulated strain diffuses radially inward with time [24], the sheared region spreads from the near-wall region to the interior bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The initial transient feature of 𝑤′is caused by the viscous decaying of shear layers along with multiple flow reflections from the outlet to the inlet due to impedance [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Other than the fact that a sizeable portion in the upstream is already compressed and partially deformed, the flow characteristics (in terms of 𝑤′) along the wall at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75 (Figure 2b) remains similar to that at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The initial CPF reaches 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75 at 𝑡′ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The inlet-based influence on flow rearrangement at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 diminishes at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Hence, 𝑤′ for wall-slip at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75 remains higher during initial time (also verifiable from the time evolution of inlet and outlet flowrates (Figure 4 in the supplementary material)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for time-dependent variations in local axial velocity 𝒘′near the fluid-wall interface at 𝝁𝒈𝟎 = 𝟏𝟎𝟎 Pa s, α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 at (a) 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏, and (b) 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟕𝟓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' with the cases for no-slip, conditional slip (τc = 𝟐 𝟑 𝝉𝒚) and shear-thinning slip (τc = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (c) Variations in slip velocity with time at τc = 0 for different axial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (d) Comparison for 𝒘′at same gel condition with 𝜹 = 𝟒 × 𝟏𝟎−𝟔, at τc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For wall-slip scenario, an altered compressional deformation and wall-stress causes increase in 𝑤′ with respect to no-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Here, the local flow commences when the CPF reaches 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 and 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75, irrespective of wall conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The initial increment in 𝑤′for wall-slip at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 is greater than that at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75, indicating prominence of initial inertial compression near the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Interestingly, the sudden rise in 𝑤′comprises the cumulative effect of slip and no-slip velocities at the nearby wall, which initially support enhanced compressional deformation at CPF during IPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Furthermore, if we refer to the inset of Figure 2a, we see a drop in 𝑤′ till 𝑡′ ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' From an apparent viewpoint, one may expect the initial CPF to propagate to the end of the pipeline by 𝑡′ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='252 and reflect to the inlet by 𝑡′ ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" Once the information of “no-resistance” reaches to the inlet, the inlet flow increases as a part of recurrent process during t' w' 50 100 150 200 250 300 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="04 no slip \uf074c=2/3 \uf074y \uf074c=0 (a) A Local Velocity at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1, \uf064 = 4E-4 t' w' 50 100 150 200 250 300 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="04 no slip \uf074c=2/3 \uf074y \uf074c=0 (b) Local Velocity at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75, \uf064 = 4E-4 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='012 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="012 t' us' 50 100 150 200 10 9 10 8 10 7 10 6 10 5 10 4 10 3 at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1 at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="75 (c) O(-5) - O(-6) Slip velocity, \uf064 = 4E-4 t us 0 3 6 9 10 9 10 8 10 7 10 6 10 5 10 4 t' w' 50 100 150 200 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="03 at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1 at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75 (d) Local Velocity at \uf064 = 4E-6, \uf074c=0 2 4 6 8 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='006 10 pressure wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The additional flow resistances due to wall-yield critical stress at 𝜏𝑐 = 2 3 ⁄ 𝜏𝑦 leads to greater initial drop in 𝑤′compared to 𝜏𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At very low compressibility (𝛿 = 4 × 10−6) with wall-slip, 𝑤′increases depending upon axial position during IPP (Figure 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One may also note that the rate of increase in 𝑤′in case of 𝛿 = 4 × 10−6 is almost same irrespective of the axial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Compressive deformation during IPP is negligible, and the flow is governed by the slip alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In such case, a linear pressure profile develops before shear deformation becomes significant to counter applied pressure (as shown in Figure 5 of the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This allows the gel to deform uniformly throughout the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' With increasing time, as the overall gel starts deforming, the magnitude of successive hikes in the 𝑤′ subsides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, once the entire upstream undergoes deformation, 𝑤′ at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 shows a hike, indicating accelerated flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This is consistent with the plots for the time evolution of the strain (in Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The time- evolution of strain in Figures 3(a, c) shows an instantaneous hike concomitant with the understanding from the local flow variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One may note from Figure 3c that the nominal (or actual) strain 𝛾′ for the no-slip scenario eventually becomes more than that for the wall-slip induced scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This suggests that while the wall-slip influences the initial strain, but once IPP is done with, the local strain due to overall deformation in a no-slip condition surpasses that of the wall-slip scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' To verify strain characteristics, we have checked for time-evolution of 𝛾′ at a nearby radially inward location (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One can clearly see that in all cases, strain away from the wall remains substantially lower than the nominal strain at 𝑟′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='975 for initial time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One expects that at the earliest instant when there is hardly any bulk deformation in the gel, including the locations near the wall at the inlet, the strain for no-slip condition should be more than the wall-slip induced scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This is precisely recovered at 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 at 𝑡′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1, when the pressure force induced at the inlet is limited to 20 kPa (Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In addition, it may be shown 11 that these counter-intuitive outcomes on wall-slip-effects reverses when the fluid is almost incompressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The non-periodic variations in 𝑤′in Figures 2(a, b) symbolize the slip-stick mechanism at 𝜏𝑐 = 2 3 ⁄ 𝜏𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This nonlinearity can be qualitatively explained by the wall-slip model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The present rheological model accurately predicts decreasing strain rate during the initial stage of flow startup (a decrease in strain rate at earlier times during flow startup is due to “shear localization” [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for the time evolution of strain 𝜸′ between (a) no-slip, conditional-slip and shear- thinning slip at 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏 and P = 40 kPa, (b) at a radial location 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗, 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏, and P = 40 kPa, (c) no-slip and shear-thinning slip at 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟕𝟓 and P = 40 kPa, and (d) no-slip and shear-thinning slip at 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏 and P = 20 kPa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' while the other gel conditions remaining the same as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" Furthermore, we extend these understandings for other structured materials (like shear- thinning-fluids, Kelvin-Voigt (KV) type viscoelastic solid) to establish a benchmark overview to the complex flow physics associated with the synergetic effect of wall-slip and compressibility t' \uf067' 5 10 15 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="5 2 (d) \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1, r' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="975, P = 20 kPa t' \uf067' 0 5 10 15 20 25 30 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="5 2 (b)\uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1, r' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9, P = 40 kPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="0004 t' \uf067' 4 8 12 16 20 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="6 2 (c) \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="75, r' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='975, P = 40 kPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="0004 t' \uf067' 0 4 8 12 16 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="5 2 no slip \uf074c=2/3 \uf074y \uf074c=0 (a) \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1, r' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='975, P = 40 kPa 12 during flow startup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In the process, we also exhibit robustness of our model in the presence of wall- slip through predicting continuous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For weakly compressible shear-thinning (ST) gel ( 𝛿 = 4 × 10−4) with no-slip, 𝑤′ (at 𝑟′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='975 and 𝛼𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1) is initially higher than that of the thixotropic elasto-viscoplastic fluids (TEVP) (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Unlike TEVP, the applied pressure force for ST does not require to overcome elastic forces at the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For TEVP, 𝑤′ increases drastically after gel degradation (when 𝛾′ > 𝛾𝑚𝑎𝑥′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The deformation in the present problem is directly associated with microstructural rearrangement of the gel’s network guided through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For ST, the deformation in the bulk segment of the gel (except at the outset) during IPP is high compared to TEVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Hence, at 𝑡′ > 21, 𝑤′for ST at the vicinity of the wall is affected by shearing actions between the adjacent shear layers along the bulk radial direction, which is evident from parabolic-type flow axial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Whereas in TEVP, the flow occurs in the form of a plug due to strong shear bands in the bulk [24, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' During IPP with or without wall-slip, 𝛾′ in the bulk portion of the gel remains less than the yield strain in TEVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Besides, the wall-slip causes additional flow due to extra compression at the CPF by un-attenuated pressure force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, the ST is only subjected to viscous shearing action, which causes continuous deformation in the bulk region of the gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Unlike TEVP, the redistribution of energy for gel degradation is not just localized close to the wall but is widespread for ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Consequently, the accumulation of shearing stress in the vicinity of the wall is lesser than that of the TEVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Hence, a higher magnitude of 𝑤′ occurs for TEVP in Figures 4(b, c) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' irrespective of the compressive resistances in the gel) during IPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for time-variant local axial velocity for shear-thinning and elasto-viscoplastic fluids at 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏, P = 40 kPa for the case of (a) no-slip scenario at 𝜹 = 𝟒 × 𝟏𝟎−𝟒, (b) 𝝉𝒄 = 𝟎 at 𝜹 = 𝟒 × 𝟏𝟎−𝟒, and (c) 𝝉𝒄= 0 at 𝜹 = 𝟒 × 𝟏𝟎−𝟔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For a weakly compressible KV-type material, the synergy between compressibility and wall-slip explains nature of pressure propagation and its continuous movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The pressure profiles in Figure 5a show that the compressive resistances in the flow delay IPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Despite the no- slip scenario, the CPF advances with initial inertial compression causing deformation near the wall at the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This actuates small movement in the axial direction without sustainable flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For wall- slip at 𝜏𝑐 = 0, one may note higher 𝑤′ compared to no-slip cases after the subsidence of the initial flow transients triggered from inertia-based compression at the outset (Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' After a certain time, 𝑤′ attains a velocity having contributions from the bulk flow overcoming compressive resistances at the upstream (in addition to slip-velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In KV-type material with no-slip, the pressure will eventually balance by the elastic force and flow stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, unlike the no-slip scenario, the wall-slip allows continuous movement of the KV-type material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In the slip-flow, a higher pressure drop in the upstream portion of the gel setup during IPP is observed (Figure 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For the wall-slip scenario, overcoming the reduced wall-stress requirement is enough to push the KV-type material to the outlet in a stable plug-like format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Accordingly, a scenario occurs at some 𝛿 (within 4 × 10−5 − 4 × 10−6) where the KV-type material may flow to the outlet due to wall- slip, and the flow does not occur in a no-slip scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" Thus, the wall-slip governs the flow of a t' w' 5 10 15 20 25 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="02 shear-thinning elasto-viscoplastic (a) No slip t' w' 5 10 15 20 25 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="02 (b) Slip at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1, \uf064 = 4E-4 t' w' 5 10 15 20 25 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="02 (c) Slip at \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1, \uf064 = 4E-6 14 viscoelastic-type solid material (like KV-type) at a later time during startup operation, opposite to what is seen for TEVP or shear-thinning fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (a) Effect of compressibility on flow startup during pressure propagation through a Kelvin-Voigt material at a later time 𝒕′ = 𝟐𝟎, 𝜶 = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏, and 𝝉𝒄 = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (b) Transients in local axial velocity at 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗𝟕𝟓, 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏, 𝜶 = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏, and 𝝉𝒄 = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for the time evolution of pressure propagation between no-slip and shear-thinning slip scenario (𝝉𝒄 = 𝟎) for Kelvin-Voigt material (at 𝜹 = 𝟒 × 𝟏𝟎−𝟔) (c) 𝒕′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟎𝟔𝟓 & 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟏𝟏, (d) 𝒕′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟏𝟐𝟓, 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟏𝟓 & 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟐, and (e) 𝒕′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟒, 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟒𝟐𝟓 & 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟎𝟓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Finally, a comparison of pressure propagation at various time instants is shown in Figures 6(a-d) for cases involving slip and no-slip at the FWI for weakly compressible TEVP fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At an earlier time 𝑡′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1, pressure builds up near the entrance due to inertial compression, as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The CPF diffuses further into the downstream at 𝑡′ = 1 (Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For the cases involving wall-slip, the viscous attenuation is less, and hence, the oscillations travel downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A steady decreasing pressure slope in upstream for a no-slip scenario indicates substantial bulk gel deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For wall-slip cases, this deformation is less, and the majority of the gel in the bulk region away from the wall remains intact at 𝑡′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One may note that at 𝑡′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="3, the pressure \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="8 1 \uf074c= 0, \uf064 =4E - 3 \uf074c= 0, \uf064 =4E - 6 (a) \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0065 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0065 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='011 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="011 (c) \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='025 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='025 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='035 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="035 (e) \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='04 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='04 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0425 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0425 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='05 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="05 (e) \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0125 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0125 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='015 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='015 no slip, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 \uf074c = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="02 (d) t' w' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="5 10 4 10 3 no slip, \uf064 =4E - 6, \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1 \uf074c= 0, \uf064 =4E - 6, \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 (b) 15 propagates downstream at a higher speed for the wall-slip cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In conclusion, the slip tends to dominate initial CPF movement during startup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for the time evolution of axial pressure profile at 𝝁𝒈𝟎 = 100 Pa s, 𝜶 = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟓, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 with the cases for no-slip, shear-thinning slip (τc = 0) and conditional slip scenario (τc = 2/3 τy) at 𝒕′ = (a) 1, (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3, (c) 5, and (d) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The pressure propagation mechanism at 𝛿 = 4 × 10−6 sees multiple reflections of pressure waves from the outlet with slow gel deformation for a wall-slip scenario (the mechanism is explained in detail in point 3 of the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For an energy efficient flow startup, pressure requirement estimations in the longer pipeline become intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' It is important to realize the importance of wall-slip in such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The present rheological model improves startup estimations for a longer pipeline (𝛼 = 4) and low gel degradation rate constant (k = 50) (shown in point 4 of the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Concluding Remarks: This analysis creates a benchmark for any flow showing synergy between compressibility and wall-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The study can be extended to various structural degradation kinetics involving the effects of structural buildup (often realized for wormlike micelles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" The analysis may \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="2 (c) 5 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="2 (c) 10 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 no slip \uf074c=0 \uf074c=2/3 \uf074y (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="2 no slip \uf074c=0 \uf074c=2/3 \uf074y (a) 1 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="3 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 (d) 75 16 be useful for demarcating the effect of shear banding in complex fluids where wall-slip inherently occurs [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' To date, the concept of critical stress calls for bigger clarity on what yields and what might not yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This letter hints at the importance for such clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' METHODS Experimental determination of parameters of the slip-model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (1)) Materials Examinations are carried out based on complex fluids like model oil with 5-7% wax concentration (TEVP fluid), toothpaste (yield-stress fluid) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5% Carbopol solution (Herschel-Bulkley fluid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The sample of model waxy oil is prepared by adding different macro-crystalline wax (Sasolwax 5054) concentrations ranging from 6 to 10 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='% in Dodecane solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The sample of model oil was heated 10-20 ℃ above WAT to assure complete solubility of the wax in Dodecane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The Carbopol solution is prepared by adding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='% of Carbopol powder-940 in distilled water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This mixture is then rotated at 1100-1300 rpm for 30 min to ensure a homogeneous gel formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In addition, commercially available toothpaste is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Experimentation A series of rheological experiments are performed with the Anton Paar MCR 301 rheometer to investigate the wall- slip for complex fluids with yielding behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The fluid sample is kept on the fixed bottom plate of the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Two parallel-plate geometries (smooth and rough types) with 50 mm diameter are used for all rheological experiments (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The surface roughness of the plates is measured using a Surface Profilometer: Alicona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The surface roughness of the smooth and rough plates is in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 𝜇𝑚 and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2-70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 𝜇𝑚, respectively A constant gap of 1 mm prevails between parallel plates during measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A Peltier plate controller from the bottom plate maintains the temperature of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For the case of waxy oil samples, the initial temperature is kept above WAT, and further, it is cooled to below the gelation temperature with a cooling rate of 1 ℃/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, in the case of the toothpaste and Carbopol solution, an isothermal temperature of 25 ℃ is maintained throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' After holding the sample for sufficient time, the yielding behavior of soft gelled fluid is investigated with the stress-ramp test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The stress ramp of 20 Pa/min for the case of toothpaste and waxy oil, and 6 Pa/min for Carbopol solution is applied during measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The shear stress corresponding to the sudden change in shear rate during the test is regarded as the yield 17 stress of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Additionally, the yielding behavior of the samples is investigated through the constant shear-rate method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A constant shear rate varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='001 s-1 to 10 s-1 is applied to the gelled sample till complete degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Finally, the shear resistance offered by the material against deformation is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The maximum shear stress in the constant shear-rate method is considered to be the yield stress of the material where it starts to flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (a) (b) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Parallel plate with overall and microscopic views for (a) smooth and (b) cross-hatched rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The stress-ramp experiment on rough surfaces shows initial elastic deformation followed by a sudden increase in shear rate i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' a stress plateau (Figures 8(a-d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This is the classical solid-liquid transition stress referred to as static yield stress 𝜏𝑦 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, at a later segment the sudden rise in shear stress can be attributed to the fragmentation of the 18 gel network [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For smooth surfaces, early yielding can be located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This is referred to as the critical shear stress for wall-slip 𝜏𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, the stress plateau for rough or smooth surfaces indicates the presence of shear banding in complex fluids [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Deducing data from Figures 8(a-d), we plotted 𝜏 versus the difference between the apparent shear rate 𝛾̇𝑎𝑝𝑝 (calculated for the cases of rough surfaces) and the nominal shear rate 𝛾̇ (calculated for the cases of smooth surfaces) to calculate slip-velocity 𝑢𝑠, and parameters B and m in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (1) of the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figures 8(a-d) suggest that m varies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 to 3, depending upon the type of fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For a model waxy oil, m varies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 with increasing wax concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The variable B depends on kinetic parameters, and for isothermal study it is a constant with an order varying from 10-5 m Pa-1s-1 at m =1 to 10-17 m Pa-3s-1 at m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The variation of m and B is consistent with the literature [7-9, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figures 8(a-d) indicates that 𝜏𝑐/𝜏𝑦 evolves in a manner that it satisfies the range of parameters assumed for the simulations in our letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Furthermore, the flow curves at shear-rate controlled experiments in Figures 9(a, b) can be qualitatively tallied with the results for stress-ramp experiments for the determination of 𝜏𝑦 and 𝜏𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Stress ramp experiments showing variations in stress 𝝉 with strain rate 𝜸̇ for different types of surfaces (rough and smooth) for (a) commercial toothpaste, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5% Carbopol solution, (c) 6% model waxy oil, and (d) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5% model waxy oil (a) (b) 250 50 200 40 2 Rough 100 Smooth T 20 50 10 Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='0 0 5 10 15 20 25 30 (in s-l) Y (in s\'l) (c) (d) 30 200 25 150 20 Pa) Pa) P 10 50 5 Tc 0 0 200 400 600 800 1000 0 200 400 600 800 1000 (in s"l) Y (in s"l)19 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Results for flow curve from shear-rate 𝜸̇ controlled experiments showing variation of stress 𝝉 with strain 𝜸 for different types of surfaces (rough and smooth) for (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5% Carbopol solution and (b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5% model waxy oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Governing equations, scaling and solution methodology The set of equations (1) to (5) are scaled to accommodate parameters like aspect ratio with less low simulation time while solving for series of non-linear partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The axial coordinate and radial coordinates are scaled based on the length L and radius R of the pipeline as follows: 𝑧′ = 𝑧 𝐿 , ⁄ 𝑟′ = 𝑟 𝑅 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The standardized velocity Ws used for scaling of axial and radial velocities is calculated based on the magnitude of static yield stress 𝜏𝑦 as 𝑊𝑠 = 𝑅𝜏𝑦 2𝜇0 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Therefore the axial and radial velocity components are written as 𝑢′ = 𝑢 𝜀𝑊𝑠 ⁄ and 𝑤′ = 𝑤 𝑊𝑠 ⁄ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The critical length Lc till which the flow always restart for a yield-stress fluid is defined as 𝐿𝑐 = 𝑃𝑅 2𝜏𝑦 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The dimensionless variable related to the aspect ratio of the pipeline 𝜀 = 𝑅 𝐿𝑐 ⁄ and a factor α which defined the ratio between the actual pipe length to the critical length, denoted as 𝛼 = 𝐿 𝐿𝑐 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The scaling of time 𝑡′ is done by resolving the smallest time scale phenomena, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', the compressive pressure wave propagation during initial stage of the flow restart [25, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The scaled pressure and time are represented as 𝑝′ = 𝑝 𝑃 ⁄ and 𝑡′ = 𝑡 (𝐿𝑐√𝛿 𝑊𝑠 ⁄ ) ⁄ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The viscosity is scaled based on the viscosity 𝜇′ = 𝜇 𝜇∞(𝑃 𝜏𝑦 ⁄ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Finally, dimensionless numbers like modified Reynolds number Re* and compressibility number δ are used in the present study which helps in rewriting the governing equations in non-trivial form which can be used for larger parametric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' These dimensionless numbers are 20 written as follows: 𝑅𝑒∗ = 𝜌0𝑅𝑊𝑠 𝜇∞(𝑃 𝜏𝑦 ⁄ ) , 𝛿 = 𝑃𝜅𝛩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Finally, the shear or elastic modulus G is scaled by a factor of P/2 and the dimensionless form of strain 𝛾′ remains the same as the dimensional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The governing equations are written in dimensionless forms as follows: Mass conservation equation: √𝛿 (𝜕𝑝′ 𝜕𝑡′ + √𝛿 [𝑢′ 𝜕𝑝′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝑝′ 𝜕𝑧′]) + 1 𝑟′ 𝜕(𝑟′𝑢′) 𝜕𝑟′ + 1 𝛼 𝜕𝑤′ 𝜕𝑧′ = 0 … … … … … (1𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Axial momentum conservation equation: 𝜕𝑤′ 𝜕𝑡′ + √𝛿 (𝑢′ 𝜕𝑤′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝑤′ 𝜕𝑧′) = − 1 𝛼𝜀𝑅𝑒∗ 𝜕𝑝′ 𝜕𝑧′ + 𝜀 2𝑅𝑒∗ (1 𝑟′ 𝜕(𝑟′𝜏𝑟𝑧′) 𝜕𝑟′ + 1 𝛼 𝜕(𝜏𝑧𝑧′) 𝜕𝑧′ ) … … … … … (2𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Radial momentum conservation equation: 𝜕𝑢′ 𝜕𝑡′ + √𝛿 (𝑢′ 𝜕𝑢′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝑢′ 𝜕𝑧′) = − 1 𝜀2𝑅𝑒∗ 𝜕𝑝′ 𝜕𝑟′ + 1 2𝜀𝑅𝑒∗ (1 𝑟′ 𝜕(𝑟′𝜏𝑟𝑟′) 𝜕𝑟′ + 1 𝛼 𝜕(𝜏𝑟𝑧′) 𝜕𝑧′ − 𝜏𝜃𝜃′ 𝑟′ ) … … … … (3𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Strain evolution equation for each component of strain tensor (𝛾𝑟𝑧′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝛾𝑧𝑧′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝛾𝑟𝑟′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝛾𝜃𝜃′): 1 √𝛿 𝜕𝛾𝑟𝑧′ 𝜕𝑡′ + (𝑢′ 𝜕𝛾𝑟𝑧′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝛾𝑟𝑧′ 𝜕𝑧′ ) − (𝛾𝑟𝑟′ 𝜀 𝜕𝑤′ 𝜕𝑟′ + 𝛾𝑟𝑧′ 𝛼 𝜕𝑤′ 𝜕𝑧′ + 𝛾𝑟𝑧′ 𝜕𝑢′ 𝜕𝑟′ + 𝜀𝛾𝑧𝑧′ 𝛼 𝜕𝑢′ 𝜕𝑧′) = 𝛾̇𝑟𝑧′ … … … … (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 1 √𝛿 𝜕𝛾𝑧𝑧′ 𝜕𝑡′ + (𝑢′ 𝜕𝛾𝑧𝑧′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝛾𝑧𝑧′ 𝜕𝑧′ ) − 2 (𝛾𝑟𝑧′ 𝜀 𝜕𝑤′ 𝜕𝑟′ + 𝛾𝑧𝑧′ 𝛼 𝜕𝑤′ 𝜕𝑧′) = 𝛾̇𝑧𝑧′ … … … … … (4𝑆),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 1 √𝛿 𝜕𝛾𝑟𝑟′ 𝜕𝑡′ + (𝑢′ 𝜕𝛾𝑟𝑟′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝛾𝑟𝑟′ 𝜕𝑧′ ) − 2 (𝛾𝑟𝑟′ 𝜕𝑢′ 𝜕𝑟′ + 𝜀𝛾𝑟𝑧′ 𝛼 𝜕𝑢′ 𝜕𝑧′) = 𝛾̇𝑟𝑟′ … … … … … (5𝑆),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 1 √𝛿 𝜕𝛾𝜃𝜃′ 𝜕𝑡′ + (𝑢′ 𝜕𝛾𝜃𝜃′ 𝜕𝑟′ + 𝑤′ 𝛼 𝜕𝛾𝜃𝜃′ 𝜕𝑧′ ) − 2 (𝛾𝜃𝜃 𝑢′ 𝑟′) = 𝛾̇𝜃𝜃′ … … … … (6𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' where 𝛾̇𝑟𝑧′, 𝛾̇𝑧𝑧′, 𝛾̇𝑟𝑟′, 𝛾̇𝜃𝜃′ are the components of strain rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Furthermore, the equation for dimensionless extra stress tensor is written as: 𝝉̈′ = 2𝜇′𝒅̈ ′ − 2 3 𝜇′ (𝛁′ ∙ 𝑼′)𝑰̈ + 𝐺′𝜸̈ ′ … … … … … (7𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 21 Boundary Conditions In the present problem, we consider gel degradation only after a pressured fluid is inserted at the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The boundary conditions based on Dirichlet’s and Neumann’s convention are imposed for dimensionless pressure 𝑝′, strain, radial velocity 𝑢′, axial velocity 𝑤′ and extra shear stress 𝜏′ as follows: At inlet: 𝑝′ = 1, 𝑢′ = 0, 𝜏𝑧𝑧′ = 0 and the pressurized fluid at the inlet has a strain 𝛾′ = 𝛾𝑖𝑛′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At outlet: 𝑝′ = 0, 𝑢′ = 0, 𝜏𝑧𝑧′ = 0 and constant flux condition is applied for strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝜕𝛾′ 𝜕𝑧′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Symmetric conditions prevail at the axis of the pipeline with no flow across the axis causing 𝑢′ = 0, 𝜏𝑟𝑧′ = 0 and similar to the outlet boundary, Neumann’s condition is applied for pressure and strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝜕𝑝′ 𝜕𝑟′ = 𝜕𝛾′ 𝜕𝑟′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At the upper wall, slip based boundary conditions are set: 𝑤′ = 𝑢𝑠′, 𝑢′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The wall-slip phenomenon in the present problem is isotropic in nature due to smooth walls [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Schematic diagram for flow representation Solution methodology In the present problem, we considered a uniform and orthogonal staggered grid arrangement to represent the numerical domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The governing equations are subjected to boundary conditions to solve for the primary variables (𝑢′, 𝑤′, 𝑝′) and components of strain (𝛾𝑟𝑧′, 𝛾𝑧𝑧′, 𝛾𝑟𝑟′, 𝛾𝜃𝜃′) using finite volume methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Central difference scheme is applied for spatial discretization of velocity, pressure, stress, viscosity and strain-based components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The transient formulations are done using second-order implicit method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" The staggered grid arrangement comprises of flux-related Equilibrium Conditionsatwall:w'=ug,u'=o Slip Region Fully Developed Shear-thinned Plug Region Flow Layers inlet Outlet p'=1,u'= 0, p'= 0, u' = 0,t'zz = 0 ar' Axisymmetric Boundary." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" u'= 0, trz = 0 ap'_ e22 variables like velocity at the face centers of the cell;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' whereas, the properties like pressure is calculated at the volumetric center of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The partial difference equations, thus formed after discretization, are solved using point-by-point iterative technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The convergence criteria for velocity and pressure based variables are maintained at a dimensionless value of 10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A higher value for convergence criteria can lead to the divergence of the solution and a lower value of convergence criteria incurs high simulation time in addition to possibility of round-off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' It is to be noted that the degradation rate constant k in the present study is varied from 10 to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, unless otherwise specified, the simulations are carried out primarily at 𝛿 = 4 × 10−4, k = 100 and 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Data availability The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Code availability The codes of the computer simulations are available from the corresponding author upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Lettinga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Manneville, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Competition between Shear Banding and Wall Slip in 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+page_content=', Fardin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Manneville, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Lerouge, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Shear Banding of Complex Fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rev.' metadata={'source': 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yield-stress fluid in a vertical pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Nonnewton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 257, 50–58 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Marinho, T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Apparent wall slip effects on rheometric measurements of waxy gels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 65, 257–272 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Black, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall-Slip and Polymer-Melt Flow Instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 77, 956-959 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Boukany, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Shear banding or not in entangled DNA solutions depending on the level of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 53, 73–83 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Seth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Locatelli-Champagne, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Monti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Bonnecaze, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Cloitre, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' How do soft particle glasses yield and flow near solid surfaces?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Soft Matter 8, 140–148 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall slip and the nonlinear dynamics of large amplitude oscillatory shear 23 flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 39, 697–712 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Bonn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Denn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Berthier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Divoux, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Manneville, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Yield stress materials in soft condensed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 89, 1–40 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Spikes, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Granick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Equation for slip of simple liquids at smooth solid surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Langmuir 19, 5065–5071 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Walls, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Caines, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Sanchez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Khan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Yield stress and wall slip phenomena in colloidal silica gels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 47, 847–868 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Bertola, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Bertrand, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Tabuteau, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Bonn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Coussot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall slip and yielding in pasty materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 47, 1211–1226 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Granick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Limits of the Hydrodynamic No-Slip Boundary Condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 88, 4 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Brochará, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & de Gennes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Shear-Dependent Slippage at a Polymer/Solid Interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Langmuir 8, 3033–3037 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Gennes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' De.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' On Fluid/Wall Slippage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Langmuir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 18, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3413–3414 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Mooney, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Explicit 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210–222 (1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Hatzikiriakos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall slip of molten polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Polym.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Nonnewton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 56, 221–251 (1995).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Negrão, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Franco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Pressure transmission in Bingham fluids compressed within a closed pipe.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Thermally assisted restart of gelled pipelines: A weakly compressible numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 118, 27–39 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 30.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Nonnewton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 294, 104582 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Damianou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Georgiou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Moulitsas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Combined effects of compressibility and slip in flows of a Herschel-Bulkley fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Nonnewton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 193, 89–102 (2013).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Cloitre, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Slip and flow in soft particle pastes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 92, 1–4 (2004).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall slip of molten high density polyethylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Sliding plate rheometer studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 35, 497–523 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Tikariha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Kumar, L.' metadata={'source': 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the rheological behavior of waxy crude oils as a function of flow and temperature history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 59, 703–732 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 25 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Mewis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Wagner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Thixotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Colloid Interface Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 147–148, 214–227 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' de Souza Mendes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Thompson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Acta 52, 673–694 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Mujumdar, A.' metadata={'source': 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Wave Phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (Wiley & Sons, New York, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Chang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Boger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Nguyen, Q.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 211, 115212 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Vinay, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Wachs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Frigaard, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Start-up transients and efficient computation of isothermal waxy crude oil flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Nonnewton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 143, 141–156 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Asmolov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Schmieschek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Harting, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Vinogradova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Flow past superhydrophobic surfaces with cosine variation in local slip length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' E - Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Nonlinear, Soft Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 87, 1–8 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 26 Supplementary to the article “Synergetic Effect of Wall-Slip and Compressibility During Startup Flow of Complex Fluids” by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Sanyal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Shinde, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Kumar 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Grid independence and model verification The benchmark for grid arrangement for the present problem is obtained from one of our previous studies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In the present study, we compared several grid arrangements with number of grid points varying from 100×10 to 400×100 along axial (Nz) × radial (Nr) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The solutions at 200×20 is seen to be numerically adequate with the finest grid arrangement of 400×100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In the present problem, the wall-slip velocity near the wall is compared for numerical adequacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Furthermore, the time-step of 10-4 following CFL criteria is taken as a benchmark for time-step independence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' After comparisons of results for wall-slip velocity at two time instants (𝑡′) for several values of time-step, ∆𝑡′ =10-5 is finalized as numerically adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The formulations for strain evolution based on upper-convection-derivative terms poses some intriguing issues based on the applicability of such formulation for a problem like the present one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Tikariha & Kumar [2] have shown that the upper-convection derivative based strain evolution methodology when compared to the ones adopted by [3] shows dissimilar results after the initial pressure propagation front has reached the outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, in the present study, a similar verification is carried out at a longer pipe length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One can see that the results for pressure propagation at various instants of time remains same, irrespective of the type of formulation (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The variations in results are marginal (< 1% relative deviation) in the regime of low compressibility number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Furthermore, the velocity profile (Figure 2) at a 𝛼𝑧′ = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 for a combined effect of wall-slip and elasto-viscoplastic rheology shows variations from plug-like profile during gel degradation (at 𝑡′ < 100 in Figures 2b and 2c) to a parabolic profile at a steady- state indicating Newtonian characteristics (𝑡′ = 400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This is qualitatively consistent with the 27 numerical results for the velocity profiles of Damianou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' In addition, the velocity magnitude at steady-state condition quantitatively complies with the analytical value obtained from Hagen–Poiseuille equation (𝑤𝑚𝑎𝑥 = ∆𝑃 4𝜇𝐿 𝑅2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Finally, the code is subjected to the verification of the idea inspired from the experiments of El-Gendy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' [5] which shows that the flow need not necessarily restart when the initial pressure propagation front reaches the outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One such scenario is shown in Figure 3 which occurs at 𝛼 = 2, 𝑘 = 50 and 𝜏𝑐 = 2 3 𝜏𝑦 for an elasto- viscoplastic based rheology involving slip at 𝛿 = 4 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Verification of pressure profiles from upper convection derivatives based strain evolution with strain evolution equation having only material derivative based strain evolution terms as used by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (2014) for different values of compressibility number (a) 𝜹 = 𝟒 × 𝟏𝟎−𝟒, and (b) 𝜹 = 𝟒 × 𝟏𝟎−𝟔 (at initial gel viscosity of 100 Pa s, P = 40 kPa and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="8 1 (b) \uf064 = 4E-6 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 4 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 4 25 (a) \uf064 = 4E-4 28 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Time evolution of axial velocity variations along radial direction at different fixed axial locations (for initial gel viscosity of 100 Pa s, P = 40 kPa, 𝜹 = 𝟒 × 𝟏𝟎−𝟒, and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5) at 𝝉𝒄=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (a) 𝜶𝒛′ = 𝟎, (b) 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟕𝟓 and (c) 𝜶𝒛′ = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' (a) Pressure propagation, and (b) time evolution of inlet and outlet flowrates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' at initial gel viscosity of 100 Pa s, 𝜶 = 𝟐, P = 40 kPa, k = 50, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 and τc = 2/3 τy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" Time-dependent inlet and outlet flowrate variation at 𝜹 = 𝟒 × 𝟏𝟎−𝟒 r' w' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 400 100 40 10 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="8 (a) \uf061z' = 0 r' w' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 400 100 40 10 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="8 (b) \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="75 r' w' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 400 100 40 10 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="8 (c) \uf061z' = 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="5 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="2 1 3 15 (a) t' Flowrate 20 40 60 80 100 120 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='015 inlet outlet (b) 29 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for (a) inlet and (b) outlet flowrates with time at initial gel viscosity of 100 Pa s, 𝜶 = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟓, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟒 with the cases for no-slip, shear-thinning slip (τc = 0) and conditional slip scenario (τc = 2/3 τy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Pressure propagation at very low gel compressibility at 𝜹 = 𝟒 × 𝟏𝟎−𝟔 Figures 5(a-c) show pressure propagation from inlet to the outlet with marginal pressure drop along the axial direction in the downstream for a scenario involving wall-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' For a wall-slip scenario, the overall gel deformation (combining net deformation at the bulk portion of the gel and the fluid- wall interface) during initial pressure propagation stage is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This can be understood through negligible pressure gradient along the downstream and a sudden fall in pressure to zero at the outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Following a decay of transients, the no-slip scenario shows a linear profile at 𝑡′ = 40 (Figure 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" t' inlet flowrate 50 100 150 200 250 300 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 no slip \uf074c= 0 \uf074c=2/3 \uf074y (a) 10 20 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="04 t' outlet flowrate 50 100 150 200 250 300 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 (b) 10 20 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='04 30 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for the time evolution of axial pressure profile at initial gel viscosity of 100 Pa s, 𝜶 = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟓, P = 40 kPa, k = 100, 𝜹 = 𝟒 × 𝟏𝟎−𝟔 with the cases for no-slip, shear-thinning slip (τc = 0) and conditional slip scenario (τc = 2/3 τy) at 𝒕′ = (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1, (b) 1, (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3, (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 (e) 5, (f) 10, (g) 40, (h) 75 and (i) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The reflected pressure wave causes a decrease in flow velocity compared to the flow induced from the approaching pressure front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' It is to be noted that the reflected flow traverses back easily due to the slippage region without any significant viscosity attenuation at 𝜏𝑐 = 0 (Figure 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At earlier time instants like 𝑡′ = 5 (Figure 5e) for the cases at 𝜏𝑐 = 0, the elongated trailing part in pressure propagation signal (𝛼𝑧′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='32) denotes region of almost no gel deformation, and pressure signal moves back and forth via slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At 𝛼𝑧′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3, the inertial compressive pressure wave moves forward with an amplitude lower than the ones in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' This indicates the gel movement with true velocity is causing viscous attenuation of pressure signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' A sudden drop in pressure signal at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 < 𝛼𝑧′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="32 specifies the region where the net resistive force from the gel’s \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='02 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="2 (g) 40 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 31 elastic strength is counterbalanced by the force associated with the approaching pressure wave (this region is analogous to a region of pressure front).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, for a subsequent span of time during the flow restart operation, the pressure wave continues to traverse through the slip region causing slower gel deformation and multiple reflection of pressure waves from the outlet to the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The enhanced slope in the trailing part of the pressure profiles at subsequent time instants (𝑡′ = 10 (Figure 5f) and 40 (Figure 5g)) indicates gel degradation through viscous shearing forces from the interference between the backwardly reflected and forward approaching flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Wall-slip effects for pressure estimation during flow restart in longer pipelines and in scenarios for low gel degradation constants In this case, we consider the overall aspect ratio (1/ε) of the pipeline as 400, which has a length equivalent to 4 times the critical length of the pipeline (Lc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Figure 6 shows that the initial pressure wave front reaches the outlet at 𝑡′ = 10 at 𝜏𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Whereas, the pressure does not propagate beyond 𝛼𝑧′ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 at 𝜏𝑐 = 2 3 ⁄ 𝜏𝑦 and 𝛼𝑧′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='3 at no-slip scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="2 t' = 1, no slip t' = 1, \uf074c = 2/3 \uf074y t' = 1, \uf074c = 0 t' = 10, \uf074c = 2/3 \uf074y t' = 10, no slip t' = 10, \uf074c = 0 t' = 50, \uf074c = 2/3 \uf074y t' = 50, no slip t' = 50, \uf074c = 0 t' = 200, \uf074c = 2/3 \uf074y t' = 200, no slip t' = 200, \uf074c = 0 32 Figure 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for time evolution of axial pressure propagation in longer pipeline for the cases of no- slip, 𝝉𝒄 = 𝟎, and 𝝉𝒄 = 𝟐 𝟑 ⁄ 𝝉𝒚 at the gel-wall interface for an initial gel viscosity 100 Pa s, 𝜶 =4, P = 40 kPa and 𝜹 = 𝟒 × 𝟏𝟎−𝟒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At low gel degradation rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' k = 50, the maximum yield strain required for beginning of the gel network’s disengagement increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='1 in the earlier-mentioned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Unlike the case of no-slip (Figure 7a), the wall-slip allows pressure signals to propagate to the outlet for a pipeline having an overall aspect ratio (1/ε) of 200 or α = 2 (Figures 11(b, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' At 𝜏𝑐 = 2 3 ⁄ 𝜏𝑦, the initial pressure wave diffuses by compressing the gel network with minimum flow resistance near the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' But, this does not guarantee a successful flow restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' One may see only inertial puncture at the outlet at 𝑡′ = 3 in comparison to the fully-developed flow at the outlet at 𝑡′ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The non- linear pressure signals at late time instants like 𝑡′ = 500 in Figure 7b suggests that the compressive resistances and viscous resistances will not be overcome in the long run, thereby leading to a halt in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' However, in Figure 7c, one may note the development of a linear profile at 𝑡′ = 50, indicating the occurrence of complete gel compression and a tendency for steady-state flow at a later time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="4 2 3 10 50 500 (c) \uf074c = 0 \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="4 10 50 400 (a) no-slip \uf061z' p' 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='4 2 3 10 30 500 (b) \uf074c = 2/3 \uf074y 33 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for time evolution of axial pressure propagation at low gel degradation scenario k = 50, for the cases of (a) no-slip, (b) 𝝉𝒄 = 𝟐 𝟑 ⁄ 𝝉𝒚, and (c) 𝝉𝒄 = 𝟎 at the gel-wall interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' at initial gel viscosity 100 Pa s, 𝜶 =2, P = 40 kPa and 𝜹 = 𝟒 × 𝟏𝟎−𝟒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Local shear stress variations Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Comparison for the time evolution of true local stress 𝝉𝒆 between no-slip, conditional slip (τc = 𝟐 𝟑 𝝉𝒚) and shear-thinning slip (τc =0) at 𝒓′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟗𝟕𝟓 at the axial location (a) 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟏 and (b) 𝜶𝒛′ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 𝟕𝟓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' while the other gel conditions remaining the same as the ones mentioned in Figure 1 in the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Sanyal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Tikariha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Kumar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The effects of partial preheating on pressure propagation and Flow-Restart phenomena in a clogged pipeline with a weakly compressible gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluids 33, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Tikariha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Kumar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Pressure propagation and flow restart in the multi-plug gelled pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 911, 1–26 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Kumar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=', Lawrence, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' & Sjöblom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' Mechanism of pressure propagation and weakly compressible homogeneous and heterogeneous thixotropic gel breakage to study flow restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' RSC Adv.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' The propagation of pressure in a gelled waxy oil pipeline as studied by particle imaging velocimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' AIChE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=' 58, 302-312 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content=" t' \uf074e 0 5 10 15 20 0 50 100 150 200 no slip \uf074c=2/3 \uf074y \uf074c=0 (a) \uf061z' = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content="1, \uf064 = 4E-4 t' \uf074e 0 5 10 15 20 0 50 100 150 200 (b) \uf061z = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} +page_content='75, \uf064 = 4E-6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE2T4oBgHgl3EQfcwcu/content/2301.03898v1.pdf'} diff --git a/itE4T4oBgHgl3EQfsA2l/content/tmp_files/2301.05213v1.pdf.txt b/itE4T4oBgHgl3EQfsA2l/content/tmp_files/2301.05213v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..504539f08e002964f1890f0444a597ba9be5baca --- /dev/null +++ b/itE4T4oBgHgl3EQfsA2l/content/tmp_files/2301.05213v1.pdf.txt @@ -0,0 +1,1139 @@ +Learning to Summarize Videos by Contrasting Clips +Ivan Sosnovik +UvA-Bosch Delta Lab +University of Amsterdam +i.sosnovik@uva.nl +Artem Moskalev +UvA-Bosch Delta Lab +University of Amsterdam +a.moskalev@uva.nl +Cees Kaandorp +Institute of Informatics +University of Amsterdam +cees.kaandorp@gmail.com +Arnold Smeuldes +UvA-Bosch Delta Lab +University of Amsterdam +a.w.m.smeulders@uva.nl +Abstract +Video summarization aims at choosing parts of a video +that narrate a story as close as possible to the original one. +Most of the existing video summarization approaches focus +on hand-crafted labels. se As the number of videos grows +exponentially, there emerges an increasing need for meth- +ods that can learn meaningful summarizations without la- +beled annotations. In this paper, we aim to maximally ex- +ploit unsupervised video summarization while concentrat- +ing the supervision to a few, personalized labels as an add- +on. To do so, we formulate the key requirements for the +informative video summarization. Then, we propose con- +trastive learning as the answer to both questions. To fur- +ther boost Contrastive video Summarization (CSUM), we +propose to contrast top-k features instead of a mean video +feature as employed by the existing method, which we im- +plement with a differentiable top-k feature selector. Our ex- +periments on several benchmarks demonstrate, that our ap- +proach allows for meaningful and diverse summaries when +no labeled data is provided. +1. Introduction +Video summarization aims at choosing parts of a video +that narrate a story as close as possible to the original one. +In this day and age, video streaming without personalized +recommendations is almost gone. +Current recommenda- +tions select a fixed preview provided by the distributor on +the basis of past preferences. We aim to go one step further +and to provide personalized previews. Apart from better +video selection for streaming, it also opens possibilities for +better video editing, ad creation, and edge-device software +development. +Existing approaches for video summarization focus on +Figure 1. The t-SNE of the feature space of example clips and the +learned summaries (black circles) for these clips. Finding a good +summary in the feature space does not boil down to the centroids +of corresponding feature distributions, but rather consists of find- +ing samples that informatively describe the whole input sequence. +supervised summarization [14, 24, 35, 45, 46, 48]. With a +growing number of videos, supervised summarization may +still be somewhat affordable for the distributor. +When +the number of videos grows exponentially, the supervised +model is not sustainable. And, from the standpoint of the +user labeling can be applied only in very moderate amounts. +In this paper, we aim to maximally exploit unsupervised +video summarization while concentrating the supervision +on a few personalized labels as an add-on. +Unsupervised video summarization techniques were de- +veloped in the pre-deep-learning era, when no large labeled +datasets were available [15]. They are still being used these +days as labeling the full spectrum of possible videos is no +longer possible [26]. Regardless of the method of video +analysis, with deep learning or not, the main reasoning for +1 +arXiv:2301.05213v1 [cs.CV] 12 Jan 2023 + +Figure 2. The main building blocks of our framework for video summarization. Feature extractor 𝑓 transforms an arbitrary-length sequence +of frames into a sequence of features. They are then used to predict a set of scores by using a function 𝑔. The summary extractor block +uses both sequences to extract a set of 𝑘 frames with the highest scores. +selecting a good summary is left unchanged. The summary +must be a compressed representation of the original video +while being closer in content to the source than to other +videos [12, 30]. Two core questions remain: how to sum- +marize a video while preserving most information in the +video, and how to measure distances between two videos +on the basis of their content? Formulating the video sum- +marization in this way, in this paper, we propose contrastive +learning [4] as the answer to both questions. Contrastive +learning was designed to handle compression and metric +learning in one go. We propose that neural networks can +be trained to optimize a contrastive loss which represents +the core requirements of a good summary. +Contrastive learning has been successfully applied in im- +age and video classification [4, 8, 36]. In classification, the +contrastive loss is evaluated by comparing descriptive fea- +ture vectors of equal size [22]. In video summarization, the +comparison is between a vector describing the full video +and a vector describing the summary. As a consequence, the +vectors will not be equal in size, and hence cannot be com- +pared directly by common contrastive losses. To overcome +the inequality in size, the common approach for comparing +vector representations of videos and their summaries is to +use their time-averaged representations [2]. In this work, +we note that such an approach is invariant to a wide range +of transformations and does not account for important mo- +ments of high information in the video. When taking an +average, the summary is adequate when the video develops +slowly like a game of snooker or a sit-com interview but +expected to be less adequate when there are short moments +of great significance. While various architectural solutions +were proposed to improve over quality by average [35], we +propose that a combination of a well-chosen loss function +and training approach suffices to avoid the unwanted invari- +ances for a wide range of backbones. +In this paper, we focus on developing a simple, flexible, +yet efficient recipe for contrastive training of deep-learning- +based video summarizers. We start from the principle of +maximum information preservation. We demonstrate that +the ensuing loss function and maximization process pre- +serves important information while avoiding undesired in- +variances. Our main contributions are the following: +• From the requirements of video summaries we propose +a method for contrastive learning of video summaries +with no need for labeling. +• We propose implementations of the main building +blocks which are required to convert any video- +analysis network into a summarizer. +• We demonstrate the advantage of contrastive video +summarizers on popular benchmarks for a set of back- +bone architectures over their original training methods. +We also demonstrate how it can be used for video high- +light selection with a slight modification. +2. Related Work +Supervised Methods +With the rise of deep learning, a +wide range of papers has considered video summarization +as a regression problem. In such a paradigm a neural net- +work is used to take frames of the video and predict their +importance scores so that the top-scored frames form the +summary. In [45] the authors use Recurrent Neural Net- +works (RNNs) to combine the temporal information from +the video with the content of each frame to successfully +predict frames’ scores. Alternatively, in [35] and [10] con- +volutional and attention-based architecture was proposed to +improve the quality of predictions. To effectively combine +information about videos from multiple scales, hierarchi- +cal models were proposed [46, 47]. +By using hierarchi- +cal RNNs, models benefit from considering the video as a +whole, as a set of short clips and as a sequence of individ- +ual frames at the same time. It allows to create summaries +2 + +Feature Extractor +Score Predictor +Summary Extractor +Video +Frame features +Scores +Summary +top-k +gFigure 3. An illustration of the proposed contrastive training pipeline. Given two videos, the features are first calculated for them and then +a summary is extracted. Both summaries and original videos are projected by a neural network ℎ to a hidden space afterward. The whole +pipeline is trained to attract the projections of summaries to the projections of the original videos and to repel them from other summaries +and videos. +of less-contract granularity than before. While these meth- +ods demonstrated the great success of deep neural networks +for video summarization, manually labeled annotations are +required for their training. It makes it impossible the scal- +ing of such methods to long videos, movies and streams +of videos that are constantly being uploaded on the major +video services. For these reasons, we focus on methods that +do not rely on human-annotated labels. +Unsupervised Methods +Early-day methods for video +summarization relied on heuristics designed by a human. +The heuristics were designed to satisfy the main require- +ments for video summaries such as representativeness and +diversity, justified in [9,31]. In [9,23,30] the authors clus- +tered frames and use the centroids to form a summary. The +authors of [6,28] formulate video summarization as a sparse +dictionary selection problem. Later, in the deep-learning +era, video summarization was approached from the perspec- +tive of adversarial training [20, 27] or in the reinforcement +learning paradigm [48]. We draw inspiration from the pre- +deep-learning era methods. By starting from the reasoning +of video summarization, we demonstrate that we can sat- +isfy the main requirements by formulating it as a contrastive +learning problem that we can easily solve. +Contrastive Learning +Contrastive learning is an ap- +proach for performing self-supervised pretraining of a +model by using a pre-text task. The model learns to at- +tract representations that are meant to be close, and are +thus called positive, and repel them from negative repre- +sentations which are meant to be distant enough to distin- +guish between different objects [4,19,29,40]. Various meth- +ods have been proposed for learning image-level [3, 5] and +spatio-temporal models [2,11,16]. The current application +of contrastive learning methods for video summarization is +rather limited due to special architectural solutions dictated +by the domain. In this work, we demonstrate an approach +for contrastive learning for video summarization that does +not rely on any specific backbones and allows one to use +any model and framework of their choice. +Video Highlight Selection +Another popular approach for +creating a compressed visual representation of videos is +video highlight selection. While the summary has a fixed +length, the highlights are not bounded in length but have +a lower bound for the importance scores. Various meth- +ods have been proposed for solving this problem both from +the supervised perspective [1, 21, 25, 39, 42–44], as well as +in the unsupervised manner [2]. In this paper, we demon- +strate that with a slight modification of our video summa- +rization framework, we can outperform modern video high- +light selection models without significant transformations +of the original pipeline. +3. Method +3.1. Summary Requirements +Video summarization is a very subjective task, as a man- +ually labeled summary is biased towards the personal pref- +erences of the annotator, assessor [38]. However, it is pos- +sible to select several properties of a good summary that we +would consider as summary requirements. They also give +us hint on how to build an efficient model for video summa- +rization. +Representativeness +The composed summary should de- +liver the same message as the original video. As the sum- +mary is a compressed representation of the source, the loss +of the original information is inevitable. However, we re- +quire a good summary to contain all the information neces- +sary to distinguish between the original video and all other +videos [27]. With no loss of generality, we can assume that +each video contains a finite set of sub-videos each of which +tells a separate narrative. Thus, the desired summary is a +3 + +Projection +Repel +h +Summary +Attract +hcombination of sub-videos that is as close as possible to all +of them at the same time, as well as distant enough from +all sub-videos of other original videos. We suggest to learn +summaries by selecting a set of sub-videos which we call +clips which once they are projected to some hidden space +minimize a variant of the triplet loss. As we want to de- +velop a model for unsupervised summarization, it leads us +to the framework of contrastive learning [4]. +Sparsity +The original videos may come from various +sources: be it video news, a video blog, several-hours-long +online streams or a TV show. For all cases, the desired +summary would be just several seconds long as it is the +average amount of time a user can spend before deciding +whether to watch it or to skip it. It leads us to the require- +ment of the sparsity of the resulting summary. While for +short videos the desired summary may be around 15% of +the length [18], this ratio may drop significantly for longer +videos. Thus, our model should be capable of choosing the +very top segments of the video with a significant distinction +from the rest. The problem of ranking items and selecting +the top of them cannot be overestimated as it has significant +limitations in the realm of deep learning especially when it +comes to a very sparse output [17,33,41]. In our approach, +we should not directly rely on any heuristics for performing +such an operation and should seek an as accurate as possible +algorithmic implementation of it. +Diversity +Another important property of a video sum- +mary is the diversity among its frames. We may assume that +for some videos and for some datasets it is possible to create +a summary that will contain a lot of very similar frames and +clips. Although it is possible for a user to understand what +the video is about just by taking a look at one frame, it is +still desired to have a summary with a higher diversity of vi- +sual information. If we consider two models which satisfy +the above-mentioned requirements, we want to select the +model which selects diverse summaries over uniform sum- +maries. It shows us that the function which we will use to +measure the distance between videos should not be invari- +ant to the spread of the frames. In other words, it should +take into account not only a single frame and the general +content of videos but also the variations inside them. +3.2. Contrastive Summarization +A wide range of trainable video summarizers can be de- +composed into the following three blocks: features extrac- +tor 𝑓 , score predictor 𝑔 and summary extractor (see Figure +2). From the summary requirement, we generated several +requirements for the video summarization pipeline. And +none of them are related to the feature extractor or the score +predictor. Thus, we assume that these two blocks are the +free parameters of our framework. Once a feature extractor +Figure 4. Illustration of frame selection based on the scores 𝑠1 +and 𝑠2. Left: the original step function 𝑥max = 𝑥1 if 𝑠1 > 𝑠2 +and 𝑥max = 𝑥2 otherwise. Right: a relaxed version with a smooth +replacement for the step function. +and a score predictor are chosen, we train their parameters +by performing a variant of contrastive learning. +During training, we consider two videos (see Figure 3). +Each of the videos is processed with the feature extractor +and the score predictor functions. After that, for each of +the videos a summary is generated. We choose the param- +eters of the networks 𝑓 and 𝑔 by training them to generate +video and summary embeddings s.t. the summary attracted +to its source video is repelled from any other videos and +summaries. It is done by minimizing the following loss [4]: +L = +∑︁ +𝑧,𝑧+ +− log +exp(dist(𝑧, 𝑧+)/𝜏) +� +𝑧− exp(dist(𝑧, 𝑧−)/𝜏) +(1) +where 𝑧, 𝑧−, 𝑧+ are embeddings for the anchor video, its neg- +ative and positive pairs. We calculate this loss by iterating +over all possible sets of such videos and their summaries. +The parameter 𝜏 is the smoothing factor of the loss func- +tions. +It is a hyperparameter of our approach. +In order +to perform the training of such a pipeline successfully, we +need to define the distance function dist(·, ·) +3.3. Clip-Contrastive Distance Function +Let us consider two sequences of 𝐷-dimensional vectors +represented as matrices: +X = {𝑋𝑖 𝑗}𝐷,𝑡 +𝑖, 𝑗=1 +Y = {𝑌𝑖 𝑗}𝐷,𝑇 +𝑖, 𝑗=1 +(2) +where 𝑡 and 𝑇 and the lengths of the sequences. In our +case, these matrices are clip features for the summary and +the original video and thus we assume 𝑡 < 𝑇. +A com- +mon approach for calculating the distance between two se- +quences [2] is to compare their features averaged it time: +¯x = � +𝑗 𝑋𝑖 𝑗/𝑡′ and ¯y = � +𝑘 𝑌𝑖𝑘/𝑇. It can be done for exam- +ple by calculating the scalar product of these vectors: +dist(X, Y ) = +� ∑︁ +𝑗 +𝑋𝑖 𝑗/𝑡′, +∑︁ +𝑘 +𝑌𝑖𝑘/𝑇 +� += +1 +𝑇 · 𝑡 +∑︁ +𝑖 𝑗𝑘 +𝑋𝑖 𝑗𝑌𝑖𝑘 +(3) +4 + +max +max +S1 = S2 +SFigure 5. Left: a sample video as a set of short clips forms a circle in some hidden space. Right: Three sets of summaries that yield +the same distance function when they are time-averaged. However, these summaries have different distances from the whole video if the +distance function is given as in Equation 4 which leads to a uniform distribution of clips. +The main drawback of such a distance function is that +it compares the videos just by calculating the discrepancy +between the average value of their clips. Thus, it does not +take into account the diversity of clips within the video. +We suggest the following procedure for calculating the +distance between two videos. Let us consider a parameter +𝑛 which we call the length of a sub-video. We consider +each of the videos as a distribution of all possible sub-videos +of the lengths 𝑛. And then we calculate the mathematical +expectation of the distance calculated between a sub-video +from the first video and a sub-video from the second video +as follows: +dist𝑛(X, Y ) = Ex′∼𝑞𝑛 (X)Ey′∼𝑞𝑛 (Y ) < x′, y′ > += Ex′∼𝑞𝑛 (X)Ey′∼𝑞𝑛 (Y ) +𝑛×𝐷 +∑︁ +𝑗=1 +𝑥′ +𝑗𝑦′ +𝑗 +(4) +where 𝑞(X) is distribution of all possible sub-videos from +X which have 𝑛 clips inside. Such a distance function will +degrade to Equation 3 if we consider 𝑛 = 1. For all other +cases, it will take into account not only the difference be- +tween the mean values of the video embeddings but also +their distributions. +3.4. Differentiable Summary Selection +In order to perform end-to-end training of the proposed +pipeline, we must make all of the steps differentiable. The +feature extractor, the score predictor and the projector are +parametrized with neural networks and are thus differen- +tiable. A more sophisticated part of the pipeline is the mod- +ule, which selects clips with the highest scores, the top-𝑘 +frame selector. Given a set of frames {𝑥1, 𝑥2, . . . , 𝑥𝑁 } and a +set of corresponding scores {𝑠1, 𝑠2, . . . 𝑠𝑁 , } the top-𝑘 selec- +tor outputs a set of 𝑘 frames which have the highest scores. +Ranking a set of frames according to their scores is +equivalent to choosing the frame with the highest score, +then removing it from the set and repeating the operation +again and again. Choosing the maximum, or the top-1 frame +can be formalized as follows: +𝑥max = +∑︁ +𝑗 +𝑥 𝑗𝟙[𝑠 𝑗 > 𝑠𝑖, ∀𝑖 ≠ 𝑗] +(5) +where 𝟙[. . . ] is the indicator function. The value of 𝑥max +changes with jumps from 𝑥1 to 𝑥2 and so on when the cor- +responding scores dominate the other scores. If we fix all +the scores but just one, and then vary it from −∞ to ∞, the +value of 𝑥max will change just once and this change will be +a jump (see Figure 4). Thus, the gradient of 𝑥max with re- +spect to the varying score will remain 0 everywhere except +for the point of the jump, where the gradient is undefined. +Therefore, using this gradient value for back-propagation is +not possible. +By following [17] we use a relaxation of this step func- +tion. Equation 5 can be approximated as follows +𝑥max ≈ +∑︁ +𝑗 +𝑥 𝑗 +exp(𝛼𝑠 𝑗) +� +𝑖 exp(𝛼𝑠𝑖) += +∑︁ +𝑗 +𝑥 𝑗 · SoftMax(𝛼𝑠) 𝑗 +(6) +The parameter 𝛼 can be interpreted as the inverse of +the width of the transition region and if 𝛼 → ∞, then +SoftMax(𝛼𝑠) → 𝟙[. . . ]. +To rank the set of frames, we step-by-step select the max- +imum element by using Equation 6 and then subtract from +the maximum score a large number, so that the same frame +will not be selected on the next step. In order to minimize +the computational complexity of such an operation, we fol- +low the approach proposed in [34] and compare pairs of +frames. Thus, the processing time growth logarithmically +with the number of frames. +5 + +Original Video +SummaryModel +Source +Supervised +TVSum +SumMe +𝐹1 +𝜏 +𝜌 +𝐹1 +𝜏 +𝜌 +vsLSTM +[45] + +54.2 +- +- +37.6 +- +- +dppLSTM +[45] + +54.7 +- +- +38.6 +- +- +VASNet +[10] + +61.4 +0.16 +0.17 +49.7 +0.16 +0.17 +MSVA +[13] + +62.8 +0.19 +0.21 +54.4 +0.20 +0.23 +CSUM vsLSTM, +Ours + +59.0 +- +- +41.0 +- +- +CSUM dppLSTM +Ours + +60.5 +- +- +44.2 +- +- +CSUM VASNet +Ours + +62.7 +0.17 +0.17 +52.1 +0.16 +0.17 +CSUM MSVA +Ours + +63.9 +0.19 +0.20 +58.2 +0.22 +0.23 +Table 1. Experimental results on the TVSum and SumMe dataset. The reported metrics are F1-score, Spearman and Kendall correlation +coefficients. We compare various backbone models with default training regimes and the same model trained with our contrastive approach. +The best results are in bold. +4. Experiments +In this section, we evaluate the quality of video summa- +rizations learned with the proposed method. We conduct +experiments on several datasets and with several backbone +models to demonstrate that the proposed method general- +izes well for various video summarization setups. Next, we +present qualitative examples of extracted video summariza- +tions. Finally, we provide an ablation study on the hyper- +parameters of our method. +Datasets +We conduct experiments with 3 datasets: TV- +Sum [38], SumMe [18] and YouTube Highlights [39] +datasets. The TVSum dataset consists of 50 videos from +10 categories from [37]. In TVSum each video has frame- +level importance scores annotated by 20 users. Importance +scores range from 1 to 5, where 5 denotes the highest im- +portance. The SumMe dataset includes 25 short videos of +various events such as cooking or sports. Each video is at- +tributed with frame-level importance scores. The YouTube +Highlights dataset contains videos divided into 6 categories +with around 1000 videos of various lengths available for +each domain. For each video, there is a ground truth high- +light in a form of a sequence of consecutive frames summa- +rizing the content of the video in the best way. +Evaluation +To quantitatively evaluate the quality of ex- +tracted summaries we employ 5-fold cross-validation with +an average F1-score across the splits. The cross-validation +splits are the same as in [13]. The average F1-score over +videos in the dataset is reported. As noted in [32] F1-score +has certain limitations. We thus also adopt Spearman’s cor- +relation (𝜌) and Kendall correlation (𝜏) coefficients between +the summaries predicted by the models and ground truth +summaries. For the YouTube Highlights dataset we per- +form a summary evaluation as a task of highlight detection +in time. We thus employ mean average precision (mAP) as +a known detection metric. The final mAP score is computed +over [0.5:0.05:0.95] IoU thresholds. +Backbone models +To demonstrate that our method gen- +eralizes for various setups, we conduct experiments with +several known backbone models: +Video-LSTM and bi- +directional Video-LSTM [45], LSTM with attention [10], +Multi-Source Visual Attention model [13] and multi-modal +Transformers [25]. For our experiments, we leave the back- +bone architecture unchanged and only modify the training +pipeline of the models. +4.1. Summarization performance +We start with summarization experiments on the TVSum +and SumMe datasets. Here we evaluate the proposed con- +trastive learning approach with various feature extraction +backbone models. We use the proposed differentiable top- +k summary extractor during training. During the inference +stage, we simply select 𝑁 frames with the highest predicted +scores. The results are reported in Table 1. +As can be seen from Table 1, using the proposed ap- +proach results in significant improvement for all of the +baseline models. +Notably, video LSTM (v-LSTM) en- +joys a 4.8% improvement in F1-score on the TVSum +dataset, given that the proposed contrastive training does +not use labels compared to its default supervised training +regime. Also, our approach outperforms the previous best- +performing unsupervised method VASNet [10] by 1.1% on +TVSum and by up to 2.4% on SumMe, when the perfor- +mance is measured with the F1-score. In terms of Spear- +man and Kendall correlation coefficients, our unsupervised +method performs on par with the supervised models. +This experiment demonstrates that our contrastive learn- +ing approach generalizes well for various backbone archi- +tectures and for various datasets. Without using any labels, +6 + +Figure 6. Top row: A visual example of the video summary extracted with our method. Bottom row: human-annotated ground truth +importance scores and the importance scores trained with our contrastive learning method. +Model +Source +Supervised +Audio +Dog +Gym. +Park. +Skat. +Ski. +Surf. +Avg +LSVM +[39] + + +60.0 +41.0 +61.0 +62.0 +36.0 +61.0 +53.6 +LIM-S +[42] + + +57.9 +41.7 +67.0 +57.8 +48.6 +65.1 +56.4 +SL-Module +[43] + + +70.8 +53.2 +77.2 +72.5 +66.1 +76.2 +69.3 +CHD +[2] + + +60.6 +71.1 +74.2 +49.8 +68.2 +68.5 +65.4 +CSUM UMT +Ours + + +60.9 +70.2 +73.8 +63.2 +70.0 +71.4 +68.3 +CSUM UMT +Ours + + +64.8 +73.6 +79.9 +70.5 +71.5 +80.0 +73.3 +MINI-Net +[21] + + +58.2 +61.7 +70.2 +72.2 +58.7 +65.1 +64.4 +TCG +[44] + + +55.4 +62.7 +70.9 +69.1 +60.1 +59.8 +63.0 +Joint-VA +[1] + + +64.5 +71.9 +80.8 +62.0 +73.2 +78.3 +71.8 +UMT +[25] + + +65.9 +75.2 +81.6 +71.8 +72.3 +82.7 +74.9 +CSUM UMT +Ours + + +66.1 +75.1 +81.6 +71.9 +73.0 +82.8 +75.1 +Table 2. Experimental results on the YouTube Highlights benchmark. The reported metric is mAP in percents. We compare both the +methods which use the audio information from the video and the methods which rely on the visual features only. For both categories the +best performing models are in bold. +we are able either to match or to outperform existing super- +vised methods. +4.2. Ablation studies +In this section, we ablate the top-k selection algorithm +and the window parameter 𝑛 in Equation 4. We also investi- +gate if our top-k selector can robustly distribute importance +scores regardless of the number of input frames. +For top-k differentiable selection ablation, we compare +Sinkhorn [41], Perturbed top-k [7] and Successive Halv- +ing [34] algorithms. As can be seen from Table 3 the choice +of top-k influences the final performance with Successive +Halving delivering the best results for the MSVA back- +bone [13] on the TVSum dataset. We thus chose to use +Successive Halving in all of the experiments. +We next ablate the window parameter 𝑛 in Equation 4. +Intuitively, 𝑛 is responsible for the granularity of the result- +ing video summarization, where lower values of 𝑛 result in +higher granularity. As can be seen in Figure 7, the sum- +marization quality benefits from higher values of 𝑛. That +indicates that good video summaries should not be of the +highest granularity. Thus, we use 𝑛 = 10 in all of the exper- +iments. +Finally, we investigate if our differentiable top-k selec- +tor is robust with respect to the number of input frames. In +Figure 8 we report how the normalized 𝐿2-error between +the feature maps of top-10 frames selected with our method +and ground truth feature maps depends on the number of in- +put frames. The results suggest that even when the number +of input frames is huge, the error does not exceed 0.06. It +indicates that the used differentiable top-k selector is robust +with respect to the number of input frames. +4.3. Highlight detection +We consider highlight detection as a special case of the +summarization task, i.e. the highlight is a top-1 summary +7 + +Busines +KimacHD +CE +NEWS +4.583 +1 12 31 是 +272.0 +cleharthcharg +ground truth +predictionTop-𝑘 method +𝐹1-score +Sinkhorn [41] +63.1 ± 0.4 +Perturbed [7] +62.9 ± 0.5 +Successive Halving [34] +63.4 ± 0.3 +Table 3. 𝐹1-score of the MSVA model [13] with different top-k se- +lection mechanism on the TVSum dataset. A Successive Halving +algorithm performs the best. +Figure 7. 𝐹1-score of the MSVA backbone [13] on the TVSum +dataset for various values of 𝑛 used in Equation 4. +extraction coupled with surrounding context frames. Prac- +tically, to detect a highlight from a full-video, we prepos- +sess summarization scores with Gaussian smoothing to en- +force temporal continuity. After that, we extract a top-score +frame with the surrounding frames with high enough scores +to serve as one highlight. In the video summarization for- +mulation, a clip is selected if it is among the fixed number +of top-rated clips. For highlight detection, we select parts of +the video which have score higher than a hyperparameter Θ. +We choose Θ by maximizing the mAP metric on a holdout +set. +We conduct experiments in both supervised and unsuper- +vised scenarios. For supervised highlight detection, we first +pre-train the models with the proposed contrastive approach +for 20 epochs and then fine-tune it for 50 epochs using the +loss described in [25]. Evaluating fine-tuned representation +is a standard procedure in contrastive learning [4, 5]. For +unsupervised highlight detection, we directly use the scores +after 20 epochs of contrastive training. +We present the results for the cases when audio fea- +tures are available and when they are not. In the super- +vised scenario, as can be seen from Table 2, our method +(CSUM UMT) outperforms the competitive approaches or +performs on par. In particular, for the no-audio case our +method delivers more than 5% improvement relative to the +best-performing non-contrastive method [43]. With the au- +dio information included, our method slightly outperforms +Figure 8. Normalized 𝐿2-error between the feature maps of top-10 +frames selected with the proposed method and ground truth feature +maps. On the x-axis is the total number of frames to extract the +summary from. +the baseline UMT model, when the only modification being +made is the contrastive pre-training used. In the unsuper- +vised case, our method delivers more than 4% increase in +mAP score with respect to the previous best performing un- +supervised method from [2]. Also, the mAP score of our +unsupervised model is only 1% behind [43], which fully re- +lies on training with labels. +We conclude that our contrastive approach is very com- +petitive with existing methods, even when comparing our +unsupervised with previous supervised results. +4.4. Qualitative evaluation +In Figure 6 we present an example of the video summa- +rization of a sequence from the SumMe dataset trained with +our contrastive framework and differentiable top-k. We can +see that the predicted importance score can detect the re- +gions of both low and high significance. +5. Discussion +In this work, we propose a novel approach for unsuper- +vised video summarization. We start by formulating the +requirements for a good video summary: representatives, +sparsity, and diversity. +We observe that the contrastive +learning framework naturally includes representatives and +diversity. For sparsity, we propose a differentiable top-k +selector based on predicted frame-level scores, where the +importance is inherently distributed only among top-k input +frames. This allows stepping away from comparing mean +feature vectors, which may result in sub-optimal solution +space, during the contrastive learning stage. Our approach +does not rely on a specific kind of backbone; we experimen- +tally show that it generalizes well for various architectures +and summarization scenarios. +8 + +64.0 +e +63.5 +f-score +63.0 +62.5 +TVSum +62.0 +61.5 +61.0 +2 +4 +6 +8 +10 +n0.06 +0.05 +0.04 +Error +0.03 +0.02 +0.01 +0.00 +102 +103 +104 +# framesReferences +[1] Taivanbat Badamdorj, Mrigank Rochan, Yang Wang, and Li +Cheng. Joint visual and audio learning for video highlight +detection. +In Proceedings of the IEEE/CVF International +Conference on Computer Vision, pages 8127–8137, 2021. 3, +7 +[2] Taivanbat Badamdorj, Mrigank Rochan, Yang Wang, and Li +Cheng. 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In Proceedings of the +AAAI Conference on Artificial Intelligence, volume 32, 2018. +1, 3 +10 + diff --git a/itE4T4oBgHgl3EQfsA2l/content/tmp_files/load_file.txt b/itE4T4oBgHgl3EQfsA2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..beb132396c55a06554aa545bb233a03d6ea903e5 --- /dev/null +++ b/itE4T4oBgHgl3EQfsA2l/content/tmp_files/load_file.txt @@ -0,0 +1,602 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf,len=601 +page_content='Learning to Summarize Videos by Contrasting Clips Ivan Sosnovik UvA-Bosch Delta Lab University of Amsterdam i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='sosnovik@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='nl Artem Moskalev UvA-Bosch Delta Lab University of Amsterdam a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='moskalev@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='nl Cees Kaandorp Institute of Informatics University of Amsterdam cees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='kaandorp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='com Arnold Smeuldes UvA-Bosch Delta Lab University of Amsterdam a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='smeulders@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='nl Abstract Video summarization aims at choosing parts of a video that narrate a story as close as possible to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Most of the existing video summarization approaches focus on hand-crafted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' se As the number of videos grows exponentially, there emerges an increasing need for meth- ods that can learn meaningful summarizations without la- beled annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this paper, we aim to maximally ex- ploit unsupervised video summarization while concentrat- ing the supervision to a few, personalized labels as an add- on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' To do so, we formulate the key requirements for the informative video summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Then, we propose con- trastive learning as the answer to both questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' To fur- ther boost Contrastive video Summarization (CSUM), we propose to contrast top-k features instead of a mean video feature as employed by the existing method, which we im- plement with a differentiable top-k feature selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Our ex- periments on several benchmarks demonstrate, that our ap- proach allows for meaningful and diverse summaries when no labeled data is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Introduction Video summarization aims at choosing parts of a video that narrate a story as close as possible to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this day and age, video streaming without personalized recommendations is almost gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Current recommenda- tions select a fixed preview provided by the distributor on the basis of past preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We aim to go one step further and to provide personalized previews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Apart from better video selection for streaming, it also opens possibilities for better video editing, ad creation, and edge-device software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Existing approaches for video summarization focus on Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The t-SNE of the feature space of example clips and the learned summaries (black circles) for these clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Finding a good summary in the feature space does not boil down to the centroids of corresponding feature distributions, but rather consists of find- ing samples that informatively describe the whole input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' supervised summarization [14, 24, 35, 45, 46, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' With a growing number of videos, supervised summarization may still be somewhat affordable for the distributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' When the number of videos grows exponentially, the supervised model is not sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' And, from the standpoint of the user labeling can be applied only in very moderate amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this paper, we aim to maximally exploit unsupervised video summarization while concentrating the supervision on a few personalized labels as an add-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Unsupervised video summarization techniques were de- veloped in the pre-deep-learning era, when no large labeled datasets were available [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' They are still being used these days as labeling the full spectrum of possible videos is no longer possible [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Regardless of the method of video analysis, with deep learning or not, the main reasoning for 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='05213v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='CV] 12 Jan 2023 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The main building blocks of our framework for video summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Feature extractor 𝑓 transforms an arbitrary-length sequence of frames into a sequence of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' They are then used to predict a set of scores by using a function 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The summary extractor block uses both sequences to extract a set of 𝑘 frames with the highest scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' selecting a good summary is left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The summary must be a compressed representation of the original video while being closer in content to the source than to other videos [12, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Two core questions remain: how to sum- marize a video while preserving most information in the video, and how to measure distances between two videos on the basis of their content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Formulating the video sum- marization in this way, in this paper, we propose contrastive learning [4] as the answer to both questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Contrastive learning was designed to handle compression and metric learning in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We propose that neural networks can be trained to optimize a contrastive loss which represents the core requirements of a good summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Contrastive learning has been successfully applied in im- age and video classification [4, 8, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In classification, the contrastive loss is evaluated by comparing descriptive fea- ture vectors of equal size [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In video summarization, the comparison is between a vector describing the full video and a vector describing the summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As a consequence, the vectors will not be equal in size, and hence cannot be com- pared directly by common contrastive losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' To overcome the inequality in size, the common approach for comparing vector representations of videos and their summaries is to use their time-averaged representations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this work, we note that such an approach is invariant to a wide range of transformations and does not account for important mo- ments of high information in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' When taking an average, the summary is adequate when the video develops slowly like a game of snooker or a sit-com interview but expected to be less adequate when there are short moments of great significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' While various architectural solutions were proposed to improve over quality by average [35], we propose that a combination of a well-chosen loss function and training approach suffices to avoid the unwanted invari- ances for a wide range of backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this paper, we focus on developing a simple, flexible, yet efficient recipe for contrastive training of deep-learning- based video summarizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We start from the principle of maximum information preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We demonstrate that the ensuing loss function and maximization process pre- serves important information while avoiding undesired in- variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Our main contributions are the following: From the requirements of video summaries we propose a method for contrastive learning of video summaries with no need for labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We propose implementations of the main building blocks which are required to convert any video- analysis network into a summarizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We demonstrate the advantage of contrastive video summarizers on popular benchmarks for a set of back- bone architectures over their original training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We also demonstrate how it can be used for video high- light selection with a slight modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Related Work Supervised Methods With the rise of deep learning, a wide range of papers has considered video summarization as a regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In such a paradigm a neural net- work is used to take frames of the video and predict their importance scores so that the top-scored frames form the summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In [45] the authors use Recurrent Neural Net- works (RNNs) to combine the temporal information from the video with the content of each frame to successfully predict frames’ scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Alternatively, in [35] and [10] con- volutional and attention-based architecture was proposed to improve the quality of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' To effectively combine information about videos from multiple scales, hierarchi- cal models were proposed [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' By using hierarchi- cal RNNs, models benefit from considering the video as a whole, as a set of short clips and as a sequence of individ- ual frames at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It allows to create summaries 2 Feature Extractor Score Predictor Summary Extractor Video Frame features Scores Summary top-k gFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' An illustration of the proposed contrastive training pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Given two videos, the features are first calculated for them and then a summary is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Both summaries and original videos are projected by a neural network ℎ to a hidden space afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The whole pipeline is trained to attract the projections of summaries to the projections of the original videos and to repel them from other summaries and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' of less-contract granularity than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' While these meth- ods demonstrated the great success of deep neural networks for video summarization, manually labeled annotations are required for their training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It makes it impossible the scal- ing of such methods to long videos, movies and streams of videos that are constantly being uploaded on the major video services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For these reasons, we focus on methods that do not rely on human-annotated labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Unsupervised Methods Early-day methods for video summarization relied on heuristics designed by a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The heuristics were designed to satisfy the main require- ments for video summaries such as representativeness and diversity, justified in [9,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In [9,23,30] the authors clus- tered frames and use the centroids to form a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The authors of [6,28] formulate video summarization as a sparse dictionary selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Later, in the deep-learning era, video summarization was approached from the perspec- tive of adversarial training [20, 27] or in the reinforcement learning paradigm [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We draw inspiration from the pre- deep-learning era methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' By starting from the reasoning of video summarization, we demonstrate that we can sat- isfy the main requirements by formulating it as a contrastive learning problem that we can easily solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Contrastive Learning Contrastive learning is an ap- proach for performing self-supervised pretraining of a model by using a pre-text task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The model learns to at- tract representations that are meant to be close, and are thus called positive, and repel them from negative repre- sentations which are meant to be distant enough to distin- guish between different objects [4,19,29,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Various meth- ods have been proposed for learning image-level [3, 5] and spatio-temporal models [2,11,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The current application of contrastive learning methods for video summarization is rather limited due to special architectural solutions dictated by the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this work, we demonstrate an approach for contrastive learning for video summarization that does not rely on any specific backbones and allows one to use any model and framework of their choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Video Highlight Selection Another popular approach for creating a compressed visual representation of videos is video highlight selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' While the summary has a fixed length, the highlights are not bounded in length but have a lower bound for the importance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Various meth- ods have been proposed for solving this problem both from the supervised perspective [1, 21, 25, 39, 42–44], as well as in the unsupervised manner [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In this paper, we demon- strate that with a slight modification of our video summa- rization framework, we can outperform modern video high- light selection models without significant transformations of the original pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Summary Requirements Video summarization is a very subjective task, as a man- ually labeled summary is biased towards the personal pref- erences of the annotator, assessor [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' However, it is pos- sible to select several properties of a good summary that we would consider as summary requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' They also give us hint on how to build an efficient model for video summa- rization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Representativeness The composed summary should de- liver the same message as the original video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As the sum- mary is a compressed representation of the source, the loss of the original information is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' However, we re- quire a good summary to contain all the information neces- sary to distinguish between the original video and all other videos [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' With no loss of generality, we can assume that each video contains a finite set of sub-videos each of which tells a separate narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, the desired summary is a 3 Projection Repel h Summary Attract hcombination of sub-videos that is as close as possible to all of them at the same time, as well as distant enough from all sub-videos of other original videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We suggest to learn summaries by selecting a set of sub-videos which we call clips which once they are projected to some hidden space minimize a variant of the triplet loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As we want to de- velop a model for unsupervised summarization, it leads us to the framework of contrastive learning [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Sparsity The original videos may come from various sources: be it video news, a video blog, several-hours-long online streams or a TV show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For all cases, the desired summary would be just several seconds long as it is the average amount of time a user can spend before deciding whether to watch it or to skip it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It leads us to the require- ment of the sparsity of the resulting summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' While for short videos the desired summary may be around 15% of the length [18], this ratio may drop significantly for longer videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, our model should be capable of choosing the very top segments of the video with a significant distinction from the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The problem of ranking items and selecting the top of them cannot be overestimated as it has significant limitations in the realm of deep learning especially when it comes to a very sparse output [17,33,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In our approach, we should not directly rely on any heuristics for performing such an operation and should seek an as accurate as possible algorithmic implementation of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Diversity Another important property of a video sum- mary is the diversity among its frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We may assume that for some videos and for some datasets it is possible to create a summary that will contain a lot of very similar frames and clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Although it is possible for a user to understand what the video is about just by taking a look at one frame, it is still desired to have a summary with a higher diversity of vi- sual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' If we consider two models which satisfy the above-mentioned requirements, we want to select the model which selects diverse summaries over uniform sum- maries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It shows us that the function which we will use to measure the distance between videos should not be invari- ant to the spread of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In other words, it should take into account not only a single frame and the general content of videos but also the variations inside them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Contrastive Summarization A wide range of trainable video summarizers can be de- composed into the following three blocks: features extrac- tor 𝑓 , score predictor 𝑔 and summary extractor (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' From the summary requirement, we generated several requirements for the video summarization pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' And none of them are related to the feature extractor or the score predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, we assume that these two blocks are the free parameters of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Once a feature extractor Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Illustration of frame selection based on the scores 𝑠1 and 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Left: the original step function 𝑥max = 𝑥1 if 𝑠1 > 𝑠2 and 𝑥max = 𝑥2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Right: a relaxed version with a smooth replacement for the step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' and a score predictor are chosen, we train their parameters by performing a variant of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' During training, we consider two videos (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Each of the videos is processed with the feature extractor and the score predictor functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' After that, for each of the videos a summary is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We choose the param- eters of the networks 𝑓 and 𝑔 by training them to generate video and summary embeddings s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' the summary attracted to its source video is repelled from any other videos and summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It is done by minimizing the following loss [4]: L = ∑︁ 𝑧,𝑧+ − log exp(dist(𝑧, 𝑧+)/𝜏) � 𝑧− exp(dist(𝑧, 𝑧−)/𝜏) (1) where 𝑧, 𝑧−, 𝑧+ are embeddings for the anchor video, its neg- ative and positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We calculate this loss by iterating over all possible sets of such videos and their summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The parameter 𝜏 is the smoothing factor of the loss func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It is a hyperparameter of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In order to perform the training of such a pipeline successfully, we need to define the distance function dist(·, ·) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Clip-Contrastive Distance Function Let us consider two sequences of 𝐷-dimensional vectors represented as matrices: X = {𝑋𝑖 𝑗}𝐷,𝑡 𝑖, 𝑗=1 Y = {𝑌𝑖 𝑗}𝐷,𝑇 𝑖, 𝑗=1 (2) where 𝑡 and 𝑇 and the lengths of the sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In our case, these matrices are clip features for the summary and the original video and thus we assume 𝑡 < 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' A com- mon approach for calculating the distance between two se- quences [2] is to compare their features averaged it time: ¯x = � 𝑗 𝑋𝑖 𝑗/𝑡′ and ¯y = � 𝑘 𝑌𝑖𝑘/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It can be done for exam- ple by calculating the scalar product of these vectors: dist(X, Y ) = � ∑︁ 𝑗 𝑋𝑖 𝑗/𝑡′, ∑︁ 𝑘 𝑌𝑖𝑘/𝑇 � = 1 𝑇 · 𝑡 ∑︁ 𝑖 𝑗𝑘 𝑋𝑖 𝑗𝑌𝑖𝑘 (3) 4 max max S1 = S2 SFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Left: a sample video as a set of short clips forms a circle in some hidden space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Right: Three sets of summaries that yield the same distance function when they are time-averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' However, these summaries have different distances from the whole video if the distance function is given as in Equation 4 which leads to a uniform distribution of clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The main drawback of such a distance function is that it compares the videos just by calculating the discrepancy between the average value of their clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, it does not take into account the diversity of clips within the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We suggest the following procedure for calculating the distance between two videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Let us consider a parameter 𝑛 which we call the length of a sub-video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We consider each of the videos as a distribution of all possible sub-videos of the lengths 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' And then we calculate the mathematical expectation of the distance calculated between a sub-video from the first video and a sub-video from the second video as follows: dist𝑛(X, Y ) = Ex′∼𝑞𝑛 (X)Ey′∼𝑞𝑛 (Y ) < x′, y′ > = Ex′∼𝑞𝑛 (X)Ey′∼𝑞𝑛 (Y ) 𝑛×𝐷 ∑︁ 𝑗=1 𝑥′ 𝑗𝑦′ 𝑗 (4) where 𝑞(X) is distribution of all possible sub-videos from X which have 𝑛 clips inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Such a distance function will degrade to Equation 3 if we consider 𝑛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For all other cases, it will take into account not only the difference be- tween the mean values of the video embeddings but also their distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Differentiable Summary Selection In order to perform end-to-end training of the proposed pipeline, we must make all of the steps differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The feature extractor, the score predictor and the projector are parametrized with neural networks and are thus differen- tiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' A more sophisticated part of the pipeline is the mod- ule, which selects clips with the highest scores, the top-𝑘 frame selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Given a set of frames {𝑥1, 𝑥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' , 𝑥𝑁 } and a set of corresponding scores {𝑠1, 𝑠2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 𝑠𝑁 , } the top-𝑘 selec- tor outputs a set of 𝑘 frames which have the highest scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Ranking a set of frames according to their scores is equivalent to choosing the frame with the highest score, then removing it from the set and repeating the operation again and again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Choosing the maximum, or the top-1 frame can be formalized as follows: 𝑥max = ∑︁ 𝑗 𝑥 𝑗𝟙[𝑠 𝑗 > 𝑠𝑖, ∀𝑖 ≠ 𝑗] (5) where 𝟙[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' ] is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The value of 𝑥max changes with jumps from 𝑥1 to 𝑥2 and so on when the cor- responding scores dominate the other scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' If we fix all the scores but just one, and then vary it from −∞ to ∞, the value of 𝑥max will change just once and this change will be a jump (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, the gradient of 𝑥max with re- spect to the varying score will remain 0 everywhere except for the point of the jump, where the gradient is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Therefore, using this gradient value for back-propagation is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' By following [17] we use a relaxation of this step func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Equation 5 can be approximated as follows 𝑥max ≈ ∑︁ 𝑗 𝑥 𝑗 exp(𝛼𝑠 𝑗) � 𝑖 exp(𝛼𝑠𝑖) = ∑︁ 𝑗 𝑥 𝑗 · SoftMax(𝛼𝑠) 𝑗 (6) The parameter 𝛼 can be interpreted as the inverse of the width of the transition region and if 𝛼 → ∞, then SoftMax(𝛼𝑠) → 𝟙[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' To rank the set of frames, we step-by-step select the max- imum element by using Equation 6 and then subtract from the maximum score a large number, so that the same frame will not be selected on the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In order to minimize the computational complexity of such an operation, we fol- low the approach proposed in [34] and compare pairs of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, the processing time growth logarithmically with the number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 5 Original Video SummaryModel Source Supervised TVSum SumMe 𝐹1 𝜏 𝜌 𝐹1 𝜏 𝜌 vsLSTM [45] \x13 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 dppLSTM [45] \x13 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 VASNet [10] \x17 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='17 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='17 MSVA [13] \x13 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='21 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='23 CSUM vsLSTM, Ours \x17 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 CSUM dppLSTM Ours \x17 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 CSUM VASNet Ours \x17 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='17 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='17 CSUM MSVA Ours \x17 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='20 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='23 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Experimental results on the TVSum and SumMe dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The reported metrics are F1-score, Spearman and Kendall correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We compare various backbone models with default training regimes and the same model trained with our contrastive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The best results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Experiments In this section, we evaluate the quality of video summa- rizations learned with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We conduct experiments on several datasets and with several backbone models to demonstrate that the proposed method general- izes well for various video summarization setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Next, we present qualitative examples of extracted video summariza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Finally, we provide an ablation study on the hyper- parameters of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Datasets We conduct experiments with 3 datasets: TV- Sum [38], SumMe [18] and YouTube Highlights [39] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The TVSum dataset consists of 50 videos from 10 categories from [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In TVSum each video has frame- level importance scores annotated by 20 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Importance scores range from 1 to 5, where 5 denotes the highest im- portance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The SumMe dataset includes 25 short videos of various events such as cooking or sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Each video is at- tributed with frame-level importance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The YouTube Highlights dataset contains videos divided into 6 categories with around 1000 videos of various lengths available for each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For each video, there is a ground truth high- light in a form of a sequence of consecutive frames summa- rizing the content of the video in the best way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Evaluation To quantitatively evaluate the quality of ex- tracted summaries we employ 5-fold cross-validation with an average F1-score across the splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The cross-validation splits are the same as in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The average F1-score over videos in the dataset is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As noted in [32] F1-score has certain limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We thus also adopt Spearman’s cor- relation (𝜌) and Kendall correlation (𝜏) coefficients between the summaries predicted by the models and ground truth summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For the YouTube Highlights dataset we per- form a summary evaluation as a task of highlight detection in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We thus employ mean average precision (mAP) as a known detection metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The final mAP score is computed over [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='05:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='95] IoU thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Backbone models To demonstrate that our method gen- eralizes for various setups, we conduct experiments with several known backbone models: Video-LSTM and bi- directional Video-LSTM [45], LSTM with attention [10], Multi-Source Visual Attention model [13] and multi-modal Transformers [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For our experiments, we leave the back- bone architecture unchanged and only modify the training pipeline of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Summarization performance We start with summarization experiments on the TVSum and SumMe datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Here we evaluate the proposed con- trastive learning approach with various feature extraction backbone models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We use the proposed differentiable top- k summary extractor during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' During the inference stage, we simply select 𝑁 frames with the highest predicted scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The results are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As can be seen from Table 1, using the proposed ap- proach results in significant improvement for all of the baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Notably, video LSTM (v-LSTM) en- joys a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8% improvement in F1-score on the TVSum dataset, given that the proposed contrastive training does not use labels compared to its default supervised training regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Also, our approach outperforms the previous best- performing unsupervised method VASNet [10] by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1% on TVSum and by up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4% on SumMe, when the perfor- mance is measured with the F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In terms of Spear- man and Kendall correlation coefficients, our unsupervised method performs on par with the supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' This experiment demonstrates that our contrastive learn- ing approach generalizes well for various backbone archi- tectures and for various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Without using any labels, 6 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Top row: A visual example of the video summary extracted with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Bottom row: human-annotated ground truth importance scores and the importance scores trained with our contrastive learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Model Source Supervised Audio Dog Gym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Skat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Ski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Avg LSVM [39] \x13 \x17 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 LIM-S [42] \x13 \x17 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 SL-Module [43] \x13 \x17 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3 CHD [2] \x17 \x17 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 CSUM UMT Ours \x17 \x17 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3 CSUM UMT Ours \x13 \x17 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3 MINI-Net [21] \x13 \x13 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 TCG [44] \x13 \x13 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 Joint-VA [1] \x13 \x13 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 UMT [25] \x13 \x13 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 CSUM UMT Ours \x13 \x13 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Experimental results on the YouTube Highlights benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The reported metric is mAP in percents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We compare both the methods which use the audio information from the video and the methods which rely on the visual features only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For both categories the best performing models are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' we are able either to match or to outperform existing super- vised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Ablation studies In this section, we ablate the top-k selection algorithm and the window parameter 𝑛 in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We also investi- gate if our top-k selector can robustly distribute importance scores regardless of the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For top-k differentiable selection ablation, we compare Sinkhorn [41], Perturbed top-k [7] and Successive Halv- ing [34] algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As can be seen from Table 3 the choice of top-k influences the final performance with Successive Halving delivering the best results for the MSVA back- bone [13] on the TVSum dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We thus chose to use Successive Halving in all of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We next ablate the window parameter 𝑛 in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Intuitively, 𝑛 is responsible for the granularity of the result- ing video summarization, where lower values of 𝑛 result in higher granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' As can be seen in Figure 7, the sum- marization quality benefits from higher values of 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' That indicates that good video summaries should not be of the highest granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Thus, we use 𝑛 = 10 in all of the exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Finally, we investigate if our differentiable top-k selec- tor is robust with respect to the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In Figure 8 we report how the normalized 𝐿2-error between the feature maps of top-10 frames selected with our method and ground truth feature maps depends on the number of in- put frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' The results suggest that even when the number of input frames is huge, the error does not exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' It indicates that the used differentiable top-k selector is robust with respect to the number of input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Highlight detection We consider highlight detection as a special case of the summarization task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' the highlight is a top-1 summary 7 Busines KimacHD CE NEWS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='583 1 12 31 是 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 cleharthcharg ground truth predictionTop-𝑘 method 𝐹1-score Sinkhorn [41] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 Perturbed [7] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 Successive Halving [34] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='3 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 𝐹1-score of the MSVA model [13] with different top-k se- lection mechanism on the TVSum dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' A Successive Halving algorithm performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 𝐹1-score of the MSVA backbone [13] on the TVSum dataset for various values of 𝑛 used in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' extraction coupled with surrounding context frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Prac- tically, to detect a highlight from a full-video, we prepos- sess summarization scores with Gaussian smoothing to en- force temporal continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' After that, we extract a top-score frame with the surrounding frames with high enough scores to serve as one highlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In the video summarization for- mulation, a clip is selected if it is among the fixed number of top-rated clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For highlight detection, we select parts of the video which have score higher than a hyperparameter Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We choose Θ by maximizing the mAP metric on a holdout set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We conduct experiments in both supervised and unsuper- vised scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For supervised highlight detection, we first pre-train the models with the proposed contrastive approach for 20 epochs and then fine-tune it for 50 epochs using the loss described in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Evaluating fine-tuned representation is a standard procedure in contrastive learning [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For unsupervised highlight detection, we directly use the scores after 20 epochs of contrastive training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We present the results for the cases when audio fea- tures are available and when they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In the super- vised scenario, as can be seen from Table 2, our method (CSUM UMT) outperforms the competitive approaches or performs on par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In particular, for the no-audio case our method delivers more than 5% improvement relative to the best-performing non-contrastive method [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' With the au- dio information included, our method slightly outperforms Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Normalized 𝐿2-error between the feature maps of top-10 frames selected with the proposed method and ground truth feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' On the x-axis is the total number of frames to extract the summary from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' the baseline UMT model, when the only modification being made is the contrastive pre-training used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' In the unsuper- vised case, our method delivers more than 4% increase in mAP score with respect to the previous best performing un- supervised method from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Also, the mAP score of our unsupervised model is only 1% behind [43], which fully re- lies on training with labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We conclude that our contrastive approach is very com- petitive with existing methods, even when comparing our unsupervised with previous supervised results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Qualitative evaluation In Figure 6 we present an example of the video summa- rization of a sequence from the SumMe dataset trained with our contrastive framework and differentiable top-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We can see that the predicted importance score can detect the re- gions of both low and high significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Discussion In this work, we propose a novel approach for unsuper- vised video summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We start by formulating the requirements for a good video summary: representatives, sparsity, and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' We observe that the contrastive learning framework naturally includes representatives and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' For sparsity, we propose a differentiable top-k selector based on predicted frame-level scores, where the importance is inherently distributed only among top-k input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' This allows stepping away from comparing mean feature vectors, which may result in sub-optimal solution space, during the contrastive learning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' Our approach does not rely on a specific kind of backbone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' we experimen- tally show that it generalizes well for various architectures and summarization scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content=' 8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 e 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 f-score 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 TVSum 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='0 2 4 6 8 10 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='04 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} +page_content='00 102 103 104 # 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+page_content=' 1, 3 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE4T4oBgHgl3EQfsA2l/content/2301.05213v1.pdf'} diff --git a/mdAzT4oBgHgl3EQfqP0Q/content/tmp_files/2301.01624v1.pdf.txt b/mdAzT4oBgHgl3EQfqP0Q/content/tmp_files/2301.01624v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c906e741ef81c8a787960b8e39ae7f63c3fafe0a --- /dev/null +++ b/mdAzT4oBgHgl3EQfqP0Q/content/tmp_files/2301.01624v1.pdf.txt @@ -0,0 +1,972 @@ +arXiv:2301.01624v1 [math.NT] 21 Dec 2022 +Pattern Recognition Experiments on +Mathematical Expressions +David Naccache1 and Ofer Yifrach-Stav1 +DIÉNS, ÉNS, CNRS, PSL University, Paris, France +45 rue d’Ulm, 75230, Paris cedex 05, France +ofer.friedman@ens.fr, david.naccache@ens.fr +Abstract. We provide the results of pattern recognition experiments +on mathematical expressions. +We give a few examples of conjectured results. None of which was thor- +oughly checked for novelty. We did not attempt to prove all the relations +found and focused on their generation. +1 +Introduction +Pattern recognition is a process that involves identifying rules in data +and matching them with particular case information. Pattern recognition +can be seen as a type of machine learning, as it uses machine learning +algorithms to recognize patterns in data. This process is characterized by +the ability to learn from data, recognize familiar patterns, and recognize +patterns even if they are partially visible. +Very schematically, there are three main types of pattern recognition +heuristics: statistical pattern recognition, syntactic pattern recognition, +and neural pattern recognition. +– Statistical pattern recognition involves using particular case data to +learn from examples and generalize rules to new observations. +– Syntactic pattern recognition (a.k.a structural pattern recognition), in- +volves identifying patterns based on simpler sub-patterns called prim- +itives. For example, opcodes can be seen as primitives that connect to +form programs. +– Neural pattern recognition relies on artificial neural networks, which +are made up of many simple processors and their connections. These +networks can learn complex nonlinear input-output relationships and +adapt to data through sequential training procedures. +Most pattern recognition heuristics proceed by two steps: +– An Explorative Stage that seeks to identify patterns + +– A Descriptive Stage that categorizes patterns found during exploration +In this work we provide the results of the explorative stage of syntactic +pattern recognition on mathematical expressions. Given the nature of the +objects we work on (conjectures) the descriptive stage is left to a human. +We give a few examples of conjectured results. None of which was thor- +oughly checked for novelty. We did not attempt to prove all the relations +found and focused on their generation. +2 +The Pattern Recognition Algorithm +The pattern recognition algorithm has two components called the gener- +alizer and the identifier. +The generalizer departs from a known continued fraction or a math- +ematical expression (a particular case) and automatically parameterizes +parts of it. The parameterized parts are target ingredients tagged by the +user. For each set of particular parameter values (taken over search space), +approximated values of the formula are collected for later analysis. +Target ingredients are replaced by progressions, denoted by µu(i), +which can be constant, (alternating) arithmetic, geometric, harmonic or +exponential depending on the parameter choices. Those are captured by +the general formula: +µu(i) = u4iu5 + (u0 + iu1)u3ui +2 +For instance, the Ramanujan Machine Project [2,4,5] re-discovered an +already known relation involving eπ. Namely, that the continued fraction +defined by bn = n2 + 4 and an = 2n + 1 converges to: +2 (eπ + 1) +eπ − 1 += 1 + +12 + 4 +3 + +22 + 4 +5 + +32 + 4 +7 + 42 + 4 +9 + ... +A natural tagging query of this identity for search by the user might +hence be: +2 + +Q(u) = µu(0) + +µv(0) +µu(1) + +µv(1) +µu(2) + +µv(2) +µu(3) + +µv(3) +µu(4) + ... +With +u = {Q, Q, 1, 1, 0, 0} and v = {Z, 0, 1, 1, Q, N} +That is: +µu(i) = (Q + iQ) and µv(i) = QiN + Z +When this is done, the program varies the progressions’ parameters +over the chosen search spaces and collects sequences of resulting values. +The tests that we list here are of course non limitative and many other +variants can be added to the proposed heuristic. +Remark 1. Obviously, we are quickly limited by the increasing complexity +due to nested loops running over the parameters of the expressions (i.e. +the uis). +Remark 2. At the risk of overlooking some gold nuggets, when we explore +Q we start by exploring N and if the search is conclusive, we refine it by +increments of 1/6 which have the advantage of exploring units, halves and +thirds at the cost of a small multiplicative factor of 6. If interesting results +are found with increments of 6 the step is refined to 1/30 and to Farey +sequences. +The sequences obtained by varying those parameters are fed into the +identifier for possible recognition. To detect conjectures the identifier per- +forms a number of tests on the obtained sequences. Tests belong to two +categories: morphological tests and serial tests. Morphological tests are +applied to very few individual results and try to spot their characteristics. +Serial tests are applied to more results and seek to discover relationships +between them. +Algebraic number identification (ANI): Collect 10 convergence +limits Q0, Q1, . . . Q9 and, using LLL [3], check if any of those Qis is the root +of a small degree (≤ 10) polynomial. If so, check that RNI failed before +returning true to avoid multiple alerts as rationals are also algebraic. This +3 + +is a morphological test. The degree 10 was chosen arbitrarily and can be +changed at wish (provided that the precision is matched to the degree). +Rational number identification (RNI): Collect 10 convergence +limits Q0, Q1, . . . Q9 and, using LLL, check if any of those Qis is a good +approximation of a rational number having a (abnormally) small numer- +ator and a small denominator. This is a morphological test. +Constant presence identification (CPI): Collect 10 convergence +limits Q0, Q1, . . . Q9. Consider the 45 pairs P1, P2 formed from those Qis. +Using LLL, check the assumption that there is at least one pair of the +form: +P1 = a1 + b1U +c1 + d1U +and P2 = a2 + b2U +c2 + d2U +Where U ̸∈ Q and a1, b1, c1, d1, a2, b2, c2, d2 ∈ Z. +Solving for U and equating we get: +a2b1 − a1b2 + (b2c1 − a2d1)P1 + (a1d2 − b1c2)P2 + (c2d1 − c1d2)P1P2 = 0 +Hence, when called with on input 1, P1, P2, P1P2 LLL will return an +abnormally short vector if the coefficients are small (as is usually the case +in remarkable identities). This is a morphological test. +Constant to exponent identification (CEI): Collect 10 conver- +gence limits Q0, Q1, . . . Q9. Consider the 7 quadruples P1, P2, P3, P4 formed +by successive Qis1. +Here we assume that at successive ranks the limits are of the form: +Pk = ak + bkU k +ck + dkU k +Which implies that: +U k = ak − ckPk +dkPk − bk +If follows that: +U = (ak+1 − ck+1Pk+1)(dkPk − bk) +(dk+1Qk+1 − bk+1)(ak − ckQk) +(ak+3 − ck+3Pk+3)(dk+2Pk+2 − bk+2) +(dk+3Pk+3 − bk+3)(ak+2 − ck+2Pk+2) = (ak+1 − ck+1Pk+1)(dkPk − bk) +(dk+1Pk+1 − bk+1)(ak − ckPk) +1 namely: {0, 1, 2, 3},{1, 2, 3, 4},{2, 3, 4, 5},{3, 4, 5, 6},{4, 5, 6, 7},{5, 6, 7, 8},{6, 7, 8, 9} +4 + +Let: +S1 = {Pk, Pk+1, Pk+2, Pk+3} +S2 = {PkPk+1, PkPk+2, Pk+1Pk+2, PkPk+3, Pk+1Pk+3, Pk+2Pk+3} +S3 = {PkPk+1Pk+2, PkPk+1Pk+3, PkPk+2Pk+3, Pk+1Pk+2Pk+3} +S = S1 ∪ S2 ∪ S3 ∪ {1, PkPk+1Pk+2Pk+3} +When called with on input S LLL will return an abnormally short vec- +tor (as is usually the case in remarkable identities). This is a morphological +test. +Remark 3. Both CPI and CEI can be generalized to detect the presence +of multiple unknown constants in an expression (i.e. U1, U2, . . .) or even +the presence of common constants in different continued fractions. We did +not implement this generalization. Following those tests we can compute +a numerical approximation of U and attempt to look it up2. +Known constant identification (KCI): Let L be the following set +of usual constants: +L = {1, √π, π, π2, π3, ζ(3), ζ(5), ζ(7), √e, e, e2, e3, φ2, γ, G, ln 2, ln 3, ln 5} +Collect 10 convergence limits Q0, Q1, . . . Q9. Check using LLL if any +of the Qi is a number of the form: +Qi +� +j +ajLj = +� +j +bjLi for a1, a2, . . . , b1, b2 . . . ∈ Z +If the solution only involves 1, a false is returned. Note that as L +increases the required precision must also be increased to prevent spot- +ting artefacts. In practice we (manually) select only a subset of L before +running the KCI test according to the nature of the constants appearing +the in the particular case. Note that KCI and CPI can have overlapping +responses. +Rational fraction progression (RFP): In this test we seek to see if +when all ui except one (say ¯u) are kept constant, the continued fraction’s +limit Q(¯u) is a ratio of two polynomials in ¯u with integer coefficients. This +is done by a non linear model fit. The fit residuals serve as a measure of +the verdict’s likelihood. This is a serial test. +2 e.g. on https://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html +5 + +Exponential function progression (EFP): In this test we seek +to see if when all ui except one (say ¯u) are kept constant, the continued +fraction’s limit Q(¯u) is a function of the form ba¯u with rational coefficients. +This is done by a non linear model fit and rationality detection on a, b. +The fit residuals serve as a measure of the verdict’s likelihood. If ab = 0 +return false to avoid reporting the same result as the RFP. This is a serial +test. +Inverse exponential progression (IEP): In this test we seek to see +if when all ui except one (say ¯u) are kept constant, the continued fraction’s +limit Q(¯u) is a function of the form ba1/¯u with rational coefficients. This +is done by a non linear model fit and rationality detection on a, b. The fit +residuals serve as a measure of the verdict’s likelihood. If ab = 0 return +false to avoid reporting the same result as the RFP. This is a serial test. +Power plus constant progression (PCP): In this test we seek to +see if when all ui except one (say ¯u) are kept constant, the continued frac- +tion’s limit Q(¯u) is a function of the form b¯ua+c with rational coefficients. +This is done by a non linear model fit and rationality detection on a, b, c. +The fit residuals serve as a measure of the verdict’s likelihood. If b = 0 +return false to avoid reporting the same result as the RFP. This is a serial +test. +Root plus constant progression (RCP): In this test we seek to see +if when all ui except one (say ¯u) are kept constant, the continued fraction’s +limit Q(¯u) is a function of the form b a√¯u+c with rational coefficients. This +is done by a non linear model fit and rationality detection on a, b, c. The +fit residuals serve as a measure of the verdict’s likelihood. If ab = 0 return +false to avoid reporting the same result as the RFP. This is a serial test. +3 +Continued Fractions Converging to 2u(euπ+1)/(euπ−1) +It appears that the relation: +2 (eπ + 1) +eπ − 1 += 1 + +12 + 4 +3 + +22 + 4 +5 + +32 + 4 +7 + 42 + 4 +9 + ... +is the first in an infinite family: +6 + +2u (euπ + 1) +euπ − 1 += 1 + +12 + 4u2 +3 + +22 + 4u2 +5 + +32 + 4u2 +7 + 42 + 4u2 +9 + ... +ANI +RNI +CPI +CEI +KCI +RFP +EFP +IEP +PCP +RCP +✗ +✓ +✓ +✓ +✗ +✗ +✗ +✗ +✗ +✓ +Table 1: Test Results +Indeed, (RCP) linear variations in u cause identifiable O(√u) varia- +tions in the limit. This is because very quickly: +lim +u→∞ +euπ + 1 +euπ − 1 = 1 +This has the somewhat adverse effect of making the RNI positive very +quickly as well. +The final form is detected thanks to the CEI test. +By-product: +Because this holds for u ∈ C∗, we get a few seemingly +“mysterious” corollary identities such as: +2 (e + 1) +π(e − 1) = 1 + +12 + 4/π2 +3 + +22 + 4/π2 +5 + +32 + 4/π2 +7 + 42 + 4/π2 +9 + ... +6 ln 2 +π += 1 + +12 + 4 ln2 2/π2 +3 + +22 + 4 ln2 2/π2 +5 + +32 + 4 ln2 2/π2 +7 + 42 + 4 ln2 2/π2 +9 + ... +7 + +Implementation +1 f[x_, {m_, d_}] := m/(d + x); +2 For[t = 0, t <= 5, +3 +den = Table[2 n + 1, {n, 1, 20000}]; +4 +num = Table[n^2 + (2 t)^2, {n, 1, 20000}]; +5 +r = 1 + (Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]); +6 +e = 2 t (1 + (E^Pi)^t)/((E^Pi)^t − 1); +7 +Print[{e, 2 n + 1, n^2 + (2 t)^2, N[{r, e}, 20]}]; +8 +t += 1/2]; +4 +Continued Fractions Converging to Polynomial Roots +It is very well known that: +√ +5 − 1 +2 += 1 +1 + +1 +1 + +1 +1 + +1 +1 + +1 +1 + +1 +1 + +1 +1 + +1 +1 + · · · +We tag3: +Q(u) = 1+ µu(0) +µu(0) + +µu(0) +µu(0) + +µu(0) +µu(0) + +µu(0) +µu(0) + +µu(0) +µu(0) + +µu(0) +µu(0) + +µu(0) +µu(0) + · · · +With: +u = {Q, 0, 1, 1, 0, 0} ⇒ µu(i) = Q +It appears that for u ∈ Q/[−4, 0] LLL identifies that the limit is a root +of a second degree polynomial, namely: +Q(u) = 1 + u +u + +u +u + +u +u + +u +u + +u +u + +u +u + +u +u + · · · +Q(u)2 + u(Q(u) − 1) = 0 +Which is trivial to prove by pushing the u into the continued fraction. +The CPI is positive because for u = 1 and u = 5 the respective values +of Q(u) comprise the common value +√ +5. +3 Adding a 1+ by commodity which does not change anything about the infinite +convergence. +8 + +ANI +RNI +CPI +CEI +KCI +RFP +EFP +IEP +PCP +RCP +✓ +✗ +✓ +✓ +✗ +✗ +✗ +✗ +✗ +✗ +Table 2: Test Results +Implementation +1 f[x_, {m_, d_}] := m/(d + x); +2 For[L = −20, L <= 20 , +3 +If[−4 <= L <= 0, L = 2/3]; +4 +num = den = Table[L, {n, 1, 200}]; +5 +r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]; +6 +Print[{L, N[r^2 + L (r − 1)]}]; +7 +L += 2/3]; +5 +Continued Fractions Converging to e2/κ +The following relations are well-known4: +e = 2 + 1 +1 + +1 +2 + +1 +1 + +1 +1 + +1 +4 + +1 +1 + +1 +1 + +1 +6 + · · · +√e = 1 + 1 +1 + +1 +1 + +1 +5 + +1 +1 + +1 +1 + +1 +9 + +1 +1 + +1 +1 + +1 +13 + · · · +3√e = 1 + 1 +2 + +1 +1 + +1 +1 + +1 +8 + +1 +1 + +1 +1 + +1 +14 + +1 +1 + +1 +1 + +1 +20 + · · · +We hence tag the ones as constants, the progression as arithmetic and +let the algorithm monitor the evolution of the limits. +Let bn = 1. Define µ(u) = κ(u + 1/2) − 1 for κ ∈ R and: +an = +� +µ(n/3) = κ(2n+3) +6 +− 1 +if n mod 3 ≡ 0 +1 +otherwise. +In other words, an is the sequence: +an = {µ(0), 1, 1, µ(1), 1, 1, µ(2), 1, 1, µ(3), 1, 1, µ(4), 1, 1, · · ·} +Then we detect that the continued fraction generated by an, bn con- +verges to e2/κ. +4 https://link.springer.com/content/pdf/bbm:978-94-91216-37-4/1.pdf +9 + +ANI +RNI +CPI +CEI +KCI +RFP +EFP +IEP +PCP +RCP +✗ +✗ +✗ +✓ +✗ +✗ +✗ +✓ +✗ +✗ +Table 3: Test Results +The CEI is positive because, for instance (e2/κ)2 = e2/κ′ implies that +2/κ = κ′ which is satisfied for several pairs of integer values. +Implementation +1 f[x_, {m_, d_}] := m/(d + x); +2 For[k = −10, k <= 10, +3 +phi = Table[k n + k/2 − 1, {n, 0, 2000 − 1}] ; +4 +num = Table[1, {n, 1, 2000}]; +5 +den = Take[ +6 +Flatten[Table[{phi[[i]], {1, 1}}, {i, 1, Floor[2000/3] + 1}]], {1, +7 +2000}]; +8 +r = 1 + (Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]); +9 +v = E^(2/k); +10 +Print[{k, v, N[{r, v}, 20]}]; +11 +k += 1/2]; +6 +Continued Fractions Involving Catalan’s Constant +It is well known that: +2G = 2 − 12 +3 + +22 +1 + +22 +3 + +42 +1 + +42 +3 + +62 +1 + +62 +3 + +82 +1 + +82 +3 + · · · +We define: +∆(u, v) = 1 +2v × +�12 +u + +22 +v + +22 +u + +42 +v + +42 +u + +62 +v + +62 +u + +82 +v + +82 +u + · · · +� +For all the following we observe that ∆(u, v) = ∆(v, u). +6.1 +For u = 1 +An exploration for u = {0, N, N, N, Z, 0} reveals that for u0 = 0, u1 = +2, u2 = 1, u3 = 2, u4 = −1, u5 = 0 we get identities when v = 4i2 − 1 with +the convergence values given in Table 4: +10 + +u +i +v = 4i2 − 1 +∆(u, 4i2 − 1) = ∆(1, 4i2 − 1) +1 +0 +-1 +1 − G +1 +1 +3 +−8/9 + G +1 +2 +15 +209/225 − G +1 +3 +35 +−10016/11025 + G +1 +4 +63 +91369/99225 − G +1 +5 +99 +−10956424/12006225 + G +1 +6 +143 +1863641881/2029052025 − G +Table 4: The first convergence values for u = 1 +Where the general formula for i > 1 is: +∆(1, 4i2 − 1) = (−1)i+1 +�i−1 +� +k=0 +(−1)k +(2k + 1)2 − G +� +ANI +RNI +CPI +CEI +KCI +RFP +EFP +IEP +PCP +RCP +✗ +✗ +✓ +✗ +✓ +✗ +✗ +✗ +✗ +✗ +Table 5: Test Results +Implementation +1 f[x_, {m_, d_}] := m/(d + x); +2 For[i = 1, i < 40, +3 +{u, v} = {1, 4 i^2 − 1}; +4 +num = Take[ +5 +Prepend[Flatten[Table[{(2 n)^2, (2 n )^2}, {n, 1, 100000}]], 1] , +6 +100000]; +7 +den = Flatten[Table[{u, v}, {n, 1, 100000/2}]]; +8 +r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]/2/v; +9 +val = (−1)^(i + 1) (Sum[(−1)^k/(2 k + 1)^2, {k, 0, i − 1}] − Catalan); +10 +Print[{i, v, val, N[{val, r}, 30]}]; +11 +i++]; +Remark 4. Note that the denominators of the numbers: +η(i) = +i−1 +� +k=0 +(−1)k +(2k + 1)2 +11 + +are interesting by their own right. At first sight they might seem perfect +squares but in reality some may contain very small prime factors to an +odd power. +6.2 +For u = 3 +The exploration in this section is interesting. It was done manually but we +would have never had the idea to probe in that specific direction without +the insight for the case u = 1 produced in the previous section. +u +i +f(i) +∆(3, f(i)) +3 +0 +1 +∆(1, −1) +3 +1 +5 +∆(1, +3) +3 +2 +21 +∆(1, 35) +3 +3 +33 +∆(1, 63) +3 +4 +65 +∆(1, 143) +3 +5 +85 +∆(1, 255) +Table 6: The first convergence values for u = 3 +The sequence f(i) is nearly the absolute value of the OEIS sequence +A0063095: +1, 5, 21, 33, 65, 85, 133, 161, 261, 341, 481, 533, 645, 705, 901, ✘✘✘ +❳❳❳ +12803 , 1281, +1541, 1633, 1825, ✘✘✘ +❳❳❳ +14615 , ✘✘✘ +❳❳❳ +11537, 2581, 3201, 3333 . . . +An unexplained phenomenon occurs for the “abnormally larger” OEIS +sequence A006309 values 12803, 14615, 11537 that remains unmatched by +any η(i) value. We do not have an explanation for this phenomenon that +requires further research. +Implementation +The following implementation was purposely left unoptimized for the +sake of clarity. We start by generating the target values for u = 3 and +store them in an array. Then we re-generate the values for u = 1 and +match the array’s contents. +5 https://oeis.org/A006309. +12 + +1 AbsA006309 = +2 +Abs[{1, 5, −21, 33, −65, 85, −133, 161, 261, −341, −481, 533, −645, +3 +705, 901, −12803, −1281, −1541, 1633, −1825}]; +4 t = {}; +5 f[x_, {m_, d_}] := m/(d + x); +6 For[i = 1, i <= Length[AbsA006309], +7 +{u, v} = {3, AbsA006309[[i]]}; +8 +num = Take[ +9 +Prepend[Flatten[Table[{(2 n)^2, (2 n)^2}, {n, 1, 40000}]], 1], +10 +40000]; +11 +den = Flatten[Table[{u, v}, {n, 1, 40000/2}]]; +12 +r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]/2/ +13 +v; +14 +AppendTo[t, {AbsA006309[[i]], N[r, 30]}]; +15 +i++]; +16 +17 For[j = 1, j <= Length[AbsA006309], +18 +If[t[[j, 1]] == 12803, +19 +Print["Exception, the value 12803 is skipped."], +20 +For[i = 1, i <= 1000000, +21 +{u, v} = {1, 4 i^2 − 1}; +22 +den = Flatten[Table[{u, v}, {n, 1, 40000/2}]]; +23 +r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]/ +24 +2/v; +25 +val = (−1)^(i + 1) (Sum[(−1)^k/(2 k + 1)^2, {k, 0, i − 1}] − +26 +Catalan); +27 +If[Abs[t[[j, 2]] − r] < 10^(−6), +28 +Print[{i, N[r, 30], N[t[[j, 2]], 30]}, "Entry ", j, ": ", val, +29 +" matched with Delta[3,", t[[j, 1]], "]"]; +30 +i = Infinity]; +31 +i++]]; +32 +j++]; +6.3 +Subsequent u values. +Table 7 provides some additional examples for various u, v combinations. +6.4 +Variations in the numerator. +Let, for instance, (u, v) = (1, 3). Removing the 1/(2v) factor in ∆ and +replacing the (2n)2 by (n − i)2 we get convergence to: +1, 4 +5, 31 +51, 16 +33, 355 +883, 11524 +33599, 171887 +575075, 10147688 +38326363, . . . +With the limits being quickly reached after a constant number of terms +in the continued fraction. +13 + +u +v +∆(u, v) +5 +7 +∆(1, 15) +5 +39 +∆(1, 143) +5 +51 +∆(1, 255) +7 +9 +∆(1, 35) +9 +11 +∆(1, 63) +11 +13 +∆(1, 99) +13 +15 +∆(1, 143) +Table 7: Other convergence values. +Implementation +1 For[i = 1, i < 20, +2 +f[x_, {m_, d_}] := m/(d + x); +3 +4 +num = Take[ +5 +Prepend[Flatten[Table[{(n − i)^2, (n − i)^2}, {n, 1, 400}]], 1] , +6 +400]; +7 +den = Flatten[Table[{1, 3}, {n, 1, 400/2}]]; +8 +r = (Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]); +9 +Print[r]; +10 +i += 1] +7 +Generalized Cloître Series +In an unpublished note [1], Benoît Cloître gives a beautiful BBP formula +for π2 based on the identity: +∞ +� +k=1 +cos(ikπ) (2 cos(jπ))k +k2 += (ℓπ)2 +Here are some i, j, ℓ combinations detected automatically: +A simple rule allowing to generate many identities consists in fixing a +factional step 1/u, letting i = κ/u for π/3 ≤ i ≤ 2π/3 and calculating the +limit for {i, j} = {κu, 2 − κu} (e.g. Table 8). However, limits for which +i + j ̸= 2 exist as well (e.g. Table 9). +8 +Conclusion & further research +The results given in this paper show that pattern matching can obviously +be of help in detecting new mathematical conjectures. The very basic pro- +cesses described in the previous sections can be improved and generalized +14 + +i/ℓ +j/ℓ +1/ℓ +11 +5 +8 +14 +6 +10 +23 +9 +16 +26 +10 +18 +22/5 +8/5 +3 +31 +9 +20 +28 +8 +18 +19 +5 +12 +16 +4 +10 +13 +3 +8 +Table 8: Example relations for which i + j = 2 +i/ℓ +j/ℓ +1/ℓ +76 +16 +30 +46 +10 +18 +41 +9 +16 +26 +6 +10 +21 +5 +8 +Table 9: Example relations for which i + j ̸= 2 +ANI +RNI +CPI +CEI +KCI +RFP +EFP +IEP +PCP +RCP +✗ +✗ +✓ +✓ +✓ +✗ +✗ +✗ +✗ +✗ +Table 10: Test Results +15 + +in a number of ways. The first is obviously an enriching of the collection of +tests. The second is deeper exploration which is highly dependent on the +computational capabilities at hand. Finally the interpretation of results +and the early pruning of less probable branches in the potential conjecture +tree can also bring efficiency and pertinence in the discovered relations. +References +1. B. Cloître. A BBP formula for π2 in golden base. Unpublished manuscript. Undated, +https://les-mathematiques.net/phorum/file.php/download/2/3584/BBPbasePHI.pdf. +2. N. B. David, G. Nimri, U. Mendlovic, Y. Manor, and I. Kaminer. On the connection +between irrationality measures and polynomial continued fractions, 2021. +3. A. Lenstra, H. Lenstra, and L. Lovász. Factoring polynomials with rational coeffi- +cients. Math. Ann., 261:515–534, 1982. +4. G. Raayoni, S. Gottlieb, Y. Manor, G. Pisha, Y. Harris, U. Mendlovic, D. Haviv, +Y. Hadad, and I. Kaminer. Generating conjectures on fundamental constants with +the Ramanujan Machine. Nature, 590(7844):67–73, Feb 2021. +5. G. Raayoni, G. Pisha, Y. Manor, U. Mendlovic, D. Haviv, Y. Hadad, and I. Kaminer. +The Ramanujan Machine: Automatically generated conjectures on fundamental con- +stants. CoRR, abs/1907.00205, 2019. +16 + diff --git a/mdAzT4oBgHgl3EQfqP0Q/content/tmp_files/load_file.txt b/mdAzT4oBgHgl3EQfqP0Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..809c17e473d6ebbc545514c079ffc9c845a73746 --- /dev/null +++ b/mdAzT4oBgHgl3EQfqP0Q/content/tmp_files/load_file.txt @@ -0,0 +1,438 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf,len=437 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='01624v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='NT] 21 Dec 2022 Pattern Recognition Experiments on Mathematical Expressions David Naccache1 and Ofer Yifrach-Stav1 DIÉNS, ÉNS, CNRS, PSL University, Paris, France 45 rue d’Ulm, 75230, Paris cedex 05, France ofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='friedman@ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='fr, david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='naccache@ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='fr Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We provide the results of pattern recognition experiments on mathematical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We give a few examples of conjectured results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' None of which was thor- oughly checked for novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We did not attempt to prove all the relations found and focused on their generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 1 Introduction Pattern recognition is a process that involves identifying rules in data and matching them with particular case information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pattern recognition can be seen as a type of machine learning, as it uses machine learning algorithms to recognize patterns in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This process is characterized by the ability to learn from data, recognize familiar patterns, and recognize patterns even if they are partially visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Very schematically, there are three main types of pattern recognition heuristics: statistical pattern recognition, syntactic pattern recognition, and neural pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' – Statistical pattern recognition involves using particular case data to learn from examples and generalize rules to new observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' – Syntactic pattern recognition (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='a structural pattern recognition), in- volves identifying patterns based on simpler sub-patterns called prim- itives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' For example, opcodes can be seen as primitives that connect to form programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' – Neural pattern recognition relies on artificial neural networks, which are made up of many simple processors and their connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' These networks can learn complex nonlinear input-output relationships and adapt to data through sequential training procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Most pattern recognition heuristics proceed by two steps: – An Explorative Stage that seeks to identify patterns – A Descriptive Stage that categorizes patterns found during exploration In this work we provide the results of the explorative stage of syntactic pattern recognition on mathematical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Given the nature of the objects we work on (conjectures) the descriptive stage is left to a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We give a few examples of conjectured results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' None of which was thor- oughly checked for novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We did not attempt to prove all the relations found and focused on their generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2 The Pattern Recognition Algorithm The pattern recognition algorithm has two components called the gener- alizer and the identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The generalizer departs from a known continued fraction or a math- ematical expression (a particular case) and automatically parameterizes parts of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The parameterized parts are target ingredients tagged by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' For each set of particular parameter values (taken over search space), approximated values of the formula are collected for later analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Target ingredients are replaced by progressions, denoted by µu(i), which can be constant, (alternating) arithmetic, geometric, harmonic or exponential depending on the parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Those are captured by the general formula: µu(i) = u4iu5 + (u0 + iu1)u3ui 2 For instance, the Ramanujan Machine Project [2,4,5] re-discovered an already known relation involving eπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Namely, that the continued fraction defined by bn = n2 + 4 and an = 2n + 1 converges to: 2 (eπ + 1) eπ − 1 = 1 + 12 + 4 3 + 22 + 4 5 + 32 + 4 7 + 42 + 4 9 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' A natural tagging query of this identity for search by the user might hence be: 2 Q(u) = µu(0) + µv(0) µu(1) + µv(1) µu(2) + µv(2) µu(3) + µv(3) µu(4) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' With u = {Q, Q, 1, 1, 0, 0} and v = {Z, 0, 1, 1, Q, N} That is: µu(i) = (Q + iQ) and µv(i) = QiN + Z When this is done, the program varies the progressions’ parameters over the chosen search spaces and collects sequences of resulting values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The tests that we list here are of course non limitative and many other variants can be added to the proposed heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Obviously, we are quickly limited by the increasing complexity due to nested loops running over the parameters of the expressions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' the uis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' At the risk of overlooking some gold nuggets, when we explore Q we start by exploring N and if the search is conclusive, we refine it by increments of 1/6 which have the advantage of exploring units, halves and thirds at the cost of a small multiplicative factor of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' If interesting results are found with increments of 6 the step is refined to 1/30 and to Farey sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The sequences obtained by varying those parameters are fed into the identifier for possible recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' To detect conjectures the identifier per- forms a number of tests on the obtained sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Tests belong to two categories: morphological tests and serial tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Morphological tests are applied to very few individual results and try to spot their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Serial tests are applied to more results and seek to discover relationships between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Algebraic number identification (ANI): Collect 10 convergence limits Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Q9 and, using LLL [3], check if any of those Qis is the root of a small degree (≤ 10) polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' If so, check that RNI failed before returning true to avoid multiple alerts as rationals are also algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This 3 is a morphological test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The degree 10 was chosen arbitrarily and can be changed at wish (provided that the precision is matched to the degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Rational number identification (RNI): Collect 10 convergence limits Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Q9 and, using LLL, check if any of those Qis is a good approximation of a rational number having a (abnormally) small numer- ator and a small denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a morphological test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Constant presence identification (CPI): Collect 10 convergence limits Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Consider the 45 pairs P1, P2 formed from those Qis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Using LLL, check the assumption that there is at least one pair of the form: P1 = a1 + b1U c1 + d1U and P2 = a2 + b2U c2 + d2U Where U ̸∈ Q and a1, b1, c1, d1, a2, b2, c2, d2 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Solving for U and equating we get: a2b1 − a1b2 + (b2c1 − a2d1)P1 + (a1d2 − b1c2)P2 + (c2d1 − c1d2)P1P2 = 0 Hence, when called with on input 1, P1, P2, P1P2 LLL will return an abnormally short vector if the coefficients are small (as is usually the case in remarkable identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a morphological test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Constant to exponent identification (CEI): Collect 10 conver- gence limits Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Consider the 7 quadruples P1, P2, P3, P4 formed by successive Qis1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Here we assume that at successive ranks the limits are of the form: Pk = ak + bkU k ck + dkU k Which implies that: U k = ak − ckPk dkPk − bk If follows that: U = (ak+1 − ck+1Pk+1)(dkPk − bk) (dk+1Qk+1 − bk+1)(ak − ckQk) (ak+3 − ck+3Pk+3)(dk+2Pk+2 − bk+2) (dk+3Pk+3 − bk+3)(ak+2 − ck+2Pk+2) = (ak+1 − ck+1Pk+1)(dkPk − bk) (dk+1Pk+1 − bk+1)(ak − ckPk) 1 namely: {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='{2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='{3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='{4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='{5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='{6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 9} 4 Let: S1 = {Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+3} S2 = {PkPk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' PkPk+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+1Pk+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' PkPk+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+1Pk+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+2Pk+3} S3 = {PkPk+1Pk+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' PkPk+1Pk+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' PkPk+2Pk+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pk+1Pk+2Pk+3} S = S1 ∪ S2 ∪ S3 ∪ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' PkPk+1Pk+2Pk+3} When called with on input S LLL will return an abnormally short vec- tor (as is usually the case in remarkable identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a morphological test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Both CPI and CEI can be generalized to detect the presence of multiple unknown constants in an expression (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=') or even the presence of common constants in different continued fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We did not implement this generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Following those tests we can compute a numerical approximation of U and attempt to look it up2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Known constant identification (KCI): Let L be the following set of usual constants: L = {1, √π, π, π2, π3, ζ(3), ζ(5), ζ(7), √e, e, e2, e3, φ2, γ, G, ln 2, ln 3, ln 5} Collect 10 convergence limits Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Check using LLL if any of the Qi is a number of the form: Qi � j ajLj = � j bjLi for a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' , b1, b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' ∈ Z If the solution only involves 1, a false is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Note that as L increases the required precision must also be increased to prevent spot- ting artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' In practice we (manually) select only a subset of L before running the KCI test according to the nature of the constants appearing the in the particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Note that KCI and CPI can have overlapping responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Rational fraction progression (RFP): In this test we seek to see if when all ui except one (say ¯u) are kept constant, the continued fraction’s limit Q(¯u) is a ratio of two polynomials in ¯u with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is done by a non linear model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The fit residuals serve as a measure of the verdict’s likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a serial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' on https://wayback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='cecm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='ca/projects/ISC/ISCmain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='html 5 Exponential function progression (EFP): In this test we seek to see if when all ui except one (say ¯u) are kept constant, the continued fraction’s limit Q(¯u) is a function of the form ba¯u with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is done by a non linear model fit and rationality detection on a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The fit residuals serve as a measure of the verdict’s likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' If ab = 0 return false to avoid reporting the same result as the RFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a serial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Inverse exponential progression (IEP): In this test we seek to see if when all ui except one (say ¯u) are kept constant, the continued fraction’s limit Q(¯u) is a function of the form ba1/¯u with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is done by a non linear model fit and rationality detection on a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The fit residuals serve as a measure of the verdict’s likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' If ab = 0 return false to avoid reporting the same result as the RFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a serial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Power plus constant progression (PCP): In this test we seek to see if when all ui except one (say ¯u) are kept constant, the continued frac- tion’s limit Q(¯u) is a function of the form b¯ua+c with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is done by a non linear model fit and rationality detection on a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The fit residuals serve as a measure of the verdict’s likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' If b = 0 return false to avoid reporting the same result as the RFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a serial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Root plus constant progression (RCP): In this test we seek to see if when all ui except one (say ¯u) are kept constant, the continued fraction’s limit Q(¯u) is a function of the form b a√¯u+c with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is done by a non linear model fit and rationality detection on a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The fit residuals serve as a measure of the verdict’s likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' If ab = 0 return false to avoid reporting the same result as the RFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is a serial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3 Continued Fractions Converging to 2u(euπ+1)/(euπ−1) It appears that the relation: 2 (eπ + 1) eπ − 1 = 1 + 12 + 4 3 + 22 + 4 5 + 32 + 4 7 + 42 + 4 9 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' is the first in an infinite family: 6 2u (euπ + 1) euπ − 1 = 1 + 12 + 4u2 3 + 22 + 4u2 5 + 32 + 4u2 7 + 42 + 4u2 9 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' ANI RNI CPI CEI KCI RFP EFP IEP PCP RCP ✗ ✓ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✓ Table 1: Test Results Indeed, (RCP) linear variations in u cause identifiable O(√u) varia- tions in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' This is because very quickly: lim u→∞ euπ + 1 euπ − 1 = 1 This has the somewhat adverse effect of making the RNI positive very quickly as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The final form is detected thanks to the CEI test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' By-product: Because this holds for u ∈ C∗, we get a few seemingly “mysterious” corollary identities such as: 2 (e + 1) π(e − 1) = 1 + 12 + 4/π2 3 + 22 + 4/π2 5 + 32 + 4/π2 7 + 42 + 4/π2 9 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6 ln 2 π = 1 + 12 + 4 ln2 2/π2 3 + 22 + 4 ln2 2/π2 5 + 32 + 4 ln2 2/π2 7 + 42 + 4 ln2 2/π2 9 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7 Implementation 1 f[x_, {m_, d_}] := m/(d + x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2 For[t = 0, t <= 5, 3 den = Table[2 n + 1, {n, 1, 20000}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 num = Table[n^2 + (2 t)^2, {n, 1, 20000}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5 r = 1 + (Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6 e = 2 t (1 + (E^Pi)^t)/((E^Pi)^t − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7 Print[{e, 2 n + 1, n^2 + (2 t)^2, N[{r, e}, 20]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 t += 1/2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 Continued Fractions Converging to Polynomial Roots It is very well known that: √ 5 − 1 2 = 1 1 + 1 1 + 1 1 + 1 1 + 1 1 + 1 1 + 1 1 + 1 1 + · · · We tag3: Q(u) = 1+ µu(0) µu(0) + µu(0) µu(0) + µu(0) µu(0) + µu(0) µu(0) + µu(0) µu(0) + µu(0) µu(0) + µu(0) µu(0) + · · · With: u = {Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 0} ⇒ µu(i) = Q It appears that for u ∈ Q/[−4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 0] LLL identifies that the limit is a root of a second degree polynomial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' namely: Q(u) = 1 + u u + u u + u u + u u + u u + u u + u u + · · · Q(u)2 + u(Q(u) − 1) = 0 Which is trivial to prove by pushing the u into the continued fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The CPI is positive because for u = 1 and u = 5 the respective values of Q(u) comprise the common value √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3 Adding a 1+ by commodity which does not change anything about the infinite convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 ANI RNI CPI CEI KCI RFP EFP IEP PCP RCP ✓ ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ Table 2: Test Results Implementation 1 f[x_, {m_, d_}] := m/(d + x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2 For[L = −20, L <= 20 , 3 If[−4 <= L <= 0, L = 2/3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 num = den = Table[L, {n, 1, 200}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5 r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6 Print[{L, N[r^2 + L (r − 1)]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7 L += 2/3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5 Continued Fractions Converging to e2/κ The following relations are well-known4: e = 2 + 1 1 + 1 2 + 1 1 + 1 1 + 1 4 + 1 1 + 1 1 + 1 6 + · · · √e = 1 + 1 1 + 1 1 + 1 5 + 1 1 + 1 1 + 1 9 + 1 1 + 1 1 + 1 13 + · · · 3√e = 1 + 1 2 + 1 1 + 1 1 + 1 8 + 1 1 + 1 1 + 1 14 + 1 1 + 1 1 + 1 20 + · · · We hence tag the ones as constants, the progression as arithmetic and let the algorithm monitor the evolution of the limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Let bn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Define µ(u) = κ(u + 1/2) − 1 for κ ∈ R and: an = � µ(n/3) = κ(2n+3) 6 − 1 if n mod 3 ≡ 0 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' In other words, an is the sequence: an = {µ(0), 1, 1, µ(1), 1, 1, µ(2), 1, 1, µ(3), 1, 1, µ(4), 1, 1, · · ·} Then we detect that the continued fraction generated by an, bn con- verges to e2/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='com/content/pdf/bbm:978-94-91216-37-4/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='pdf 9 ANI RNI CPI CEI KCI RFP EFP IEP PCP RCP ✗ ✗ ✗ ✓ ✗ ✗ ✗ ✓ ✗ ✗ Table 3: Test Results The CEI is positive because, for instance (e2/κ)2 = e2/κ′ implies that 2/κ = κ′ which is satisfied for several pairs of integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Implementation 1 f[x_, {m_, d_}] := m/(d + x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2 For[k = −10, k <= 10, 3 phi = Table[k n + k/2 − 1, {n, 0, 2000 − 1}] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 num = Table[1, {n, 1, 2000}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5 den = Take[ 6 Flatten[Table[{phi[[i]], {1, 1}}, {i, 1, Floor[2000/3] + 1}]], {1, 7 2000}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 r = 1 + (Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 9 v = E^(2/k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 10 Print[{k, v, N[{r, v}, 20]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 11 k += 1/2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6 Continued Fractions Involving Catalan’s Constant It is well known that: 2G = 2 − 12 3 + 22 1 + 22 3 + 42 1 + 42 3 + 62 1 + 62 3 + 82 1 + 82 3 + · · · We define: ∆(u, v) = 1 2v × �12 u + 22 v + 22 u + 42 v + 42 u + 62 v + 62 u + 82 v + 82 u + · · · � For all the following we observe that ∆(u, v) = ∆(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='1 For u = 1 An exploration for u = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 0} reveals that for u0 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' u1 = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' u2 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' u3 = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' u4 = −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' u5 = 0 we get identities when v = 4i2 − 1 with the convergence values given in Table 4: 10 u i v = 4i2 − 1 ∆(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4i2 − 1) = ∆(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4i2 − 1) 1 0 1 1 − G 1 1 3 −8/9 + G 1 2 15 209/225 − G 1 3 35 −10016/11025 + G 1 4 63 91369/99225 − G 1 5 99 −10956424/12006225 + G 1 6 143 1863641881/2029052025 − G Table 4: The first convergence values for u = 1 Where the general formula for i > 1 is: ∆(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4i2 − 1) = (−1)i+1 �i−1 � k=0 (−1)k (2k + 1)2 − G � ANI RNI CPI CEI KCI RFP EFP IEP PCP RCP ✗ ✗ ✓ ✗ ✓ ✗ ✗ ✗ ✗ ✗ Table 5: Test Results Implementation 1 f[x_,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' {m_,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' d_}] := m/(d + x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2 For[i = 1, i < 40, 3 {u, v} = {1, 4 i^2 − 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 num = Take[ 5 Prepend[Flatten[Table[{(2 n)^2, (2 n )^2}, {n, 1, 100000}]], 1] , 6 100000];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7 den = Flatten[Table[{u, v}, {n, 1, 100000/2}]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]/2/v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 9 val = (−1)^(i + 1) (Sum[(−1)^k/(2 k + 1)^2, {k, 0, i − 1}] − Catalan);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 10 Print[{i, v, val, N[{val, r}, 30]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 11 i++];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Note that the denominators of the numbers: η(i) = i−1 � k=0 (−1)k (2k + 1)2 11 are interesting by their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' At first sight they might seem perfect squares but in reality some may contain very small prime factors to an odd power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='2 For u = 3 The exploration in this section is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' It was done manually but we would have never had the idea to probe in that specific direction without the insight for the case u = 1 produced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' u i f(i) ∆(3, f(i)) 3 0 1 ∆(1, −1) 3 1 5 ∆(1, 3) 3 2 21 ∆(1, 35) 3 3 33 ∆(1, 63) 3 4 65 ∆(1, 143) 3 5 85 ∆(1, 255) Table 6: The first convergence values for u = 3 The sequence f(i) is nearly the absolute value of the OEIS sequence A0063095: 1, 5, 21, 33, 65, 85, 133, 161, 261, 341, 481, 533, 645, 705, 901, ✘✘✘ ❳❳❳ 12803 , 1281, 1541, 1633, 1825, ✘✘✘ ❳❳❳ 14615 , ✘✘✘ ❳❳❳ 11537, 2581, 3201, 3333 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' An unexplained phenomenon occurs for the “abnormally larger” OEIS sequence A006309 values 12803, 14615, 11537 that remains unmatched by any η(i) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We do not have an explanation for this phenomenon that requires further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Implementation The following implementation was purposely left unoptimized for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' We start by generating the target values for u = 3 and store them in an array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Then we re-generate the values for u = 1 and match the array’s contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5 https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='org/A006309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 12 1 AbsA006309 = 2 Abs[{1, 5, −21, 33, −65, 85, −133, 161, 261, −341, −481, 533, −645, 3 705, 901, −12803, −1281, −1541, 1633, −1825}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4 t = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5 f[x_, {m_, d_}] := m/(d + x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6 For[i = 1, i <= Length[AbsA006309], 7 {u, v} = {3, AbsA006309[[i]]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 num = Take[ 9 Prepend[Flatten[Table[{(2 n)^2, (2 n)^2}, {n, 1, 40000}]], 1], 10 40000];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 11 den = Flatten[Table[{u, v}, {n, 1, 40000/2}]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 12 r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]/2/ 13 v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 14 AppendTo[t, {AbsA006309[[i]], N[r, 30]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 15 i++];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 16 17 For[j = 1, j <= Length[AbsA006309], 18 If[t[[j, 1]] == 12803, 19 Print["Exception, the value 12803 is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' "], 20 For[i = 1, i <= 1000000, 21 {u, v} = {1, 4 i^2 − 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 22 den = Flatten[Table[{u, v}, {n, 1, 40000/2}]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 23 r = Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]/ 24 2/v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 25 val = (−1)^(i + 1) (Sum[(−1)^k/(2 k + 1)^2, {k, 0, i − 1}] − 26 Catalan);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 27 If[Abs[t[[j, 2]] − r] < 10^(−6), 28 Print[{i, N[r, 30], N[t[[j, 2]], 30]}, "Entry ", j, ": ", val, 29 " matched with Delta[3,", t[[j, 1]], "]"];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 30 i = Infinity];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 31 i++]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 32 j++];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='3 Subsequent u values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Table 7 provides some additional examples for various u, v combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='4 Variations in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Let, for instance, (u, v) = (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Removing the 1/(2v) factor in ∆ and replacing the (2n)2 by (n − i)2 we get convergence to: 1, 4 5, 31 51, 16 33, 355 883, 11524 33599, 171887 575075, 10147688 38326363, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' With the limits being quickly reached after a constant number of terms in the continued fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 13 u v ∆(u, v) 5 7 ∆(1, 15) 5 39 ∆(1, 143) 5 51 ∆(1, 255) 7 9 ∆(1, 35) 9 11 ∆(1, 63) 11 13 ∆(1, 99) 13 15 ∆(1, 143) Table 7: Other convergence values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Implementation 1 For[i = 1, i < 20, 2 f[x_, {m_, d_}] := m/(d + x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3 4 num = Take[ 5 Prepend[Flatten[Table[{(n − i)^2, (n − i)^2}, {n, 1, 400}]], 1] , 6 400];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 7 den = Flatten[Table[{1, 3}, {n, 1, 400/2}]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 r = (Fold[f, Last@num/Last@den, Reverse@Most@Transpose@{num, den}]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 9 Print[r];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 10 i += 1] 7 Generalized Cloître Series In an unpublished note [1], Benoît Cloître gives a beautiful BBP formula for π2 based on the identity: ∞ � k=1 cos(ikπ) (2 cos(jπ))k k2 = (ℓπ)2 Here are some i, j, ℓ combinations detected automatically: A simple rule allowing to generate many identities consists in fixing a factional step 1/u, letting i = κ/u for π/3 ≤ i ≤ 2π/3 and calculating the limit for {i, j} = {κu, 2 − κu} (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' However, limits for which i + j ̸= 2 exist as well (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 8 Conclusion & further research The results given in this paper show that pattern matching can obviously be of help in detecting new mathematical conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The very basic pro- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='cesses described in the previous sections can be improved and generalized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='i/ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='j/ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='1/ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='22/5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='8/5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='Table 8: Example relations for which i + j = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='i/ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='j/ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='1/ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='Table 9: Example relations for which i + j ̸= 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='ANI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='RNI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='CPI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='CEI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='KCI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='RFP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='EFP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='IEP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='PCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='RCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='Table 10: Test Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='in a number of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The first is obviously an enriching of the collection of tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The second is deeper exploration which is highly dependent on the computational capabilities at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Finally the interpretation of results and the early pruning of less probable branches in the potential conjecture tree can also bring efficiency and pertinence in the discovered relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Cloître.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' A BBP formula for π2 in golden base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Unpublished manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Undated, https://les-mathematiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='net/phorum/file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='php/download/2/3584/BBPbasePHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' David, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Nimri, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Mendlovic, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Manor, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Kaminer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' On the connection between irrationality measures and polynomial continued fractions, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Lenstra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Lenstra, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Lovász.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Factoring polynomials with rational coeffi- cients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=', 261:515–534, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Raayoni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Gottlieb, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Generating conjectures on fundamental constants with the Ramanujan Machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Nature, 590(7844):67–73, Feb 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Raayoni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Pisha, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Manor, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Mendlovic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Haviv, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Hadad, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' Kaminer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' The Ramanujan Machine: Automatically generated conjectures on fundamental con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' CoRR, abs/1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content='00205, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfqP0Q/content/2301.01624v1.pdf'} diff --git a/o9FKT4oBgHgl3EQfyC7z/content/tmp_files/2301.11906v1.pdf.txt b/o9FKT4oBgHgl3EQfyC7z/content/tmp_files/2301.11906v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6de71b167c61d906945e6c7e7cec2e67588830e6 --- /dev/null +++ b/o9FKT4oBgHgl3EQfyC7z/content/tmp_files/2301.11906v1.pdf.txt @@ -0,0 +1,832 @@ +arXiv:2301.11906v1 [math.CA] 27 Jan 2023 +A SUFFICIENT CONDITION FOR HAAR MULTIPLIERS +IN TRIEBEL-LIZORKIN SPACES +GUSTAVO GARRIG´OS +ANDREAS SEEGER +TINO ULLRICH +In memory of Guido Weiss +Abstract. We consider Haar multiplier operators Tm acting on Sobolev +spaces, and more generally Triebel-Lizorkin spaces F s +p,q(R), for indices +in which the Haar system is not unconditional. When m depends only +on the Haar frequency, we give a sufficient condition for the boundedness +of Tm in F s +p,q, in terms of the variation norms ∥m∥Vu, which is optimal +in u (up to endpoints) when p, q > 1. +1. Introduction +Consider the classical Haar system in R, +(1.1) +H = +� +hj,µ : +j ≥ −1, µ ∈ Z +� +, +where, if h = +1[0,1/2) − +1[1/2,1), we let +hj,µ(x) = h(2jx − µ) , +for +j = 0, 1, 2, . . . , µ ∈ Z, +while for j = −1 we let +h−1,µ = +1[µ,µ+1), +µ ∈ Z. +We shall refer to the elements of the family Hj = {hj,µ : µ ∈ Z} as Haar +functions of frequency 2j. +Let F s +p,q denote the usual Triebel-Lizorkin space in R; see [17]. It is known +from the work of Triebel [19, Theorem 2.9.ii] that H is an unconditional +basis of F s +p,q(R) when s belongs to the range +(1.2) +max +� +1/p − 1, 1/q − 1 +� +< s < min +� +1/p, 1/q, 1 +� +. +That this range is actually optimal was shown by the last two authors in +[13, 14]. +More recently, we proved in [3] that H is a Schauder basis of +F s +p,q(R) (with respect to natural enumerations) in the larger range +(1.3) +1/p − 1 < s < min +� +1/p, 1 +� +, +(for all 0 < q < ∞), +Date: January 30, 2023. +2010 Mathematics Subject Classification. 46E35, 46B15, 42C40. +Key words and phrases. Haar basis, Triebel-Lizorkin spaces, multipliers, variation +norms. +1 + +2 +G. GARRIG´OS +A. SEEGER +T. ULLRICH +while at the endpoints (see [5]) the property holds if and only if +(1.4) +s = 1/p − 1 +and +1/2 < p ≤ 1, +also for all 0 < q < ∞. These regions are depicted in Figure 1 below. +unconditional +1 +p +s +1 +2 +1 +1 +q +1 +q −1 +−1 +Schauder +1 +p +s +1 +2 +1 +−1 +Figure 1. Parameter domain for H to be an unconditional +basis (left figure) or a Schauder basis (right figure) in F s +p,q(R). +We shall mainly be interested in values of the parameters outside the +region of unconditionality. In that range, it becomes a natural question to +find sufficient conditions on a sequence {mj,µ} so that the mapping +f �−→ +� +j≥0 +� +µ∈Z +mj,µ 2j⟨f, hj,µ⟩hj,µ, +defined say for f ∈ span H , extends as a bounded linear operator in the +space F s +p,q. +In this paper we regard this problem in the special case when the sequence +is constant in each frequency level, namely, if m = {m(j)}j≥0, we consider +the operators +Tmf = +� +j≥0 +m(j) Djf, +where Dj denotes the orthogonal projection onto the space generated by Hj, +that is +Djf = +� +µ∈Z +2j⟨f, hj,µ⟩ hj,µ, +j ≥ 0. +It is well known that one can write +Dj = Ej+1 − Ej, +where Ej is the conditional expectation operator defined by +(1.5) +Ejf(x) = +� +µ∈Z +1Ij,µ(x) 2j +� +Ij,µ +f(y)dy , + +HAAR MULTIPLERS +3 +associated with the dyadic intervals Ij,µ = [µ2−j, (µ + 1)2−j), µ ∈ Z. +The uniform boundedness of the operators EN in F s +p,q (and Bs +p,q) has +been throughly studied in the papers [3, 4, 5]. In particular, it is shown in +those papers that H is a Schauder basis of F s +p,q (with respect to natural +enumerations) if and only if +sup +N≥0 +��EN +�� +F sp,q→F sp,q < ∞ +and +span H is dense in F s +p,q, +and this in turn is equivalent to (s, p, q) belonging to the ranges in (1.3) and +(1.4). In those cases, an elementary summation by parts argument and the +σ-triangle inequality, with σ = min{1, p, q}, imply that +(1.6) +∥Tmf∥F sp,q ≲ ∥m∥ℓ∞ + +� ∞ +� +j=1 +|m(j) − m(j − 1)|σ� 1 +σ , +for all f ∈ span H with ∥f∥F sp,q ≤ 1. +We shall next formulate a stronger multiplier result which involves the +Wiener space notion of sequences of bounded u-variation. We recall how +these are defined. If u ≥ 1, we let Vu(m) be the u-variation of the sequence +{m(j)}j≥0, defined by +Vu(m) = sup +� N +� +n=1 +|m(jn) − m(jn−1)|u�1/u +with the supremum taken over all finite strings of numbers {j0, . . . , jN} +satisfying jn−1 < jn for 1 ≤ n ≤ N, and jn ∈ N ∪ {0}. Note that if u = 1 +we simply have +V1(m) = +∞ +� +j=1 +|m(j) − m(j − 1)|. +We denote by Vu the space of all m : N ∪ {0} → C for which +∥m∥Vu := ∥m∥∞ + Vu(m) < ∞. +In particular, if 1 ≤ u1 ≤ u2 < ∞, it holds +V1 ֒→ Vu1 ֒→ Vu2 ֒→ ℓ∞, +As an example, observe that m(n) = 1/(n + 1)α belongs to V1 for all α > 0, +while the alternate sequence M(n) = (−1)nm(n) belongs to Vu iff α > 1/u. +We wish to find, in the region of exponents (s, p, q) where H is a con- +ditional basis of F s +p,q, the largest possible u for which m ∈ Vu implies the +boundedness of the operator Tm in F s +p,q. The examples given in [13], based +on multipliers taking the values 0 and 1 (suitable characteristic functions of +finite sets of integers), show that for 1/u < s − 1/q there are m ∈ Vu such +that the corresponding operators Tm are unbounded on F s +p,q; see also §3.3 +below. Our main result in this note shows that, in the case 1 < p, q < ∞, +boundedness holds in the complementary range, except perhaps at the end- +point. + +4 +G. GARRIG´OS +A. SEEGER +T. ULLRICH +Theorem 1.1. Let 1 < p < q < ∞ and 1/q ≤ s < 1/p. Then +∥Tm∥F sp,q→F sp,q + ∥Tm∥F −s +p′q′→F −s +p′q′ ≤ C∥m∥Vu, +if +1 +u > s − 1 +q . +Remark 1.2. The appearance of the variation norms is inspired by a result of +Coifman, Rubio de Francia and Semmes [2] on Fourier multipliers (which is +based on the square function result of Rubio de Francia [8], see also [15, 6]). +However the variation spaces come up in quite different ways in [2] where +the variation norm is taken over dyadic intervals [2j, 2j+1), with a bound +uniformly in j. This has no analogue in our situation as for each interval +[2j, 2j+1) there is only one Haar frequency; instead our conditions involve +the variation norms in the parameter j. +2. Subspaces of Vu +As in [2], in order to analyze functions in Vu it is convenient to consider +certain subspaces Ru of Vu built on convex combinations of characteristic +functions of unions of disjoint dyadic intervals. This is sketched in [2], but +for the convenience of the reader we give a detailed exposition in the setting +of variation spaces for functions on the integers. +For 1 ≤ u < ∞, let ru be the class of functions g : N0 → C which are of +the form +g = +� +ν +aνχIν, +with +( +� +ν +|aν|u)1/u ≤ 1, +where the Iν are mutually disjoint intervals. Then Ru is the space of all +sequences of the form +(2.1) +m = +� +l +clgl, +with +gl ∈ ru +and +� +|cl| < ∞. +The norm ∥m∥Ru is defined as the infimum of � +l |cl| over all representations +as in (2.1). These definitions (for functions on the real line) can be found in +[2]. The following result is a discrete analogue of [2, Lemme 2], whose proof +is sketched for completeness. +Proposition 2.1. For ε > 0, and 1 ≤ u < ∞ we have +(2.2) +Ru ⊂ Vu ⊂ Ru+ε +with continuous embedding. +Proof. Consider first g ∈ ru, with g = � +ν aνχIν. It is straightforward to see +that Vu(g) ≤ 2∥a∥ℓu, thus Ru ⊂ Vu. +For the second inclusion assume that f ∈ Vu, with +Vu(f) = 1, +for some u < ∞. This implies that limn→∞ f(n) exists and is finite. + +HAAR MULTIPLERS +5 +Let w(0) = 0, and for n ≥ 1, let w(n) be the u-th power of the u-variation +of f over [0, n], that is +w(n) = +sup +0≤n0 0 we have +� 2j−1 +� +µ=0 +|2j⟨hj,µ, ρ⟩|u+ε� +1 +u+ε ≤ 3 · 2− 1 +u 2− j +u 2 +j +u+ε =: cj,ε. +Since w is increasing it is clear that the functions +n �→ +1Ileft +j,µ +� +w(n) +� +, +n �→ +1Iright +j,µ +� +w(n) +� +are characteristic functions of intervals restricted to the integers. For fixed +j these intervals are also mutually disjoint, so we see that +gj,ε := +1 +2cj,ε +� +ρj ◦ w +� +∈ ru+ε. +Since Cε := 2 � +j≥0 |cj,ε| < ∞, it then follows that +f = ρ ◦ w = +� 1 +0 +ρ + +∞ +� +j=0 +2cj,ε gj,ε ∈ Ru+ε, +with ∥f∥Ru+ε ≤ 1 + Cε. +□ +Remark 2.2. Observe that the previous proof actually shows that, if m ∈ +Vu and ε > 0, then one can write m = �∞ +j=0 cjmj with mj ∈ ru+ε and +�∞ +j=1 |cj|σ < ∞, for all σ > 0. + +HAAR MULTIPLERS +7 +3. The proof of Theorem 1.1 +We shall actually prove a stronger result than Theorem 1.1, which pro- +vides optimal boundedness (up to endpoints) in a slightly larger region of +indices. Given a fixed q > 1, we denote by Tq the open triangle in the plane +(1/p, s) with vertices (1, 1), (1/q, 1/q) and (1 + 1/q, 1/q); see Figure 2. +1/p +s +1 +2 +1 +q + 1 +1 +1 +q +1 +q −1 +−1 +Figure 2. In red, the region Tq. +For this region we give the following result, which includes Theorem 1.1 +as a special case. +Theorem 3.1. Let 1 < q < ∞ and (1/p, s) ∈ Tq. Then, for all u ≥ 1 such +that 1/u > s − 1/q, it holds +(3.1) +∥Tmf∥F sp,q→F sp,q ≤ c ∥m∥Vu, +m ∈ Vu. +Moreover, a necessary condition for (3.1) to hold for all such m is that +1/u ≥ s − 1/q. +In view of Proposition 2.1, we shall first consider sequences from the class +ru, that is multipliers m of the form +(3.2) +m[a, I] = +� +ν +aν +1Iν, +where I = {Iν} is a family of disjoint intervals and a = {aν} a sequence in +ℓu. For these multipliers we have the following result. +Proposition 3.2. Let 1 < q < ∞ and (1/p, s) ∈ Tq. Then, for all u ≥ 1 +such that 1/u > s − 1/q, there exists c = c(p, q, s, u) > 0 such that +(3.3) +∥Tm[a,I]f∥F sp,q ≤ c ∥a∥ℓu∥f∥F sp,q, +∀ f ∈ F s +p,q(R), a ∈ ℓu, +for every multiplier m[a, I] defined as in (3.2). Moreover, a necessary con- +dition for (3.3) to hold for all such m[a, I] is that 1/u ≥ s − 1/q. +Remark 3.3. We emphasize that the constant c in (3.3) does not depend on +the family of disjoint intervals I = {Iν}. + +8 +G. GARRIG´OS +A. SEEGER +T. ULLRICH +In the next subsections we shall prove Proposition 3.2. For the sufficiency +part we shall use complex interpolation applied to the bilinear operator +(a, f) �−→ T [a, f] := Tm[a,I]f. +For simplicity we shall remove the dependence on I in the subsequent no- +tation, as it will be clear from the proofs that the involved constants do not +depend on it. +3.1. Interpolation with varying q and p = 1. We first prove an inequality for +p = 1 which is efficient for s near 1, namely +(3.4) +∥T [a, f]∥F s +1,q ≲ ∥a∥ℓu∥f∥F s +1,q, +s < 1, +1/u > 1 − 1/q. +Since the Haar system is an unconditional basis on F s +1,1 = Bs +1,1, 0 < s < 1 +(see Theorem 2.9 in [19]) we have +∥T [a, f]∥F s +1,1 ≲ ∥a∥ℓ∞∥f∥F s +1,1, +0 < s < 1. +Next, the uniform boundedness of the operators EN in F s +1,q1 (see [3, Corollary +1.3]) and the trivial estimate in (1.6) (with σ = 1) imply that for any +q1 ∈ (q, ∞) +∥T [a, f]∥F s +1,q1 ≲ ∥a∥1∥f∥F s +1,q1, +0 < s < 1. +By complex interpolation we then obtain (3.4) for 1/u = (1−1/q)/(1−1/q1), +which after choosing q1 large enough implies (3.4) whenever 1/u > 1 − 1/q. +3.2. Interpolation with fixed q. Let q ∈ (1, ∞) be fixed, and let (1/p, s) ∈ Tq. +We shall prove (3.3) by interpolating sufficiently close to the upper vertex +(1, 1) and the lower segment (1/p1, 1/q) of Tq. +1/p +s +1 +2 +1 +q + 1 +1 +1 +q +F 1 +1,q +F 1/q +p1,q +F s +p,q +Figure 3. Interpolation strategy for points (1/p, s) ∈ Tq. +Let P = (1/p, s) ∈ Tq and ε1 > ε0 > 0 be sufficiently small, to be chosen. +Let P0 = (1, s0) with s0 = 1 − ε0. Draw a line through P0 and P, and let + +HAAR MULTIPLERS +9 +P1 = (1/p1, s1) be the intersection with the horizontal line s1 = 1/q − ε1. +That is, +P = (1 − θ) · P0 + θ · P1, +with +θ = s0 − s +s0 − s1 +. +Choosing ε0, ε1 sufficiently small we can guarantee that s1 < s < s0 (and +hence θ ∈ (0, 1)), and that P1 lies in the green region. Next, take +1 +u0 +:= 1 − 1 +q + ε1 − ε0 > 1 − 1 +q. +From the previous step and the unconditional basis property we have +T : ℓu0 × F s0 +1,q → F s0 +1,q +and +T : ℓ∞ × F s1 +p1,q → F s1 +p1,q. +Using complex interpolation this yields +(3.5) +∥T [a, f]∥F sp,q ≲ ∥a∥ℓu∥f∥F s +1,q, +with +1 +u = 1 − θ +u0 ++ θ +∞ = s − s1 +s0 − s1 +· 1 +u0 += s − 1 +q + ε1. +Letting ε1 ց 0 we deduce the validity of (3.5) whenever 1 +u > s − 1 +q. This +completes the proof of the sufficient condition in Proposition 3.2. +3.3. Necessary condition. Suppose first that 1 < p < q with 1/q < s < 1/p. +Then, the example constructed in [13, §5] gives a multiplier of the form +m = +1E, so that card E = 2N (with the elements in E being N-separated), +and with the property that +(3.6) +∥Tm∥F sp,q→F sp,q ≳ 2N(s− 1 +q ). +Since we can write m in the form (3.2) (with Iν = {ν} and aν = 1, for +ν ∈ E), then the validity of (3.3) will imply that +2N(s− 1 +q ) ≲ ∥Tm∥F sp,q→F sp,q ≲ ∥a∥ℓu = 2N/u. +Thus, we must necessarily have 1/u ≥ s − 1/q. +Arguing by interpolation as in [13, §7] one can show that (3.6) (with an +ε loss) continues to hold for all (1/p, s) with +(3.7) +max{1/q, 1/p − 1} < s < min{1/p, 1} +which is a larger region than Tq; see Figure 2. +To be more precise, let P1 = (1/p, s) belong to the open quadrilateral +defined by (3.7), where we assume p ≤ 1. We shall interpolate close to the +points shown in Figure 4. +Namely, given ε > 0, let P0 = (1 +q, 1 +q − ε). Draw a segment from P0 to +P1, and consider the convex combination of P0 and P1 with first coordinate +(1 + ε)−1; i.e let θ ∈ (0, 1) and sθ be such that +(3.8) +� +1 +1+ε, sθ +� += (1 − θ) +�1 +q, 1 +q − ε +� ++ θ +�1 +p, s +� +. + +10 +G. GARRIG´OS +A. SEEGER +T. ULLRICH +1/p +s +1 +2 +1 +q + 1 +1 +1 +q +F s +p,q +F 1/q +q,q +F sθ +1,q +Figure 4. +Then, by complex interpolation we have +∥Tm∥F +sθ +1+ε,q ≲ ∥Tm∥1−θ +F +1 +q −ε +q,q +∥Tm∥θ +F sp,q +By unconditionality, ∥Tm∥ +F +1 +q −ε +q,q +≲ 1, so we arrive at +∥Tm∥F sp,q ≳ ∥Tm∥1/θ +F +sθ +1+ε,q ≳ 2 +N +θ (sθ− 1 +q ), +the last bound due to (3.6). Now, solving for sθ in (3.8) we see that +sθ − 1 +q = +� +s − 1 +q +� +θ − (1 − θ)ε. +Thus, +∥Tm∥F sp,q ≳ 2N [(s− 1 +q )− 1−θ +θ +ε]. +So, if (3.3) was true, arguing as above we would arrive at +1 +u ≥ (s − 1 +q) − 1 − θ +θ +ε, +which letting ε ց 0 leads to 1 +u ≥ s − 1 +q. +3.4. Conclusion of the proof of Theorem 3.1. Let q > 1 and let (1/p, s) ∈ Tq +be fixed. Let 1/u > s − 1/q and m ∈ Vu with u ≥ 1. Then, for some u1 > u +we also have 1/u1 > s − 1/q. By Remark 2.2 we can write m = �∞ +j=0 cjmj +with mj ∈ ru1 and �∞ +j=0 |cj|σ ≲ ∥m∥σ +Vu, with σ = min{1, p}. Then, using +the σ-triangle inequality, we have +∥Tmf∥σ +F sp,q ≤ +∞ +� +j=0 +|cj|σ ∥Tmjf∥σ +F sp,q, +f ∈ F s +p,q. +By Proposition 3.2, ∥Tmjf∥F sp,q ≲ ∥f∥F sp,q, for all j ≥ 0, so we conclude that +∥Tm∥F sp,q→F sp,q ≲ ∥m∥Vu. +□ + +HAAR MULTIPLERS +11 +Remark 3.4. When p > 1, the assertion +∥Tm∥F −s +p′,q′→F −s +p′,q′ ≲ ∥m∥Vu +stated in Theorem 1.1 follows from (3.1) by duality. So, when q > 1, the +condition on Vu is optimal (up to endpoints) also in the lower triangle on +the left of Figure 1. +Remark 3.5. When 1/2 < p ≤ 1, we did not state any result for the right +upper triangle in Figure 2. It is also possible to obtain, by complex inter- +polation, a sufficient condition for multipliers of the form m[a, I] in terms +of ∥a∥ℓu, although in this range the value of u will no longer match the +necessary condition from §3.3. +Remark 3.6. When 1/2 < q ≤ 1, one can also prove by interpolation, for +multipliers of the form m[a, I], that +∥a∥ℓu < ∞, +1 +u > 1 +q − 1 − s, +is a sufficient condition in the open triangle with vertices (0, −1), (1/q, 1/q− +1) and (1/q − 1, 1/q − 1); see Figure 5 below. This matches the necessary +condition from the examples in [13, §5] (except for the endpoint). In the +remaining part of the figure, however, the sufficient condition obtained by +interpolation will be weaker than this one. +1/p +s +1 +2 +1 +1 +q −1 +−1 +Figure 5. Parameter domain for the cases 1/2 < q < 1. +Remark 3.7. It may be interesting to note that even for the special case of +Sobolev spaces Hs +p = F s +p,2, 1 < p < 2, 1/2 ≤ s < 1/p and u < +2 +2s−1 the +use of Triebel-Lizorkin spaces F s +p,q with q ̸= 2 is crucial. Such interpola- +tion arguments were used for multiplier transformations in other contexts +to establish endpoint results on Lorentz spaces Lp,2, see [10, 11] for basic +versions, and [12, 16] for more advanced versions. + +12 +G. GARRIG´OS +A. SEEGER +T. ULLRICH +References +[1] M. Bruneau, Variation totale d’une fonction. (French) Lecture Notes in Mathematics, +Vol. 413. Springer-Verlag, Berlin-New York, 1974. +[2] R. Coifman, J.-L. Rubio de Francia, S. Semmes. Multiplicateurs de Fourier de Lp(R) +et estimations quadratiques. C. R. Acad. Sci. Paris S´er. I Math. 306 (1988), no. 8, +351–354. +[3] Gustavo Garrig´os, Andreas Seeger, T. Ullrich. The Haar system as a Schauder basis +in spaces of Hardy-Sobolev type. Jour. Fourier Anal. Appl., 24 (5) (2018), 1319–1339. +[4] +. Basis properties of the Haar system in limiting Besov spaces. In Geometric +Aspects of Harmonic Analysis, P. Ciatti, A. Martini (eds), Springer Indam Series 45 +(2021), pp 361–424. Also available as preprint arXiv:1901.09117. +[5] +. The Haar system in Triebel-Lizorkin spaces: Endpoint results. J. Geom. Anal. +31 (2021), no. 9, 9045–9089. +[6] Michael T. Lacey. Issues related to Rubio de Francia’s Littlewood-Paley inequality. +New York Journal of Mathematics. NYJM Monographs, 2. State University of New +York, University at Albany, Albany, NY, 2007. 36 pp. +[7] Jaak Peetre. On spaces of Triebel-Lizorkin type. Ark. Mat. 13 (1975),123–130. +[8] J.L. Rubio de Francia. A Littlewood-Paley inequality for arbitrary intervals. Rev. Mat. +Iberoamericana 1 (2) (1985), 1–14. +[9] Thomas Runst, Winfried Sickel. Sobolev spaces of fractional order, Nemytskij oper- +ators, and nonlinear partial differential equations, volume 3 of de Gruyter Series in +Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1996. +[10] Andreas Seeger. A limit case of the H¨ormander multiplier theorem. Monatsh. Math. +105 (2) (1988), 151–160. +[11] +. Estimates near L1 for Fourier multipliers and maximal functions. Arch. +Math. (Basel) 53 (1989), no. 2, 188–193. +[12] Andreas Seeger, Terence Tao. +Sharp Lorentz space estimates for rough operators. +Math. Ann. 320 (2001), no. 2, 381–415. +[13] Andreas Seeger, Tino Ullrich. Haar projection numbers and failure of unconditional +convergence in Sobolev spaces. Math. Z. 285 (2017), 91 – 119. +[14] +. Lower bounds for Haar projections: Deterministic Examples. Constr. Appr. +46 (2017), 227–242. +[15] Per Sj¨olin. A note on Littlewood-Paley decompositions with arbitrary intervals. J. +Approx. Theory 48 (3) (1986), 328–334. +[16] Terence Tao, James Wright. +Endpoint multiplier theorems of Marcinkiewicz type. +Rev. Mat. Iberoamericana 17 (2001), no. 3, 521–558. +[17] H. Triebel.Theory of function spaces. Birkh¨auser Verlag, Basel, 1983. +[18] +. Theory of function spaces II. Monographs in Mathematics, 84. Birkh¨auser +Verlag, Basel, 1992. +[19] +. Bases in function spaces, sampling,discrepancy, numerical integration. EMS +Tracts in Mathematics, 11. European Mathematical Society (EMS), Z¨urich, 2010. + +HAAR MULTIPLERS +13 +Gustavo Garrig´os, Department of Mathematics, University of Murcia, 30100 +Espinardo, Murcia, Spain +Email address: gustavo.garrigos@um.es +Andreas Seeger, Department of Mathematics, University of Wisconsin, 480 +Lincoln Drive, Madison, WI,53706, USA +Email address: seeger@math.wisc.edu +Tino Ullrich, Fakult¨at f¨ur Mathematik, Technische Universit¨at Chemnitz, +09107 Chemnitz, Germany +Email address: tino.ullrich@mathematik.tu-chemnitz.de + diff --git a/o9FKT4oBgHgl3EQfyC7z/content/tmp_files/load_file.txt b/o9FKT4oBgHgl3EQfyC7z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29192a8e442883b2047b7218e653a9c661a82d8d --- /dev/null +++ b/o9FKT4oBgHgl3EQfyC7z/content/tmp_files/load_file.txt @@ -0,0 +1,392 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf,len=391 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='11906v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='CA] 27 Jan 2023 A SUFFICIENT CONDITION FOR HAAR MULTIPLIERS IN TRIEBEL-LIZORKIN SPACES GUSTAVO GARRIG´OS ANDREAS SEEGER TINO ULLRICH In memory of Guido Weiss Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We consider Haar multiplier operators Tm acting on Sobolev spaces, and more generally Triebel-Lizorkin spaces F s p,q(R), for indices in which the Haar system is not unconditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' When m depends only on the Haar frequency, we give a sufficient condition for the boundedness of Tm in F s p,q, in terms of the variation norms ∥m∥Vu, which is optimal in u (up to endpoints) when p, q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Introduction Consider the classical Haar system in R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='1) H = � hj,µ : j ≥ −1, µ ∈ Z � , where, if h = 1[0,1/2) − 1[1/2,1), we let hj,µ(x) = h(2jx − µ) , for j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' , µ ∈ Z, while for j = −1 we let h−1,µ = 1[µ,µ+1), µ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We shall refer to the elements of the family Hj = {hj,µ : µ ∈ Z} as Haar functions of frequency 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Let F s p,q denote the usual Triebel-Lizorkin space in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' It is known from the work of Triebel [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='ii] that H is an unconditional basis of F s p,q(R) when s belongs to the range (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='2) max � 1/p − 1, 1/q − 1 � < s < min � 1/p, 1/q, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' That this range is actually optimal was shown by the last two authors in [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' More recently, we proved in [3] that H is a Schauder basis of F s p,q(R) (with respect to natural enumerations) in the larger range (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='3) 1/p − 1 < s < min � 1/p, 1 � , (for all 0 < q < ∞), Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' 46E35, 46B15, 42C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Haar basis, Triebel-Lizorkin spaces, multipliers, variation norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' 1 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' GARRIG´OS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' SEEGER T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' ULLRICH while at the endpoints (see [5]) the property holds if and only if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='4) s = 1/p − 1 and 1/2 < p ≤ 1, also for all 0 < q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' These regions are depicted in Figure 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' unconditional 1 p s 1 2 1 1 q 1 q −1 −1 Schauder 1 p s 1 2 1 −1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Parameter domain for H to be an unconditional basis (left figure) or a Schauder basis (right figure) in F s p,q(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We shall mainly be interested in values of the parameters outside the region of unconditionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' In that range, it becomes a natural question to find sufficient conditions on a sequence {mj,µ} so that the mapping f �−→ � j≥0 � µ∈Z mj,µ 2j⟨f, hj,µ⟩hj,µ, defined say for f ∈ span H , extends as a bounded linear operator in the space F s p,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' In this paper we regard this problem in the special case when the sequence is constant in each frequency level, namely, if m = {m(j)}j≥0, we consider the operators Tmf = � j≥0 m(j) Djf, where Dj denotes the orthogonal projection onto the space generated by Hj, that is Djf = � µ∈Z 2j⟨f, hj,µ⟩ hj,µ, j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' It is well known that one can write Dj = Ej+1 − Ej, where Ej is the conditional expectation operator defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='5) Ejf(x) = � µ∈Z 1Ij,µ(x) 2j � Ij,µ f(y)dy , HAAR MULTIPLERS 3 associated with the dyadic intervals Ij,µ = [µ2−j, (µ + 1)2−j), µ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' The uniform boundedness of the operators EN in F s p,q (and Bs p,q) has been throughly studied in the papers [3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' In particular, it is shown in those papers that H is a Schauder basis of F s p,q (with respect to natural enumerations) if and only if sup N≥0 ��EN �� F sp,q→F sp,q < ∞ and span H is dense in F s p,q, and this in turn is equivalent to (s, p, q) belonging to the ranges in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' In those cases, an elementary summation by parts argument and the σ-triangle inequality, with σ = min{1, p, q}, imply that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='6) ∥Tmf∥F sp,q ≲ ∥m∥ℓ∞ + � ∞ � j=1 |m(j) − m(j − 1)|σ� 1 σ , for all f ∈ span H with ∥f∥F sp,q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We shall next formulate a stronger multiplier result which involves the Wiener space notion of sequences of bounded u-variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We recall how these are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' If u ≥ 1, we let Vu(m) be the u-variation of the sequence {m(j)}j≥0, defined by Vu(m) = sup � N � n=1 |m(jn) − m(jn−1)|u�1/u with the supremum taken over all finite strings of numbers {j0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' , jN} satisfying jn−1 < jn for 1 ≤ n ≤ N, and jn ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Note that if u = 1 we simply have V1(m) = ∞ � j=1 |m(j) − m(j − 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We denote by Vu the space of all m : N ∪ {0} → C for which ∥m∥Vu := ∥m∥∞ + Vu(m) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' In particular, if 1 ≤ u1 ≤ u2 < ∞, it holds V1 ֒→ Vu1 ֒→ Vu2 ֒→ ℓ∞, As an example, observe that m(n) = 1/(n + 1)α belongs to V1 for all α > 0, while the alternate sequence M(n) = (−1)nm(n) belongs to Vu iff α > 1/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' We wish to find, in the region of exponents (s, p, q) where H is a con- ditional basis of F s p,q, the largest possible u for which m ∈ Vu implies the boundedness of the operator Tm in F s p,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' The examples given in [13], based on multipliers taking the values 0 and 1 (suitable characteristic functions of finite sets of integers), show that for 1/u < s − 1/q there are m ∈ Vu such that the corresponding operators Tm are unbounded on F s p,q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' see also §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Our main result in this note shows that, in the case 1 < p, q < ∞, boundedness holds in the complementary range, except perhaps at the end- point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' GARRIG´OS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' SEEGER T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' ULLRICH Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Let 1 < p < q < ∞ and 1/q ≤ s < 1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Then ∥Tm∥F sp,q→F sp,q + ∥Tm∥F −s p′q′→F −s p′q′ ≤ C∥m∥Vu, if 1 u > s − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' The appearance of the variation norms is inspired by a result of Coifman, Rubio de Francia and Semmes [2] on Fourier multipliers (which is based on the square function result of Rubio de Francia [8], see also [15, 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' However the variation spaces come up in quite different ways in [2] where the variation norm is taken over dyadic intervals [2j, 2j+1), with a bound uniformly in j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' This has no analogue in our situation as for each interval [2j, 2j+1) there is only one Haar frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' instead our conditions involve the variation norms in the parameter j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Subspaces of Vu As in [2], in order to analyze functions in Vu it is convenient to consider certain subspaces Ru of Vu built on convex combinations of characteristic functions of unions of disjoint dyadic intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' This is sketched in [2], but for the convenience of the reader we give a detailed exposition in the setting of variation spaces for functions on the integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' For 1 ≤ u < ∞, let ru be the class of functions g : N0 → C which are of the form g = � ν aνχIν, with ( � ν |aν|u)1/u ≤ 1, where the Iν are mutually disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Then Ru is the space of all sequences of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='1) m = � l clgl, with gl ∈ ru and � |cl| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' The norm ∥m∥Ru is defined as the infimum of � l |cl| over all representations as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' These definitions (for functions on the real line) can be found in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' The following result is a discrete analogue of [2, Lemme 2], whose proof is sketched for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' For ε > 0, and 1 ≤ u < ∞ we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content='2) Ru ⊂ Vu ⊂ Ru+ε with continuous embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' Consider first g ∈ ru, with g = � ν aνχIν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' It is straightforward to see that Vu(g) ≤ 2∥a∥ℓu, thus Ru ⊂ Vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' For the second inclusion assume that f ∈ Vu, with Vu(f) = 1, for some u < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' This implies that limn→∞ f(n) exists and is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FKT4oBgHgl3EQfyC7z/content/2301.11906v1.pdf'} +page_content=' HAAR MULTIPLERS 5 Let w(0) = 0, and for n ≥ 1, let w(n) be the u-th power of the u-variation of f over [0, n], that is w(n) = sup 0≤n0 2 cannot occur for so-called geometrically admissible +algebras A. In particular, this result holds for finite-dimensional central simple associative, Lie or +Jordan algebras; see Section 4.2. More precisely, we have the following result; see Theorem 4.1. +Theorem A. Let A be a geometrically admissible algebra over a field k of characteristic 0, e.g. a +finite-dimensional central simple Lie, associative, or Jordan algebra over k. +The double D(A[[z]], δ) of a non-degenerate topological D-bialgebra (A[[z]], δ) is isomorphic to +A((z)) × A[z]/znA[z] for some n ∈ {0, 1, 2}. +■ +Let us note that, if A is a Lie algebra, Theorem A is one of the main results in [MSZ10]. However, +our proof is independent of the proof in [MSZ10] and is based on the geometrization of A-lattices +(see Section 4.1). This method was already used in order to give a new proof of the Belavin- +Drinfeld trichotomy of non-degenerate solutions of the CYBE (1.2) in [Abe21] and the classification +of topological Lie bialgebras in [AMSZ22]. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +3 +In Section 5, we will show that every non-degenerate topological D-bialgebra (A[[z]], δ) is of the +form a(z) �→ r(x, y)a(x)(1) − a(y)(2)r(x, y) for a solution +(1.3) +r(x, y) = λ(x)ynγ +x − y ++ t(x, y) ∈ (A ⊗ A)[[x, y]][(x − y)−1] +of the so-called A-classical Yang-Baxter equation (A-CYBE) +(1.4) +r13(z1, z3)r12(z1, z2) − r12(z1, z2)r23(z2, z3) + r23(z2, z3)r13(z1, z3) = 0. +Here, λ ∈ k[[x]]×, t ∈ (g ⊗ g)[[x, y]] and γ ∈ A ⊗ A is a canonical A-invariant element determined +by the algebra metric β of A; see Section 5.1 for details. In particular, the classification of non- +degenerate topological D-bialgebras (A[[z]], δ) up to isomorphism is equivalent to the classification +of solutions of the A-CYBE (1.4) of the form (1.3) up to a certain type of equivalence relation. +The A-CYBE already appeared in several special cases in literature. If A = g is a Lie algebra, +(1.4) is exactly the usual CYBE (1.2). For constant r ∈ A ⊗ A, (1.4) was examined in [Agu01] +for an associative algebra A and in [Zhe99] for a Jordan algebra A. An endomorphism version of +(1.4) for an associative algebra A and meromorphic functions r = r(x, y) in two complex variables +was related to pairs of compatible associative algebra structures in [OS08]. We also point out +that solutions of the associative Yang-Baxter equation introduced by Polishchuk in [Pol02, Eq. +(0.1)] which do not depend on the first parameter are precisely difference depending meromorphic +solutions of the A-CYBE for associative algebras A. +In Section 6, we will see that non-degenerate topological D-bialgebras can be categorized rather +explicitly by the form of their associated solution of the A-CYBE if A is a so-called strongly +geometrically admissible k-algebra; see Subsection 6.1. The most important examples are finite- +dimensional simple Lie, Jordan, and associative algebras over an algebraically closed field of char- +acteristic 0; see Section 6.2. More precisely, we have the following result; see Theorem 6.1. +Theorem B. Let k be an algebraically closed of characteristic 0 and A be a unital strongly ge- +ometrically admissible algebra (e.g. a finite-dimensional, central, simple associative or Jordan al- +gebra). Furthermore, let (A[[z]], δ) be a non-degenerate topological D-bialgebra in some category of +k-algebras closed under taking subalgebras. +Up to isomorphism of topological D-bialgebras, δ = δr for a solution of the A-CYBE of precisely +one of the following forms: +(1) r is trigonometric in the sense that there exists σ ∈ Autk-alg(A) of order m ∈ N and t ∈ +L(A, σ) ⊗ L(A, σ) such that +r(x, y) = +1 +exp (x − y) − 1 +m−1 +� +j=0 +exp +�x − y +m +� +γj + t +� +exp +� x +m +� +, exp +� y +m +�� +. +Here, γj ∈ A ⊗ A is uniquely determined by γ = �m−1 +j=0 γj and (σ ⊗ 1)γj = εjγj for some +primitive m-th root of unity ε ∈ k, where γ ∈ A ⊗ A is the canonical A-invariant element; +(2) r is rational in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r(x, y) = +γ +x−y + t(x, y); +(3) r is quasi-trigonometric in the sense that there exists a polynomial t ∈ (A ⊗ A)[x, y] such that +r(x, y) = +yγ +x−y + t(x, y); +(4) r is quasi-rational in the sense that there exists a polynomial t ∈ (A ⊗ A)[x, y] such that +r(x, y) = y2γ +x−y + t(x, y). +In particular, every solution of the A-CYBE (1.4) of the form (1.3) is, up to equivalence, of one +of the above forms. +■ +The analog of Theorem B for a Lie algebra A = g was proven in [AMSZ22] and can be seen as a +generalization of the Belavin-Drinfeld trichotomy for non-degenerate r-matrices from [BD83]. A +consequence of this result is, that all topological Lie bialgebras (g[[z]], δ) are classified. The proof +of Theorem B is, under consideration of Theorem A, similar to the proof of its Lie algebra analog + +4 +RASCHID ABEDIN +in [AMSZ22]. More precisely, it proceeds by refining the algebro-geometric methods already used +to proof Theorem A. Namely, we can assign a particular type of geometric data, called geometric +A-CYBE datum (see Section 6.3), to any Lagrangian subalgebra W ⊆ Dn(A) complementary to +A[[z]]; see Section 6.4. If this assignment is done, Theorem B is a consequence of the classification +results for sheaves of algebras on the (punctured) affine line presented in Proposition 6.9, which is +a consequence of the results from [Pia05]; see [Abe22, Theorem 6.1.1] for details. +Let us point out that the unitality assumption in Theorem B is actually rather weak, since +strongly geometrically admissible power associative algebras which are not anti-commutative are +automatically unital; see Remark 6.3. The most interesting strongly geometrically admissible anti- +commutative algebras are precisely Lie algebras, where the analog of Theorem B is already known +as mentioned above. +We conclude this paper, by using Theorem B to classify all topological associative D-bialgebras +(A[[z]], δ) for A associative and D(A[[z]], δ) ≇ D(A[[z]], 0). It turns out that the trigonometric and +quasi-trigonometric cases do not occur. More precisely, we obtain the following result in Section +7. +Theorem C. Let A be a finite-dimensional simple associative k-algebra over an algebraically closed +field k of characteristic 0, i.e. A ∼= Mn(k), and (A[[z]], δ) be a topological associative D-bialgebra +such that D(A[[z]], δ) ≇ D(A[[z]], 0). +Then, up to isomorphism, δ = δr where r is either the rational or the quasi-rational solution of +the A-CYBE determined by an associative Stolin pair (S, B) of class k ∈ 0, n − 1; see Section 7.4 +and Section 7.5 for details. +In particular, every solution of the A-CYBE (1.4) of the form (1.3) is, up to equivalence, of one +of the above forms. +■ +Let us remark that meromorphic solutions of the Mn(C)-CYBE which depend on the difference of +their variables and have diagonal residue γ were already shown to be rational up to equivalence in +[Pol09, Theorem 0.2]. +We were unable to provide examples of quasi-trigonometric and trigonometric solutions of the +A-CYBE for non-associative unital strongly geometrically admissible algebras A as well. We con- +jecture that, similar to the associative case in Theorem C, the unitality obstructs the existence of +these solutions. +Acknowledgments. I thank Ivan Shestakov for explaining several facts about non-associative +algebras to me. This work was supported by the DFG grant AB 940/1–1. It was also supported +as a part of NCCR SwissMAP, a National Centre of Competence in Research, funded by the Swiss +National Science Foundation (grant number 205607). +Notation and conventions. If the reader is unsure about the meaning of symbols or names +which are ambiguously used in literature, we refer to the Appendix A. There we have tried to give +an overview on our conventions. +2. Introduction to D-bialgebras +2.1. Survey on Manin triples. Throughout this paper k is a field and all algebras, vector spaces, +tensor products, etc. are understood over k if not stated otherwise. Let us remark that for us an +R-algebra over an (unital, commutative, associative) ring R satisfies no additional assumptions: an +R-algebra A = (A, µ) consists of an R-module A equipped with an R-linear map µ: A ⊗R A → A, +called multiplication map. We write ab := µ(a ⊗ b) for a, b ∈ A if no confusion arises. +2.1.1. Metric algebras. Let R be a ring and A be an R-algebra. We call a map β : A × A → R +algebra metric if it is non-degenerate, symmetric, associative, and R-bilinear. +Thereby, “non- +degenerate” means that the canonical map A → HomR(A, R) defined by a �→ β(a, −) is injective + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +5 +and “associative” means that +(2.1) +β(ab, c) = β(a, bc) +holds for all a, b, c ∈ A. We call a pair (A, β) metric R-algebra if A is an R-algebra equipped with +an algebra metric β : A × A → R. Moreover, two metric algebras (A1, β1) and (A2, β2) are called +isomorphic, written (A1, β1) ∼= (A2, β2), if there exists an R-algebra isomorphism ϕ: A1 → A2 +such that β2(ϕ(a), ϕ(b)) = β1(a, b) for all a, b ∈ A1. +2.1.2. Manin pairs and Manin triples. A Manin pair ((M, β), N) consists of a metric k-algebra +(M, β) and a Lagrangian subalgebra N ⊆ M. In other words, N ⊥ = N ⊆ M is a subalgebra. +A Manin triple ((M, β), M+, M−) consists of a metric k-algebra (M, β) and subalgebras M± ⊆ +M such that M = M+ ⊕ M− and M± ⊆ M ⊥ +±. +It is easy to see that for any Manin triple +((M, B), M+, M−), M ⊥ +± = M± already holds. In particular, ((M, β), M+) and ((M, β), M−) are +automatically Manin pairs. +Remark 2.1. +In literature, Manin pairs and triples are usually only defined for Lie algebras. The +definition given here is a straight-forward generalization to arbitrary algebras. +♦ +2.1.3. Manin triples and comultiplication maps. Recall that a k-coalgebra C is a k-vector space +equipped with a k-linear map δ: C → C ⊗ C, called comultiplication map. +The restriction of +δ∗ : (C ⊗ C)∗ → C∗ to C∗ ⊗ C∗ ⊆ (C ⊗ C)∗ always defines k-algebra structure on C∗, hence the +name. Explicitly, the multiplication f1f2 ∈ C∗ of two maps f1, f2 ∈ C∗ is defined by +(2.2) +f1f2(a) := (f1 ⊗ f2)δ(a) +for all a ∈ C. +Let us note that for an infinite-dimensional algebra A, A∗ is not necessarily a coalgebra, since +the dual A∗ → (A⊗A)∗ of the multiplication map might fail to have values in A∗ ⊗A∗ ⊆ (A⊗A)∗. +Two k-coalgebras (C1, δ1) and (C2, δ2) are called isomorphic, written C1 ∼= C2, if there exists +an isomorphism ϕ: C1 → C2 of vector spaces satisfying (ϕ ⊗ ϕ)δ1 = δ2ϕ. +We say that a comultiplication map δ: M+ → M+ ⊗ M+ is determined by a Manin triple +((M, β), M+, M−) if +(2.3) +β⊗2(δ(a), b1 ⊗ b2) = B(a, b1b2) +holds for all a ∈ M+ and b1, b2 ∈ M−. Here, β⊗2(a1 ⊗ a2, b1 ⊗ b2) := β(a1, b1)β(a2, b2). The +name stems from the fact that, if �δ: M+ → M+ ⊗ M+ is another comultiplication determined by +((M, β), M+, M−), we have δ = �δ. +2.1.4. Isomorphism of Manin pairs and Manin triples. We call two Manin pairs (resp. Manin +triples) +((M1, β1), N1) and ((M2, β2), N2) +(resp. ((M1, β1), M1,+, M1,−) and ((M2, β2), M2,+, M2,−)) +isomorphic if there exists an isomorphism ϕ: (M1, β1) → (M2, β2) of metric algebras such that +ϕ(N1) = N2 (resp. ϕ(M1,±) = M2,±). In this case, we write ((M1, β1), N1) ∼= ((M2, β2), N2) (resp. +((M1, β1), M1,+, M1,−) ∼= ((M2, β2), M2,+, M2,−)). +Assume that ((Mi, βi), Mi,+, Mi,−) determines a comultiplication δi on Mi,+ for i ∈ {1, 2}. +Then ((M1, β1), M1,+, M1,−) ∼= ((M2, β2), M2,+, M2,−) via an isomorphism ϕ: (M1, β1) → (M2, β2) +implies that +β2((ϕ ⊗ ϕ)δ1(a), b1 ⊗ b2) = β1(δ1(a), ϕ−1(b1) ⊗ ϕ−1(b2)) = β1(a, ϕ−1(b1)ϕ−1(b2)) += β1(a, ϕ−1(b1b2)) = β2(ϕ(a), b1b2) = β2(δ2(ϕ(a)), b1 ⊗ b2) +holds for all a ∈ M1,+ and b1, b2 ∈ M2,−. Consequently, (ϕ ⊗ ϕ)δ1 = δ2ϕ holds, so M1,+ ∼= M2,+ +holds both as algebras and coalgebras. + +6 +RASCHID ABEDIN +2.2. D-bialgebras. Let us call a pair (A, δ) consisting of a k-algebra A and a comultiplication +map δ: A → A⊗A bialgebra. In particular, we do not assume any compatibility conditions between +multiplication and comultiplication of A in this definition. +To any bialgebra (A, δ), there is a unique k-algebra structure on D(A, δ) := A ⊕ A∗ such that +((D(A, δ), ev), A, A∗) is a Manin triple determining δ, where: +• The multiplication of A∗ is defined by the comultiplication δ in the sense of Subsection 2.1.3; +• ev: D(A, δ) × D(A, δ) → k is the evaluation pairing +(2.4) +ev(a + f, b + g) = f(b) + g(a) , +a, b ∈ A and f, g ∈ A∗. +Explicitly, the multiplication on D(A, δ) is determined by: +• A, A∗ ⊆ D(A, δ) are subalgebras; +• The identities +ev(af, b) = ev(f, ba) = f(ba) = ev(fRa, b) +ev(af, g) = ev(a, fg) = (f ⊗ g)δ(a) = ev((f ⊗ 1)δ(a), g) +(2.5) +for a, b ∈ A, f, g ∈ A∗ yield af = (f ⊗ 1)δ(a) + fRa and similarly fa = (1 ⊗ f)δ(a) + fLa holds. +Here, R, L: A → End(A) denote the right and left multiplication maps respectively, i.e. +(2.6) +Rab = ba = Lba +for all a, b ∈ A. +The k-algebra D(A, δ) associated to a bialgebra (A, δ) is called called classical double of (A, δ). +Let Algk be the category of k-algebras, i.e. the category with k-algebras as objects and k- +algebra homomorphisms as morphisms. Furthermore, let C be a full subcategory of Algk closed +under taking subalgebras. For instance, C can be any subcategory of equation based k-algebras +like the category of Lie algebras, associative algebras or Jordan algebras. +We call a bialgebra (A, δ) D-bialgebra in C if D(A, δ) is an algebra in C. Observe that, by +construction, A, A∗ ∈ C and ((D(A, δ), ev), A, A∗) is a Manin triple determining δ. +Let us point out that D-bialgebras in Algk are exactly bialgebras. +2.2.1. Isomorphism of D-bialgebras. Let C be a full subcategory of Algk closed under taking sub- +algebras. Two D-bialgebras (A1, δ1) and (A2, δ2) in C are called isomorphic, written (A1, δ1) ∼= +(A2, δ2), if there exists a k-linear map ϕ: A1 → A2 which is both an isomorphism of k-algebras +and k-coalgebras, i.e. if the identities +(2.7) +ϕ(a1a2) = ϕ(a1)ϕ(a2) and (ϕ ⊗ ϕ)δ1(a) = δ2(ϕ(a)) +hold for all a, a1, a2 ∈ A1 . +Lemma 2.2. Let C be a full subcategory of Algk closed under taking subalgebras and (A1, δ1),(A2, δ2) +be two D-bialgebras in C. Then +(A1, δ1) ∼= (A2, δ2) ⇐⇒ ((D(A1, δ1), ev), A1, A∗ +1) ∼= ((D(A2, δ2), ev), A2, A∗ +2). +Proof. The fact that ((D(A1, δ1), ev), A1, A∗ +1) ∼= ((D(A2, δ), ev), A2, A∗ +2) implies (A1, δ1) ∼= (A2, δ2) +was already mentioned in Section 2.1.4. On the other hand, let ϕ: A1 → A2 define the isomorphism +(A1, δ1) ∼= (A2, δ2). It is easy to see that �ϕ(a + f) := ϕ(a) + fϕ−1, where a ∈ A1, f ∈ A∗ +1, defines +an isomorphism ((D(A1, δ1), ev), A1, A∗ +1) ∼= ((D(A2, δ), ev), A2, A∗ +2). +■ +2.3. Examples. In [Zhe97], the D-bialgebra structures for the most important categories C of +algebras where discussed. Let us give a short outline of their explicit descriptions. In the following, +we write for any elements a, a1, . . . , an in some k-algebra A +a(i)(a1 ⊗ · · · ⊗ an) = a1 ⊗ · · · ⊗ aai ⊗ · · · ⊗ an +(a1 ⊗ · · · ⊗ an)a(i) = a1 ⊗ · · · ⊗ aia ⊗ · · · ⊗ an. +(2.8) + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +7 +2.3.1. D-bialgebras in the category of Lie algebras. Recall that a Lie bialgebra (L, δ) consists of a +Lie algebra L equipped with a linear map δ: L → L ⊗ L such that: +• δ is a 1-cocycle, i.e. for all a, b ∈ L +δ(ab) = (a(1) + a(2))δ(b) + δ(a)(b(1) + b(2)); +• The restriction of δ∗ to L∗ ⊗ L∗ ⊆ (L ⊗ L)∗ is a Lie bracket. +It is well-known that for a linear map δ: L → L ⊗ L on a Lie algebra L the double D(L, δ) is again +a Lie algebra if and only if (L, δ) is a Lie bialgebra. Therefore, a D-bialgebra in the category of +Lie algebras is exactly a Lie bialgebra. +2.3.2. D-bialgebras in the category of associative algebras. An infinitesimal bialgebra (A, δ) consists +of a Lie algebra A equipped with a cobracket δ: A → A ⊗ A such that: +• δ is a 1-cocycle, i.e. for all a, b ∈ A +δ(ab) = a(1)δ(b) + δ(a)b(2); +• The restriction of δ∗ to A∗ ⊗ A∗ ⊆ (A ⊗ A)∗ is an associative multiplication. +It was shown by Aguiar [Agu01] that there is classical double like construction for infinitesimal +algebras, this time by giving D(A, δ) := (A ⊕ A∗) ⊕ (A ⊗ A∗) an associative algebra structure. In +general, these are not related to Manin triples. However, under the condition that δ is balanced, +i.e. if for all a1, a2 ∈ A +(2.9) +a(1) +1 τδ(a2) + a(2) +2 δ(a1) = δ(a1)a(1) +2 ++ τδ(a2)a(2) +1 +holds, the double can be reduced to Dred(A, δ) = A⊕ A∗. It turns out that Dred(A, δ) = D(A, τδ), +so τδ is an associative D-bialgebra structure, i.e. a D-bialgebra in the category of associative +k-algebras. Here and in the following τ(a ⊗ b) = b ⊗ a. +For every bialgebra (A, δ), we call (A, τδ) the co-opposite bialgebra of (A, δ). It is shown in +[Zhe97] that indeed all associative D-bialgebra structures are of the above form, i.e. the co-opposites +of balanced infinitesimal bialgebras are exactly associative D-bialgebras. +2.3.3. D-bialgebras in the category of Jordan algebras. The D-bialgebra structures in the category +of Jordan algebras, which we will simply call Jordan bialgebras, where found in [Zhe97, Theorem +2]: a Jordan bialgebra (J, δ) consists of a Jordan algebra J and a linear map δ: J → J ⊗ J such +that J∗ is a Jordan algebra and the following identities hold: +1 +2((δ ⊗ 1) − (1 ⊗ δ))δ(a2) = a(2)(δ ⊗ 1 − 1 ⊗ δ)δ(a) + (a(3) − a(1))(1 ⊗ τ)(δ ⊗ 1)δ(a) ++ (δ(a) ⊗ 1 − 1 ⊗ δ(a))(1 ⊗ τ)(δ(a) ⊗ 1); +(δ ⊗ 1 + 1 ⊗ δ + (1 ⊗ τ)(δ ⊗ 1))(1 ⊗ a + a ⊗ 1)δ(a) = 2a(2)(1 ⊗ δ)δ(a) + a(1)(1 ⊗ τ)(δ ⊗ 1)δ(a) ++ (1 ⊗ δ(a))(1 ⊗ τ)(δ(a) ⊗ 1) + (δ ⊗ 1)δ(a2); +δ(a2b) − δ(a2)b(1) − δ(b)(a2)(2) + 2δ(b)(a ⊗ a) − 2δ(ab)a(1) + 2(δ(a)b(1))a(1) ++ 2(δ(a)b(2))a(2) − 2δ(a)(ab)(2) = 0. +3. Non-degenerate topological D-bialgebra structures on power series algebras +3.1. Topological D-bialgebra structures on power series algebras. Let A be a finite- +dimensional k-algebra and let us equip A[[z]] (resp. (A⊗A)[[x, y]]) with the (z)-adic (resp. (x, y)-adic) +topology; see Appendix A for definition of (·)[[z]] and (·)[[x, y]]. Note that if δ: A[[z]] → (A⊗A)[[x, y]] +is a continuous linear map and if (·)∨ denotes taking the continuous dual space, A[[z]]∨ is naturally +a k-algebra with the multiplication defined by +(3.1) +A[[z]]∨ ⊗ A[[z]]∨ ∼= (A ⊗ A)[[x, y]]∨ +δ∨ +−→ A[[z]]∨. + +8 +RASCHID ABEDIN +We call a pair (A[[z]], δ) as above topological bialgebra. +For any topological bialgebra (A[[z]], δ), there is a unique k-algebra structure on D(A[[z]], δ) = +A[[z]] ⊕ A[[z]]∨ such that ((D(A[[z]], δ), ev), A[[z]], A[[z]]∨) is a Manin determining δ. Here, the eval- +uation pairing ev: D(A[[z]], δ) × D(A[[z]], δ) → k is defined analogous to (2.4). The multiplication +map of D(A, δ) satisfying these conditions can be explicitly determined in the same way as in the +non-topological setting in Section 2.2. The algebra D(A[[z]], δ) is called classical double of (A[[z]], δ). +Let C be a full subcategory of Algk closed under taking subalgebras. We call a topological +bialgebra (A[[z]], δ) topological D-bialgebra in C if D(A[[z]], δ) is an algebra in C. Observe that if +(A[[z]], δ) is a topological D-bialgebra in C, we have A, A[[z]], A[[z]]∨ ∈ C. +It is easy to see that (A[[z]], δ) is a topological D-bialgebra in C if and only if (A[[z]]∨, µ∨) is a +usual D-bialgebra in C, where µ: (A ⊗ A)[[x, y]] → A[[z]] is the multiplication map. Therefore, we +can describe topological D-bialgebras in the most important categories of algebras using the same +axioms as in Section 2.3. +3.1.1. Isomorphism of topological D-bialgebras on series. Let A be a k-algebra and C be a full +subcategory of Algk closed under taking subalgebras and such that A[[z]] ∈ C. Two topological D- +bialgebras (A1[[z]], δ1) and (A2[[z]], δ2) in C are called isomorphic, written (A1[[z]], δ1) ∼= (A2[[z]], δ2), +if there exists a continuous linear map ϕ: A1[[z]] → A2[[z]] which is both an isomorphism of algebras +and coalgebras, i.e. if for every a, a1, a2 ∈ A1 the identities +(3.2) +ϕ(a1a2) = ϕ(a1)ϕ(a2) and (ϕ ⊗ ϕ)δ1(a) = δ2(ϕ(a)) +hold. Here, in the latter equation ϕ ⊗ ϕ was continuously extended from an automorphism of +A[[z]] ⊗ A[[z]] to an automorphism of (A ⊗ A)[[x, y]]. +Lemma 3.1. Let A be a k-algebra and C be a full subcategory of Algk closed under taking subal- +gebras and (A1[[z]], δ1),(A2[[z]], δ2) be two D-bialgebras in C. Then (A1[[z]], δ1) ∼= (A2[[z]], δ2) if and +only if +((D(A1[[z]], δ1), ev), A1[[z]], A1[[z]]∨) ∼= ((D(A2[[z]], δ2), ev), A2[[z]], A2[[z]]∨) +via an isomorphism D(A1[[z]], δ1) ∼= D(A2[[z]], δ2) which restricts to a continuous isomorphism +A1[[z]] → A2[[z]]. +Proof. Repeat the arguments in the proof of Lemma 3.1 under consideration of the fact that any +continuous linear isomorphism A1[[z]] → A2[[z]] has a continuous inverse, since A1[[z]] is linearly +compact. +■ +Remark 3.2. +Let A be a finite-dimensional central simple k-algebra. Then [AMSZ22, Theorem +3.3] states that for every φ ∈ Autk-alg(A[[z]]) exists a ϕ ∈ Autk[[z]]-alg(A[[z]]) ⊆ End(A)[[z]] and +u ∈ zk[[z]]× such that +(3.3) +φ(a)(z) = ϕ(z)a(u(z)) +for all a ∈ A[[z]]. As a consequence, every k-algebra automorphism of A[[z]] is continuous in the +(z)-adic topology. In particular, in this case Lemma 3.1 can be refined for A = A1 = A2 as +(A[[z]], δ1) ∼= (A[[z]], δ2) ⇐⇒ ((D(A[[z]], δ1), ev), A[[z]], A[[z]]∨) ∼= ((D(A[[z]], δ2), ev), A[[z]], A[[z]]∨). +♦ +3.2. Non-degenerate topological D-bialgebra structures. Consider Manin triples of the form +(3.4) +((Dn(A), β(n,λ)), A[[z]], W), +where: +• (A, β) is a finite-dimensional metric k-algebra (recall the definition from Section 2.1); +• n ∈ N and Dn(A) := A((z)) × A[z]/znA[z]; + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +9 +• A[[z]] is identified with the image of the diagonal embedding A[[z]] → Dn(A) defined by +a �−→ (a, [a]). +Here, for any a ∈ A[[z]], [a] := a + znA[[z]] ∈ A[[z]]/znA[[z]] = A[z]/znA[z]. +• λ ∈ k[[z]]× and β(n,λ) is given by +(3.5) +β(n,λ)((a1, [a2]), (b1, [b2])) = res0 +1 +znλ(β(a1, b1) − β(a2, b2)), +where β was extended to a k((z))-bilinear form A((z)) × A((z)) → k((z)) on the right-hand side. +It is easy to see that all triples of the form (3.4) are indeed Manin triples in the sense of Definition +2.1.2. +Let C be a full subcategory of Algk that is closed under taking subalgebras. We call a topological +D-bialgebra (A[[z]], δ) in C non-degenerate if and only if there exist n ∈ N and λ ∈ k[[z]]× such that +((D(A[[z]], δ), ev), A[[z]]) ∼= ((Dn(A), β(n,λ)), A[[z]]) as Manin pairs (see Section 2.1.4). +In other +words, (A[[z]], δ) is non-degenerate if and only if there exist n ∈ N and λ ∈ k[[z]]× such that +Dn(A) ∈ C and δ is determined by the Manin triple ((Dn(A), β(n,λ)), A[[z]], W) for an appropriate +W ⊆ Dn(A). +Remark 3.2 implies that, if A is central and simple. the classification of non-degenerate topolog- +ical D-bialgebra structures on A[[z]] up to isomorphisms of topological D-bialgebras is equivalent +to the classification of Manin triples of the form (3.4) up to isomorphisms of Manin triples. +3.3. Connection to trace extensions of k[[z]]. A trace extension (R, t) of k[[z]] consists of a +commutative and associative k-algebra extension R ⊇ k[[z]] equipped with a linear map t: R → k, +called trace map, such that: +(1) (a, b) �→ βt(a, b) := t(ab) is an algebra metric making k[[z]] ⊆ R a Lagrangian subalgebra; +(2) For all continuous (in the (z)-adic topology) linear maps f : k[[z]] → k exists an a ∈ R such +that f(b) = t(ab) for all b ∈ k[[z]]. +In other words, trace extensions (R, t) are in bijection with Manin pairs ((R, βt), k[[z]]) for which +R is associative and commutative and βt satisfies (2). +Two trace extensions (R1, t1) and (R2, t2) are called isomorphic, written (R1, t1) ∼= (R2, t2), if +their associated Manin pairs are isomorphic. In other words, (R1, t1) ∼= (R2, t2) if there exists an +algebra isomorphism ϕ: R1 → R2 such that ϕ(k[[z]]) = k[[z]] and t2ϕ = t1. Observe that ϕ|k[[z]] is +automatically continuous. +Trace extensions were classified up to isomorphism in [MSZ10, Proposition 2.9]: +Proposition 3.3. Let (R, t) be a trace extension of k[[z]]. Then precisely one of the following cases +occurs: +(1) (R, t) ∼= (R∞, t∞), where R∞ := k[[z]] ⊕ Spank{ak | k ∈ N} with multiplication defined by +ajak = 0 and +ajzk = +� +aj−k +if k ⩽ j, +0 +otherwise +and t∞ is the unique trace map on R∞ defined by t(aj) = δj0 for j ∈ N. +(2) There exists n ∈ N and λ ∈ k[[z]]× such that (R, t) ∼= (Rn, t(n,λ)), where Rn := k((z))×k[z]/(zn) +and +t(n,λ)(a, [b]) := res0 +1 +znλ(a − b). +Here, a ∈ k((z)), b ∈ k[[z]], [b] := b + znk[z] ∈ k[[z]]/znk[[z]] and k[[z]] is identified with its image +via the embedding a �→ (a, [a]). +■ +A trace extension (R, t) of k[[z]] is called trivial if (R, t) ∼= (R∞, t∞). + +10 +RASCHID ABEDIN +If (A, β) is a finite-dimensional metric k-algebra and (R, t) is a trace extension of k[[z]], then +(A ⊗ R, β ⊗ t) is a metric k-algebra. Here, +(3.6) +(β ⊗ t)(a1 ⊗ b1, a2 ⊗ b2) := β(a1, a2)t(b1b2) +for all a1, a2 ∈ A, b1, b2 ∈ R. +Observe that ((A ⊗ Rn, β ⊗ t(n,λ)), A ⊗ k[[z]]) ∼= ((Dn(A), β(n,λ)), A[[z]]) holds for all n ∈ N0 and +λ ∈ k[[z]]×. Therefore, Proposition 3.3 states that, if (R, t) is a non-trivial trace extension of k[[z]], +there exists n ∈ N and λ ∈ k[[z]]× such that ((A ⊗ R, β ⊗ t), A ⊗ k[[z]]) ∼= ((Dn(A), β(n,λ)), A[[z]]) +as Manin pairs. In particular, the Manin triples considered in Section 3.2 are exactly those which +arise by finding Lagrangian subalgebras W in (A ⊗ R, β ⊗ t) complementary to A ⊗ k[[z]] for any +non-trivial trace extension (R, t). +3.4. Non-triangular topological Lie D-bialgebra structures are non-degenerate. Let A +be a finite-dimensional k-algebra and C be a full subcategory of Algk closed under taking subalge- +bras such that A ⊗ R∞ ∈ C. Then the zero map +δ = 0: A[[z]] → (A ⊗ A)[[x, y]] +defines a topological D-bialgebra structure in C with double (D(A[[z]], δ), ev) ∼= (A ⊗ R∞, β ⊗ t∞). +We say that a topological D-bialgebra structure δ: A[[z]] → (A ⊗ A)[[x, y]] is triangular if +(3.7) +((D(A[[z]], δ), ev), A[[z]]) ∼= ((A ⊗ R∞, β ⊗ t∞), A ⊗ k[[z]]). +The origin of the name is explained in Remark 5.6. +Let k be algebraically closed of characteristic 0. If C is the category of Lie algebras over k and +A = g ∈ C is simple, it was shown in [MSZ10, Corollary 2.2, Lemma 2.3 and Proposition 2.8] +(see also [AMSZ22, Corollary 3.10]) that any non-triangular topological Lie bialgebra (g[[z]], δ) is +non-degenerate in the sense of Section 3.2. +We will prove an analog of this results for the case that C is the category of associative algebras +in Section 7.1. +4. Categorization of non-degenerate D-bialgebra structures +In this section, we will show that, up to isomorphism of D-bialgebras and for a large class of +central simple k-algebras A, all non-degenerate topological D-bialgebras (A[[z]], δ) are determined +by a Manin triple of the form (3.4) for some n ∈ {0, 1, 2} and λ = 1. The main method we use +to prove this result is the geometrization of A-lattices developed in [Abe21, Section 2.3] (see also +[Abe22, Section 1.3]), which we will recall in Subsection 4.1. The precise formulation of the above +mentioned result is then given in Subsection 4.2 and the reminder of this section will be devoted +to its proof. +Throughout the reminder of this paper, we assume that k is a field of characteristic 0. +4.1. Geometrization of lattices. Let A be a finite-dimensional, central, simple k-algebra. We +call a subalgebra W ⊆ A((z)) satisfying +(4.1) +dim(A[[z]] ∩ W) < ∞ and dim(A((z))/(A[[z]] + W)) < ∞ +A-lattice. Furthermore, we call a pair (O, W) consisting of an A-lattice W ⊆ A((z)) and a unital +subalgebra O ⊆ {f ∈ k((z)) | fW ⊆ W} of finite codimension ringed A-lattice. +Let us fix a ringed A-lattice (O, W). The graded k-algebra +(4.2) +gr(O) := +∞ +� +j=0 +tj � +O ∩ z−jk[[z]] +� +⊆ O[t] +defines an irreducible projective curve X := Proj(gr(O)) over k of arithmetic genus +(4.3) +h1(OX) = dim(k((z))/(k[[z]] + O)). + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +11 +The k-rational smooth point p = (t) of X satisfies D+(t) = X\{p}. Furthermore, there is canonical +isomorphism c: �OX,p → k[[z]] such that the induced isomorphism Q( �OX,p) → k((z)) on quotient +fields, which will be denoted again by c, has the property c(Γ(X \ {p}, OX)) = O. +Consider the graded gr(O)-algebra +(4.4) +gr(W) := +� +j∈Z +tj(W ∩ z−jA[[z]]) ⊆ W[t, t−1] +defined by W. Then the quasi-coherent sheaf A on X = Proj(gr(O)) associated to gr(W) is a +coherent torsion-free OX-algebra. This sheaf comes equipped with an c-equivariant isomorphism +ζ : � +Ap → A[[z]] such that the induced isomorphism Q( � +Ap) → A((z)), which will be denoted again +by ζ, has the property ζ(Γ(X \ {p}, A)) = W. The dimensions of the cohomology of A can be +calculated by +(4.5) +h0(A) = dim(A[[z]] ∩ W) and h1(A) = dim(A((z))/(A[[z]] + W)). +4.2. Geometrically admissible algebra metrics and the main theorem. Let (A, β) be a +metric k-algebra and let us denote the k((z))-bilinear extension of β by the same symbol, i.e. +(4.6) +β : A((z)) × A((z)) → k((z)) , +�� +k∈Z +akzk, +� +k∈Z +bkzk +� +�−→ +� +k,ℓ∈Z +β(ak, bℓ)zk+ℓ. +We call (A, β) geometrically admissible if: +(1) A is finite-dimensional, central, and simple; +(2) For any ringed A-lattices (O, W) and any maximal ideal m ⊆ O such that Wm is free as +Om-module, we have β(Wm, Wm) ⊆ Om. +4.2.1. Examples. Let (A, β) be a finite-dimensional, central, simple, metric k-algebra, (O, W) be +a ringed A-lattice, and m ⊆ O be a regular maximal ideal of O such that Wm is a free Om- +module. Then Wm is of rank d := dim(A), so we can choose an Om-basis {bi}d +i=1 ⊆ Wm and write +bibj = �d +k=1 Ck +ijbk for {Ck +ij}d +i,j,k=1 ⊆ Om. Observe that {bi}d +i=1 ⊆ Wm ⊆ A((z)) is also a k((z))-basis +of A((z)). +(1) Assume A is a Lie algebra. Then β is a scalar multiple of the Killing form of A since A is +simple. As a consequence, the extension (4.6) of β is equal to λK for the Killing form K of +A((z)) and some λ ∈ k×. Therefore, +β(bi, bj) = λCℓ +ikCk +jℓ ∈ Om +holds, so β is geometrically admissible. +(2) Assume that A is power associative and not anti-commutative, e.g. if A is associative or Jordan. +Then the existence of an algebra metric β on A implies that A is a non-commutative Jordan +algebra; see e.g. [BK66, Kapitel I, Satz 6.5]. Moreover, [She71, Theorem 1] implies that A is +not nil, so there exists λ ∈ k× such that +β(a, b) = λ +2 (Tr(Rab) + Tr(Lab)), +for all a, b ∈ A; see e.g. [Sch55]. Here, R, L: A → End(A) are the right and left multiplication +maps respectively. Therefore, +β(bi, bj) = λ +2 +d +� +k,ℓ=1 +Cℓ +ij(Ck +kℓ + Ck +ℓk) ∈ O +holds, so β is geometrically admissible. + +12 +RASCHID ABEDIN +4.2.2. Geometrically admissible metrics and geometrization of lattices. Let (A, β) be a geomet- +rically admissible metric k-algebra and (O, W) be a ringed A-lattice. The following results are +true: +(1) For all regular maximal ideals m ⊆ O, we have β(Wm, Wm) ⊆ Om. +(2) Let N be the integral closure of O. +Then N can be understood as a subalgebra of k((z)) +and V := NW ⊆ A((z)) is an A-lattice. +Consider the geometric datum ((X, A), (p, z, ζ)) +constructed from the ringed A-lattice (N, V ) in Section 4.1. Then there exists a unique pairing +βA : A × A → OX such that +(4.7) +Γ(U, A) × Γ(U, A) +βA � +ζ×ζ +� +Γ(U, OX) +c +� +A((z)) × A((z)) +β +� k((z)) +commutes for all U ⊆ X open. +(3) The pairing βA gives rise to a short exact sequence +0 −→ A −→ A∗ −→ C −→ 0 +for a torsion sheaf C. Here, A∗ = HomOX(A, OX) is the sheaf of morphisms from A to OX. +4.2.3. Proof of Subsection 4.2.2.(1). By definition, Om is a regular local ring. +Therefore, the +torsion-free Om-module Wm is free, so β(Wm, Wm) ⊆ Om holds since β is geometrically admissible. +4.2.4. Proof of Subsection 4.2.2.(2). Since the quotient field of O is a subalgebra of k((z)), we have +N ⊆ k((z)). Furthermore, since O has Krull dimension one, dim(N/O) < ∞ and dim(V/W) < ∞. +In particular, V is an A-lattice and (N, V ) is a ringed A-lattice. +Every closed point q ∈ X \ {q} ∼= Spec(N) corresponds to a maximal ideal mq ⊆ N. Since N is +integrally closed of dimension one, mq is regular. Combined with c(OX,q) = Omq and ζ(Aq) = Wmq +for all q ∈ X \ {p}, we obtain +β(ζ(Aq), ζ(Aq)) ⊆ OX,q +from (1). This implies that for all U ⊆ X \ {p} open +β(ζ(Γ(U, A)), ζ(Γ(U, A))) ⊆ c(Γ(U, OX)) +holds. +Combined with the fact that ζ(Γ(U, A)) = ζ(Γ(U \ {p}, A)) ∩ A[[z]] holds for all open +neighbourhoods U ⊆ X of p, we can simply define βA via the diagrams (4.7). +4.2.5. Proof of Subsection 4.2.2.(3). Note that the fiber of βA at p can be identified with β. In +particular, this fiber is non-degenerate. Let A → A∗ be the canonical morphism induced by βA. +Then the fact that βA|p is non-degenerate translates to the fact that A|p → A∗|p is an isomorphism. +In particular, the kernel and cokernel of A → A∗ are torsion. The observation that the kernel, as +a torsion subsheaf of the torsion-free sheaf A, is vanishing concludes the proof. +4.3. The categorization theorem. The rest of this section is dedicated to the proof of the +following theorem. +Theorem 4.1. Let (A, β) be a geometrically admissible algebra over a field k of characteristic 0, +n ∈ N, and λ ∈ k[[z]]×. Furthermore, let ((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple associated +to this datum in Subsection 3.2. +Then n ∈ {0, 1, 2} and λ = 1 up to isomorphism in the sense that +((Dn(A), β(n,λ)), A[[z]], W) ∼= ((Dn(A), β(n,1)), A[[z]], � +W) +for an appropriate � +W ⊆ Dn(A). + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +13 +In particular, if C is a full subcategory of Algk closed under taking subalgebras, a non-degenerate +topological D-bialgebra (A[[z]], δ) in C satisfies +((D(A[[z]], δ), ev), A[[z]]) ∼= ((Dn(A), β(n,1)), A[[z]]) +for an appropriate n ∈ {0, 1, 2}. +The proof proceeds in several steps. We being by collecting several algebraic properties of Manin +triples of the form (3.4) in Section 4.4. Using the geometrization method from Section 4.1, we pass +from these Manin triples to certain geometric data. The application of algebro-geometric tools +then concludes the proof of the refinement of Theorem 4.1 in Section 4.5. +Remark 4.2. +If A is a Lie algebra and k is algebraically closed, Theorem 4.1 coincides with +[MSZ10, Theorem 2.10]. However, our proof is independent of the proof of [MSZ10, Theorem +2.10]. In other words, we give a new proof of this result. +♦ +Remark 4.3. +The assumption on characteristic could be weakened by careful analysis of the follow- +ing steps. For instance, the geometrization in Section 4.1 works over fields where the characteristic +does not divide the dimension of A. Furthermore, most geometric methods used below are adapt- +able to fields of non-zero characteristic. However, for sake of clarity, we shall not pursue this level +of generality. +♦ +4.4. Algebraic properties of Manin triples of the form (3.4). Let (A, β) be a finite-dimensional +metric k-algebra, n ∈ N0, λ ∈ k[[z]]×, and +((Dn(A), β(n,λ)), A[[z]], W) +be the Manin triple associated to this datum in Subsection 3.2. Furthermore, let W+ (resp. W−) +be the projection of +W ⊆ Dn(A) = A((z)) × A[z]/znA[z] +onto A((z)) (resp. A[z]/znA[z]). +The following results are true: +(1) W ⊥ +± ⊆ W± with respect to the bilinear forms β± +(n,λ) defined by +(4.8) +β+ +(n,λ)(a1, a2) := res0 +1 +znλβ(a1, a2) and β− +(n,λ)([b1], [b2]) := res0 +1 +znλβ(b1, b2), +where a1, a2 ∈ A((z)) and [b1], [b2] ∈ A[z]/znA[z] = A[[z]]/znA[[z]] are the classes of b1, b2 ∈ A[[z]]. +(2) A((z)) = A[[z]] + W+ and dim(A[[z]] ∩ W+) < ∞; +(3) W+/W ⊥ ++ × W−/W ⊥ +− = (A[[z]] ∩ (W+ × W−)) ⊕ W/(W ⊥ ++ × W ⊥ +− ) is a finite-dimensional Manin +triple, so dim(W+/W ⊥ ++ ) = dim(W−/W ⊥ +− ) < ∞. Here, we recall that A[[z]] is considered as a +subalgebra of Dn(A) = A((z)) × A[z]/znA[z] via the diagonal embedding; +(4) If n > 0, we have dim(A[[z]] ∩ W+) > 0. +4.4.1. Proof of Subsection 4.4.(1). Follows immediately from the fact that (3.5) and (4.8) implies +(4.9) +W ⊥ ++ × W ⊥ +− = (W+ × W−)⊥ ⊆ W ⊥ = W ⊆ W+ × W−. +4.4.2. Proof of Subsection 4.4.(2). Observe that A[[z]] + W = A((z)) × A[z]/znA[z] implies A((z)) = +A[[z]] + W+. Therefore, {0} = (A[[z]] + W+)⊥ = znA[[z]] ∩ W ⊥ ++ since +A[[z]]⊥ = znλA[[z]] = znA[[z]] +with respect to β+ +(n,λ). This implies that A[[z]] ∩ W ⊥ ++ can be embedded into A[[z]]/znA[[z]] and is +therefore finite-dimensional. Consequently, the dimension of A[[z]] ∩ W+ is finite if the quotient +(A[[z]]∩W+)/(A[[z]]∩W ⊥ ++ ) is finite-dimensional. The latter space can be identified with a subspace +of W+/W ⊥ ++ . Therefore, Subsection 4.4.(2) follows from Subsection 4.4.(3). + +14 +RASCHID ABEDIN +4.4.3. Proof of Subsection 4.4.(3). The kernel K of the projection W → W+ contains {0} × W ⊥ +− +by virtue of (4.9). On the other hand, any element of K is of the form (0, a) for some a ∈ W−, so +for all (w+, w−) ∈ W +(4.10) +0 = β(n,λ)((0, a), (w+, w−)) = −β− +(n,λ)(a, w−) +holds, implying a ∈ W ⊥ +− and hence K = {0} × W ⊥ +− . We obtain an isomorphism +W/(W ⊥ ++ × W ⊥ +− ) −→ W+/W ⊥ ++ . +A similar argument yields W/(W ⊥ ++ × W ⊥ +− ) ∼= W−/W ⊥ +− . Therefore, we obtain an isomorphism +W+/W ⊥ ++ → W−/W ⊥ +− . In particular, +dim(W+/W ⊥ ++ ) = dim(W−/W ⊥ +− ) ⩽ dim(A[z]/znA[z]) < ∞. +Considering W ⊆ W+ × W−, the identity A[[z]] ⊕ W = A((z)) × A[z]/znA[z] is equivalent to +(4.11) +W+ × W− = (A[[z]] ∩ (W+ × W−)) ⊕ W. +Quoiting out W ⊥ ++ × W ⊥ +− concludes the proof. +4.4.4. Proof of Subsection 4.4.(4). Assume that n > 0 and A[[z]] ∩ W+ = {0}. Then +A[[z]] ∩ (W+ × W−) = {0} +and (4.11) imply +(4.12) +W = W+ × W− = W ⊥ ++ × W ⊥ +− . +For any a ∈ A[z] exists b ∈ A[[z]] and w± ∈ W± such that +(0, [a]) = (b, [b]) + (w+, w−) ∈ A[[z]] ⊕ (W+ × W−). +Therefore, w+ = −b ∈ A[[z]] ∩ W+ = {0} results in [a] = w− ∈ W−. Since a ∈ A[z] was arbitrary, +we conclude W− = A[z]/znA[z], which contradicts W ⊥ +− = W− in (4.12). +4.5. Geometric properties of Manin triples over series. We are now in the position to proof +Theorem 4.1. More precisely, we proof the following refinement of this theorem. +Let (A, β) be a geometrically admissible algebra over a field k of characteristic 0, n ∈ N0, +λ ∈ k[[z]]×, and ((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple constructed in Subsection 3.2. +Furthermore, let W+ ⊆ A((z)) be the image of W under the projection Dn(A) → A((z)) and +consider M := {f ∈ k((z)) | fW+ ⊆ W+}. +Then n ∈ {0, 1, 2} and, up to isomorphism of Manin triples, λ = 1 and precisely one of the +following cases occurs: +(1) n = 0 and M is integrally closed satisfying dim(k((z))/(k[[x]] + M)) = 1; +(2) n = 0 and k[u′, uu′] ⊆ M for u ∈ z−1k[[z]]× satisfying u ̸= z−1; +(3) n = 0 and k[z−2, z−3] ⊆ M; +(4) n = 1 and M = k[z−1]; +(5) n = 2 and M = k[z−1]. +4.6. Proof of the results in Section 4.5 (and by proxy of Theorem 4.1). We use similar +arguments as in the proof of [Abe21, Theorem 3.6] or [AMSZ22, Section 8.4]. +Let N be the integral closure of M. Subsection 4.2.2.(2) states that V := NW+ is an A-lattice. +Let ((X, A), (p, c, ζ)) be the geometric datum associated to the ringed A-lattice (N, V ) in Section +4.1. By virtue of Subsection 4.2.2.(3), we have a short exact sequence +0 −→ A−→A∗ −→ C −→ 0, +(4.13) +where C is a torsion sheaf. The associated long exact sequence in cohomology reads +(4.14) +0 −→ H0(A)−→H0(A∗) −→ H0(C) −→ H1(A)−→H1(A∗) −→ H1(C) −→ 0. +The identities H1(A) = 0 = H1(C) imply that H1(A∗) = 0. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +15 +The Riemann-Roch theorem for A and A∗ combined with the fact that h1(OX) = g reads +0 ⩽ h0(A) − h1(A) = deg(det(A)) + (1 − g)rank(A), +0 ⩽ h0(A∗) − h1(A∗) = −deg(det(A)) + (1 − g)rank(A), +where we used that det(A∗) = det(A)∗ implies deg(det(A∗)) = −deg(det(A)). We conclude g ⩽ 1. +4.6.1. The case g = 1. Assume g = 1, then X is an elliptic curve. Then the sheaf Ω1 +X of regular +1-forms on X satisfies Ω1 +X ∼= OX. Therefore, 0 = h1(A∗) = h0(A) because of Serre duality. In +particular, by (4.5), W+ ∩ g[[z]] ⊆ V ∩ g[[z]] = {0}, so Subsection 4.4.(4) implies n = 0. Moreover, +W+ ⊕ g[[z]] = g((z)) = V ⊕ g[[z]] and W+ ⊆ V imply V = W+, so M = N is integrally closed. Since +h1(OX) = dim(k((z))/(k[[z]] + M)) = 1 by virtue of (4.3), we are in case (1). +4.6.2. The case g = 0. Note that g = 0 means k((z)) = k[[z]] + N by virtue of (4.3). Since +N ∩ k[[z]] = H0(OX) = k, +we can see that N = k[u] for the unique u ∈ (z−1 + zk[[z]]) ∩ N ̸= {0}. Let N ⊥ be the orthogonal +complement of N with respect to the bilinear form R(n,λ) : k((z)) × k((z)) → k defined by +(4.15) +(f, g) �−→ res0 +1 +znλfg. +Since (A, β) is geometrically admissible, we have β(V, V ) ⊆ N for β from (4.6), so for all f ∈ N ⊥ +and a, b ∈ W+ ⊆ V we have +(4.16) +β+ +(n,λ)(fa, b) = res0 +1 +znλfβ(a, b) = R(n,λ)(f, β(a, b)) = 0. +Here, we recall that β+ +(n,λ) was defined in (4.8). +In particular, fa ∈ W ⊥ ++ so fa ∈ W+ since +W ⊥ ++ ⊆ W+ by Subsection 4.4.(1). Therefore, N ⊥ ⊆ {f ∈ k((x)) | fW+ ⊆ W+} = M ⊆ N. The +identity +(4.17) +R(n,λ)(znλu′, uk) = res0u′uk = +1 +k + 1res0 +� +uk+1�′ = 0 +for all k ∈ N yields znλu′ ∈ N ⊥ ⊆ M. +4.6.3. The case (n, g) = (0, 0). Since R(n,λ) is associative and v := λu′ ∈ N ⊥ we have the inclusion +vN ⊆ N ⊥. Furthermore, since v ∈ N ⊥ ⊆ N = k[u], we obtain k[v, vu] ⊆ k + vN ⊆ k + N ⊥. Since +all three spaces are of codimension one in N = k[u] we conclude +(4.18) +k[v, vu] = k + vN = k + N ⊥ ⊆ {f ∈ k((z)) | fW+ ⊆ W+} = M. +4.6.4. Case (n, g) = (1, 0). Since v := zλu′ ∈ N ⊥ ∩ z−1k[[z]]× ⊆ N ∩ z−1k[[z]]× = k×u + k, we have +M = N = k[v]. +4.6.5. Case (n, g) = (2, 0). Since z2λu′ ∈ N ⊥ ∩ k[[z]] ⊆ N ∩ k[[z]] = k we have au′ = −z−2λ−1 for +some a ∈ k×. Consequently, res0z−2λ−1 = res0au′ = 0 and N = k[u] = N ⊥ ⊆ M by (4.17). +4.6.6. Case n ⩾ 3. The fact that z3λu′ ∈ N ⊥ ∩zk[[z]] = {0} is a contradiction. In particular, there +cannot exist any Manin triple of the form ((Dn(A), β(n,λ)), A[[z]], W) for n ⩽ 3. + +16 +RASCHID ABEDIN +4.6.7. Concluding the proof. As a metric algebra (Dn(A), β(n,λ)) ∼= (A⊗Rn, β ⊗t(n,λ)); see Section +3.3. It is shown in [AMSZ22, Proposition 3.12] that (Rn, t(n,λ)) ∼= (Rn, t(n,1/(1+azn−1))) as trace +extensions for a = res0z−nλ−1 ∈ k. +In particular, since n ⩽ 2 and for n = 2 the identity +res0z−nλ−1 = 0 holds, we obtain λ = 1 up to isomorphism in all cases. +If (n, g) = (0, 0), this means that k[u′, uu′] ⊆ M ⊆ N = k[u]. In particular, since by definition +u ∈ (z−1 + zb + z2k[[z]]) for some b ∈ k, we see that u′ + u2 − 3b ∈ k[u] ∩ zk[[z]] = {0}. If b = 0 this +formal differential equation has the unique solution u = z−1 and we are in case (3) and if b ̸= 0 we +are in case (2). +If (n, g) = (1, 0), we have zu′ ∈ M = N = k[u], so zu′ = −u. The only solution to this equation +is again z−1 and we are in case (4). Finally, if (n, g) = (2, 0), we have u′ = −z−2 so u = z−1 again +and we are in case (5). +5. Non-degenerate D-bialgebra structures and the classical Yang-Baxter +equation +5.1. Series of type (n, λ). Let (A, β) be a finite-dimensional metric k-algebra. Choose a basis +{bi}d +i=1 of A and consider its dual basis {b∗ +i }d +i=1, i.e. β(bi, b∗ +j) = δij. The tensor γ = �d +i=1 b∗ +i ⊗ bi +is independent of the choice of {bi}d +i=1 ⊆ A, symmetric, and satisfies +(5.1) +a(1)γ = γa(2) or, equivalently, a(2)γ = γa(1). +The first identity follows from the fact that +β⊗2(a(1)γ, bj ⊗ b∗ +k) = +d +� +i=1 +β(ab∗ +i , bj)β(bi, b∗ +k) = β(ab∗ +k, bj) = β(b∗ +k, bja) = β(γa(2), bj ⊗ b∗ +k). +holds for all j, k ∈ 1, d, and the second follows from the first by using the symmetry of γ. We call +γ canonical A-invariant element of (A, β). +Let us note that the canonical embedding (A ⊗ A)[[x, y]] → (Dn(A) ⊗ A)[[y]] extends to +(5.2) +(A ⊗ A)[[x, y]][(x − y)−1] −→ (Dn(A) ⊗ A)((y)) +by writing +(5.3) +1 +x − y = +n−1 +� +k=0 +(0, −[x]n−k−1)yk−n + +∞ +� +k=0 +(x−k−1, 0)yk ∈ (k((x)) × k[x]/(xn))((y)). +Indeed, this is appropriate since we can calculate +((x, [x]) − y) +�n−1 +� +k=0 +(0, −[x]n−k−1)yk−n + +∞ +� +k=0 +(x−k−1, 0)yk +� += +n−1 +� +k=0 +(0, −[x]n−k)yk−n − +n +� +k=1 +(0, −[x]n−k)yk−n + (1, 0) = (1, 1) +(5.4) +inside (k((x)) × k[x]/(xn))((y)). +In particular, for any n ∈ N we obtain +ynγ +x − y = +n−1 +� +k=0 +d +� +i=1 +(0, −[b∗ +i xn−1−k]) ⊗ biyk + +∞ +� +k=n +d +� +i=1 +(b∗ +i xn−1−k, 0) ⊗ biyk += +∞ +� +k=0 +d +� +i=1 +wk,i ⊗ biyk ∈ (Dn(A) ⊗ A)[[y]]. +(5.5) + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +17 +For any λ ∈ k[[z]]× and s ∈ (A ⊗ A)[[x, y]], we identify the expression +r(x, y) = ynλ(x)γ +x − y ++ s(x, y) ∈ (A ⊗ A)[[x, y]][(x − y)−1] +(5.6) +with its series in (Dn(A) ⊗ A)[[y]] and say that r is a series of type (n, λ). +Remark 5.1. +Every r(x, y) = +a(x,y)γ +x−y ++ s(x, y) for a ∈ k[[x, y]] such that a(z, z) ̸= 0 and any +s ∈ (A ⊗ A)[[x, y]] has a unique representation as a series of type (n, λ). Indeed, chose (n, λ) such +that a(z, z) = znλ(z). Then a(x, y) − ynλ(x) = (x − y)b(x, y) for some b ∈ k[[x, y]], so +(5.7) +r(x, y) = ynλ(x)γ +x − y ++ b(x, y)γ + s(x, y) +is a series of type (n, λ). +In the construction of b we used the following easy fact: for any k-vector space V +(5.8) +f ∈ V [[x, y]], f(z, z) = 0 =⇒ f(x, y) = (x − y)g(x, y) for some g ∈ V [[x, y]] +holds. +♦ +Note that we have a linear automorphism of (A ⊗ A)[[x, y]][(x − y)−1] defined by +(5.9) +a(x, y) �−→ a(x, y) := −τ(a(y, x)) +where τ(a ⊗ b) = b ⊗ a is applied coefficient-wise. For any series r of type (n, λ), r is again a series +of type (n, λ) by Remark 5.1. We call r skew-symmetric if r = r. +5.2. The (generalized) classical Yang-Baxter equation with coefficients in arbitrary +algebras. As in the last section, (A, β) is a finite-dimensional metric k-algebra, {bi}d +i=1 and {b∗ +i }d +i=1 +are basis of A satisfying β(b∗ +i , bj) = δij, and γ := �d +i=1 b∗ +i ⊗ bi. +Furthermore, let U be the +unitalization of A, i.e. U = A ⊕ k with multiplication +(5.10) +(a1, u1)(a2, u2) = (a1a2 + u1a2 + u2a1, u1u2) +for all a1, a2 ∈ A and u1, u2 ∈ k. +For any s ∈ (A ⊗ A)[[x, y]][(x − y)−1], let us define the expressions +(5.11) +s12(z1, z2), s13(z1, z3), s23(z2, z3) ∈ (U ⊗ U ⊗ U)[[z1, z2, z3]] +� +1 +(z1 − z2)(z1 − z3)(z2 − z3) +� +coefficient-wise via +(5.12) +t12 = t ⊗ 1, t13 = a ⊗ 1 ⊗ b, t23 = 1 ⊗ t ∈ U ⊗ U ⊗ U +for t = a ⊗ b ∈ A ⊗ A. +Let us point out that for example (a1 ⊗ a2)13(b1 ⊗ b2)12 = a1b1 ⊗ b2 ⊗ a2 ∈ A ⊗ A ⊗ A. This +and similar identities imply that for all s1, s2 ∈ (A ⊗ A)[[x, y]][(x − y)−1] +(5.13) +s13 +1 (z1, z3)s12 +2 (z1, z2), s12 +1 (z1, z2)s23 +2 (z2, z3), and s23 +1 (z2, z3)s13 +2 (z1, z3) +are elements of +(A ⊗ A ⊗ A)[[z1, z2, z3]] +� +1 +(z1 − z2)(z1 − z3)(z2 − z3) +� +. +Furthermore, if s1, s2 are of the form (5.6) and we write +sǫ = +� +k∈Z +d +� +i=1 +sǫ;k,i(x) ⊗ biyk ∈ (Dn(A) ⊗ A)((y)), +for ǫ ∈ {1, 2} for the associated series via (5.2), we get +s13 +1 (z1, z3)s12 +2 (z1, z2) = +� +k,ℓ∈N +d +� +i,j=1 +s1;k,i(z1)s2;ℓ,j(z1) ⊗ bjzℓ +2 ⊗ bizk +3 ∈ (Dn(A) ⊗ A ⊗ A)[[z2, z3]]. + +18 +RASCHID ABEDIN +Similar formulas for s12 +1 (z1, z2)s23 +2 (z2, z3) and s23 +1 (z2, z3)s13 +2 (z1, z3) hold. +For any r ∈ (A ⊗ A)[[x, y]][(x − y)−1], we call the equation GCYB(r) = 0 the A-generalized +classical Yang-Baxter equation (short: A-GCYBE), where +(5.14) +GCYB(r) := r13r12 − r12r23 + r23r13 +Here, (·) was defined in (5.9). +Similarly, we call the equation CYB(r) = 0 the A-classical Yang-Baxter equation (short: A- +CYBE), where +(5.15) +CYB(r) := r13r12 − r12r23 + r23r13. +If A is a Lie algebra, these are exactly the usual (generalized) classical Yang-Baxter in two-formal +parameter. If A is associative, the A-classical Yang-Baxter equation is a formal variant of the +associative version of the CYBE used in [OS08], which is itself a spectral parameter generalization +of the associative CYBE discussed in e.g. [Agu01]. +5.3. Solutions of the A-(G)CYBE and subspaces of Dn(A). Series of type (n, λ) can be seen +as generating series of certain subspaces of Dn(A). More precisely, we have the following result, +which is a generalization of known statements in the Lie algebra case; see e.g. [GC83; Skr13; +AMS22]. +Theorem 5.2. Let (A, β) be a finite-dimensional metric k-algebra, {bi}d +i=1 and {b∗ +i }d +i=1 be ba- +sis of A satisfying β(b∗ +i , bj) = δij, γ := �d +i=1 b∗ +i ⊗ bi, and n ∈ N. +To any series r(x, y) = +�∞ +k=0 +�d +i=1 rk,i(x) ⊗ biyk ∈ (Dn(A) ⊗ A)[[y]], we can define a linear subspace +(5.16) +A(r) := Spank{rk,i | k ∈ N, i ∈ 1, d} ⊆ Dn(A). +For any fixed λ ∈ k[[z]]× the following results are true: +(1) r �→ A(r) defines a bijection between series r of type (n, λ) (i.e. of the from (5.6)) and subspaces +W ⊆ Dn(A) satisfying Dn(A) = A[[z]] ⊕ W. +(2) For any series r of type (n, λ), the identity A(r)⊥ = A(r) holds, where (·)⊥ is meant with +respect to β(n,λ) from (3.5) and (·) is defined in (5.9). +(3) For any series r of type (n, λ), the identity +(5.17) +GCYB(r) = ϕ +holds for the unique element ϕ ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]] determined by +(5.18) +β⊗3 +(n,λ)(v1 ⊗ v2 ⊗ v3, ϕ) = β(n,λ)(v1, v3v2) +for all v1 ∈ A(r), v2, v3 ∈ A(r). +The proof of Theorem 5.2 is postponed to Subsection 5.3.1. +A direct consequence of Theorem 5.2.(1)&(3) is that r �→ A(r) defines a bijection between +solutions r of the A-GCYBE (5.14) of type (n, λ) and subalgebras W ⊆ Dn(A) satisfying Dn(A) = +A[[z]]⊕W. Combined with Theorem 5.2.(2), we obtain a bijection between skew-symmetric solutions +r of the A-CYBE (5.15) of type (n, λ) and Manin triples ((Dn(A), β(n,λ)), A[[z]], W). We will see +that, if A is simple, any solution r of the A-CYBE of type (n, λ) is already skew-symmetric. +Therefore, we have the following consequence of Theorem 5.2. +Corollary 5.3. Let (A, β) be a finite-dimensional simple metric k-algebra, n ∈ N, and λ ∈ k[[z]]×. +Then r �→ A(r) defines a bijection between solutions of the A-CYBE (5.15) r of type (n, λ) and +Manin triples ((Dn(A), β(n,λ)), A[[z]], W). +The proof will be given in Subsection 5.3.2. +Since Manin triples of the form (3.4) exist only for n ⩽ 2 by virtue of Theorem 4.1, Theorem 5.3 +gives the same restriction for solutions of the A-CYBE for any geometrically admissible k-algebra +(A, β). To be precise, we have the following result. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +19 +Corollary 5.4. Let (A, β) be a finite-dimensional simple metric k-algebra, n ∈ N and λ ∈ k[[z]]×. +If r ∈ (Dn(A) ⊗ A)[[y]] is a solution of the A-CYBE of type (n, λ), we have n ∈ {0, 1, 2}. +5.3.1. Proof of Theorem 5.2. The proof of (1) and (2) is completely analogous to the proof in the +case that A is a Lie algebra in [AMS22, Theorem 3.6], so it remains to prove (3). +Let us begin by proving, that +(5.19) GCYB(r) ∈ (A⊗A⊗A)[[z1, z2, z3]] for all series r ∈ (A⊗A)[[x, y]][(x−y)−1] of type (n, λ). +To this end, let r(x, y) = ynλ(x)γ +x−y ++ s(x, y) be a series of type (n, λ). Clearly, T1 := GCYB(s) is an +element of (A ⊗ A ⊗ A)[[z1, z2, z3]]. Since a(1)γ = γa(2) for all a ∈ A we have γ13γ12 = γ12γ23 = +γ23γ13. Therefore, if we write w := ynλ(x)γ +x−y +, we have +(z1 − z2)(z1 − z3)(z2 − z3)GCYB (w) += (z2z3)nλ(z1)(λ(z1)(z2 − z3) − λ(z2)(z1 − z3) + λ(z3)(z1 − z2))γ13γ12. +This expression is zero if z1 = z2, z1 = z3 or z2 = z3, so +(5.20) +T2 := GCYB (w) ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]] +Now let us turn to +GCYB(r) = GCYB (w + s) = T1 + T2 ++ w13s12 + s13w12 − w12s23 − s12w23 + w23s13 + s23w13 += T1 + T2 + (s13w12 − w12s23) +� +�� +� +:=T3 +− (s12w23 − w23s13) +� +�� +� +:=T4 ++ (s23w13 + w13s12) +� +�� +� +:=T5 +Write s(x, y) = � +k∈N +�d +i=1 si,j +k,ℓxkyℓbi ⊗ bj and note that s(x, y) = − � +k∈N +�d +i=1 si,j +k,ℓxℓykbj ⊗ bi. +Using zka(1)γ − γa(2)zk = 0 for all a ∈ A we see that +(5.21) +T3 = λ(z1)zn +2 +z1 − z2 +� +k,ℓ∈N +d +� +i,j=1 +si,j +k,ℓ(zk +1b(1) +i γ − γb(2) +i zk +2) ⊗ bjzℓ +3 ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]]. +by virtue of (5.8). +Similarly, under consideration of w = λ(y)xnγ +x−y +, we get +(5.22) T4 = +1 +z1 − z2 +� +k,ℓ∈N +d +� +i,j=1 +sk,ℓ +i,j bizk +1 ⊗(λ(z2)zn +3 zℓ +2b(1) +j γ−γb(2) +j zℓ +3λ(z3)zn +2 ) ∈ (A⊗A⊗A)[[z1, z2, z3]] +Using a(2)γ = γa(1) and the notation θa(b ⊗ c) = b ⊗ a ⊗ c for all a ∈ A, we obtain +(5.23) +T5 = λ(z1)zn +3 +z1 − z3 +� +k,ℓ∈N +d +� +i,j=1 +si,j +k,ℓθbjzℓ +2(−zk +3b(2) +i γ + γb(1) +i zk +1) ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]]. +Summarized, we have GCYB(r) = T1 + T2 + T3 + T4 + T5 ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]]. +Let us now write +r(x, y) = ynλ(x)γ +x − y ++ s(x, y) = +� +k∈N +d +� +i=1 +rk,i(x) ⊗ biyk + +20 +RASCHID ABEDIN +and observe that +(5.24) +GCYB(r) = +� +k,ℓ∈N +d +� +i,j=1 +rℓ,jrk,i ⊗ bizk +2 ⊗ bjzℓ +3 +− +� +k∈N +d +� +i=1 +rk,i ⊗ +� +zk +2b(1) +i r(z2, z3) − r(z2, z3)b(2) +i zk +3 +� +holds. Here, we used the fact that the embedding (5.2) induces a commutative diagram +(5.25) +(A ⊗ A ⊗ A)[[z1, z2, z3]] +� +� +(Dn(A) ⊗ A ⊗ A)[[z2, z3]] +(A ⊗ A ⊗ A)[[z1, z2, z3]] +� +1 +z1−z3 +� +�❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +. +Applying β⊗3 +(n,λ)(rk1,i1 ⊗ rk2,i2 ⊗ rk3,i3, −) to (5.24), where r = �∞ +k=0 +�d +i=1 rk,i ⊗ biyk yields +(5.26) +β⊗3 +(n,λ)(rk1,i1 ⊗ rk2,i2 ⊗ rk3,i3, GCYB(r)) = β(n,λ)(rk1,i1, rk3,i3rk2,i2) +since rk1,i1 ∈ A(r) = A(r)⊥. +This concludes the proof, since {rk,i | k ∈ N0, i ∈ 1, d} (resp. +{rk,i | k ∈ N0, i ∈ 1, d}) is a basis of A(r) (resp. A(r)). +5.3.2. Proof of Corollary 5.3. Under consideration of Theorem 5.2 and the remarks after this +theorem, it remains to prove that, if A is simple, any solution r of the A-CYBE of type (n, λ) is +automatically skew-symmetric. +The equality CYB(r) = 0 implies that A(r) ⊆ Dn(A) is a subalgebra by rewriting the CYBE +similarly to (5.24). Therefore, (3) implies that r solves the A-GCYBE (5.14). This implies +(5.27) +0 = CYB(r) − GCYB(r) = (r23 − r23)r13 +Multiplying by z1 − z3 and setting z1 = z3 we obtain (r23 − r23)γ13 = 0. +This implies that +(r − r)b(2) +i += 0 for all i ∈ 1, d. Since A is simple, we have aA = {0} implies a = 0. Therefore, +r = r, which concludes the proof. +5.4. Connection to topological D-bialgebra structures. Let r ∈ (Dn(A)⊗A)[[y]] be a solution +of the A-CYBE (5.15) of type (n, λ). The identity CYB(r) = 0 can be rewritten as +∞ +� +k,ℓ=0 +d +� +i,j=1 +rk,i(z1)rℓ,j(z1) ⊗ bjzℓ +2 ⊗ bizk +3 += +∞ +� +k=0 +d +� +i=1 +rk,i(z1) ⊗ +�� +bizk +2 +�(1) r(z2, z3) − r(z2, z3) +� +bizk +3 +�(2)� +(5.28) +Therefore, under consideration of (5.8), we can deduce that +(5.29) +δr(a)(x, y) := r(x, y)a(x)(1) − a(y)(2)r(x, y) = − +� +a(x)(1)r(x, y) − r(x, y)a(y)(2) +� +defines a continuous linear map δr : A[[z]] → (A ⊗ A)[[x, y]]. Applying +(5.30) +β⊗3 +(n,λ)(bi1zk1 +1 ⊗ rk3,i3 ⊗ rk2,j2, −) +to (5.28) results in +(5.31) +β(n,λ) +� +bi1zk1, rk2,i2rk3,j3 +� += β⊗2 +(n,λ) +� +δr +� +bi1zk1� +, rk2,i2 ⊗ rk3,j3 +� + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +21 +This proves that δr is determined by ((Dn(A), β(n,λ)), A[[z]], A(r)) and thus defines a topological +D-bialgebra structure in any full subcategory C of Algk that is closed under taking subalgebras +and contains Dn(A). Therefore, +((Dn(A), β(n,λ)), A[[z]], A(r)) ∼= (D(A[[z]], δr), ev), A[[z]], A[[z]]∨), +so (A[[z]], δr) is a non-degenerate topological D-bialgebra in C. In fact, the results in Section 5.3 +imply that every non-degenerate topological D-bialgebra structure in C is of this form. +5.5. Equivalence of solutions of the A-CYBE. Let n ∈ N and (A, β) be a finite-dimensional +metric k-algebra. We call two solutions r1, r2 ∈ (A ⊗ A)[[x, y]][(x − y)−1] of the A-CYBE (5.15) +equivalent, written r1 ∼ r2, if there exists ϕ ∈ Autk[[z]]-alg(A[[z]]) and u ∈ zk[[z]]× such that +(5.32) +(ϕ(x) ⊗ ϕ(y))r1(u(x), u(y)) = r2(x, y). +Lemma 5.5. Let n ∈ N, (A, β) be a finite-dimensional, central, simple, metric k-algebra, λǫ ∈ +k[[z]]× and let rǫ ∈ (A⊗A)[[x, y]][(x−y)−1] be a solution of the A-CYBE of type (n, λǫ) for ǫ ∈ {1, 2}. +Then the following statements are equivalent: +• r1 and r2 are equivalent. +• (A[[z]], δr1) ∼= (A[[z]], δr2) as topological D-bialgebra structures in any full subcategory C of Algk +that is closed under taking subalgebras and contains Dn(A). +• ((Dn(A), β(n,λ1)), A[[z]], A(r1)) ∼= ((Dn(A), β(n,λ2)), A[[z]], A(r2)). +Proof. The equivalence of the latter two items is already dealt with in Remark 3.2. +For the +equivalence of the first two items, recall that any φ ∈ Autk-alg(A[[z]]) is of the form φ(a)(z) = +ϕ(z)a(u(z)) for some ϕ ∈ Autk[[z]]-alg(A[[z]]) and u ∈ zk[[z]]×; see [AMSZ22, Theorem 3.3]. Now, its +easy to see that +(5.33) +(ϕ(x) ⊗ ϕ(y))r1(u(x), u(y)) = r2(x, y). +is equivalent to (φ ⊗ φ)δr1φ−1 = δr2, which means that (A[[z]], δr1) ∼= (A[[z]], δr2) as topological +D-bialgebras. +■ +5.6. Solutions of the A-CYBE and triangular D-bialgebra structures. It is also possible +to consider solutions to the A-CYBE (5.15) of the form r ∈ (A⊗ A)[[x, y]]. Namely, the assignment +r �→ A(r) defines a bijection between: +• skew-symmetric solutions r ∈ (A ⊗ A)[[x, y]] of the A-CYBE (5.15) and +• subspaces W ⊆ A ⊗ R∞ such that ((A ⊗ R∞, β ⊗ t∞), A[[z]], W) is a Manin triple. +Moreover, (A[[z]], δr) is a topological D-bialgebra structure (in any category of algebras closed under +taking subalgebras that contains A ⊗ R∞) determined by ((A ⊗ R∞, β ⊗ t∞), A[[z]], W). Therefore, +(5.34) +((A ⊗ R∞, β ⊗ t∞), A[[z]], W) ∼= ((Dn(A[[z]], δr), ev), A[[z]], A[[z]]∨), +so δr is a triangular topological D-bialgebra structure. On the other hand, all triangular topological +D-bialgebra structures on A[[z]] are of this form. +Recall that a Lie bialgebra structure (L, δ) is called triangular if δ = δr for some skew-symmetric +solution r ∈ L ⊗ L of the CYBE. If (g[[z]], δ) is a topological Lie bialgebra structure for some Lie +algebra g, it is natural to replace g[[z]] ⊗ g[[z]] by its completion (g ⊗ g)[[x, y]] in this definition. In +particular, it is natural to call δ triangular if δ = δr for a skew-symmetric solution r ∈ (g⊗ g)[[x, y]] +of the CYBE. The D-bialgebra structures satisfying (5.34) are then called triangular in analogy to +their Lie counterparts. +6. Refined categorization of non-degenerate topological D-bialgebras +In this section, we refine Theorem 4.1 for so-called strongly geometrically admissible algebras +over algebraically closed fields of characteristic 0. The main result of this section, Theorem 6.1, +can be seen as an analog of the main results from [AMSZ22] for a large class of non-Lie algebras. +The proof relies on refining the geometric approach already used in the proof of Theorem 4.1. + +22 +RASCHID ABEDIN +Throughout the remainder of this paper, k is an algebraically closed field of characteristic 0. +6.1. The main theorem. We call a metric k-algebra (A, β) strongly geometrically admissible if +(1) (A, β) is geometrically admissible in the sense of Subsection 4.2; +(2) For any ringed A-lattices (O, W) and any maximal ideal m ⊆ O such that +• Wm is free as Om-module and +• the pairing Wm × Wm → Om induced by β is perfect, +we have W/mW ∼= A. +As we will see in Corollary 6.8 below, many central simple k-algebras are strongly geometrically +admissible, e.g. all finite-dimensional simple associative, Lie and Jordan algebras. +The rest of this section is dedicated to proving the following result. +Theorem 6.1. Let us fix the following notation: +• k is an algebraically closed field of characteristic 0; +• (A, β) is a unital strongly geometrically admissible metric k-algebra (e.g. a finite-dimensional +simple Jordan or associative k-algebra) and γ ∈ A ⊗ A is its canonical A-invariant element (see +Subsection 5.1); +• ((Dn(A), β(n,λ)), A[[z]], W) is the Manin triple associated to (A, β) as well as some n ∈ N and +λ ∈ k[[z]]×in Subsection 3.2; +• r is the solution of the A-CYBE associated to the Manin triple ((Dn(A), β(n,λ)), A[[z]], W) via +Theorem 5.3. +Precisely one of the following cases occurs: +(1) n = 0 and r is either: +(a) Trigonometric in the sense that there exists a β-orthogonal σ ∈ Autk-alg(A) of order m ∈ N +and s ∈ L(A, σ) ⊗ L(A, σ) such that r is equivalent to +1 +exp (x − y) − 1 +m−1 +� +j=0 +exp +�x − y +m +� +γj + s +� +exp +� x +m +� +, exp +� y +m +�� +. +Here, L(A, σ) ⊆ A[�v, �v−1] is the loop algebra twisted by σ (see Proposition 6.9 for the +definition) and γj ∈ A ⊗ A is uniquely determined by γ = �d +j=1 γj and (σ ⊗ 1)γj = εjγj +for some primitive m-th root of unity ε ∈ k; +(b) Rational in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r is equivalent to +γ +x − y + t(x, y). +(2) n = 1 and r is quasi-trigonometric in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r +is equivalent to +yγ +x − y + t(x, y). +(3) n = 2 and r is quasi-rational in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r is +equivalent to +y2γ +x − y + t(x, y). +In particular, every solution of the A-CYBE (5.15) of type (n, λ) is, up to equivalence, either +trigonometric, rational, quasi-trigonometric or quasi-rational. +Corollary 6.2. Let k be an algebraically closed field of characteristic 0, (A, β) be a strongly geo- +metrically admissible k-algebra, and C be a full subcategory of Algk closed under taking subalgebras +and satisfying Dn(A) ∈ C. +Then every non-degenerate topological D-bialgebra (A[[z]], δ) in C satisfies, up to isomorphism, +δ = δr for a solution r of the A-CYBE which is either trigonometric, rational, quasi-trigonometric, +or quasi-rational. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +23 +Remark 6.3. +Let us note that the unitality assumption in Theorem 6.1 is actually a rather weak +assumption. Indeed, if a strongly geometrically admissible algebra is power-associative and not +anti-commutative, it is a non-nil (see [She71]) trace-admissible algebra. These are automatically +unital; see [Alb49]. +♦ +The proof of Theorem 6.1 is again based on the geometrization scheme from Subsection 4.1. +However, to refine the geometric approach already used in the proof of Theorem 4.1, we need +to establish some facts about ´etale locally trivial sheaves of algebras in Subsection 6.2. There +we also explain how examples of strongly geometrically admissible algebras can be constructed +using the notion of rigidity. The results from Subsection 6.2 and the refinement of Theorem 4.1 +in Subsection 4.5 are then used to associate more explicit geometric data to Manin triples of the +form (3.4). Namely, so-called geometric A-CYBE data, which will be defined in Subsection 6.3. +We will assign such a datum to any Manin triple of the form (3.4) in Subsection 6.4. Theorem 6.1 +is then a consequence of the classification results for sheaves of algebras from Proposition 6.9. +6.2. ´Etale locally trivial sheaves of algebras. Let A be a sheaf of algebras on a k-scheme +X. We call A ´etale A-locally free at a point p ∈ X, for some k-algebra A, if there exists an ´etale +morphism f : Y → X such that p ∈ f(Y ) and f ∗A is isomorphic to A ⊗ OY as OY -algebras. +Furthermore, A is called ´etale A-locally free if A is ´etale A-locally free at all points of X. Let us +remark that an ´etale A-locally free sheaf of algebras is automatically quasi-coherent and, if A is +finite-dimensional, coherent. +´Etale local triviality can actually be checked on fibers by virtue of the following result, which is +an algebro-geometric version of [Kir78], see [Abe21, Theorem 2.10]. +Proposition 6.4. Let k be an algebraically closed field of characteristic 0, X be a reduced k-scheme +of finite-type, A be a finite-dimensional k-algebra, and A be a quasi-coherent sheaf of algebras on +X. Then A is ´etale A-locally free if and only if A|p ∼= A for all closed points p ∈ X. +It turns out that a sheaf of algebras which can be ´etale trivialized by a unital algebra is automat- +ically unital, i.e. we have the following result. +Lemma 6.5. Let A be a unital algebra over a field k and A be an ´etale A-locally free sheaf of +algebras on a k-scheme X. Then A is unital. In particular, h0(A) > 0. +Proof. Let U ⊆ X be an open subset and assume that U has an affine open covering {Ui}i∈I such +that Γ(Ui, A) is unital for all i ∈ I. Since Γ(D(f), A) = Γ(Ui, A)f and Γ(Ui, A) → Γ(D(f), A) as +well as Γ(D(f), A) → Γ(D(fg), A) are unital for all f, g ∈ Γ(Ui, OX) and i ∈ I, a gluing argument +shows that Γ(Ui ∩ Uj, A) and Γ(Ui, A) → Γ(Ui ∩ Uj, A) are unital. Therefore, a second gluing +argument implies that Γ(U, A) is unital. A similar consideration shows that Γ(U, A) → Γ(V, A) is +unital for all V ⊆ U. We conclude that A is unital if and only if every p ∈ X has an affine open +neighbourhood U such that Γ(U, A) is unital. +For every p ∈ X, We can chose an irreducible affine open neighbourhood U of p, an irreducible +affine scheme U ′, and a surjective ´etale morphism f : U ′ → U such that there exists an isomorphism +ψ: B ⊗R S → A ⊗ S of S-algebras, where B := Γ(U, A), R := Γ(U, OX), and S := Γ(U ′, OU′). The +element ψ−1(1) ∈ B ⊗R S is a unit and ψ is unital. Since f is faithfully flat of finite type, we can +recover B from B ⊗R S as +(6.1) +B = {a ∈ B ⊗R S | φ(a ⊗ 1) = 1 ⊗ a} +where φ: (B ⊗R S) ⊗R S → S ⊗R (B ⊗R S) is defined by φ((b ⊗ s) ⊗ t) = s ⊗ (b ⊗ t); see e.g. [Mil80, +Remark 2.21]. In particular, B can be identified with an subalgebra of the unital algebra A ⊗ S. +Since φ is an isomorphism, it is unital. Therefore, B = Γ(U, A) contains the unit of B ⊗R S. Thus, +the argument in the beginning of this proof implies that A is unital. +Now h0(A) > 0 follows from the fact that 1 ∈ H0(A). +■ + +24 +RASCHID ABEDIN +6.2.1. Rigidity and strongly geometrically admissible algebras. Consider the affine variety Alg(d, k) = +Hom(kd ⊗ kd, kd) of all possible multiplication maps on kd. There is a natural action of the group +of invertible d × d-matrices GL(d, k) given by +(g · ϑ)(v ⊗ w) = g−1ϑ(gv ⊗ gw) +∀g ∈ GL(d, k), ϑ ∈ Alg(d, k), v, w ∈ kd. +(6.2) +The orbit of a multiplication map under this action corresponds to the isomorphism class of the +associated algebra. +Let M ⊆ Alg(d, k) be a GLn(d, k)-invariant affine subvariety and write A ∈ M for an algebra +A = (kd, µ), if µ ∈ M. A k-algebra A = (kd, µ) is called M-rigid if A ∈ M and the orbit +(6.3) +O(A) := {A′ = (kd, µ′) ∈ M | µ′ = gµ for some g ∈ GL(d, k)} +contains an open neighbourhood of A in M. +A sufficient condition for ´etale local triviality is the rigidity of the fiber, as the following result, +which is a algebro-geometric version of a generalization of [Kir83, Lemma 2.1], states. +Proposition 6.6. Let k be an algebraically closed field of characteristic 0 and M be a GL(d, k)- +stable subvariety of Alg(d, k). Furthermore, let A be a locally free sheaf of algebras on a reduced +k-scheme X such that A|q ∈ M for all q ∈ X closed. +If A|p is M-rigid for some closed point p ∈ X, A is ´etale A|p-locally free in p. In particular, +A|q ∼= A|p for all closed points q ∈ X in some neighbourhood of p. +The proof is a straight forward adaptation of the proof of [Abe21, Theorem 2.11] to this setting. A +consequence of Lemma 6.6 is the following important criterion for strong geometric admissibility. +Proposition 6.7. Let k be an algebraically closed field of characteristic 0, M ⊆ Alg(d, k) be a +GLn(d, k)-invariant affine subvariety, and (A, β) metric k-algebra in M. +Then (A, β) is strongly geometrically admissible if: +(1) (A, β) is geometrically admissible in the sense of Subsection 4.2; +(2) For any ringed A-lattices (O, W) and any maximal ideal m ⊆ O such that +• Wm is free as Om-module and +• the pairing Wm × Wm → Om induced by β is perfect +the k-algebra W/mW is M-rigid. +Proof. Let ((X, A), (p, c, ζ)) is the geometric datum associated to (O, W) in Subsection 4.1. Fur- +thermore, let U be the set of closed points q ∈ X such that +• Aq is a free OX,q-module; +• The restriction ζ(Aq) × ζ(Aq) → c(OX,q) of β from (4.6) is non-degenerate. +Then A|q is M-rigid for all q ∈ U by assumption. Combining Proposition 6.6 and p ∈ U, U is a +non-empty open subset of the set of closed points of X. In particular, U is connected since X is +irreducible. +Furthermore, every q ∈ U has an open neighbourhood U ′ ⊆ U such that A|q ∼= A|q′ holds for all +q′ ∈ U ′ by virtue of Proposition 6.6. The connectedness of U therefore implies that A|q ∼= A|p ∼= A +for all q ∈ U. This implies that (A, β) is strongly geometrically admissible. +■ +Consider M ∈ {Lied, Assd, Jord} where Lied, Assd, Jord ⊆ Alg(d, k) are the varieties of d-dimensional +Lie, associative, and Jordan algebras respectively. In Subsection 4.2.1, we discussed that any sim- +ple A ∈ M has an, up to multiplication by a scalar, unique algebra metric β and that the metric +algebra (A, β) is geometrically admissible. +Let (O, W) be a ringed A-lattice and m ⊆ O be a maximal ideal such that Wm is free. +If +the restriction Wm × Wm → Om of β from (4.6) is non-degenerate, the k-algebra W/mW ∈ M +inherits an algebra metric from β. +This algebra metric can be explicitly described using the +formula in Subsection 4.2.1 and we see from this description that W/mW is semi-simple, i.e. a +direct sum of simple subalgebras. If M = Lied we use Cartan’s criterion for semi-simplicity and + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +25 +if M ∈ {Assd, Jord} this is a consequence of general results on trace-admissible algebras; see e.g. +[Alb49]. +Since semi-simple algebras in M are rigid (see [HG88] for the case that M ∈ {Lied, Assd} and +[Fin90] for the case that M = Jord), we see that (A, β) satisfies the conditions of Proposition 6.7. +Therefore, we obtain the following result. +Corollary 6.8. Any finite-dimensional simple Lie, associative, or Jordan algebra over an alge- +braically closed field of characteristic 0 is strongly geometrically admissible if equipped with its (up +to scalar multiple) unique algebra metric. +6.2.2. Sheaves of algebras on one-dimensional affine algebraic groups. Recall that over an alge- +braically closed field of characteristic 0, a connected affine algebraic group over k of dimension one +is either isomorphic to the affine line or the punctured affine line. Let us conclude this subsection +with a classification of all sheaves of algebras with constant fibers on these schemes; see [Abe22, +Theorem 6.1.1] for a proof. +Proposition 6.9. Let A be a finite-dimensional algebra over an algebraically closed field k of +characteristic 0. +(1) Let B be a k[v, v−1]-algebra satisfying B/(v − λ)B ∼= A for all λ ∈ k×. Then there exists +σ ∈ Autk-alg(A) of order m ∈ N such that +B ∼= L(A, σ) := {a ∈ A[�v, �v−1] | a (exp (2πi/m) �v) = σ(a(�v))} +as k[v, v−1]-algebras. Here, the k[v, v−1]-module structure of L(A, σ) is defined by �vm = v. +(2) Let B be an k[z]-algebra satisfying B/(z − λ)B ∼= A for all λ ∈ k. Then B ∼= A[z] as k[z]- +algebras. +6.3. Geometric A-CYBE datum. We call a triple (X, (A, βA)) geometric A-CYBE datum for +a finite-dimensional k-algebra A if: +• X is an irreducible cubic plane curve over k; +• A is a coherent sheaf of algebras on X such that: +(1) H0(A) = 0 = H1(A); +(2) A|p ∼= A for all smooth closed p ∈ X; +• βA : A × A → OX is a symmetric, perfect, associative OX-bilinear form. +The name “geometric A-CYBE datum” will become clear in Subsection 6.3.1. +Remark 6.10. +It is well-known that any irreducible plane cubic curve X over k is defined by an +equation y2 = x3 + ax + b and precisely one of the following cases occurs: +(1) X is smooth if and only if 4b3 + 27a2 ̸= 0, in which case X is an elliptic curve. +(2) X has a unique nodal singularity if 4b3 = −27a2 ̸= 0. In this case, X \ {s} is isomorphic to +the punctured affine line Spec(k[v, v−1]). +(3) X has a unique cuspidal singularity if a = b = 0. In this case, X \ {s} is isomorphic to the +affine line Spec(k[z]). +Let us note that, up to isomorphism, irreducible cubic plane curves are precisely irreducible pro- +jective curves over k of arithmetic genus 1. +♦ +The following lemma is important for the identification of geometric A-CYBE data below. +Lemma 6.11. Let k be an algebraically closed field of characteristic 0, X be an irreducible plane +cubic curve over k, and F be coherent sheaf on X satisfying: +• H1(F) = 0; +• There exists a non-degenerate symmetric OX-bilinear form βF : F × F → OX. +Then βF is perfect. In particular, βF|p is non-degenerate for all p ∈ X. +Proof. Since βF is non-degenerate, we have a short exact sequence +(6.4) +0 −→ F +βa +F +−→ F∗ −→ C −→ 0, + +26 +RASCHID ABEDIN +where C := Cok(βa +F) is a torsion sheaf. We obtain the long exact sequence in cohomology +0 −→ H0(F)−→H0(F∗) −→ H0(C) −→ H1(F)−→H1(F∗) −→ H1(C) −→ 0. +(6.5) +The dualizing sheaf of any irreducible cubic plane curve is trivial, so Serre duality implies that +h0(F∗) = h1(F) = 0 and thus H0(C) = 0. Since C is a torsion sheaf, we see that C = 0, so βa +F is an +isomorphism. +■ +6.3.1. Geometric solutions of the A-CYBE. In this subsection, we will repeat the construction of +geometric solutions of the usual CYBE from [BG18] in our general setting. This will result in a +construction of geometric solutions of a geometric A-CYBE from a geometric A-CYBE datum. In +particular, this explains the name “geometric A-CYBE datum”. +Let A be a finite-dimensional k-algebra, (X, (A, βA)) be a geometric A-CYBE datum, and +C ⊆ X be the set of smooth points. Fix a global section η ∈ H0(ωX) of the dualizing sheaf ωX of +X. Let us remark that ωX can be identified with the sheaf of so-called Rosenlicht-regular 1-forms; +see e.g. [Con00, Section 5.2]. Consider the diagonal residue sequence +0 −→ OX×C −→ OX×C(∆) +resη +∆ +−→ δ∗OC −→ 0. +(6.6) +Here, ∆ ⊆ X × C is the image of δ: C → X × C defined by p �→ (p, p). Furthermore, resη +∆ is +determined by (u1 − u2)−1 �→ µ locally around any closed point p ∈ C, where: +• u is a local parameter of p defined on an affine open subset U of C; +• ωC and OX×C(−∆) are locally generated by du and +u1 − u2 := u ⊗ 1 − 1 ⊗ u ∈ Γ(U, OX) ⊗ Γ(U, OX) ∼= Γ(U × U, OX×X) +respectively, after potentially shrinking U; +• ηp = µdu holds for some uniquely determined µ ∈ Γ(U, OX). +The tensor product of (6.6) with A ⊠ A|C := pr∗ +1A ⊗OX×C pr∗ +2A|C for the canonical projections +X +pr1 +←− X × C +pr2 +−→ C gives the short exact sequence +0 −→ A ⊠ A|C −→ A ⊠ A|C(∆) −→ δ∗(A|C ⊗OC A|C) −→ 0. +(6.7) +Using the K¨unneth formula and H0(A) = 0 = H1(A) results in +H0(A ⊠ A|C) = H0(A) ⊗ H0(A|C) = 0 and +H1(A ⊠ A|C) = +� +H1(A) ⊗ H0(A|C) +� +⊕ +� +H0(A) ⊗ H1(A|C) +� += 0. +(6.8) +Therefore, the long exact sequence in cohomology induced by (6.7) results in an isomorphism +R: H0(A ⊠ A|C(∆)) → H0(A|C ⊗ A|C). +The pairing βA of A induces an isomorphism B : A|C ⊗OC A|C → EndOC(A|C) defined by +(6.9) +a ⊗ b �−→ βA(b, −)a for all U ⊆ C affine open a, b ∈ Γ(U, A). +Combined with R, we obtain an isomorphism Φ := BR: H0(A ⊠ A|C(∆)) → EndOC(A|C). +Consider the section ρ := Φ−1(idA|C) ∈ H0(A ⊠ A|C(∆)). Then, if U ⊆ C is any affine open +subset such that η = µ−1du for µ, u ∈ Γ(U, OX, we can write +(6.10) +ρ|U×U = (1 ⊗ µ)χ +u1 − u2 ++ s +for some s ∈ Γ(U × U, A ⊠ A), where χ ∈ Γ(U × U, A ⊠ A) is any preimage of idA|U under the +surjective map +Γ(U × U, A ⊠ A) −→ Γ(U, A ⊗ A) → EndOU (A|U). +One should think of (6.10) as an analog of the standard form (5.6) of (n, λ)-type series. +By repeating the arguments in [BG18, Theorem 3.11 and Theorem 4.3], we can see that +(6.11) +ρ13ρ12 − ρ12ρ23 + ρ13ρ23 = 0. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +27 +Here, the summands on the left-hand side can be understood as a rational section of A⊠ A⊠ A by +adapting the notations from Subsection 5.2 to this geometric setting. In particular, ρ is a solution +of a geometric version of the A-CYBE. +6.4. Geometrization of Manin triples. Recall that k is an algebraically closed field of char- +acteristic 0. Let (A, β) be a strongly geometrically admissible k-algebra, n ∈ N, λ ∈ k[[z]]×, and +((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple associated to that data in Subsection 3.2. Recall +from Theorem 4.1 that n ∈ {0, 1, 2} and that we may assume λ = 1. +The goal of this section is to assign a geometric A-CYBE datum to ((Dn(A), β(n,1)), A[[z]], W). +6.4.1. Geometrization in case n = 0. There exists a particular O ⊆ M := {f ∈ k((z)) | fW ⊆ W} +such that dim(k((z))/(k[[z]] + O)) = 1; see Subsection 4.5. Namely, the integral closure N of M +either satisfies dim(k((z))/(k[[z]] + N)) = 1 or N = k[u] for some u ∈ z−1 + zk[[z]]×, and then +(6.12) +O := +� +N +if dim(k((z))/(k[[z]] + N)) = 1, +k[u′, u′u] +if N = k[u]. +Applying the geometrization procedure form Section 4.1 to (O, W) gives a geometric datum +((X, A), (p, c, ζ)). Observe that A satisfies ζ : � +Ap +∼ += +−→ A[[z]] and H0(A) = 0 = H1(A) (combine +A((z)) = A[[z]] ⊕ W with (4.5)). +If X is smooth it is a smooth irreducible cubic plane curve. If O = k[u′, u′u] for u ̸= z−1, we +have −u′ = u2 − a for some a ∈ k \ {0}. This equation is equivalent to u′,2(a − u′) = u′,2u2, so +putting y = u′u and x = u′, we see that X is a nodal irreducible cubic plane curve. Finally, if +O = k[z−2, z−3], X is clearly a cuspidal irreducible cubic plane curve. +In order to see that (X, A) gives rise to a geometric A-CYBE datum, we have to construct an +appropriate OX-bilinear map βA : A × A → OX. +If X is smooth (which is equivalent to the fact that O is integrally closed), the geometrically +admissible metric β defines a pairing βA : A × A → OX; see Subsection 4.2.2.(2). +Let us assume that X is singular, i.e. O = k[u′, u′u]. Since β is geometrically admissible, it +induces a pairing W × W → N = k[u]. Since W is Lagrangian, the image under this pairing +lies in the kernel of res0 restricted to k[u]. It is easy to see that this kernel is equal to O, so the +coefficient-wise application of β defines a map W × W → O. It is now straight forward to see that +for every U ⊆ X, the commutative diagram +(6.13) +Γ(U, A) × Γ(U, A) +� +ζ×ζ +� +Γ(U, OX) +c +� +A((z)) × A((z)) +β +� k((z)) +defines a pairing βA : A × A → OX. +Lemma 6.12. The triple (X, (A, βA)) is a geometric A-CYBE datum. In particular, A|q ∼= A for +all smooth closed q ∈ X. +Proof. Since by construction the fiber of βA is β at p ∈ X, the kernel of the canonical morphism +A → A∗ is a torsion subsheaf of the torsion free sheaf A, hence zero. +In other words, βA is +non-degenerate. By virtue of Lemma 6.11, βA is perfect. +It remains to prove that A|q ∼= A for all smooth closed q ∈ X. Observe that A|p ∼= A already +holds, so we may assume q ̸= p. Let m ⊆ O be the maximal ideal associated to +q ∈ X \ {p} ∼= Spec(O). +Then βA,q is identified with the restriction Wm ×Wm → Om of β from (4.6) through Aq ∼= Wm and +OX,q ∼= Om. Since A is strongly geometrically admissible (recall the definition from Subsection +6.1) and βA,q is perfect, we obtain A|q ∼= W/mW ∼= A. +■ + +28 +RASCHID ABEDIN +Let us note the following important consequence of Lemma 6.12. +Proposition 6.13. The following results are true. +(1) If X is elliptic, A is non-unital. +(2) Let X be singular and ρ be the section constructed in Subsection 6.3.1 from (X, (A, βA)) and +η := dv ∈ H0(ωX), where v := u/u′ is the local generator of p associated to the representation +(6.12). Then image of ρ under the Taylor expansion +H0(A ⊠ A|C(∆)) −→ lim +←− +k +� +Γ(X \ {p}, A) ⊗ � +Ap/mp � +Ak +p +� ζ⊗ζ +−→ (A((z)) ⊗ A)[[z]] +at X × {p} trivialized by (c, ζ) is equivalent to the solution r of the A-CYBE associated to the +Manin triple ((A((z)), β(0,0)), A[[z]], W). +Proof of (1). Assume X is elliptic and A is unital. According to Lemma 6.12 and Proposition 6.4, +A is A-´etale locally free. By virtue of Lemma 6.5, this contradicts h0(A) = 0. Therefore, A is +non-unital if X is elliptic. +■ +Proof of (2). The proof is a straight forward repetition of the the proof of [Abe21, Theorem 3.17] +(see also [Abe22, Theorem 3.3.3]). +■ +6.4.2. Geometrization in case n = 1. Let W+ (resp. W−) be the projection of +W ⊆ D1(A) = A((z)) × A +onto the A((z)) (resp. A) component. By virtue of Section 6.5.(4), k[z−1]W+ ⊆ W+ and we can +consider the geometrization ((Y, W), (p, c, ζ)) of (k[z], W+), where Y = P1 and s− := p = (z). Let +s+ ∈ P1 be the point corresponding to the ideal (z−1) ⊆ k[z−1] via c(Γ(P1 \ {s−}, OX)) = k[z−1]. +Let the sheaf of algebras V on P1 be defined as the pull-back +(6.14) +V +� +� +W− +� +W +� W|s− ∼= A +where A, W−, and W|s− are understood as skyscraper sheaves at s−. In other words, V fits into +the short exact sequence +(6.15) +0 −→ V −→ W ⊕ W− −→ A −→ 0. +Since the morphism W− → A is injective, the morphism V → W is too and we can identify V with +a subsheaf of W. Let βW : W × W → OP1 be the pairing induced by β in Subsection 4.2.2.(2) and +βV : V × V → OP1 be the restriction to V. +Lemma 6.14. The following is true: +(1) H0(V) ∼= ι(g[[x]]) ∩ (W+ × W−), H1(V) = 0 and V|P1\{s−} = W|P1\{s−}; +(2) There exist canonical surjective morphisms θ± : V|s± → W±/W ⊥ +± such that +βV|s±(a, b) = β± +(1,1)(θ±(a), θ±(b)) +holds for all a, b ∈ V|s±. +Proof of (1). The restriction of the short exact sequence (6.15) to P1 \ {s−} results in V|P1\{s−} = +W|P1\{s−}. Since the first cohomology group of torsion sheaves vanishes and A[[z]] + W+ = A((z)) +implies H1(W) = 0 because of (4.5), the long exact sequence of (6.15) in cohomology reads +(6.16) +0 −→ H0(V) −→ H0(W) ⊕ W− −→ A −→ H1(V) −→ 0. +In particular, H0(W) ∼= A[[z]] ∩ W+ implies +H0(V) ∼= A[[z]] ∩ (W+ × W−). + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +29 +The image W + of A[[z]] ∩ W+ → A under the canonical map A[[z]] → A satisfies W + + W− = A. +Therefore, H0(W) ⊕ W− → A in (6.16) is surjective. Consequently, H1(V) = 0. +■ +Proof of (2). The algebra W+ is a torsion-free as k[z−1]-module, so it is a free k[z−1]-module. +Since β is geometrically admissible, this implies β(W+, W+) ⊆ k[z−1]. In particular, +(6.17) +β+ +(1,1)(z−1a, b) = res0z−2β(a, b) = 0 +for all a, b ∈ W+, so z−1W+ ⊆ W ⊥ ++ . Therefore, we have a surjective morphism +θ+ : V|s+ ∼= W+/z−1W+ −→ W+/W ⊥ ++ +intertwining the corresponding forms. +On the other hand, the construction of V as pull-back gives a canonical map V → W− which +is surjective since W → W|s− is surjective. This morphism factors through a surjective morphism +V|s− → W− which respects the forms and this map induces θ−. +■ +Let X be an irreducible cubic plane curve with nodal singularity s and chose the normalization +ν : P1 → X in such a way that ν−1(s) = {s+, s−}. Let us understand W±/W ⊥ +± as skyscraper sheaf +at s± and let θ be the direct image under ν of the morphism +(6.18) +V −→ V|s+ × V|s− +(θ+,θ−) +−→ +W+/W ⊥ ++ × W−/W ⊥ +− +for θ± from Lemma 6.14. Let A be defined as pull-back +(6.19) +A +� +� +W/(W ⊥ ++ × W ⊥ +− ) +� +ν∗V +θ +� ν∗(W+/W ⊥ ++ × W−/W ⊥ +− ) +where W/(W ⊥ ++ × W ⊥ +− ) is viewed as skyscraper sheaves at s ∈ X. Again, this is equivalent to the +short exact sequence +(6.20) +0 −→ A −→ ν∗V ⊕ (W/(W ⊥ ++ × W ⊥ +− )) −→ W+/W ⊥ ++ × W−/W ⊥ +− −→ 0. +Therefore, A can be identified with a subsheaf of ν∗V. Let βA : A × A → ν∗OP1 be the restriction +of ν∗βV to A, where we recall that βV : V × V → OP1 is obtained by restriction from βW. +Lemma 6.15. The datum (X, (A, βA)) is a geometric A-CYBE datum. In particular, A|q ∼= A +for all smooth closed q ∈ X. +Proof. The long exact sequence in cohomology of (6.20) is given by +(6.21) +0 −→ H0(A) −→ H0(V) ⊕ (W/(W ⊥ ++ × W ⊥ +− )) −→ W+/W ⊥ ++ × W−/W ⊥ +− −→ H1(A) −→ 0. +Here, we used that the first cohomology group of torsion sheaves vanishes and H1(V) = 0; see +Lemma 6.14.(1). The canonical map H0(V) → W+/W ⊥ ++ × W−/W ⊥ +− thereby coincides with the +inclusion +A[[z]] ∩ (W+ × W−) −→ W+/W ⊥ ++ × W−/W ⊥ +− +under the identification H0(V) ∼= A[[z]] ∩ (W+ × W−). Therefore, Subsection 4.4.(3) implies that +the middle arrow in (6.21) is an isomorphism. Consequently, H0(A) = 0 = H1(A), so property (1) +of in the definition of a geometric A-CYBE datum in Section 6.3 is satisfied. +Next, we want to see that βA actually takes values in OX ⊆ ν∗OP1. Let a, b ∈ A|s and a±, b± ∈ +W± be representatives of the images of a, b under the canonical maps A|s → V|s± → W±/W ⊥ +± . +Then +(6.22) +βA|s(a, b) = (β+ +(1,1)(a+, b+), β− +(1,1)(a−, b−)) ∈ k × k ∼= ν∗OP1|s + +30 +RASCHID ABEDIN +holds, since the θ± : V|s± → W±/W ⊥ +± intertwine the forms β± +(1,1) and βV|s±. The definition of A +implies that (a+, a−), (b+, b−) ∈ W, and the Lagrangian property of W gives +(6.23) +0 = β(1,1)((a+, a−), (b+, b−)) = β+ +(1,1)(a+, b+) − β− +(1,1)(a−, b−). +We obtain βA|s(a, b) ∈ {(λ, λ) | λ ∈ k}. This implies that βA takes values in OX. +Since βW|s− can be identified with the algebra metric β on W|s− ∼= A, βW, and consequently +βA, is non-vanishing on an open subset of X. Combined with Proposition 6.6 this implies that +there exists a closed point q ∈ P1 \ {s+, s−} such that A ∼= W|q ∼= A|ν(q) and βA|ν(q) is a non- +zero associative bilinear form on this space. In particular, βA|q is automatically non-degenerate. +Therefore, the kernel of the canonical morphism A → A∗ induced by βA is a torsion subsheaf of +the torsion free sheaf A, hence zero. Consequently, βA is non-degenerate. +Lemma 6.11 now states that βA is perfect. +Consequently, for all closed q ∈ P1 \ {s+, s−} +the bilinear form βW,q, which can be identified with βA,ν(q) via Wq ∼= Aν(q), is perfect. Since +(A, β) is strongly geometrically admissible, this implies that W+/(z − a)W+ ∼= W|q ∼= A|ν(q) +is isomorphic to A. Here, for a ∈ k× the maximal ideal (z − a) ⊆ k[z, z−1] defines the point +q ∈ P1\{s+, s−} ∼= Spec(k[z, z−1]). In conclusion, (X, (A, βA)) is a geometric A-CYBE datum. +■ +Proposition 6.16. There exists a ϕ ∈ Autk[[z]]-alg(A[[z]]), unique +{tk,i ∈ A[z] | k ∈ N, i ∈ 1, n}, +and N ∈ N such that +(6.24) +ϕ(W) = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} +and tk,i = 0 for all k ⩾ N, i ∈ 1, n. Here, the wk,i were defined in (5.5). +Proof. Lemma 6.15 implies W|q ∼= A|ν(q) ∼= A for all q ∈ P1 \ {s+, s−}. Combined with W|s− ∼= A, +this implies that B := ζ(Γ(P1 \ {s+}, W)) ⊆ A[[z]] is a free k[z] = c(Γ(P1 \ {s+}, OP1))-algebra +satisfying B/(z − λ)B ∼= A for all λ ∈ k. Therefore, B ∼= A[z] by virtue of Theorem 6.9.(2). +Completing said automorphism in the (z)-adic topology yields ϕ ∈ Autk[[x]]-alg(A[[z]]) with the +property ϕ(B) = A[z]. Since W is a sheaf, we have +(6.25) +ϕ(W+) = ϕ(ζ(Γ(P1 \ {s−}, W)) ⊆ ϕ(ζ(Γ(P1 \ {s+, s−}, W))) = ϕ(B)[z−1] = A[z, z−1]. +This, combined with the fact that W+ is a free k[z−1]-module, implies that ϕ(W+) ⊆ zN−1A[z−1] +holds for a sufficiently large integer N ∈ N. Consequently, +(6.26) +z−NA[z−1] ⊆ ϕ(W+)⊥ ⊆ ϕ(W+) ⊆ zN−1A[z−1]. +Since A[[z]] ⊕ ϕ(W+) = D1(A), we can now write +(6.27) +ϕ(W) = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} +for uniquely determined {tk,i ∈ A[[z]] | k ∈ N, i ∈ 1, n}. Equation (6.26) now implies that tk,i ∈ A[z] +and tk,i = 0 for all k ⩾ N. +■ +6.4.3. Geometrization in case n = 2. Similar to the previous case, +W ⊆ D2(A) = A((z)) × A[z]/x2A[z] +and we denote by W+ (resp. W−) the projection of W to A((z)) (resp. A[z]/z2A[z]). +Lemma 6.17. The following facts are true. +(1) W = W+ × W−; +(2) W+ ∩ z2A[[z]] = {0}, so W+ ∩ A[[z]] can be identified with a subalgebra of A[z]/z2A[z]; +(3) (W+ ∩ A[[z]]) ⊕ W− = A[z]/z2A[z]. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +31 +Proof. For (1), observe that β(W+, W+) ⊆ k[z−1] holds since (A, β) is geometrically admissible and +W+ is a free k[z−1]-algebra by virtue of Section 4.5.(5). Therefore, z−2β(W+, W+) ⊆ z−2k[z−1] +implies β(2,1)(a, b) = res0z−2β(a, b) = 0 for all a, b ∈ W+. Consequently, W+ ⊆ W ⊥ ++ . Together +with W ⊥ ++ ⊆ W+ we arrive at W+ = W ⊥ ++ . Subsection 4.4.(3) implies W− = W ⊥ +− , so +W ⊥ ++ × W ⊥ +− ⊆ W ⊆ W+ × W− +concludes the proof of (1). +The identities {0} = (A[[z]] + W+)⊥ = z2A[[z]] ∩ W ⊥ ++ = z2A[[z]] ∩ W+ imply (2). Part (3) now +follows from (2) and A[[z]] ⊕ (W+ × W−) = A((z)) × A[z]/z2A[z]. +■ +Consider the geometrization ((Y, W), (p, c, ζ)) of (k[z−1], W+), where as in the last section we have +Y = P1. Let X be an irreducible plane cubic curve with cuspidal singularity s and chose the +normalization ν : P1 → X in such a way that ν(p) = s. The isomorphism ζ : � +Wp → A[[z]] implies +that +(6.28) +ν∗W|s ∼= ζ(� +Wp)/z2ζ(� +Wp) = A[z]/z2A[z]. +This yields a surjective morphism ν∗W → A[z]/z2A[z]. Let A be the sheaf of algebras defined as +the pull-back +(6.29) +A +� +� +W− +� +ν∗W +� A[z]/z2A[z] +where A[z]/z2A[z] and W− are understood as skyscraper sheaves at s. Equivalently, A fits into +the short exact sequence +(6.30) +0 −→ A −→ ν∗W ⊕ W− −→ A[x]/z2A[z] −→ 0. +Let βW : W×W → OP1 be the pairing induced by β in Subsection 4.2.2 and let βA : A×A → ν∗OP1 +be the the restriction of ν∗βW to A ⊆ ν∗W. +Lemma 6.18. The datum (X, (A, βA)) is a geometric A-CYBE datum. In particular, A|q ∼= A for +all smooth closed q ∈ X. Furthermore, there exists ϕ ∈ Autk((z))-alg(A((z))) such that the identity +ϕ(W+) = A[z−1] holds. +Proof. The global section of ν∗W → A[z]/z2A[z] coincides with the canonical morphism +A[[z]] ∩ W+ → A[z]/z2A[z] +if H0(W) is identified with A[[z]] ∩ W+. Therefore, the middle arrow in the long exact sequence in +cohomology +(6.31) +0 −→ H0(A) −→ H0(W) ⊕ W− −→ A[z]/z2A[z] −→ H1(A) −→ 0 +of (6.30) is an isomorphism by virtue of Lemma 6.17.(3). Here we used again that: +• H1(W) = 0 by virtue of Subsection 4.4.(2) and (4.5); +• The first cohomology group of torsion sheaves vanishes. +Consequently, H0(A) = 0 = H1(A). +Let us now show that βA : A × A → ν∗OX takes values in OX. For any a, b ∈ A|s we have +(6.32) +ν∗βW|s(a, b) = β(a1, b1) + [z](β(a1, b2) + β(a2, b1)) ∈ k[z]/(z2), +where a1 + [z]a2 and b1 + [z]b2 ∈ A[z]/z2A[z] are the images of a and b respectively under +A|s → ν∗W|s ∼= A[z]/z2A[z]. +By definition of A, a1 +[z]a2, b1 +[z]b2 ∈ W− and β(a1, b2)+β(a2, b1) = 0 since W− ⊆ A[z]/z2A[z] +is Lagrangian. Therefore, βA|s(a, b) = β(a1, b1) ∈ k, implying that βA takes values in OX ⊆ ν∗OP1. + +32 +RASCHID ABEDIN +Repeating the arguments in the end of the proof of Lemma 6.15, we can deduce that (X, (A, βA)) +is a geometric A-CYBE datum and A|q ∼= A for all smooth closed q ∈ X. Now (6.29) implies that +W|q ∼= A|ν(q) ∼= A for all q ∈ P1 \ {p}. Consequently, +W+ = ζ(Γ(P1 \ {p}, W)) ⊆ A((z)) +is a free k[z−1] = c(Γ(P1 \ {p}, OP1))-algebra satisfying W+/(z−1 − λ)W+ ∼= A for all λ ∈ k. +Therefore, W+ ∼= A[z−1] by virtue of Theorem 6.9.(2). This induces the automorphism ϕ. +■ +We can now copy the arguments of Proposition 6.16 to deduce that. +Proposition 6.19. There exists a ϕ ∈ Autk[[z]]-alg(A[[z]]), a set +{tk,i ∈ A[z] | k ∈ N, i ∈ 1, n}, +and a natural number N ∈ N such that +(6.33) +ϕ(W) = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} +and tk,i = 0 for all k ⩾ N, i ∈ 1, n. Here, the wk,i were defined in (5.5). +6.5. Proof of Theorem 6.1. Recall the notation and statement of Theorem 6.1: +• k is an algebraically closed field of characteristic 0; +• (A, β) is a unital strongly geometrically admissible metric k-algebra and γ ∈ A ⊗ A is its +canonical A-invariant element; +• ((Dn(A), β(n,λ)), A[[z]], W) is a Manin triple of the form 3.4 for some n ∈ N and λ ∈ k[[z]]×. +• r is the solution of the A-CYBE associated to the Manin triple ((Dn(A), β(n,λ)), A[[z]], W) via +Theorem 5.3. +Then precisely one of the following cases occurs: +(1) If n = 0, the curve X from the A-CYBE datum (X, A) of ((Dn(A), βn), A[[z]], W) constructed +in Subsection 6.4.1 is either a nodal or cuspidal irreducible cubic plane curve. Furthermore: +(a) X is nodal if and only if r is trigonometric in the sense of Theorem 6.1; +(b) X is cuspidal if and only if r is rational in the sense in the sense of Theorem 6.1; +(2) n = 1 if and only if r is quasi-trigonometric in the sense of Theorem 6.1; +(3) n = 2 if and only if r is quasi-rational in the sense of Theorem 6.1. +6.5.1. Proof of (1). First of all, since A is unital, X cannot be elliptic by virtue of Proposition +6.13.(1). Therefore, X is either a nodal or a cuspidal irreducible plane cubic curve. Let s ∈ X be +the unique singularity in both cases. +Let η and ρ be as in Proposition 6.13.(2) and chose isomorphisms +(6.34) +C := X \ {s} +f +−→ +� +Spec(k[v, v−1]) +if X is nodal; +Spec(k[z]) +if X is cuspidal +such that +(6.35) +η = +� +v−1dv +if X is nodal; +dz +if X if cuspidal. +In both cases we can chose U = C in (6.10) in order to obtain +(6.36) +ρ|C×C = (1 ⊗ µ)χ +u1 − u2 ++ s +where u = v (resp. u = z) and µ = v (resp. µ = 1) if X is nodal (resp. if X is cuspidal). Recall +that s is some element in H0(A|C ⊠ A|C) = H0(A|C) ⊗ H0(A|C) and χ is some preimage of idA|C +under +H0(A|C ⊠ A|C) −→ H0(A|C ⊗OC A|C) −→ EndOC(A|C). + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +33 +Using Lemma 6.12 and Theorem 6.9 we can see that there exists a f ♯-equivariant isomorphism +(6.37) +H0(A|C) +φ1 +−→ +� +L(A, σ) +if X is nodal; +A[z] +if X is cuspidal, +where in the nodal case σ ∈ Autk-alg(A) is of finite order. Here, f ♯ is the map k[v, v−1] → Γ(C, OX) +(resp. k[z] → Γ(C, OX)) defined by f if X is nodal (resp. cuspidal). Let us conclude the proof of +(1) in a case by case fashion. +Case (a): X is nodal. Let Aj := {a ∈ A | σ(a) = εja} for the m-th root of unity ε ∈ k from +Theorem 6.1. Note that βA induces an algebra metric L(A, σ) × L(A, σ) → k[v, v−1] defined by +the coefficient-wise application of β. In particular, since v = �vm and +(6.38) +β(�vka, �vℓ) = β(a, b)�vk+ℓ ∈ k[v, v−1] +holds for all a ∈ Ak, b ∈ Aℓ, we have β(Ak, Aℓ) = {0} if k + ℓ /∈ mZ. Furthermore, +β(σ(a), b) = εkβ(a, b) = εk+ℓ−ℓβ(a, b) = β(a, σ−1(b)) +holds for k + ℓ ∈ mZ. Combined, we see that σ is orthogonal with respect to β. +Since σ is orthogonal with respect to β, it is easy to see that γ = �m−1 +j=0 γj ∈ �m−1 +j=0 (Aj ⊗ A−j). +We can choose χ as the preimage of +(6.39) +m−1 +� +j=0 +� �v +�w +�j +γj ∈ L(A, σ) ⊗ L(A, σ) +under the isomorphism φ1 ⊗ φ1 : H0(A|C)⊗ H0(A|C) = H0(A|C ⊠ A|C) → L(A, σ)⊗ L(A, σ). Then +(6.40) +(φ1 ⊗ φ1)ρ|C×C = +1 +(v/w) − 1 +� �v +�w +�j +γj + t +holds for t := (φ1 × φ1)s ∈ L(A, σ) ⊗ L(A, σ). +Let exp be the completion of k[v, v−1] → k[[z]], v �→ exp(z) with respect to the ideal (v − 1) +and φ2 ∈ Autk-alg(A[[z]]) be the exp-equivariant isomorphism obtained by completing the map +L(A, σ) → A[[z]], f �→ f(exp(z/m)) at the same ideal. Using Proposition 6.13.(2), we can see that +the automorphism φ := φ2φ1ζ−1 ∈ Autk-alg(A[[z]]) satisfies +(6.41) +(φ ⊗ φ)r(x, y) = +1 +exp (x − y) − 1 +m−1 +� +j=0 +exp +�x − y +m +� +γj + s +� +exp +� x +m +� +, exp +� y +m +�� +. +This concludes the proof in the nodal case. +■ +Case (b): X is cuspidal. We can chose χ ∈ H0(A|C)⊗H0(A|C) as the preimage of γ ∈ (A⊗A)[x, y] +under the isomorphism φ1 ⊗ φ1. Then +(6.42) +(φ1 ⊗ φ1)ρ|C×C = +γ +x − y + t +holds for t := (φ1 ⊗ φ1)s ∈ (A ⊗ A)[x, y]. +Let φ2 ∈ Autk[[z]]-alg(A[[z]]) be the completion of A[z] → A[[z]]. Using Proposition 6.13.(2), we +can see that +(6.43) +(φ ⊗ φ)r = +γ +x − y + t +holds for φ := φ2φ1ζ−1 ∈ Autk[[z]]-alg(A[[z]]). This concludes the proof in the cuspidal case. +■ + +34 +RASCHID ABEDIN +6.5.2. Proof of (2) and (3). By virtue of Proposition 6.16 and Proposition 6.19 there exist +{tk,i ∈ A[z] | k ∈ N, i ∈ 1, n} +and N ∈ N such that, up to isomorphism of Manin triples, +(6.44) +W = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} +and tk,i = 0 for all k ⩾ N. Here, the wk,i ∈ Dn(A) are defined in (5.5). The solution r of the +A-CYBE of W can now be determined by +(6.45) +r(x, y) = +∞ +� +k=0 +d +� +i=1 +(wk,i + tk,i) ⊗ biyk = ynγ +x − y + t(x, y), +where t = �N +k=0 +�d +i=1 tk,i(x) ⊗ biyk ∈ (A ⊗ A)[x, y]. +7. Classification of associative D-bialgebra structures over series +7.1. Non-triangular topological associative D-bialgebras on series are non-degenerate. +The final goal of this paper is the classification of all non-triangular topological associative D- +bialgebra structures on A[[z]] (i.e. topological D-bialgebra structures in the category of associative +algebras) for any finite-dimensional simple associative algebra A over an algebraically closed field +k of characteristic 0. Recall that these are exactly the co-opposites of (non-triangular) topological +balanced infinitesimal D-bialgebra structures on A[[z]]. Therefore, the classification of the latter is +equivalent. +In order to use Theorem 6.1, we begin by proving that, as in the case of a simple Lie algebra +over k, these are all non-degenerate. +Proposition 7.1. Let k be algebraically closed of characteristic 0 and (A, β) be a finite-dimensional, +simple, associative, metric k-algebra, i.e. A ∼= Mn(k) is the space of n × n-matrices with entries in +k and β is a scalar multiple of the algebra metric defined by the trace of matrices. +Any non-triangular topological associative D-bialgebra structure δ: A[[z]] → (A ⊗ A)[[x, y]] is +non-degenerate in the sense of Section 3.2. +7.1.1. Proof of Proposition 7.1. Let us begin by proving. +Lemma 7.2. Let k be algebraically closed of characteristic 0 and A be a finite dimensional asso- +ciative k-algebra. Every associative algebra B containing A as subalgebra is isomorphic to A ⊗ R +for some unital associative k-algebra R. Furthermore, if B is equipped with an algebra metric �β, +then for all a, b ∈ A and r, s ∈ R +(7.1) +�β(a ⊗ f, b ⊗ g) = β(a, b)t(rs) +for some t: R → k such that the associated pairing (r, s) �→ t(rs) is an algebra metric of R. +Proof. The algebra B splits into a direct sum of irreducible A-bimodules: B = � +i∈I AriA, where +I := {r ∈ B | ArA is irreducible}/ ∼ for r ∼ s if ArA = AsA and i �→ ri is some choice function +I → R. The modules AriA are all isomorphic to A itself, so B ∼= A ⊗ R as A-bimodule, for the +vector space R over k with basis {ri}i∈I. Let us write the original copy of A in B as A ⊗ 1 for +some distinguished element 1 ∈ R and note that (a ⊗ 1)(b ⊗ r) = ab ⊗ r for all a, b ∈ A and +r ∈ R by construction. Consider (1 ⊗ ri)(1 ⊗ rj) = � +k∈I ak ⊗ rk, where only finitely many ak are +non-zero. Now [a ⊗ 1, 1 ⊗ ri] = 0 = [a ⊗ 1, 1 ⊗ rj] implies [a, ak] = 0 for all a ∈ A, k ∈ I. Therefore, +ak ∈ k1 ⊆ A for all k ∈ I, so +(7.2) +(1 ⊗ ri)(1 ⊗ rj) = 1 ⊗ +� +k∈I +Ck +ijrk + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +35 +for some {Ck +ij}k∈I ⊆ k which are almost all 0. In particular, R is a k-algebra with multiplication +determined by (1 ⊗ r)(1 ⊗ s) = 1 ⊗ rs. Then 1 ∈ R is a unit and since B is associative, R is too. +For the second part of the statement, note that �β(a ⊗ 1, b ⊗ 1) = λβ(a, b) for some λ ∈ k×, so +(7.3) +t(r) := 1 +nλ +�β(1 ⊗ 1, 1 ⊗ r) +is the desired map t: R → k. +■ +Lemma 7.3. Let R be an alternative algebra over a field of characteristic larger then 3 equipped +with a linear map t: R → k such that (r, s) �→ t(rs) is an algebra metric. Assume there exists a +reduced, commutative, associative subalgebra S ⊆ R satisfying S⊥ ⊆ S. +The algebra R is commutative and associative. +Proof. Let p, q ∈ S and r, s ∈ R be arbitrary elements. The identities +(7.4) +t(p(qr)) = t((pq)r) = t((qp)r) = t(r(qp)) = t((rq)p) = t(p(rq)) +show that t(p[q, r]) = 0. As a consequence we see that [S, R] ⊆ S⊥ ⊆ S. Furthermore, since R is +alternative, the subalgebra k[q, r] ⊆ R is associative and we see that +0 = [q, [q, r2]] = [q, [q, r]r + r[q, r]] = [q, [q, r]]r + [q, r]2 + [q, r]2 + r[q, [q, r]] = 2[q, r]2, +where we used that [q, r], [q, r2] ∈ S implies [q, [q, r2]] = 0 = [q, [q, r]]. Since R is reduced, we +deduce that [S, R] = 0. Consequently, +(7.5) +t(q[r, s]) = t([qr, s]) = 0 +so [R, R] ⊆ S⊥ ⊆ S. Combined with [R, S] = 0, this implies [[r, s], s] = 0 = [[r, sr], s], so +(7.6) +[r, s]2 = [[r, s]r, s] = [[r, sr], s] = 0 +holds. Since R is reduced, we deduce that [R, R] = 0 and the fact that any unital commutative +associative algebra over a field of characteristic larger 3 is associative concludes the proof. +■ +We can now proof Proposition 7.1. By virtue of Lemma 7.2, we have D(A[[z]], δ) ∼= A ⊗ R for some +unital associative k-algebra R and ev(a ⊗ r, b ⊗ s) = β(a, b)t(rs) for some t: R → k which defines +an algebra metric. Since A[[z]] ⊆ D(A[[z]], δ) is a Lagrangian subalgebra, k[[z]] ⊆ R is a Lagrangian +subalgebra. Therefore, Lemma 7.3 implies that R is commutative. It is now easy to see that (R, t) +is a trace extension of k[[z]] in the sense of Section 3.3 and Proposition 3.3 concludes the proof. +7.2. Categorization of topological associative D-algebra structures on series. Let k be an +algebraically closed field of characteristic 0. Recall that any finite-dimensional simple associative k- +algebra is isomorphic to the algebra A = Mn(C) of n×n-matrices with entries in k and the bilinear +form β : A × A → A defined by the trace (a, b) �→ tr(ab) is strongly geometrically admissible; see +Corollary 6.8. +Theorem 6.1 states that we have four different types of non-triangular associative topological +D-bialgebra structures on A[[z]]. Namely, those associated to solutions of the A-CYBE which are +either trigonometric, rational, quasi-trigonometric, or quasi-rational. +In Subsection 7.3, we will show that there are no trigonometric nor quasi-trigonometric solutions +of the A-CYBE. So we are left with two different types of non-triangular associative topological +D-bialgebra structures on A[[z]]. Namely, those associated to solutions of the A-CYBE which are +either rational or quasi-rational. +In the remainder of this section, we will establish the structure theory of (quasi-)rational solu- +tions of the A-CYBE by combining the methods from [Agu01] with the approach of [Sto91] to the +structure theory of (quasi-)rational solutions of the sln(C)-CYBE. + +36 +RASCHID ABEDIN +7.3. Absence of (quasi-)trigonometric solutions of the A-CYBE. Let k be an algebraically +closed field of characteristic 0, A = Mn(k) be the k-algebra of n × n-matrices, and β be the trace +pairing of A. In this section, we prove the following result. +Theorem 7.4. There are no quasi-trigonometric nor trigonometric solutions of the A-CYBE. +We will thereby proceed in two steps. First, we show that (quasi-)trigonometric solutions of +the A-CYBE define certain subalgebras of A[v, v−1] × A[v, v−1]. Then, using the classification of +trigonometric solutions of the sln(k)-CYBE from [BD83] in the formulation of [AM21; AB21], we +prove that these subalgebras cannot exist. +7.3.1. (Quasi-)trigonometric solutions of the A-CYBE and subalgebras of L := A[v, v−1]. Consider +L := A[v, v−1]. Let us prove that any (quasi-)trigonometric solution r of the A-CYBE defines a +subspace Wr ⊆ L × L such that: +(1) Wr is a subalgebra complementary to the diagonal D := {(a, a) | a ∈ L}, i.e. L × L = D ⊕ Wr; +(2) Wr is Lagrangian with respect to the algebra metric �β on L × L defined by +(7.7) +�β((a1, a2), (b1, b2)) := res0 +1 +v (β(a1(v), b1(v)) − β(a2(v), b2(v)) +for a1, a2, b1, b2 ∈ L, where β(a(v), b(v)) is the coefficient-wise trace of a(v)b(v) ∈ L; +(3) Wr is commensurable with V := A[z]×A[z−1] in the sense that dim((Wr +V )/(Wr ∩V )) < ∞. +Construction of Wr. Since all automorphisms of A are inner, L(A, σ) ∼= A[v, v−1] for all σ ∈ +Autk-alg(A) of finite order (see [Pia05]). Therefore, both trigonometric and quasi-trigonometric +solutions of the A-CYBE are described by expressions of the form +(7.8) +r(v, w) = +wγ +v − w + t(v, w) for some t ∈ (A ⊗ A)[v, v−1, w, w−1] +such that r(exp(x), exp(y)) is a solution of the A-CYBE. We construct a subspace Wr to r in a +similar fashion as subalgebras were associated to solutions of the A-CYBE in Section 5.3.1. +Note that the natural embedding L ⊗ L → (L ⊗ A)((w±1)) extends to +(7.9) +(L ⊗ L)[(v − w)−1] −→ (L ⊗ A)((w±1)) +by interpreting (v − w)−1 as +(7.10) +� +k∈N +v−k−1wk ∈ k[v, v−1]((w)) and − +� +k∈N +vkw−k−1 ∈ k[v, v−1]((w−1)) +respectively. These embeddings can be understood as the Laurent expansions in w = 0 and w = ∞ +respectively. +Let us consider an r of the form (7.8), chose an orthonormal basis {bi}d +i=1 ⊆ A with respect to +the trace pairing β, and let +(7.11) +� +k∈N +d +� +i=1 +r+ +k,i(v) ⊗ biwk ∈ (L ⊗ A)((w−1)) and +� +k∈N +d +� +i=1 +r− +k,i(v) ⊗ biwk ∈ (L ⊗ A)((w)) +be the Laurent expansions of r in w = ∞ and w = 0 respectively. +If t = � +k∈Z +�d +i=1 tk,i(v) ⊗ wkbi, where only finitely many tk,i are non-zero, and +(7.12) +w− +k,i := +� +biv−k +k > 0 +0 +k ⩽ 0 and w+ +k,i := +� +0 +k > 0 +−biv−k +k ⩽ 0 +we have r± +k,i = w± +k,i + tk,i for k ∈ Z and i ∈ 1, d. +Let us define +(7.13) +Wr := Spank{(r+ +k,i, r− +k,i) | k ∈ Z, i ∈ 1, n}. +Clearly, Wr is commensurable with V , so we have to verify that conditions (a) and (b) hold. +■ + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +37 +Wr satisfies (a). It is easy to see that L × L = D ⊕ Wr, so we have to show that Wr ⊆ L × L is a +subalgebra. Similar to Section 5.2, we can define for every +s ∈ (L ⊗ L)[(v − w)−1] = (A ⊗ A)[v, v−1, w, w−1, (v − w)−1] +the expression +(7.14) +CYB(s) = s13s12 − s12s13 + s23s12 ∈ (L ⊗ L ⊗ L) +� +1 +(v1 − v2)(v1 − v3)(v2 − v3) +� +. +by using the notations (5.12) coefficient-wise. +Then s satisfies the CYB(s) = 0 if and only if +s(exp(x), exp(y)) satisfies the usual A-CYBE (5.15). In particular, we can see that CYB(s) = 0 +implies already that s is skew-symmetric. +Similar arguments as in the Section 5.3.1 show that if r is a skew-symmetric, we have +(7.15) +CYB(r) ∈ L ⊗ L ⊗ L. +Therefore, we can rewrite this expression using the Laurent expansions (7.9) in v3 = ∞ and v3 = 0 +to obtain +CYB±(r) = +� +k,ℓ∈Z +d +� +i,j=1 +r± +ℓ,jr± +k,i ⊗ bizk +2 ⊗ bjzℓ +3 +− +� +m∈N +d +� +i=1 +r± +k,i ⊗ +� +zk +2b(1) +i r(z2, z3) − r(z2, z3)b(2) +i zk +3 +� +. +(7.16) +If CYB(r) = 0, then CYB+(r) = 0 = CYB−(r) and (7.16) implies that Wr ⊆ L × L is a +subalgebra. +■ +Wr satisfies (b). The fact that r is skew-symmetric is equivalent to t = t − γ, which means +(7.17) +tℓ,j +k,i = −tk,i +ℓ,j − δijδk0δℓ0 +if t = � +k,ℓ∈Z tℓ,j +k,ibjvℓ ⊗ biwk. Furthermore, the identities +�β((w+ +k,i, w− +k,i), (w+ +ℓ,j, w− +ℓ,j)) = δijδk0δℓ0 and �β((w+ +k,i, w− +k,i), (bjvℓ, bjvℓ)) = −δijδkℓ +(7.18) +are easily verified. +This implies that, if t is identified with its image under L ⊗ L ∼= D ⊗ D, +�β((r+ +k,i, r− +k,i), (r+ +ℓ,j, r− +ℓ,j)) = �β((w+ +k,i, w− +k,i), (w+ +ℓ,j, w− +ℓ,j)) +� +�� +� +=δijδk0δℓ0 ++ �β((tk,i, tk,i), (tℓ,j, tℓ,j)) +� +�� +� +=0 ++ �β((w+ +k,i, w− +k,i), (tℓ,j, tℓ,j)) +� +�� +� +tk,i +ℓ,j ++ �β((w+ +k,i, w− +k,i), (tℓ,j, tℓ,j)) +� +�� +� +tℓ,j +k,i += tℓ,j +k,i + tk,i +ℓ,j + δijδkℓ = 0. +(7.19) +We conclude that Wr ⊆ W ⊥ +r . This, L × L = D ⊕ Wr, and D⊥ = D imply Wr = W ⊥ +r . +■ +7.3.2. Proof of Theorem 7.4. Let π: A → g := sln(k) be the surjective Lie algebra homomorphism +defined by a �→ a − tr(a) +n +and let its coefficient-wise extension L → L := g[v, v−1] be denoted by +the same symbol. +Observe that for an r of the form (7.8), which defines a trigonometric (resp. quasi-trigonometric) +solution of the A-CYBE, (π ⊗ π)r defines a trigonometric (resp. quasi-trigonometric) solution of +the g-CYBE. +We can assume that in the orthonormal basis {bi}d +i=1 of A, we have bd = +1 +√n ∈ A. +Using +this choice, it is straight forward to see that our construction of W := Wr is consistent with the +construction of the subalgebra W := W(π⊗π)r associated to (π ⊗ π)r in [AM21; AB21] in the sense + +38 +RASCHID ABEDIN +that W = (π ⊗ π)W. Furthermore, it is shown there, that using the classification of trigonometric +solutions of the g-CYBE from [BD83], there exists g ∈ SLn(k) such that +(7.20) +Adg(W±) = N± ⊕ (W± ∩ h) ⊕ (W± ∩ N∓). +Here, we used the following notation: +• g = n+ ⊕ h ⊕ n− is the triangular decomposition into the subalgebras n+, n−, h ⊆ g of upper, +lower, trace-less diagonal matrices and N± := n± ⊕ z±1g[z±1]; +• W± := pr±(W) for the projections pr± : L × L → L defined by (a+, a−) �→ a±. +Since Adg leaves k[v, v−1] = Ker(π) ⊆ L point-wise fixed for every g ∈ SLn(k), this implies that +the subalgebra W± := pr±(W) ⊆ L satisfies +(7.21) +Adg(W±) = N± ⊕ (W± ∩ H) ⊕ (W± ∩ N∓). +Here, N± := n± ⊕ z±1A[z±1] and H ⊆ A is the subalgebra of diagonal matrices. +Since W is Lagrangian with respect to �β, we can deduce that W± ⊆ A[v, v−1] is coisotropic with +respect to the bilinear form β+ +(1,1) from (4.8). In particular, the subalgebras H± := H ∩ W± ⊆ H +satisfy H⊥ +± ⊆ H± with respect to the trace pairing. However, H± are both Artinian k-algebras +without nilpotents, hence H± ∼= kℓ± for some ℓ± ⩽ n as algebras. Observe that any embedding +k → H is of the form a �→ ahi1 + · · · + ahik, where hi is the diagonal matrix with 1 at the i-th row +and column as only non-zero entry. Therefore, H = H± ⊕ H⊥ +±. This combined with H⊥ +± ⊆ H± +implies H± = H. +Now, let us note that we can deduce +W+/W ⊥ ++ × W−/W ⊥ +− = {(a, a) | a ∈ L} ⊕ W/(W ⊥ ++ × W ⊥ +− ) +by following the same arguments as in the proof of Subsection 4.4.(3). This implies that +W/(W ⊥ ++ × W ⊥ +− ) = {(a, θ(a)) | a ∈ W+/W ⊥ ++ } +holds for some k-algebra isomorphism θ: W+/W ⊥ ++ → W−/W ⊥ +− . Consequently, +W = {(a, b) ∈ W+ × W− | θ(a) = b}. +In particular, since 1 ∈ H ⊆ W±/W ⊥ +± and θ is unital since it is an isomorphism, we have (1, 1) ∈ W. +But this contradicts W ∩ D = {0}. In conclusion, W cannot exist. +7.4. Structure theory of rational D-bialgebra structures over A. As in the previous sub- +section, let k be an algebraically closed field of characteristic 0, A = Mn(k) be the k-algebra of +n × n-matrices, and β be the trace pairing of A. +The assignment r �→ A(r) defines a bijection between rational solutions of the A-CYBE and +Lagrangian subalgebras W ⊆ A((z)) satisfying A[[z]] ⊕ W = A((z)) and z−NA[z−1] ⊆ A(r) ⊆ +zNA[z−1] for some sufficiently large N ∈ N0. Therefore, we can apply the associative analog of +the maximal order theory developed in [Sto91; Sto95] to study these solutions. More precisely, we +have the following result. +Proposition 7.5. Let W ⊆ A((z)) be a subalgebra satisfying z−NA[z−1] ⊆ W ⊆ zNA[z−1] for +some N ∈ N. Then there exists g ∈ SL(n, k((z))) such that Ad(g)W ⊆ A[z−1] +Proof. First of all, W ⊆ A[z, z−1] and it suffices to prove that z−NA[z−1] ⊆ W ⊆ zNA[z−1] for +some N ∈ N implies the existence of g ∈ SL(n, k[z, z−1]) such that Ad(g)W ⊆ A[z−1]. +Without loss of generality, we may assume that k[z−1] ⊆ W, since we can pass to the algebra +k[z−1]W + k[z−1] which contains W and satisfies z−NA[z−1] ⊆ k[z−1]W + k[z−1] ⊆ zNA[z−1]. +Now, W = π(W) ⊕ k[z−1] as vector spaces, where π: A[z, z−1] → g[z, z−1] is the coefficient-wise +application of a �→ a − tr(a) +n +∈ g := sln(k). +The subalgebra π(W) ⊆ g[z, z−1] satisfies z−Ng[z−1] ⊆ π(W) ⊆ zNg[z−1]. By virtue of [Sto95, +Theorem 4’] and the description of maximal orders for g = sln(k) from [Sto91], there exists g ∈ + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +39 +SL(n, k[z, z−1]) such that Ad(g)π(W) ⊆ g[z−1]. Since Ad(g)k[z−1] = k[z−1], this implies that +Ad(g)W ⊆ A[z−1]. +■ +By virtue of Proposition 7.5, for every rational solution r of the A-CYBE exists g ∈ SL(n, k((z))) +such that Ad(g)A(r) ⊆ A[z−1]. By virtue of e.g. [Sto91, 2.2 Sauvage Lemma], there exists d = +diag(zd1, . . . , zdn) such that g = g−dg+ for g+ ∈ SL(n, k[[z]]) and g− ∈ SL(n, k[z−1]). Therefore, +up to equivalence, A(r) ⊆ d−1A[z−1]d. The fact that A[[z]] + A(r) = A((z)) holds implies that +0 ⩽ di ⩽ 1 for all i ∈ 1, n. Thus, after reordering the indices, we can assume that d = dk := +(1, . . . , 1, z, . . ., z), where z appears k-times on the right hand side. +We call r rational solution of type k, if A(r) ⊆ Nk := d−1 +k A[z−1]dk, where we remark that +(7.22) +Nk := +�� +A +B +C +D +� +∈ L = Mn(k[z, z−1]) +����� +A∈Mn−k(k[z−1]), B∈zM(n−k)×k(k[z−1]) +C∈z−1Mk×(n−k)(k[z−1]), D∈Mk(k[z−1]) +� +. +We will now show that these solutions are parametrized by associative versions of Stolin pairs, +which parameterize rational solutions of the g-CYBE. To this end, let +(7.23) +Pk := +�� +A +B +0 +D +� +∈ A = Mn(k) +����� A ∈ Mn−k(k), B ∈ M(n−k)×k(k), and D ∈ Mk(k) +� +Then (S, χ) is called associative Stolin pair of type k if S ⊆ A is a subalgebra and χ: S × S → k +is a bilinear form such that +• S + Pk = A; +• χ is a Connes 2-cocycle, i.e. χ is skew-symmetric and +χ(a1a2, a3) + χ(a2a3, a1) + χ(a3a1, a2) = 0 +holds for all a1, a2, a3 ∈ S, and χ restricts to a non-degenerate bilinear form on S ∩ Pk. +By adjusting the arguments in [Sto91], we will prove the following result. +Theorem 7.6. Rational solutions of the A-CYBE of type k are in bijection with associative Stolin +pairs of type k. +Remark 7.7. +Let us note that Stolin pairs of type 0 are simply subalgebras S ⊆ A which admit +a non-degenerate Connes 2-cocycle. +Theorem 7.6.(1) states that these are in bijection with rational solutions r of the A-CYBE +satisfying A(r) ⊆ A[z−1]. It is easy to see that r(x, y) = +γ +x−y + t for a constant tensor t ∈ A ⊗ A. +Then r is a solution of the A-CYBE if and only if t is a skew-symmetric solution of the A-CYBE. +The fact that these are in bijection with Stolin pairs of type 0 is actually exactly [Agu01, Proposition +2.7]. +♦ +7.4.1. Proof of Theorem 7.6. It suffices to prove that there is a bijection between Lagrangian +subalgebras W ⊆ Nk such that A((z)) = A[[z]] ⊕ W and Stolin pairs of type k. +Observe that the image of A[[z]] ∩ Nk in Dǫ := Nk/z−2Nk ∼= A[ǫ]/ǫ2A[ǫ] = A ⊕ ǫA is exactly +Pk ⊕ ǫP ⊥ +k and Dǫ inherits the algebra metric +(7.24) +βǫ(a1 + ǫa2, b1 + ǫb2) := β(a1, b2) + β(a2, b1) +from A((z)). +Since z−2Nk = N ⊥ +k ⊆ W ⊥ = W ⊆ Nk holds, we can see that W �→ W/z−2Nk defines a bijection +between Lagrangian subalgebras W ⊆ Nk such that A((z)) = A[[z]]⊕W and Lagrangian subalgebras +V ⊆ Dǫ such that Dǫ = (Pk ⊕ ǫP ⊥ +k ) ⊕ V . Therefore, it suffices to establish a bijection between the +latter Lagrangian subalgebras and associative Stolin pairs of type k. +Let V ⊆ Dǫ be any Lagrangian subspace and S be the image of V under ǫ �→ 0. Then +(7.25) +ǫS⊥ = (S ⊕ ǫA)⊥ ⊆ V ⊥ = V ⊆ S ⊕ ǫA. + +40 +RASCHID ABEDIN +A dimension argument implies that the mapping ǫ �→ 0 defines an isomorphism V/ǫS⊥ → S. In +other words, there exists a linear map f : S �→ A/S⊥ such that V/ǫS⊥ = {a + ǫf(a) | a ∈ S}. +Consider the bilinear form on S defined by χ(a, b) := β(f(a), b) for a, b ∈ S. Observe that, since +β pairs S and A/S⊥ non-degenerately, f is completely determined by χ. Furthermore, χ is skew- +symmetric since +(7.26) +0 = βǫ(a + ǫf(a), b + ǫf(b)) = χ(a, b) + χ(b, a). +Note that V can be uniquely reconstructed from S and f and hence from the pair (S, χ). This es- +tablishes a bijection between Lagrangian subspaces V ⊆ Dǫ and pairs (S, χ) consisting of subspaces +S ⊆ A equipped with a skew-symmetric bilinear form χ. +It remains to prove that V ⊆ Dǫ is a subalgebra satisfying Dǫ = (Pk ⊕ ǫPk) ⊕ V if and only if +(S, χ) is a Stolin pair of type k. +Observe that V ⊆ Dǫ is a subalgebra if and only if for all a, b ∈ S +(7.27) +(a + ǫf(a))(b + ǫf(b)) = ab + ǫ(f(a)b + af(b)) ∈ V/ǫS⊥. +and this is equivalent to f(ab) = f(a)b + af(b). Now let us note that +χ(a1a2, a3) + χ(a2a3, a1) + χ(a3a1, a2) = χ(a1a2, a3) − χ(a1, a2a3) − χ(a2, a3a1) += β(f(a1a2), a3) − β(f(a1), a2a3) − β(f(a2), a3a1) += β(f(a1a2), a3) − β(f(a1)a2, a3) − β(a1f(a2), a3) += β(f(a1a2) − f(a1)a2 − a1f(a2), a3), +(7.28) +where we used the skew-symmetry of χ and the associativity of β. Since S and A/S⊥ are non- +degenerately paired by β, this identity shows that f(ab) = f(a)b+af(b) for all a, b ∈ S is equivalent +to the fact that χ is a Connes 2-cocycle. +To conclude the proof, we have to show that Dǫ = (Pk ⊕ ǫPk)⊕ V is equivalent to the facts that +S + Pk = A holds and χ is non-degenerate on S ∩ Pk. +Assume first that Dǫ = (Pk ⊕ ǫPk) ⊕ V and observe that S + Pk = A immediately follows from +Dǫ = (Pk ⊕ ǫP ⊥ +k ) + V . Assume that a ∈ S ∩ Pk satisfies χ(a, b) = 0 for all b ∈ S ∩ Pk. In other +words, af ∈ (S ∩ Pk)⊥ = S⊥ + P ⊥ +k for any representative af of f(a), so af = a1 − a2 for a1 ∈ P ⊥ +k +and a2 ∈ S⊥. Then a + ǫ(af + a2) ∈ V ∩ (Pk ⊕ ǫP ⊥ +k ) = {0}. This proves that χ is non-degenerate +on S ∩ Pk. +Conversely, assume that S + Pk = A and χ is non-degenerate on S ∩ Pk. Let +a + ε(af + a⊥) = p + ǫp⊥ ∈ (Pk ⊕ ǫPk) ∩ V, +for a ∈ S, a⊥ ∈ S⊥, p ∈ Pk, p⊥ ∈ P ⊥ +k , and a representative af ∈ A of f(a) ∈ A/S⊥. +Then +a = p ∈ S∩Pk and af = p⊥−a⊥ ∈ S⊥+P ⊥ +k = (S∩Pk)⊥. This implies that χ(a, b) = βǫ(f(a), b) = 0 +for all b ∈ S ∩ Pk, so a = 0 since χ is non-degenerate on S ∩ Pk. Therefore, +af + a⊥ = p⊥ ∈ S⊥ ∩ P ⊥ +k = (S + Pk)⊥ = {0}. +Summarized, (Pk ⊕ ǫPk) ∩ V = {0} and by dimension reasoning we see that Dǫ = (Pk ⊕ ǫP ⊥ +k ) ⊕ V . +7.5. Structure theory of quasi-rational D-bialgebra structures over A. Recall that k is +an algebraically closed field of characteristic 0, A = Mn(k) is the k-algebra of n × n-matrices, and +β is the trace pairing of A. +The assignment r �→ ((D2(A), β(2,1)), A[[z]], A(r)) defines a bijection between quasi-rational solu- +tions of the A-CYBE and Manin triples ((D2(A), β(2,1)), A[[z]], W) satisfying z−NA[z−1] ⊆ W+ ⊆ +zNA[z−1] for some sufficiently large N ∈ N. +Here, W+ is the projection of W ⊆ D2(A) = +A((z)) × A[z]/z2A[z] onto A((z)). +Repeating the arguments in Section 7.4, we obtain W+ ⊆ Nk = d−1 +k A[z−1]dk for some k ∈ 1, n +up to equivalence. Here, Nk is explicitly given in (7.22). +We call a quasi-rational solution r of the A-CYBE of type k, if A(r) ⊆ Nk × A[z]/z2A[z]. + +CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS +41 +Theorem 7.8. Quasi-rational solutions of the A-CYBE of type k are in bijection with associative +Stolin pairs of type k. +Remark 7.9. +In general, the rational and quasi-rational solution of the A-CYBE associated to the +same Stolin pair have no obvious connection. +However, if (S, χ) is a Stolin pair of type 0, then the associated rational solution is r(x, y) = +γ +x−y + t for some t ∈ A ⊗ A and the associated quasi-rational solution is �r(x, y) = +xyγ +x−y + t = +y2γ +x−y − xΩ + t. In particular, r(x−1, y−1) = �r(x, y). Observe that z �→ z−1 is not an admissible +coordinate transformation of A((z)) and D2(A). +♦ +7.5.1. Proof of Theorem 7.8. Let r be a quasi-rational solution of the A-CYBE of type k and +((D2(A), β(2,1)), A[[z]], W) be the associated Manin triple. +Recall from Lemma 6.17.(1) and its proof that W = W+ ×W− for some Lagrangian subalgebras +W+ ⊆ A((z)) and W− ⊆ A[z]/z2A[z]. Since A[z−1] and consequently Nk is Lagrangian in A((z)), +where A((z)) is equipped with β+ +(2,1) from (4.8), W+ ⊆ Nk implies Nk = N ⊥ +k ⊆ W ⊥ ++ = W+ and +thus W+ = Nk. +Now Lemma 6.17.(2),(3) states that W+ ∩A[[z]] can be embedded into A[z]/z2A[z] in such a way +that (W+∩A[[z]])⊕W− = A[z]/z2A[z]. But we have seen in the proof of Theorem 7.6 that this image +of W+∩A[[z]] is precisely Pk ⊕[z]P ⊥ +k and that the decompositions (Pk ⊕[z]P ⊥ +k )⊕W− = A[z]/z2A[z] +into Lagrangian subalgebras are in bijection with Stolin pairs of type k. +Since all steps made are invertible, we obtain the desired bijection. +Appendix A. Notations and conventions +Throughout this document k denotes the base field we are working over. From Section 4 onward +it will be of characteristic 0 and from Section 6 onward it will be additionally algebraically closed. +By convention the set of natural numbers N = {0, 1, 2, . . .} include 0 and we use the notation +m, n = {m, . . . , n} for the set of natural numbers between a number m and larger number n. +Commutative algebra. In this text, rings are always unital, associative, and commutative. For a +ring R and R-modules M, N, the space of R-linear maps M → N (resp. M → M) is denoted +by HomR(M, N) (resp. EndR(M)), while the tensor product of M and N is written as M ⊗R N. +For R = k the indices are omitted. The invertible elements of R are denoted by R×, and M ∗ := +HomR(M, R) is the dual module of M. +If R is a domain, Q(R) := (R \ {0})−1R denotes its +quotient field and we write Q(M) := M ⊗R Q(R). Let f : R → �R be a morphism of rings and � +M +be an �R-module. We say that a map g : M → � +M is f-equivariant if it is a group homomorphism +satisfying g(rm) = f(r)g(m) for all r ∈ R, m ∈ M. +Non-associative algebra. Let R be a ring. +In this text, an R-algebra A satisfies no additional +assumptions if not mentioned explicitly, i.e. A = (A, µA) consists of an R-module A equipped with +a multiplication map µA : A ⊗R A → A. The left (resp. right) multiplication maps with respect to +an element a ∈ A are denoted by La (resp. Ra), i.e. +(A.1) +La(b) = ab = Rb(a) for all a, b ∈ A. +The group of invertible R-algebra endomorphisms of A, i.e. invertible R-linear maps f : A → A +satisfying fµA = µA(f ⊗ f), will be denoted by AutR-alg(A). We note that “ ⊕ ” will always +denote the direct sum of modules and not of algebras, while “ × ” is used for the latter. For any +a, a1, . . . , an ∈ A, we write +a(i)(a1 ⊗ · · · ⊗ an) = a1 ⊗ · · · ⊗ aai ⊗ · · · ⊗ an +(a1 ⊗ · · · ⊗ an)a(i) = a1 ⊗ · · · ⊗ aia ⊗ · · · ⊗ an. +(A.2) + +42 +RASCHID ABEDIN +We say that a map β : A×A → R is an algebra metric if it is a non-degenerate symmetric R-bilinear +map such that +β(ab, c) = β(a, bc) for all a, b, c ∈ A. +(A.3) +In this case, we call the pair (A, β) metric R-algebra. +Formal series. For a module M over a ring R, the module of formal power series in the formal +variable z with coefficients in M is denoted by +(A.4) +M[[z]] := +� +m = +� +k∈N +mkzk +����� mk ∈ M for k ∈ N +� +. +Furthermore, we write M[[z1, . . . , zk]] := M[[z1]] . . . [[zk]]. The R-module R[[z]] (resp. R[[z1, . . . , zk]]) +is a ring extension of R and M[[z]] (resp. M[[z1, . . . , zk]]) is an R[[z]]-module (resp. R[[z1, . . . , zk]]- +module). Then M((z)) := M[[z]][z−1] = Q(M[[z]]) is the module of formal Laurent series. We note +that if M is an R-algebra the module M[[z]] (resp. M((z))) is naturally an R[[z]]-algebra (resp. +R((z))-algebra). Elements p in M((z)) (resp. M((z1)) . . . ((zk))) will sometimes be denoted with the +formal variable (resp. variables) for convenience, i.e. p = p(z) (resp. p = p(z1, . . . , zk)). A generic +element p ∈ M((z)) is written p(z) = � +k∈Z pkzk and p′(z) = � +k∈Z kpkzk−1 denotes the formal +derivative of p. If p(z) ∈ mz−k + z−k+1M[[z]], it is said to be of order k with main part mz−k. +Finally, +M[[z1, . . . , zk]]∨ := {f ∈ M[[z1, . . . , zk]]∗ | f((z1, . . . , zk)mM[[z1, . . . , zk]]) = {0} for some m ∈ N} +is the continuous dual of M[[z1, . . . , zk]]. +Algebraic geometry. Let X = (X, OX) be a ringed space and F, G be two OX-modules. For a +morphism f : X → Y = (Y, OY ) of ringed spaces, we denote the additional structure morphism +by f ♭: OY → f∗OX and write f ♯ : f −1OY → OX for the induced morphism. The set of OX- +module homomorphisms F → G (resp. F → F) is denoted by HomOX(F, G) (resp. EndOX(F)) +while its sheaf counterpart is denoted by HomOX(F, G) (resp. EndOX(F)). In particular, we write +F∗ := HomOX(F, OX). The tensor product of F and G is written as F ⊗OX G. +Assume that X and Y are S-schemes. The fiber product of X and Y over S is denoted by +X ×S Y and F|p is the fiber of F in a point p ∈ X. If S = Spec(k), the index S is omitted and +Hn(F) denotes the n-th global cohomology group of F, while hn(F) denotes its dimension over k, +if said space is finite-dimensional. In particular, H0(F) = Γ(X, F) is the space of global sections +of F. +References +[AB21] +R. Abedin and I. Burban. “Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical +Yang–Baxter Equation”. In: Communications in Mathematical Physics (2021). +[Abe21] +R. Abedin. “Geometrization of solutions of the generalized classical Yang-Baxter equation and a new +proof of the Belavin-Drinfeld trichotomy”. In: (2021). prerpint. arXiv: 2012.05678. +[Abe22] +R. Abedin. “Algebraic geometry of the classical Yang-Baxter equation and its generalizations”. PhD +thesis. 2022. url: https://digital.ub.uni-paderborn.de/hs/content/titleinfo/6660394. +[Agu01] +M. Aguiar. “On the Associative Analog of Lie Bialgebras”. In: Journal of Algebra 244.2 (2001), +pp. 492–532. +[Agu02] +M. Aguiar. “Infinitesimal Hopf Algebras and the cd-Index of Polytopes”. In: Discrete Comput. Geom. +27.1 (2002), pp. 3–28. +[Alb49] +A. A. Albert. “A Theory of Trace-Admissible Algebras”. In: Proc. Natl. Acad. Sci. U.S.A. 35.6 (1949), +pp. 317–322. +[AM21] +R. Abedin and S. Maximov. “Classification of classical twists of the standard Lie bialgebra structure +on a loop algebra”. In: Journal of Geometry and Physics 164 (2021), p. 104149. +[AMS22] +R. Abedin, S. Maximov, and A. Stolin. Topological quasi-Lie bialgebras and (n, s)-type series. preprint. +2022. arXiv: 2211.08807. + +REFERENCES +43 +[AMSZ22] +R. Abedin, S. Maximov, A. Stolin, and E. Zelmanov. Topological Lie bialgebra structures and their +classification over g[[x]]. preprint. 2022. arXiv: 2203.01105. +[BD83] +A. Belavin and V. Drinfeld. “Solutions of the classical Yang-Baxter equation for simple Lie algebras”. +In: Funct. Anal. Appl. 16.3 (1983). +[BG18] +I. Burban and L. Galinat. “Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical +Yang–Baxter Equation”. In: Communications in Mathematical Physics 364 (2018), pp. 123–169. +[BK66] +H. Braun and M. Koecher. “Jordan-Algebren”. In: Zamm-zeitschrift Fur Angewandte Mathematik +Und Mechanik 46 (1966), pp. 558–558. +[Con00] +B. Conrad. Grothendieck Duality and Base Change. Springer, 2000. +[CP95] +V. Chari and A. Pressley. A Guide to Quantum Groups. Cambridge University Press, 1995. +[Dri83] +V. Drinfeld. “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the +classical Yang–Baxter equations”. In: Sov. Math. Dokl. 27 (1983), pp. 68–71. +[Dri88] +V. Drinfeld. “Quantum groups”. In: Journal of Soviet Mathematics 41.2 (1988), pp. 898–915. +[Fin90] +D. Finston. “Rigidity and compact real forms of semisimple complex Jordan algebras”. In: Commu- +nications in Algebra 18.10 (1990), pp. 3323–3338. +[GC83] +I. Gelfand and I. Cherednik. “The abstract Hamiltonian formalism for the classical Yang-Baxter +bundles”. In: Russian Mathematical Surveys 38.3 (1983), pp. 1–22. +[HG88] +M. Hazewinkel and M. Gerstenhaber. Deformation Theory of Algebras and Structures and Applica- +tions. Springer Netherlands, 1988. +[Kir78] +B. S. Kiranagi. “Lie Algebra Bundles”. In: Bull. Sci. Math. 102.2 (1978), pp. 57–62. +[Kir83] +B. S. Kiranagi. “Semi-simple Lie algebra bundles”. In: Bull. Math. de la Sci. Math. de la R. S. de +Roumanie 27(75).3 (1983), pp. 253–257. +[Mil80] +J. S. Milne. ´Etale Cohomology. Princeton University Press, 1980. +[MSZ10] +F. Montaner, A. Stolin, and E. Zelmanov. “Classification of Lie bialgebras over current algebras”. In: +Selecta Mathematica 16.4 (2010), pp. 935–962. +[OS08] +A. Odesskii and V. Sokolov. “Pairs of Compatible Associative Algebras, Classical Yang-Baxter Equa- +tion and Quiver Representations”. In: Commun. Math. Phys. 278.1 (2008), pp. 83–99. +[Pia05] +A. Pianzola. “Vanishing of H1 for Dedekind rings and applications to loop algebras”. In: Comptes +Rendus Mathematique 340.9 (2005), pp. 633–638. +[Pol02] +A. Polishchuk. “Classical Yang-Baxter equation and the A∞-constraint”. In: 168.1 (2002), pp. 56–95. +[Pol09] +A. Polishchuk. “Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the +Yang–Baxter Equations”. In: Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. +Manin. Ed. by Yuri Tschinkel and Yuri Zarhin. Birkh¨auser Boston, 2009. +[Sch55] +R. D. Schafer. “Noncommutative Jordan Algebras of Characteristic 0”. In: Proc. Amer. Math. Soc. +6.3 (1955), pp. 472–475. +[She71] +I. Shestakov. “Certain classes of noncommutative Jordan ring”. In: 10.4 (1971), pp. 407–448. +[Skr13] +T. Skrypnyk. “Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two ap- +proaches”. In: Journal of Mathematical Physics 54.10 (2013), p. 103507. +[Sto91] +A. Stolin. “On rational solutions of Yang-Baxter equation for sl(n)”. In: Mathematica Scandinavica +69 (1991), pp. 57–80. +[Sto95] +A. Stolin. “A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part +I”. In: Conference A on Mathematics and Theoretical Physics, at the 2nd Gauss Symposium (1995), +pp. 347–357. +[Zhe00] +V. Zhelyabin. “Jordan D-bialgebras and symplectic forms on Jordan algebras”. In: Matematicheskie +Trudy 3 (2000). +[Zhe97] +V. Zhelyabin. “Jordan bialgebras and their relation to Lie bialgebras”. In: Algebra And Logic 36.1 +(1997), pp. 1–15. +[Zhe98] +V. Zhelyabin. “Jordan bialgebras of symmetric elements and Lie bialgebras”. In: Siberian Math. J. +39.2 (1998), pp. 261–276. +[Zhe99] +V. Zhelyabin. “On a class of Jordan D-bialgebras”. In: St. Petersburg Math. J. 11 (1999). +ETH Z¨urich, Department of Mathematics, R¨amistrasse 101, 8006 Zurich, Switzerland +Email address: raschid.abedin@math.ethz.ch + diff --git a/o9FPT4oBgHgl3EQfLTSh/content/tmp_files/load_file.txt b/o9FPT4oBgHgl3EQfLTSh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..57d7dca16c9c0cb8a7f00b7534c0d3a7bd6b0878 --- /dev/null +++ b/o9FPT4oBgHgl3EQfLTSh/content/tmp_files/load_file.txt @@ -0,0 +1,2336 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf,len=2335 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13022v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='AG] 30 Jan 2023 CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS RASCHID ABEDIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this paper, we use algebro-geometric methods in order to derive classification results for so-called D-bialgebra structures on the power series algebra A[[z]] for certain central simple non-associative algebras A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non- degenerate topological Lie bialgebra structures on A[[z]] as well as of non-degenerate solutions of the usual CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A is associative, we achieve the classification of non-triangular topological balanced infini- tesimal bialgebra structures on A[[z]] as well as of all non-degenerate solutions of an associative version of the CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Introduction Background and motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A Lie bialgebra (L, δ) over a field k consists of a Lie algebra L over k equipped with a skew-symmetric 1-cocycle δ: L → L⊗L such that the dual map δ∗ : (L⊗L)∗ → L∗ restricted to L∗ ⊗ L∗ ⊆ (L ⊗ L)∗ is a Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Equivalently, given a pair (L, δ) of any (not necessarily Lie) algebra L over k and any linear map δ: L → L ⊗ L, there is a canonical k-algebra structure on D(L, δ) := L ⊕ L∗ and (L, δ) is a Lie bialgebra if and only if D(L, δ) is a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The algebra D(L, δ) is called classical double of (L, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lie bialgebras first appeared in [Dri83], where Drinfeld noticed that the Lie algebra tangent to a Poisson-Lie group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' a Lie group equipped with a Poisson structure compatible with the group operation, is a Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Later in [Dri88], he proposed to approach the quantization of Poisson-Lie group structures by deforming the universal enveloping algebra of the associated Lie bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This made Lie bialgebras integral to the field of quantum groups which arose from these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lie bialgebras have also seen application in the theory of classical integrable systems and are closely related to the classical Yang-Baxter equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [CP95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let Algk be the category of non-associative k-algebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' of vector spaces A over k equipped with any linear map A⊗A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The classical double construction can be used to define analogs of Lie bialgebras in any full subcategory C of Algk which is closed under taking subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, consider a pair (A, δ) of an algebra A ∈ Algk and a linear map δ: A → A⊗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As mentioned above, D(A, δ) = A ⊕ A∗ can be equipped with a canonical k-algebra structure and (A, δ) is called D- bialgebra in C if D(A, δ) ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see [Zhe97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that in this case A, A∗ ∈ C holds automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, D-bialgebras in the category of Lie algebras are precisely Lie bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The D-bialgebras in the category of associative algebras, or associative D-bialgebras for short, turn out to be precisely the co-opposites of balanced infinitesimal bialgebras in the sense of Aguiar [Agu01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The latter see applications in the study of compatible associative multiplications (see [OS08]) and in combinatorics (see [Agu02]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Moreover, they induce Lie bialgebra structures and are related to an associative version of the classical Yang-Baxter equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see [Agu01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The D-bialgebras in the category of Jordan algebras, called Jordan bialgebras, were introduced and discussed in the works of Zhelyabin [Zhe97;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Zhe98;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Zhe00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Zhe99], where they were related to associative D-bialgebras, Lie bialgebras, symplectic forms on Jordan algebras, and a Jordan version of the classical Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 1 2 RASCHID ABEDIN Let g be a finite-dimensional simple Lie algebra over an algebraically closed field k of char- acteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Several important infinite-dimensional Lie bialgebras, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' arbitrary Lie bialgebra structures on the polynomial Lie algebra g[z] or on a twisted loop algebra L(g, σ), can be com- pleted to so-called topological Lie bialgebra structures on the Lie algebra g[[z]] of formal power series with coefficients in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A topological Lie bialgebra (g[[z]], δ) thereby consists of a skew-symmetric 1-cocycle δ: g[[z]] → (g ⊗ g)[[x, y]] whose continuous dual δ∨ : g[[z]]∨ ⊗ g[[z]]∨ ∼= (g ⊗ g)[[x, y]]∨ → g[[z]]∨ is a Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Similar to the non-topological setting, there is a canonical multiplication on D(g[[z]], δ) = g[[z]]⊕g[[z]]∨ and (g[[x]], δ) is a topological Lie bialgebra if and only if this multiplication is a Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In [MSZ10], all possibilities for D(g[[z]], δ) were classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, they proved that either D(g[[z]], δ) ∼= D(g[[z]], 0) or D(g[[z]], δ) is isomorphic to g((z)) × g[z]/zng[z] for some n ∈ {0, 1, 2} as a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Topological Lie bialgebras are closely related to solutions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) r(x, y) = λ(x)ynγ x − y + t(x, y) of the classical Yang-Baxter equation (CYBE) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) [r12(z1, z2), r13(z1, z3)] + [r12(z1, z2), r23(z2, z3)] + [r13(z1, z3), r23(z2, z3)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, γ is a the quadratic Casimir element of g, λ ∈ k[[x]]× and t ∈ (g ⊗ g)[[x, y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, the assignment a(z) �→ [a(x)⊗1+1⊗a(y), r(x, y)] defines a 1-cocycle δr : g[[z]] → (g⊗g)[[x, y]] for r of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, r �→ δr defines a bijection between topological Lie bialgebra structures on g[[z]] with double g((z))×g[z]/zng[z] and solutions of the CYBE of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In [AMSZ22], it is shown that a topological Lie bialgebra structure is determined up to isomorphism by one of 6 types of solutions of the CYBE: degenerate, elliptic, trigonometric, rational, quasi-trigonometric, or quasi-rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' All non-degenerate solutions of the CYBE have been classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this paper, we introduce topological D-bialgebras structures on A[[z]] for any k-algebra A ∈ Algk and extend some results from [MSZ10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' AMSZ22] to these using the methods from [Abe21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' AMSZ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Moreover, we establish a connection between topological D-bialgebras and a generalization of the classical Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Content and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The notion of D-bialgebras as well as supplementary objects such as the classical double and isomorphisms of D-bialgebras will be discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In Section 3, these notions are adapted to the topological setting for power series algebras and we will introduce the main object of study: non-degenerate topological D-bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional, central, simple k-algebra equipped with a non-degenerate, as- sociative, symmetric bilinear form β : A × A → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By definition, a non-degenerate topological D-bialgebra (A[[z]], δ) has a double isomorphic to Dn(A) := A((z)) × A[z]/znA[z], where n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In Section 4, we will prove that the cases n > 2 cannot occur for so-called geometrically admissible algebras A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, this result holds for finite-dimensional central simple associative, Lie or Jordan algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, we have the following result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a geometrically admissible algebra over a field k of characteristic 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' a finite-dimensional central simple Lie, associative, or Jordan algebra over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The double D(A[[z]], δ) of a non-degenerate topological D-bialgebra (A[[z]], δ) is isomorphic to A((z)) × A[z]/znA[z] for some n ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Let us note that, if A is a Lie algebra, Theorem A is one of the main results in [MSZ10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, our proof is independent of the proof in [MSZ10] and is based on the geometrization of A-lattices (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This method was already used in order to give a new proof of the Belavin- Drinfeld trichotomy of non-degenerate solutions of the CYBE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) in [Abe21] and the classification of topological Lie bialgebras in [AMSZ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 3 In Section 5, we will show that every non-degenerate topological D-bialgebra (A[[z]], δ) is of the form a(z) �→ r(x, y)a(x)(1) − a(y)(2)r(x, y) for a solution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) r(x, y) = λ(x)ynγ x − y + t(x, y) ∈ (A ⊗ A)[[x, y]][(x − y)−1] of the so-called A-classical Yang-Baxter equation (A-CYBE) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) r13(z1, z3)r12(z1, z2) − r12(z1, z2)r23(z2, z3) + r23(z2, z3)r13(z1, z3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, λ ∈ k[[x]]×, t ∈ (g ⊗ g)[[x, y]] and γ ∈ A ⊗ A is a canonical A-invariant element determined by the algebra metric β of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, the classification of non- degenerate topological D-bialgebras (A[[z]], δ) up to isomorphism is equivalent to the classification of solutions of the A-CYBE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) up to a certain type of equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The A-CYBE already appeared in several special cases in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A = g is a Lie algebra, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) is exactly the usual CYBE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For constant r ∈ A ⊗ A, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) was examined in [Agu01] for an associative algebra A and in [Zhe99] for a Jordan algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' An endomorphism version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) for an associative algebra A and meromorphic functions r = r(x, y) in two complex variables was related to pairs of compatible associative algebra structures in [OS08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We also point out that solutions of the associative Yang-Baxter equation introduced by Polishchuk in [Pol02, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1)] which do not depend on the first parameter are precisely difference depending meromorphic solutions of the A-CYBE for associative algebras A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In Section 6, we will see that non-degenerate topological D-bialgebras can be categorized rather explicitly by the form of their associated solution of the A-CYBE if A is a so-called strongly geometrically admissible k-algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The most important examples are finite- dimensional simple Lie, Jordan, and associative algebras over an algebraically closed field of char- acteristic 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, we have the following result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed of characteristic 0 and A be a unital strongly ge- ometrically admissible algebra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' a finite-dimensional, central, simple associative or Jordan al- gebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let (A[[z]], δ) be a non-degenerate topological D-bialgebra in some category of k-algebras closed under taking subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Up to isomorphism of topological D-bialgebras, δ = δr for a solution of the A-CYBE of precisely one of the following forms: (1) r is trigonometric in the sense that there exists σ ∈ Autk-alg(A) of order m ∈ N and t ∈ L(A, σ) ⊗ L(A, σ) such that r(x, y) = 1 exp (x − y) − 1 m−1 � j=0 exp �x − y m � γj + t � exp � x m � , exp � y m �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, γj ∈ A ⊗ A is uniquely determined by γ = �m−1 j=0 γj and (σ ⊗ 1)γj = εjγj for some primitive m-th root of unity ε ∈ k, where γ ∈ A ⊗ A is the canonical A-invariant element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) r is rational in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r(x, y) = γ x−y + t(x, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) r is quasi-trigonometric in the sense that there exists a polynomial t ∈ (A ⊗ A)[x, y] such that r(x, y) = yγ x−y + t(x, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (4) r is quasi-rational in the sense that there exists a polynomial t ∈ (A ⊗ A)[x, y] such that r(x, y) = y2γ x−y + t(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, every solution of the A-CYBE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) is, up to equivalence, of one of the above forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ The analog of Theorem B for a Lie algebra A = g was proven in [AMSZ22] and can be seen as a generalization of the Belavin-Drinfeld trichotomy for non-degenerate r-matrices from [BD83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A consequence of this result is, that all topological Lie bialgebras (g[[z]], δ) are classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof of Theorem B is, under consideration of Theorem A, similar to the proof of its Lie algebra analog 4 RASCHID ABEDIN in [AMSZ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, it proceeds by refining the algebro-geometric methods already used to proof Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, we can assign a particular type of geometric data, called geometric A-CYBE datum (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3), to any Lagrangian subalgebra W ⊆ Dn(A) complementary to A[[z]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If this assignment is done, Theorem B is a consequence of the classification results for sheaves of algebras on the (punctured) affine line presented in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9, which is a consequence of the results from [Pia05];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see [Abe22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us point out that the unitality assumption in Theorem B is actually rather weak, since strongly geometrically admissible power associative algebras which are not anti-commutative are automatically unital;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The most interesting strongly geometrically admissible anti- commutative algebras are precisely Lie algebras, where the analog of Theorem B is already known as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We conclude this paper, by using Theorem B to classify all topological associative D-bialgebras (A[[z]], δ) for A associative and D(A[[z]], δ) ≇ D(A[[z]], 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It turns out that the trigonometric and quasi-trigonometric cases do not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, we obtain the following result in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional simple associative k-algebra over an algebraically closed field k of characteristic 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A ∼= Mn(k), and (A[[z]], δ) be a topological associative D-bialgebra such that D(A[[z]], δ) ≇ D(A[[z]], 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then, up to isomorphism, δ = δr where r is either the rational or the quasi-rational solution of the A-CYBE determined by an associative Stolin pair (S, B) of class k ∈ 0, n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4 and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, every solution of the A-CYBE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) is, up to equivalence, of one of the above forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Let us remark that meromorphic solutions of the Mn(C)-CYBE which depend on the difference of their variables and have diagonal residue γ were already shown to be rational up to equivalence in [Pol09, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We were unable to provide examples of quasi-trigonometric and trigonometric solutions of the A-CYBE for non-associative unital strongly geometrically admissible algebras A as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We con- jecture that, similar to the associative case in Theorem C, the unitality obstructs the existence of these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' I thank Ivan Shestakov for explaining several facts about non-associative algebras to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This work was supported by the DFG grant AB 940/1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It was also supported as a part of NCCR SwissMAP, a National Centre of Competence in Research, funded by the Swiss National Science Foundation (grant number 205607).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Notation and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If the reader is unsure about the meaning of symbols or names which are ambiguously used in literature, we refer to the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There we have tried to give an overview on our conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Introduction to D-bialgebras 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Survey on Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Throughout this paper k is a field and all algebras, vector spaces, tensor products, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' are understood over k if not stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us remark that for us an R-algebra over an (unital, commutative, associative) ring R satisfies no additional assumptions: an R-algebra A = (A, µ) consists of an R-module A equipped with an R-linear map µ: A ⊗R A → A, called multiplication map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We write ab := µ(a ⊗ b) for a, b ∈ A if no confusion arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Metric algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let R be a ring and A be an R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a map β : A × A → R algebra metric if it is non-degenerate, symmetric, associative, and R-bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Thereby, “non- degenerate” means that the canonical map A → HomR(A, R) defined by a �→ β(a, −) is injective CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 5 and “associative” means that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) β(ab, c) = β(a, bc) holds for all a, b, c ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a pair (A, β) metric R-algebra if A is an R-algebra equipped with an algebra metric β : A × A → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Moreover, two metric algebras (A1, β1) and (A2, β2) are called isomorphic, written (A1, β1) ∼= (A2, β2), if there exists an R-algebra isomorphism ϕ: A1 → A2 such that β2(ϕ(a), ϕ(b)) = β1(a, b) for all a, b ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Manin pairs and Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A Manin pair ((M, β), N) consists of a metric k-algebra (M, β) and a Lagrangian subalgebra N ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, N ⊥ = N ⊆ M is a subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A Manin triple ((M, β), M+, M−) consists of a metric k-algebra (M, β) and subalgebras M± ⊆ M such that M = M+ ⊕ M− and M± ⊆ M ⊥ ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that for any Manin triple ((M, B), M+, M−), M ⊥ ± = M± already holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, ((M, β), M+) and ((M, β), M−) are automatically Manin pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In literature, Manin pairs and triples are usually only defined for Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The definition given here is a straight-forward generalization to arbitrary algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Manin triples and comultiplication maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that a k-coalgebra C is a k-vector space equipped with a k-linear map δ: C → C ⊗ C, called comultiplication map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The restriction of δ∗ : (C ⊗ C)∗ → C∗ to C∗ ⊗ C∗ ⊆ (C ⊗ C)∗ always defines k-algebra structure on C∗, hence the name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Explicitly, the multiplication f1f2 ∈ C∗ of two maps f1, f2 ∈ C∗ is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) f1f2(a) := (f1 ⊗ f2)δ(a) for all a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us note that for an infinite-dimensional algebra A, A∗ is not necessarily a coalgebra, since the dual A∗ → (A⊗A)∗ of the multiplication map might fail to have values in A∗ ⊗A∗ ⊆ (A⊗A)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Two k-coalgebras (C1, δ1) and (C2, δ2) are called isomorphic, written C1 ∼= C2, if there exists an isomorphism ϕ: C1 → C2 of vector spaces satisfying (ϕ ⊗ ϕ)δ1 = δ2ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We say that a comultiplication map δ: M+ → M+ ⊗ M+ is determined by a Manin triple ((M, β), M+, M−) if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) β⊗2(δ(a), b1 ⊗ b2) = B(a, b1b2) holds for all a ∈ M+ and b1, b2 ∈ M−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, β⊗2(a1 ⊗ a2, b1 ⊗ b2) := β(a1, b1)β(a2, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The name stems from the fact that, if �δ: M+ → M+ ⊗ M+ is another comultiplication determined by ((M, β), M+, M−), we have δ = �δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Isomorphism of Manin pairs and Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call two Manin pairs (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Manin triples) ((M1, β1), N1) and ((M2, β2), N2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ((M1, β1), M1,+, M1,−) and ((M2, β2), M2,+, M2,−)) isomorphic if there exists an isomorphism ϕ: (M1, β1) → (M2, β2) of metric algebras such that ϕ(N1) = N2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ϕ(M1,±) = M2,±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this case, we write ((M1, β1), N1) ∼= ((M2, β2), N2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ((M1, β1), M1,+, M1,−) ∼= ((M2, β2), M2,+, M2,−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume that ((Mi, βi), Mi,+, Mi,−) determines a comultiplication δi on Mi,+ for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then ((M1, β1), M1,+, M1,−) ∼= ((M2, β2), M2,+, M2,−) via an isomorphism ϕ: (M1, β1) → (M2, β2) implies that β2((ϕ ⊗ ϕ)δ1(a), b1 ⊗ b2) = β1(δ1(a), ϕ−1(b1) ⊗ ϕ−1(b2)) = β1(a, ϕ−1(b1)ϕ−1(b2)) = β1(a, ϕ−1(b1b2)) = β2(ϕ(a), b1b2) = β2(δ2(ϕ(a)), b1 ⊗ b2) holds for all a ∈ M1,+ and b1, b2 ∈ M2,−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, (ϕ ⊗ ϕ)δ1 = δ2ϕ holds, so M1,+ ∼= M2,+ holds both as algebras and coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6 RASCHID ABEDIN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' D-bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us call a pair (A, δ) consisting of a k-algebra A and a comultiplication map δ: A → A⊗A bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, we do not assume any compatibility conditions between multiplication and comultiplication of A in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' To any bialgebra (A, δ), there is a unique k-algebra structure on D(A, δ) := A ⊕ A∗ such that ((D(A, δ), ev), A, A∗) is a Manin triple determining δ, where: The multiplication of A∗ is defined by the comultiplication δ in the sense of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ev: D(A, δ) × D(A, δ) → k is the evaluation pairing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) ev(a + f, b + g) = f(b) + g(a) , a, b ∈ A and f, g ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Explicitly, the multiplication on D(A, δ) is determined by: A, A∗ ⊆ D(A, δ) are subalgebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The identities ev(af, b) = ev(f, ba) = f(ba) = ev(fRa, b) ev(af, g) = ev(a, fg) = (f ⊗ g)δ(a) = ev((f ⊗ 1)δ(a), g) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) for a, b ∈ A, f, g ∈ A∗ yield af = (f ⊗ 1)δ(a) + fRa and similarly fa = (1 ⊗ f)δ(a) + fLa holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, R, L: A → End(A) denote the right and left multiplication maps respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) Rab = ba = Lba for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The k-algebra D(A, δ) associated to a bialgebra (A, δ) is called called classical double of (A, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let Algk be the category of k-algebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' the category with k-algebras as objects and k- algebra homomorphisms as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let C be a full subcategory of Algk closed under taking subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For instance, C can be any subcategory of equation based k-algebras like the category of Lie algebras, associative algebras or Jordan algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a bialgebra (A, δ) D-bialgebra in C if D(A, δ) is an algebra in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that, by construction, A, A∗ ∈ C and ((D(A, δ), ev), A, A∗) is a Manin triple determining δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us point out that D-bialgebras in Algk are exactly bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Isomorphism of D-bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let C be a full subcategory of Algk closed under taking sub- algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Two D-bialgebras (A1, δ1) and (A2, δ2) in C are called isomorphic, written (A1, δ1) ∼= (A2, δ2), if there exists a k-linear map ϕ: A1 → A2 which is both an isomorphism of k-algebras and k-coalgebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' if the identities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) ϕ(a1a2) = ϕ(a1)ϕ(a2) and (ϕ ⊗ ϕ)δ1(a) = δ2(ϕ(a)) hold for all a, a1, a2 ∈ A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let C be a full subcategory of Algk closed under taking subalgebras and (A1, δ1),(A2, δ2) be two D-bialgebras in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (A1, δ1) ∼= (A2, δ2) ⇐⇒ ((D(A1, δ1), ev), A1, A∗ 1) ∼= ((D(A2, δ2), ev), A2, A∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The fact that ((D(A1, δ1), ev), A1, A∗ 1) ∼= ((D(A2, δ), ev), A2, A∗ 2) implies (A1, δ1) ∼= (A2, δ2) was already mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' On the other hand, let ϕ: A1 → A2 define the isomorphism (A1, δ1) ∼= (A2, δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that �ϕ(a + f) := ϕ(a) + fϕ−1, where a ∈ A1, f ∈ A∗ 1, defines an isomorphism ((D(A1, δ1), ev), A1, A∗ 1) ∼= ((D(A2, δ), ev), A2, A∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In [Zhe97], the D-bialgebra structures for the most important categories C of algebras where discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us give a short outline of their explicit descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In the following, we write for any elements a, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , an in some k-algebra A a(i)(a1 ⊗ · · · ⊗ an) = a1 ⊗ · · · ⊗ aai ⊗ · · · ⊗ an (a1 ⊗ · · · ⊗ an)a(i) = a1 ⊗ · · · ⊗ aia ⊗ · · · ⊗ an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8) CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' D-bialgebras in the category of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that a Lie bialgebra (L, δ) consists of a Lie algebra L equipped with a linear map δ: L → L ⊗ L such that: δ is a 1-cocycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' for all a, b ∈ L δ(ab) = (a(1) + a(2))δ(b) + δ(a)(b(1) + b(2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The restriction of δ∗ to L∗ ⊗ L∗ ⊆ (L ⊗ L)∗ is a Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is well-known that for a linear map δ: L → L ⊗ L on a Lie algebra L the double D(L, δ) is again a Lie algebra if and only if (L, δ) is a Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, a D-bialgebra in the category of Lie algebras is exactly a Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' D-bialgebras in the category of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' An infinitesimal bialgebra (A, δ) consists of a Lie algebra A equipped with a cobracket δ: A → A ⊗ A such that: δ is a 1-cocycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' for all a, b ∈ A δ(ab) = a(1)δ(b) + δ(a)b(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The restriction of δ∗ to A∗ ⊗ A∗ ⊆ (A ⊗ A)∗ is an associative multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It was shown by Aguiar [Agu01] that there is classical double like construction for infinitesimal algebras, this time by giving D(A, δ) := (A ⊕ A∗) ⊕ (A ⊗ A∗) an associative algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In general, these are not related to Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, under the condition that δ is balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' if for all a1, a2 ∈ A (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9) a(1) 1 τδ(a2) + a(2) 2 δ(a1) = δ(a1)a(1) 2 + τδ(a2)a(2) 1 holds, the double can be reduced to Dred(A, δ) = A⊕ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It turns out that Dred(A, δ) = D(A, τδ), so τδ is an associative D-bialgebra structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' a D-bialgebra in the category of associative k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here and in the following τ(a ⊗ b) = b ⊗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For every bialgebra (A, δ), we call (A, τδ) the co-opposite bialgebra of (A, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is shown in [Zhe97] that indeed all associative D-bialgebra structures are of the above form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' the co-opposites of balanced infinitesimal bialgebras are exactly associative D-bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' D-bialgebras in the category of Jordan algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The D-bialgebra structures in the category of Jordan algebras, which we will simply call Jordan bialgebras, where found in [Zhe97, Theorem 2]: a Jordan bialgebra (J, δ) consists of a Jordan algebra J and a linear map δ: J → J ⊗ J such that J∗ is a Jordan algebra and the following identities hold: 1 2((δ ⊗ 1) − (1 ⊗ δ))δ(a2) = a(2)(δ ⊗ 1 − 1 ⊗ δ)δ(a) + (a(3) − a(1))(1 ⊗ τ)(δ ⊗ 1)δ(a) + (δ(a) ⊗ 1 − 1 ⊗ δ(a))(1 ⊗ τ)(δ(a) ⊗ 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (δ ⊗ 1 + 1 ⊗ δ + (1 ⊗ τ)(δ ⊗ 1))(1 ⊗ a + a ⊗ 1)δ(a) = 2a(2)(1 ⊗ δ)δ(a) + a(1)(1 ⊗ τ)(δ ⊗ 1)δ(a) + (1 ⊗ δ(a))(1 ⊗ τ)(δ(a) ⊗ 1) + (δ ⊗ 1)δ(a2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' δ(a2b) − δ(a2)b(1) − δ(b)(a2)(2) + 2δ(b)(a ⊗ a) − 2δ(ab)a(1) + 2(δ(a)b(1))a(1) + 2(δ(a)b(2))a(2) − 2δ(a)(ab)(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Non-degenerate topological D-bialgebra structures on power series algebras 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Topological D-bialgebra structures on power series algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite- dimensional k-algebra and let us equip A[[z]] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A⊗A)[[x, y]]) with the (z)-adic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (x, y)-adic) topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Appendix A for definition of (·)[[z]] and (·)[[x, y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Note that if δ: A[[z]] → (A⊗A)[[x, y]] is a continuous linear map and if (·)∨ denotes taking the continuous dual space, A[[z]]∨ is naturally a k-algebra with the multiplication defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) A[[z]]∨ ⊗ A[[z]]∨ ∼= (A ⊗ A)[[x, y]]∨ δ∨ −→ A[[z]]∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 8 RASCHID ABEDIN We call a pair (A[[z]], δ) as above topological bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any topological bialgebra (A[[z]], δ), there is a unique k-algebra structure on D(A[[z]], δ) = A[[z]] ⊕ A[[z]]∨ such that ((D(A[[z]], δ), ev), A[[z]], A[[z]]∨) is a Manin determining δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, the eval- uation pairing ev: D(A[[z]], δ) × D(A[[z]], δ) → k is defined analogous to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The multiplication map of D(A, δ) satisfying these conditions can be explicitly determined in the same way as in the non-topological setting in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The algebra D(A[[z]], δ) is called classical double of (A[[z]], δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let C be a full subcategory of Algk closed under taking subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a topological bialgebra (A[[z]], δ) topological D-bialgebra in C if D(A[[z]], δ) is an algebra in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that if (A[[z]], δ) is a topological D-bialgebra in C, we have A, A[[z]], A[[z]]∨ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that (A[[z]], δ) is a topological D-bialgebra in C if and only if (A[[z]]∨, µ∨) is a usual D-bialgebra in C, where µ: (A ⊗ A)[[x, y]] → A[[z]] is the multiplication map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we can describe topological D-bialgebras in the most important categories of algebras using the same axioms as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Isomorphism of topological D-bialgebras on series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a k-algebra and C be a full subcategory of Algk closed under taking subalgebras and such that A[[z]] ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Two topological D- bialgebras (A1[[z]], δ1) and (A2[[z]], δ2) in C are called isomorphic, written (A1[[z]], δ1) ∼= (A2[[z]], δ2), if there exists a continuous linear map ϕ: A1[[z]] → A2[[z]] which is both an isomorphism of algebras and coalgebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' if for every a, a1, a2 ∈ A1 the identities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) ϕ(a1a2) = ϕ(a1)ϕ(a2) and (ϕ ⊗ ϕ)δ1(a) = δ2(ϕ(a)) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, in the latter equation ϕ ⊗ ϕ was continuously extended from an automorphism of A[[z]] ⊗ A[[z]] to an automorphism of (A ⊗ A)[[x, y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a k-algebra and C be a full subcategory of Algk closed under taking subal- gebras and (A1[[z]], δ1),(A2[[z]], δ2) be two D-bialgebras in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (A1[[z]], δ1) ∼= (A2[[z]], δ2) if and only if ((D(A1[[z]], δ1), ev), A1[[z]], A1[[z]]∨) ∼= ((D(A2[[z]], δ2), ev), A2[[z]], A2[[z]]∨) via an isomorphism D(A1[[z]], δ1) ∼= D(A2[[z]], δ2) which restricts to a continuous isomorphism A1[[z]] → A2[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Repeat the arguments in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 under consideration of the fact that any continuous linear isomorphism A1[[z]] → A2[[z]] has a continuous inverse, since A1[[z]] is linearly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional central simple k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then [AMSZ22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3] states that for every φ ∈ Autk-alg(A[[z]]) exists a ϕ ∈ Autk[[z]]-alg(A[[z]]) ⊆ End(A)[[z]] and u ∈ zk[[z]]× such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) φ(a)(z) = ϕ(z)a(u(z)) for all a ∈ A[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As a consequence, every k-algebra automorphism of A[[z]] is continuous in the (z)-adic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, in this case Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 can be refined for A = A1 = A2 as (A[[z]], δ1) ∼= (A[[z]], δ2) ⇐⇒ ((D(A[[z]], δ1), ev), A[[z]], A[[z]]∨) ∼= ((D(A[[z]], δ2), ev), A[[z]], A[[z]]∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Non-degenerate topological D-bialgebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider Manin triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) ((Dn(A), β(n,λ)), A[[z]], W), where: (A, β) is a finite-dimensional metric k-algebra (recall the definition from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' n ∈ N and Dn(A) := A((z)) × A[z]/znA[z];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 9 A[[z]] is identified with the image of the diagonal embedding A[[z]] → Dn(A) defined by a �−→ (a, [a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, for any a ∈ A[[z]], [a] := a + znA[[z]] ∈ A[[z]]/znA[[z]] = A[z]/znA[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' λ ∈ k[[z]]× and β(n,λ) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) β(n,λ)((a1, [a2]), (b1, [b2])) = res0 1 znλ(β(a1, b1) − β(a2, b2)), where β was extended to a k((z))-bilinear form A((z)) × A((z)) → k((z)) on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that all triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) are indeed Manin triples in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let C be a full subcategory of Algk that is closed under taking subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a topological D-bialgebra (A[[z]], δ) in C non-degenerate if and only if there exist n ∈ N and λ ∈ k[[z]]× such that ((D(A[[z]], δ), ev), A[[z]]) ∼= ((Dn(A), β(n,λ)), A[[z]]) as Manin pairs (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, (A[[z]], δ) is non-degenerate if and only if there exist n ∈ N and λ ∈ k[[z]]× such that Dn(A) ∈ C and δ is determined by the Manin triple ((Dn(A), β(n,λ)), A[[z]], W) for an appropriate W ⊆ Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 implies that, if A is central and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' the classification of non-degenerate topolog- ical D-bialgebra structures on A[[z]] up to isomorphisms of topological D-bialgebras is equivalent to the classification of Manin triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) up to isomorphisms of Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Connection to trace extensions of k[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A trace extension (R, t) of k[[z]] consists of a commutative and associative k-algebra extension R ⊇ k[[z]] equipped with a linear map t: R → k, called trace map, such that: (1) (a, b) �→ βt(a, b) := t(ab) is an algebra metric making k[[z]] ⊆ R a Lagrangian subalgebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) For all continuous (in the (z)-adic topology) linear maps f : k[[z]] → k exists an a ∈ R such that f(b) = t(ab) for all b ∈ k[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, trace extensions (R, t) are in bijection with Manin pairs ((R, βt), k[[z]]) for which R is associative and commutative and βt satisfies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Two trace extensions (R1, t1) and (R2, t2) are called isomorphic, written (R1, t1) ∼= (R2, t2), if their associated Manin pairs are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, (R1, t1) ∼= (R2, t2) if there exists an algebra isomorphism ϕ: R1 → R2 such that ϕ(k[[z]]) = k[[z]] and t2ϕ = t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that ϕ|k[[z]] is automatically continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Trace extensions were classified up to isomorphism in [MSZ10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9]: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (R, t) be a trace extension of k[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then precisely one of the following cases occurs: (1) (R, t) ∼= (R∞, t∞), where R∞ := k[[z]] ⊕ Spank{ak | k ∈ N} with multiplication defined by ajak = 0 and ajzk = � aj−k if k ⩽ j, 0 otherwise and t∞ is the unique trace map on R∞ defined by t(aj) = δj0 for j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) There exists n ∈ N and λ ∈ k[[z]]× such that (R, t) ∼= (Rn, t(n,λ)), where Rn := k((z))×k[z]/(zn) and t(n,λ)(a, [b]) := res0 1 znλ(a − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, a ∈ k((z)), b ∈ k[[z]], [b] := b + znk[z] ∈ k[[z]]/znk[[z]] and k[[z]] is identified with its image via the embedding a �→ (a, [a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ A trace extension (R, t) of k[[z]] is called trivial if (R, t) ∼= (R∞, t∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 10 RASCHID ABEDIN If (A, β) is a finite-dimensional metric k-algebra and (R, t) is a trace extension of k[[z]], then (A ⊗ R, β ⊗ t) is a metric k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) (β ⊗ t)(a1 ⊗ b1, a2 ⊗ b2) := β(a1, a2)t(b1b2) for all a1, a2 ∈ A, b1, b2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that ((A ⊗ Rn, β ⊗ t(n,λ)), A ⊗ k[[z]]) ∼= ((Dn(A), β(n,λ)), A[[z]]) holds for all n ∈ N0 and λ ∈ k[[z]]×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 states that, if (R, t) is a non-trivial trace extension of k[[z]], there exists n ∈ N and λ ∈ k[[z]]× such that ((A ⊗ R, β ⊗ t), A ⊗ k[[z]]) ∼= ((Dn(A), β(n,λ)), A[[z]]) as Manin pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, the Manin triples considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 are exactly those which arise by finding Lagrangian subalgebras W in (A ⊗ R, β ⊗ t) complementary to A ⊗ k[[z]] for any non-trivial trace extension (R, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Non-triangular topological Lie D-bialgebra structures are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional k-algebra and C be a full subcategory of Algk closed under taking subalge- bras such that A ⊗ R∞ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then the zero map δ = 0: A[[z]] → (A ⊗ A)[[x, y]] defines a topological D-bialgebra structure in C with double (D(A[[z]], δ), ev) ∼= (A ⊗ R∞, β ⊗ t∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We say that a topological D-bialgebra structure δ: A[[z]] → (A ⊗ A)[[x, y]] is triangular if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) ((D(A[[z]], δ), ev), A[[z]]) ∼= ((A ⊗ R∞, β ⊗ t∞), A ⊗ k[[z]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The origin of the name is explained in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be algebraically closed of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If C is the category of Lie algebras over k and A = g ∈ C is simple, it was shown in [MSZ10, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8] (see also [AMSZ22, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10]) that any non-triangular topological Lie bialgebra (g[[z]], δ) is non-degenerate in the sense of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We will prove an analog of this results for the case that C is the category of associative algebras in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Categorization of non-degenerate D-bialgebra structures In this section, we will show that, up to isomorphism of D-bialgebras and for a large class of central simple k-algebras A, all non-degenerate topological D-bialgebras (A[[z]], δ) are determined by a Manin triple of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) for some n ∈ {0, 1, 2} and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The main method we use to prove this result is the geometrization of A-lattices developed in [Abe21, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3] (see also [Abe22, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3]), which we will recall in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The precise formulation of the above mentioned result is then given in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 and the reminder of this section will be devoted to its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Throughout the reminder of this paper, we assume that k is a field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrization of lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional, central, simple k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a subalgebra W ⊆ A((z)) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) dim(A[[z]] ∩ W) < ∞ and dim(A((z))/(A[[z]] + W)) < ∞ A-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, we call a pair (O, W) consisting of an A-lattice W ⊆ A((z)) and a unital subalgebra O ⊆ {f ∈ k((z)) | fW ⊆ W} of finite codimension ringed A-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us fix a ringed A-lattice (O, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The graded k-algebra (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) gr(O) := ∞ � j=0 tj � O ∩ z−jk[[z]] � ⊆ O[t] defines an irreducible projective curve X := Proj(gr(O)) over k of arithmetic genus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) h1(OX) = dim(k((z))/(k[[z]] + O)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 11 The k-rational smooth point p = (t) of X satisfies D+(t) = X\\{p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, there is canonical isomorphism c: �OX,p → k[[z]] such that the induced isomorphism Q( �OX,p) → k((z)) on quotient fields, which will be denoted again by c, has the property c(Γ(X \\ {p}, OX)) = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider the graded gr(O)-algebra (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) gr(W) := � j∈Z tj(W ∩ z−jA[[z]]) ⊆ W[t, t−1] defined by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then the quasi-coherent sheaf A on X = Proj(gr(O)) associated to gr(W) is a coherent torsion-free OX-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This sheaf comes equipped with an c-equivariant isomorphism ζ : � Ap → A[[z]] such that the induced isomorphism Q( � Ap) → A((z)), which will be denoted again by ζ, has the property ζ(Γ(X \\ {p}, A)) = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The dimensions of the cohomology of A can be calculated by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) h0(A) = dim(A[[z]] ∩ W) and h1(A) = dim(A((z))/(A[[z]] + W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrically admissible algebra metrics and the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a metric k-algebra and let us denote the k((z))-bilinear extension of β by the same symbol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) β : A((z)) × A((z)) → k((z)) , �� k∈Z akzk, � k∈Z bkzk � �−→ � k,ℓ∈Z β(ak, bℓ)zk+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call (A, β) geometrically admissible if: (1) A is finite-dimensional, central, and simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) For any ringed A-lattices (O, W) and any maximal ideal m ⊆ O such that Wm is free as Om-module, we have β(Wm, Wm) ⊆ Om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a finite-dimensional, central, simple, metric k-algebra, (O, W) be a ringed A-lattice, and m ⊆ O be a regular maximal ideal of O such that Wm is a free Om- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then Wm is of rank d := dim(A), so we can choose an Om-basis {bi}d i=1 ⊆ Wm and write bibj = �d k=1 Ck ijbk for {Ck ij}d i,j,k=1 ⊆ Om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that {bi}d i=1 ⊆ Wm ⊆ A((z)) is also a k((z))-basis of A((z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1) Assume A is a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then β is a scalar multiple of the Killing form of A since A is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As a consequence, the extension (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) of β is equal to λK for the Killing form K of A((z)) and some λ ∈ k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, β(bi, bj) = λCℓ ikCk jℓ ∈ Om holds, so β is geometrically admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) Assume that A is power associative and not anti-commutative, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' if A is associative or Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then the existence of an algebra metric β on A implies that A is a non-commutative Jordan algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [BK66, Kapitel I, Satz 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Moreover, [She71, Theorem 1] implies that A is not nil, so there exists λ ∈ k× such that β(a, b) = λ 2 (Tr(Rab) + Tr(Lab)), for all a, b ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Sch55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, R, L: A → End(A) are the right and left multiplication maps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, β(bi, bj) = λ 2 d � k,ℓ=1 Cℓ ij(Ck kℓ + Ck ℓk) ∈ O holds, so β is geometrically admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 12 RASCHID ABEDIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrically admissible metrics and geometrization of lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a geomet- rically admissible metric k-algebra and (O, W) be a ringed A-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The following results are true: (1) For all regular maximal ideals m ⊆ O, we have β(Wm, Wm) ⊆ Om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) Let N be the integral closure of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then N can be understood as a subalgebra of k((z)) and V := NW ⊆ A((z)) is an A-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider the geometric datum ((X, A), (p, z, ζ)) constructed from the ringed A-lattice (N, V ) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then there exists a unique pairing βA : A × A → OX such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) Γ(U, A) × Γ(U, A) βA � ζ×ζ � Γ(U, OX) c � A((z)) × A((z)) β � k((z)) commutes for all U ⊆ X open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) The pairing βA gives rise to a short exact sequence 0 −→ A −→ A∗ −→ C −→ 0 for a torsion sheaf C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, A∗ = HomOX(A, OX) is the sheaf of morphisms from A to OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By definition, Om is a regular local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, the torsion-free Om-module Wm is free, so β(Wm, Wm) ⊆ Om holds since β is geometrically admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since the quotient field of O is a subalgebra of k((z)), we have N ⊆ k((z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, since O has Krull dimension one, dim(N/O) < ∞ and dim(V/W) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, V is an A-lattice and (N, V ) is a ringed A-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Every closed point q ∈ X \\ {q} ∼= Spec(N) corresponds to a maximal ideal mq ⊆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since N is integrally closed of dimension one, mq is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with c(OX,q) = Omq and ζ(Aq) = Wmq for all q ∈ X \\ {p}, we obtain β(ζ(Aq), ζ(Aq)) ⊆ OX,q from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that for all U ⊆ X \\ {p} open β(ζ(Γ(U, A)), ζ(Γ(U, A))) ⊆ c(Γ(U, OX)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with the fact that ζ(Γ(U, A)) = ζ(Γ(U \\ {p}, A)) ∩ A[[z]] holds for all open neighbourhoods U ⊆ X of p, we can simply define βA via the diagrams (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Note that the fiber of βA at p can be identified with β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, this fiber is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A → A∗ be the canonical morphism induced by βA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then the fact that βA|p is non-degenerate translates to the fact that A|p → A∗|p is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, the kernel and cokernel of A → A∗ are torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The observation that the kernel, as a torsion subsheaf of the torsion-free sheaf A, is vanishing concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The categorization theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The rest of this section is dedicated to the proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a geometrically admissible algebra over a field k of characteristic 0, n ∈ N, and λ ∈ k[[z]]×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let ((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple associated to this datum in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then n ∈ {0, 1, 2} and λ = 1 up to isomorphism in the sense that ((Dn(A), β(n,λ)), A[[z]], W) ∼= ((Dn(A), β(n,1)), A[[z]], � W) for an appropriate � W ⊆ Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 13 In particular, if C is a full subcategory of Algk closed under taking subalgebras, a non-degenerate topological D-bialgebra (A[[z]], δ) in C satisfies ((D(A[[z]], δ), ev), A[[z]]) ∼= ((Dn(A), β(n,1)), A[[z]]) for an appropriate n ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof proceeds in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We being by collecting several algebraic properties of Manin triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Using the geometrization method from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1, we pass from these Manin triples to certain geometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The application of algebro-geometric tools then concludes the proof of the refinement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A is a Lie algebra and k is algebraically closed, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 coincides with [MSZ10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, our proof is independent of the proof of [MSZ10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, we give a new proof of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The assumption on characteristic could be weakened by careful analysis of the follow- ing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For instance, the geometrization in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 works over fields where the characteristic does not divide the dimension of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, most geometric methods used below are adapt- able to fields of non-zero characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, for sake of clarity, we shall not pursue this level of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Algebraic properties of Manin triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a finite-dimensional metric k-algebra, n ∈ N0, λ ∈ k[[z]]×, and ((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple associated to this datum in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let W+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' W−) be the projection of W ⊆ Dn(A) = A((z)) × A[z]/znA[z] onto A((z)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A[z]/znA[z]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The following results are true: (1) W ⊥ ± ⊆ W± with respect to the bilinear forms β± (n,λ) defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8) β+ (n,λ)(a1, a2) := res0 1 znλβ(a1, a2) and β− (n,λ)([b1], [b2]) := res0 1 znλβ(b1, b2), where a1, a2 ∈ A((z)) and [b1], [b2] ∈ A[z]/znA[z] = A[[z]]/znA[[z]] are the classes of b1, b2 ∈ A[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) A((z)) = A[[z]] + W+ and dim(A[[z]] ∩ W+) < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) W+/W ⊥ + × W−/W ⊥ − = (A[[z]] ∩ (W+ × W−)) ⊕ W/(W ⊥ + × W ⊥ − ) is a finite-dimensional Manin triple, so dim(W+/W ⊥ + ) = dim(W−/W ⊥ − ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, we recall that A[[z]] is considered as a subalgebra of Dn(A) = A((z)) × A[z]/znA[z] via the diagonal embedding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (4) If n > 0, we have dim(A[[z]] ∩ W+) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Follows immediately from the fact that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8) implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9) W ⊥ + × W ⊥ − = (W+ × W−)⊥ ⊆ W ⊥ = W ⊆ W+ × W−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that A[[z]] + W = A((z)) × A[z]/znA[z] implies A((z)) = A[[z]] + W+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, {0} = (A[[z]] + W+)⊥ = znA[[z]] ∩ W ⊥ + since A[[z]]⊥ = znλA[[z]] = znA[[z]] with respect to β+ (n,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that A[[z]] ∩ W ⊥ + can be embedded into A[[z]]/znA[[z]] and is therefore finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, the dimension of A[[z]] ∩ W+ is finite if the quotient (A[[z]]∩W+)/(A[[z]]∩W ⊥ + ) is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The latter space can be identified with a subspace of W+/W ⊥ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) follows from Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 14 RASCHID ABEDIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The kernel K of the projection W → W+ contains {0} × W ⊥ − by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' On the other hand, any element of K is of the form (0, a) for some a ∈ W−, so for all (w+, w−) ∈ W (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10) 0 = β(n,λ)((0, a), (w+, w−)) = −β− (n,λ)(a, w−) holds, implying a ∈ W ⊥ − and hence K = {0} × W ⊥ − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We obtain an isomorphism W/(W ⊥ + × W ⊥ − ) −→ W+/W ⊥ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A similar argument yields W/(W ⊥ + × W ⊥ − ) ∼= W−/W ⊥ − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we obtain an isomorphism W+/W ⊥ + → W−/W ⊥ − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, dim(W+/W ⊥ + ) = dim(W−/W ⊥ − ) ⩽ dim(A[z]/znA[z]) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Considering W ⊆ W+ × W−, the identity A[[z]] ⊕ W = A((z)) × A[z]/znA[z] is equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11) W+ × W− = (A[[z]] ∩ (W+ × W−)) ⊕ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Quoiting out W ⊥ + × W ⊥ − concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume that n > 0 and A[[z]] ∩ W+ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then A[[z]] ∩ (W+ × W−) = {0} and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11) imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12) W = W+ × W− = W ⊥ + × W ⊥ − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any a ∈ A[z] exists b ∈ A[[z]] and w± ∈ W± such that (0, [a]) = (b, [b]) + (w+, w−) ∈ A[[z]] ⊕ (W+ × W−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, w+ = −b ∈ A[[z]] ∩ W+ = {0} results in [a] = w− ∈ W−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since a ∈ A[z] was arbitrary, we conclude W− = A[z]/znA[z], which contradicts W ⊥ − = W− in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometric properties of Manin triples over series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We are now in the position to proof Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, we proof the following refinement of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a geometrically admissible algebra over a field k of characteristic 0, n ∈ N0, λ ∈ k[[z]]×, and ((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple constructed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let W+ ⊆ A((z)) be the image of W under the projection Dn(A) → A((z)) and consider M := {f ∈ k((z)) | fW+ ⊆ W+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then n ∈ {0, 1, 2} and, up to isomorphism of Manin triples, λ = 1 and precisely one of the following cases occurs: (1) n = 0 and M is integrally closed satisfying dim(k((z))/(k[[x]] + M)) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) n = 0 and k[u′, uu′] ⊆ M for u ∈ z−1k[[z]]× satisfying u ̸= z−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) n = 0 and k[z−2, z−3] ⊆ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (4) n = 1 and M = k[z−1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (5) n = 2 and M = k[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of the results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5 (and by proxy of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We use similar arguments as in the proof of [Abe21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6] or [AMSZ22, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let N be the integral closure of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) states that V := NW+ is an A-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let ((X, A), (p, c, ζ)) be the geometric datum associated to the ringed A-lattice (N, V ) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3), we have a short exact sequence 0 −→ A−→A∗ −→ C −→ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13) where C is a torsion sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The associated long exact sequence in cohomology reads (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14) 0 −→ H0(A)−→H0(A∗) −→ H0(C) −→ H1(A)−→H1(A∗) −→ H1(C) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The identities H1(A) = 0 = H1(C) imply that H1(A∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 15 The Riemann-Roch theorem for A and A∗ combined with the fact that h1(OX) = g reads 0 ⩽ h0(A) − h1(A) = deg(det(A)) + (1 − g)rank(A), 0 ⩽ h0(A∗) − h1(A∗) = −deg(det(A)) + (1 − g)rank(A), where we used that det(A∗) = det(A)∗ implies deg(det(A∗)) = −deg(det(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We conclude g ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The case g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume g = 1, then X is an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then the sheaf Ω1 X of regular 1-forms on X satisfies Ω1 X ∼= OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, 0 = h1(A∗) = h0(A) because of Serre duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5), W+ ∩ g[[z]] ⊆ V ∩ g[[z]] = {0}, so Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (4) implies n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Moreover, W+ ⊕ g[[z]] = g((z)) = V ⊕ g[[z]] and W+ ⊆ V imply V = W+, so M = N is integrally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since h1(OX) = dim(k((z))/(k[[z]] + M)) = 1 by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3), we are in case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The case g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Note that g = 0 means k((z)) = k[[z]] + N by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since N ∩ k[[z]] = H0(OX) = k, we can see that N = k[u] for the unique u ∈ (z−1 + zk[[z]]) ∩ N ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let N ⊥ be the orthogonal complement of N with respect to the bilinear form R(n,λ) : k((z)) × k((z)) → k defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) (f, g) �−→ res0 1 znλfg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since (A, β) is geometrically admissible, we have β(V, V ) ⊆ N for β from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6), so for all f ∈ N ⊥ and a, b ∈ W+ ⊆ V we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16) β+ (n,λ)(fa, b) = res0 1 znλfβ(a, b) = R(n,λ)(f, β(a, b)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, we recall that β+ (n,λ) was defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, fa ∈ W ⊥ + so fa ∈ W+ since W ⊥ + ⊆ W+ by Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, N ⊥ ⊆ {f ∈ k((x)) | fW+ ⊆ W+} = M ⊆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17) R(n,λ)(znλu′, uk) = res0u′uk = 1 k + 1res0 � uk+1�′ = 0 for all k ∈ N yields znλu′ ∈ N ⊥ ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The case (n, g) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since R(n,λ) is associative and v := λu′ ∈ N ⊥ we have the inclusion vN ⊆ N ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, since v ∈ N ⊥ ⊆ N = k[u], we obtain k[v, vu] ⊆ k + vN ⊆ k + N ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since all three spaces are of codimension one in N = k[u] we conclude (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='18) k[v, vu] = k + vN = k + N ⊥ ⊆ {f ∈ k((z)) | fW+ ⊆ W+} = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Case (n, g) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since v := zλu′ ∈ N ⊥ ∩ z−1k[[z]]× ⊆ N ∩ z−1k[[z]]× = k×u + k, we have M = N = k[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Case (n, g) = (2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since z2λu′ ∈ N ⊥ ∩ k[[z]] ⊆ N ∩ k[[z]] = k we have au′ = −z−2λ−1 for some a ∈ k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, res0z−2λ−1 = res0au′ = 0 and N = k[u] = N ⊥ ⊆ M by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Case n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The fact that z3λu′ ∈ N ⊥ ∩zk[[z]] = {0} is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, there cannot exist any Manin triple of the form ((Dn(A), β(n,λ)), A[[z]], W) for n ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 16 RASCHID ABEDIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As a metric algebra (Dn(A), β(n,λ)) ∼= (A⊗Rn, β ⊗t(n,λ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is shown in [AMSZ22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12] that (Rn, t(n,λ)) ∼= (Rn, t(n,1/(1+azn−1))) as trace extensions for a = res0z−nλ−1 ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, since n ⩽ 2 and for n = 2 the identity res0z−nλ−1 = 0 holds, we obtain λ = 1 up to isomorphism in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If (n, g) = (0, 0), this means that k[u′, uu′] ⊆ M ⊆ N = k[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, since by definition u ∈ (z−1 + zb + z2k[[z]]) for some b ∈ k, we see that u′ + u2 − 3b ∈ k[u] ∩ zk[[z]] = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If b = 0 this formal differential equation has the unique solution u = z−1 and we are in case (3) and if b ̸= 0 we are in case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If (n, g) = (1, 0), we have zu′ ∈ M = N = k[u], so zu′ = −u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The only solution to this equation is again z−1 and we are in case (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Finally, if (n, g) = (2, 0), we have u′ = −z−2 so u = z−1 again and we are in case (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Non-degenerate D-bialgebra structures and the classical Yang-Baxter equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Series of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a finite-dimensional metric k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Choose a basis {bi}d i=1 of A and consider its dual basis {b∗ i }d i=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' β(bi, b∗ j) = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The tensor γ = �d i=1 b∗ i ⊗ bi is independent of the choice of {bi}d i=1 ⊆ A, symmetric, and satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) a(1)γ = γa(2) or, equivalently, a(2)γ = γa(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The first identity follows from the fact that β⊗2(a(1)γ, bj ⊗ b∗ k) = d � i=1 β(ab∗ i , bj)β(bi, b∗ k) = β(ab∗ k, bj) = β(b∗ k, bja) = β(γa(2), bj ⊗ b∗ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' holds for all j, k ∈ 1, d, and the second follows from the first by using the symmetry of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call γ canonical A-invariant element of (A, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us note that the canonical embedding (A ⊗ A)[[x, y]] → (Dn(A) ⊗ A)[[y]] extends to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) (A ⊗ A)[[x, y]][(x − y)−1] −→ (Dn(A) ⊗ A)((y)) by writing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) 1 x − y = n−1 � k=0 (0, −[x]n−k−1)yk−n + ∞ � k=0 (x−k−1, 0)yk ∈ (k((x)) × k[x]/(xn))((y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Indeed, this is appropriate since we can calculate ((x, [x]) − y) �n−1 � k=0 (0, −[x]n−k−1)yk−n + ∞ � k=0 (x−k−1, 0)yk � = n−1 � k=0 (0, −[x]n−k)yk−n − n � k=1 (0, −[x]n−k)yk−n + (1, 0) = (1, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) inside (k((x)) × k[x]/(xn))((y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, for any n ∈ N we obtain ynγ x − y = n−1 � k=0 d � i=1 (0, −[b∗ i xn−1−k]) ⊗ biyk + ∞ � k=n d � i=1 (b∗ i xn−1−k, 0) ⊗ biyk = ∞ � k=0 d � i=1 wk,i ⊗ biyk ∈ (Dn(A) ⊗ A)[[y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 17 For any λ ∈ k[[z]]× and s ∈ (A ⊗ A)[[x, y]], we identify the expression r(x, y) = ynλ(x)γ x − y + s(x, y) ∈ (A ⊗ A)[[x, y]][(x − y)−1] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) with its series in (Dn(A) ⊗ A)[[y]] and say that r is a series of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Every r(x, y) = a(x,y)γ x−y + s(x, y) for a ∈ k[[x, y]] such that a(z, z) ̸= 0 and any s ∈ (A ⊗ A)[[x, y]] has a unique representation as a series of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Indeed, chose (n, λ) such that a(z, z) = znλ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then a(x, y) − ynλ(x) = (x − y)b(x, y) for some b ∈ k[[x, y]], so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) r(x, y) = ynλ(x)γ x − y + b(x, y)γ + s(x, y) is a series of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In the construction of b we used the following easy fact: for any k-vector space V (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8) f ∈ V [[x, y]], f(z, z) = 0 =⇒ f(x, y) = (x − y)g(x, y) for some g ∈ V [[x, y]] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ Note that we have a linear automorphism of (A ⊗ A)[[x, y]][(x − y)−1] defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9) a(x, y) �−→ a(x, y) := −τ(a(y, x)) where τ(a ⊗ b) = b ⊗ a is applied coefficient-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any series r of type (n, λ), r is again a series of type (n, λ) by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call r skew-symmetric if r = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The (generalized) classical Yang-Baxter equation with coefficients in arbitrary algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As in the last section, (A, β) is a finite-dimensional metric k-algebra, {bi}d i=1 and {b∗ i }d i=1 are basis of A satisfying β(b∗ i , bj) = δij, and γ := �d i=1 b∗ i ⊗ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let U be the unitalization of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' U = A ⊕ k with multiplication (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10) (a1, u1)(a2, u2) = (a1a2 + u1a2 + u2a1, u1u2) for all a1, a2 ∈ A and u1, u2 ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any s ∈ (A ⊗ A)[[x, y]][(x − y)−1], let us define the expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11) s12(z1, z2), s13(z1, z3), s23(z2, z3) ∈ (U ⊗ U ⊗ U)[[z1, z2, z3]] � 1 (z1 − z2)(z1 − z3)(z2 − z3) � coefficient-wise via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12) t12 = t ⊗ 1, t13 = a ⊗ 1 ⊗ b, t23 = 1 ⊗ t ∈ U ⊗ U ⊗ U for t = a ⊗ b ∈ A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us point out that for example (a1 ⊗ a2)13(b1 ⊗ b2)12 = a1b1 ⊗ b2 ⊗ a2 ∈ A ⊗ A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This and similar identities imply that for all s1, s2 ∈ (A ⊗ A)[[x, y]][(x − y)−1] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13) s13 1 (z1, z3)s12 2 (z1, z2), s12 1 (z1, z2)s23 2 (z2, z3), and s23 1 (z2, z3)s13 2 (z1, z3) are elements of (A ⊗ A ⊗ A)[[z1, z2, z3]] � 1 (z1 − z2)(z1 − z3)(z2 − z3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, if s1, s2 are of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) and we write sǫ = � k∈Z d � i=1 sǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='k,i(x) ⊗ biyk ∈ (Dn(A) ⊗ A)((y)), for ǫ ∈ {1, 2} for the associated series via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2), we get s13 1 (z1, z3)s12 2 (z1, z2) = � k,ℓ∈N d � i,j=1 s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='k,i(z1)s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='ℓ,j(z1) ⊗ bjzℓ 2 ⊗ bizk 3 ∈ (Dn(A) ⊗ A ⊗ A)[[z2, z3]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 18 RASCHID ABEDIN Similar formulas for s12 1 (z1, z2)s23 2 (z2, z3) and s23 1 (z2, z3)s13 2 (z1, z3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any r ∈ (A ⊗ A)[[x, y]][(x − y)−1], we call the equation GCYB(r) = 0 the A-generalized classical Yang-Baxter equation (short: A-GCYBE), where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14) GCYB(r) := r13r12 − r12r23 + r23r13 Here, (·) was defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Similarly, we call the equation CYB(r) = 0 the A-classical Yang-Baxter equation (short: A- CYBE), where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) CYB(r) := r13r12 − r12r23 + r23r13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A is a Lie algebra, these are exactly the usual (generalized) classical Yang-Baxter in two-formal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A is associative, the A-classical Yang-Baxter equation is a formal variant of the associative version of the CYBE used in [OS08], which is itself a spectral parameter generalization of the associative CYBE discussed in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Agu01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Solutions of the A-(G)CYBE and subspaces of Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Series of type (n, λ) can be seen as generating series of certain subspaces of Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, we have the following result, which is a generalization of known statements in the Lie algebra case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [GC83;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Skr13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' AMS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a finite-dimensional metric k-algebra, {bi}d i=1 and {b∗ i }d i=1 be ba- sis of A satisfying β(b∗ i , bj) = δij, γ := �d i=1 b∗ i ⊗ bi, and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' To any series r(x, y) = �∞ k=0 �d i=1 rk,i(x) ⊗ biyk ∈ (Dn(A) ⊗ A)[[y]], we can define a linear subspace (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16) A(r) := Spank{rk,i | k ∈ N, i ∈ 1, d} ⊆ Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any fixed λ ∈ k[[z]]× the following results are true: (1) r �→ A(r) defines a bijection between series r of type (n, λ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' of the from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6)) and subspaces W ⊆ Dn(A) satisfying Dn(A) = A[[z]] ⊕ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) For any series r of type (n, λ), the identity A(r)⊥ = A(r) holds, where (·)⊥ is meant with respect to β(n,λ) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) and (·) is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) For any series r of type (n, λ), the identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17) GCYB(r) = ϕ holds for the unique element ϕ ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]] determined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='18) β⊗3 (n,λ)(v1 ⊗ v2 ⊗ v3, ϕ) = β(n,λ)(v1, v3v2) for all v1 ∈ A(r), v2, v3 ∈ A(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 is postponed to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A direct consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1)&(3) is that r �→ A(r) defines a bijection between solutions r of the A-GCYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14) of type (n, λ) and subalgebras W ⊆ Dn(A) satisfying Dn(A) = A[[z]]⊕W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2), we obtain a bijection between skew-symmetric solutions r of the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) of type (n, λ) and Manin triples ((Dn(A), β(n,λ)), A[[z]], W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We will see that, if A is simple, any solution r of the A-CYBE of type (n, λ) is already skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we have the following consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a finite-dimensional simple metric k-algebra, n ∈ N, and λ ∈ k[[z]]×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then r �→ A(r) defines a bijection between solutions of the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) r of type (n, λ) and Manin triples ((Dn(A), β(n,λ)), A[[z]], W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof will be given in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since Manin triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) exist only for n ⩽ 2 by virtue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 gives the same restriction for solutions of the A-CYBE for any geometrically admissible k-algebra (A, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' To be precise, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 19 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a finite-dimensional simple metric k-algebra, n ∈ N and λ ∈ k[[z]]×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If r ∈ (Dn(A) ⊗ A)[[y]] is a solution of the A-CYBE of type (n, λ), we have n ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof of (1) and (2) is completely analogous to the proof in the case that A is a Lie algebra in [AMS22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6], so it remains to prove (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us begin by proving, that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='19) GCYB(r) ∈ (A⊗A⊗A)[[z1, z2, z3]] for all series r ∈ (A⊗A)[[x, y]][(x−y)−1] of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' To this end, let r(x, y) = ynλ(x)γ x−y + s(x, y) be a series of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Clearly, T1 := GCYB(s) is an element of (A ⊗ A ⊗ A)[[z1, z2, z3]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since a(1)γ = γa(2) for all a ∈ A we have γ13γ12 = γ12γ23 = γ23γ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, if we write w := ynλ(x)γ x−y , we have (z1 − z2)(z1 − z3)(z2 − z3)GCYB (w) = (z2z3)nλ(z1)(λ(z1)(z2 − z3) − λ(z2)(z1 − z3) + λ(z3)(z1 − z2))γ13γ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This expression is zero if z1 = z2, z1 = z3 or z2 = z3, so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='20) T2 := GCYB (w) ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]] Now let us turn to GCYB(r) = GCYB (w + s) = T1 + T2 + w13s12 + s13w12 − w12s23 − s12w23 + w23s13 + s23w13 = T1 + T2 + (s13w12 − w12s23) � �� � :=T3 − (s12w23 − w23s13) � �� � :=T4 + (s23w13 + w13s12) � �� � :=T5 Write s(x, y) = � k∈N �d i=1 si,j k,ℓxkyℓbi ⊗ bj and note that s(x, y) = − � k∈N �d i=1 si,j k,ℓxℓykbj ⊗ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Using zka(1)γ − γa(2)zk = 0 for all a ∈ A we see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='21) T3 = λ(z1)zn 2 z1 − z2 � k,ℓ∈N d � i,j=1 si,j k,ℓ(zk 1b(1) i γ − γb(2) i zk 2) ⊗ bjzℓ 3 ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' by virtue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Similarly, under consideration of w = λ(y)xnγ x−y , we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='22) T4 = 1 z1 − z2 � k,ℓ∈N d � i,j=1 sk,ℓ i,j bizk 1 ⊗(λ(z2)zn 3 zℓ 2b(1) j γ−γb(2) j zℓ 3λ(z3)zn 2 ) ∈ (A⊗A⊗A)[[z1, z2, z3]] Using a(2)γ = γa(1) and the notation θa(b ⊗ c) = b ⊗ a ⊗ c for all a ∈ A, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='23) T5 = λ(z1)zn 3 z1 − z3 � k,ℓ∈N d � i,j=1 si,j k,ℓθbjzℓ 2(−zk 3b(2) i γ + γb(1) i zk 1) ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Summarized, we have GCYB(r) = T1 + T2 + T3 + T4 + T5 ∈ (A ⊗ A ⊗ A)[[z1, z2, z3]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us now write r(x, y) = ynλ(x)γ x − y + s(x, y) = � k∈N d � i=1 rk,i(x) ⊗ biyk 20 RASCHID ABEDIN and observe that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='24) GCYB(r) = � k,ℓ∈N d � i,j=1 rℓ,jrk,i ⊗ bizk 2 ⊗ bjzℓ 3 − � k∈N d � i=1 rk,i ⊗ � zk 2b(1) i r(z2, z3) − r(z2, z3)b(2) i zk 3 � holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, we used the fact that the embedding (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) induces a commutative diagram (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='25) (A ⊗ A ⊗ A)[[z1, z2, z3]] � � (Dn(A) ⊗ A ⊗ A)[[z2, z3]] (A ⊗ A ⊗ A)[[z1, z2, z3]] � 1 z1−z3 � �❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Applying β⊗3 (n,λ)(rk1,i1 ⊗ rk2,i2 ⊗ rk3,i3, −) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='24), where r = �∞ k=0 �d i=1 rk,i ⊗ biyk yields (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='26) β⊗3 (n,λ)(rk1,i1 ⊗ rk2,i2 ⊗ rk3,i3, GCYB(r)) = β(n,λ)(rk1,i1, rk3,i3rk2,i2) since rk1,i1 ∈ A(r) = A(r)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This concludes the proof, since {rk,i | k ∈ N0, i ∈ 1, d} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' {rk,i | k ∈ N0, i ∈ 1, d}) is a basis of A(r) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Under consideration of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 and the remarks after this theorem, it remains to prove that, if A is simple, any solution r of the A-CYBE of type (n, λ) is automatically skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The equality CYB(r) = 0 implies that A(r) ⊆ Dn(A) is a subalgebra by rewriting the CYBE similarly to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, (3) implies that r solves the A-GCYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='27) 0 = CYB(r) − GCYB(r) = (r23 − r23)r13 Multiplying by z1 − z3 and setting z1 = z3 we obtain (r23 − r23)γ13 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that (r − r)b(2) i = 0 for all i ∈ 1, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since A is simple, we have aA = {0} implies a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, r = r, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Connection to topological D-bialgebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let r ∈ (Dn(A)⊗A)[[y]] be a solution of the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) of type (n, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The identity CYB(r) = 0 can be rewritten as ∞ � k,ℓ=0 d � i,j=1 rk,i(z1)rℓ,j(z1) ⊗ bjzℓ 2 ⊗ bizk 3 = ∞ � k=0 d � i=1 rk,i(z1) ⊗ �� bizk 2 �(1) r(z2, z3) − r(z2, z3) � bizk 3 �(2)� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='28) Therefore, under consideration of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8), we can deduce that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='29) δr(a)(x, y) := r(x, y)a(x)(1) − a(y)(2)r(x, y) = − � a(x)(1)r(x, y) − r(x, y)a(y)(2) � defines a continuous linear map δr : A[[z]] → (A ⊗ A)[[x, y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='30) β⊗3 (n,λ)(bi1zk1 1 ⊗ rk3,i3 ⊗ rk2,j2, −) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='28) results in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='31) β(n,λ) � bi1zk1, rk2,i2rk3,j3 � = β⊗2 (n,λ) � δr � bi1zk1� , rk2,i2 ⊗ rk3,j3 � CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 21 This proves that δr is determined by ((Dn(A), β(n,λ)), A[[z]], A(r)) and thus defines a topological D-bialgebra structure in any full subcategory C of Algk that is closed under taking subalgebras and contains Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, ((Dn(A), β(n,λ)), A[[z]], A(r)) ∼= (D(A[[z]], δr), ev), A[[z]], A[[z]]∨), so (A[[z]], δr) is a non-degenerate topological D-bialgebra in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In fact, the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 imply that every non-degenerate topological D-bialgebra structure in C is of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Equivalence of solutions of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let n ∈ N and (A, β) be a finite-dimensional metric k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call two solutions r1, r2 ∈ (A ⊗ A)[[x, y]][(x − y)−1] of the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) equivalent, written r1 ∼ r2, if there exists ϕ ∈ Autk[[z]]-alg(A[[z]]) and u ∈ zk[[z]]× such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='32) (ϕ(x) ⊗ ϕ(y))r1(u(x), u(y)) = r2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let n ∈ N, (A, β) be a finite-dimensional, central, simple, metric k-algebra, λǫ ∈ k[[z]]× and let rǫ ∈ (A⊗A)[[x, y]][(x−y)−1] be a solution of the A-CYBE of type (n, λǫ) for ǫ ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then the following statements are equivalent: r1 and r2 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A[[z]], δr1) ∼= (A[[z]], δr2) as topological D-bialgebra structures in any full subcategory C of Algk that is closed under taking subalgebras and contains Dn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ((Dn(A), β(n,λ1)), A[[z]], A(r1)) ∼= ((Dn(A), β(n,λ2)), A[[z]], A(r2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The equivalence of the latter two items is already dealt with in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For the equivalence of the first two items, recall that any φ ∈ Autk-alg(A[[z]]) is of the form φ(a)(z) = ϕ(z)a(u(z)) for some ϕ ∈ Autk[[z]]-alg(A[[z]]) and u ∈ zk[[z]]×;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see [AMSZ22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now, its easy to see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='33) (ϕ(x) ⊗ ϕ(y))r1(u(x), u(y)) = r2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' is equivalent to (φ ⊗ φ)δr1φ−1 = δr2, which means that (A[[z]], δr1) ∼= (A[[z]], δr2) as topological D-bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Solutions of the A-CYBE and triangular D-bialgebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is also possible to consider solutions to the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) of the form r ∈ (A⊗ A)[[x, y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, the assignment r �→ A(r) defines a bijection between: skew-symmetric solutions r ∈ (A ⊗ A)[[x, y]] of the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) and subspaces W ⊆ A ⊗ R∞ such that ((A ⊗ R∞, β ⊗ t∞), A[[z]], W) is a Manin triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Moreover, (A[[z]], δr) is a topological D-bialgebra structure (in any category of algebras closed under taking subalgebras that contains A ⊗ R∞) determined by ((A ⊗ R∞, β ⊗ t∞), A[[z]], W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='34) ((A ⊗ R∞, β ⊗ t∞), A[[z]], W) ∼= ((Dn(A[[z]], δr), ev), A[[z]], A[[z]]∨), so δr is a triangular topological D-bialgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' On the other hand, all triangular topological D-bialgebra structures on A[[z]] are of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that a Lie bialgebra structure (L, δ) is called triangular if δ = δr for some skew-symmetric solution r ∈ L ⊗ L of the CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If (g[[z]], δ) is a topological Lie bialgebra structure for some Lie algebra g, it is natural to replace g[[z]] ⊗ g[[z]] by its completion (g ⊗ g)[[x, y]] in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, it is natural to call δ triangular if δ = δr for a skew-symmetric solution r ∈ (g⊗ g)[[x, y]] of the CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The D-bialgebra structures satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='34) are then called triangular in analogy to their Lie counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Refined categorization of non-degenerate topological D-bialgebras In this section, we refine Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 for so-called strongly geometrically admissible algebras over algebraically closed fields of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The main result of this section, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1, can be seen as an analog of the main results from [AMSZ22] for a large class of non-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof relies on refining the geometric approach already used in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 22 RASCHID ABEDIN Throughout the remainder of this paper, k is an algebraically closed field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a metric k-algebra (A, β) strongly geometrically admissible if (1) (A, β) is geometrically admissible in the sense of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) For any ringed A-lattices (O, W) and any maximal ideal m ⊆ O such that Wm is free as Om-module and the pairing Wm × Wm → Om induced by β is perfect, we have W/mW ∼= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As we will see in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8 below, many central simple k-algebras are strongly geometrically admissible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' all finite-dimensional simple associative, Lie and Jordan algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The rest of this section is dedicated to proving the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us fix the following notation: k is an algebraically closed field of characteristic 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A, β) is a unital strongly geometrically admissible metric k-algebra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' a finite-dimensional simple Jordan or associative k-algebra) and γ ∈ A ⊗ A is its canonical A-invariant element (see Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ((Dn(A), β(n,λ)), A[[z]], W) is the Manin triple associated to (A, β) as well as some n ∈ N and λ ∈ k[[z]]×in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' r is the solution of the A-CYBE associated to the Manin triple ((Dn(A), β(n,λ)), A[[z]], W) via Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Precisely one of the following cases occurs: (1) n = 0 and r is either: (a) Trigonometric in the sense that there exists a β-orthogonal σ ∈ Autk-alg(A) of order m ∈ N and s ∈ L(A, σ) ⊗ L(A, σ) such that r is equivalent to 1 exp (x − y) − 1 m−1 � j=0 exp �x − y m � γj + s � exp � x m � , exp � y m �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, L(A, σ) ⊆ A[�v, �v−1] is the loop algebra twisted by σ (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9 for the definition) and γj ∈ A ⊗ A is uniquely determined by γ = �d j=1 γj and (σ ⊗ 1)γj = εjγj for some primitive m-th root of unity ε ∈ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (b) Rational in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r is equivalent to γ x − y + t(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) n = 1 and r is quasi-trigonometric in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r is equivalent to yγ x − y + t(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) n = 2 and r is quasi-rational in the sense that there exists t ∈ (A ⊗ A)[x, y] such that r is equivalent to y2γ x − y + t(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, every solution of the A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) of type (n, λ) is, up to equivalence, either trigonometric, rational, quasi-trigonometric or quasi-rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0, (A, β) be a strongly geo- metrically admissible k-algebra, and C be a full subcategory of Algk closed under taking subalgebras and satisfying Dn(A) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then every non-degenerate topological D-bialgebra (A[[z]], δ) in C satisfies, up to isomorphism, δ = δr for a solution r of the A-CYBE which is either trigonometric, rational, quasi-trigonometric, or quasi-rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 23 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us note that the unitality assumption in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 is actually a rather weak assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Indeed, if a strongly geometrically admissible algebra is power-associative and not anti-commutative, it is a non-nil (see [She71]) trace-admissible algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' These are automatically unital;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see [Alb49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 is again based on the geometrization scheme from Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, to refine the geometric approach already used in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1, we need to establish some facts about ´etale locally trivial sheaves of algebras in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There we also explain how examples of strongly geometrically admissible algebras can be constructed using the notion of rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The results from Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 and the refinement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5 are then used to associate more explicit geometric data to Manin triples of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, so-called geometric A-CYBE data, which will be defined in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We will assign such a datum to any Manin triple of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 is then a consequence of the classification results for sheaves of algebras from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ´Etale locally trivial sheaves of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a sheaf of algebras on a k-scheme X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call A ´etale A-locally free at a point p ∈ X, for some k-algebra A, if there exists an ´etale morphism f : Y → X such that p ∈ f(Y ) and f ∗A is isomorphic to A ⊗ OY as OY -algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, A is called ´etale A-locally free if A is ´etale A-locally free at all points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us remark that an ´etale A-locally free sheaf of algebras is automatically quasi-coherent and, if A is finite-dimensional, coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ´Etale local triviality can actually be checked on fibers by virtue of the following result, which is an algebro-geometric version of [Kir78], see [Abe21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0, X be a reduced k-scheme of finite-type, A be a finite-dimensional k-algebra, and A be a quasi-coherent sheaf of algebras on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then A is ´etale A-locally free if and only if A|p ∼= A for all closed points p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It turns out that a sheaf of algebras which can be ´etale trivialized by a unital algebra is automat- ically unital, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a unital algebra over a field k and A be an ´etale A-locally free sheaf of algebras on a k-scheme X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then A is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, h0(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let U ⊆ X be an open subset and assume that U has an affine open covering {Ui}i∈I such that Γ(Ui, A) is unital for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since Γ(D(f), A) = Γ(Ui, A)f and Γ(Ui, A) → Γ(D(f), A) as well as Γ(D(f), A) → Γ(D(fg), A) are unital for all f, g ∈ Γ(Ui, OX) and i ∈ I, a gluing argument shows that Γ(Ui ∩ Uj, A) and Γ(Ui, A) → Γ(Ui ∩ Uj, A) are unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, a second gluing argument implies that Γ(U, A) is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A similar consideration shows that Γ(U, A) → Γ(V, A) is unital for all V ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We conclude that A is unital if and only if every p ∈ X has an affine open neighbourhood U such that Γ(U, A) is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For every p ∈ X, We can chose an irreducible affine open neighbourhood U of p, an irreducible affine scheme U ′, and a surjective ´etale morphism f : U ′ → U such that there exists an isomorphism ψ: B ⊗R S → A ⊗ S of S-algebras, where B := Γ(U, A), R := Γ(U, OX), and S := Γ(U ′, OU′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The element ψ−1(1) ∈ B ⊗R S is a unit and ψ is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since f is faithfully flat of finite type, we can recover B from B ⊗R S as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) B = {a ∈ B ⊗R S | φ(a ⊗ 1) = 1 ⊗ a} where φ: (B ⊗R S) ⊗R S → S ⊗R (B ⊗R S) is defined by φ((b ⊗ s) ⊗ t) = s ⊗ (b ⊗ t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Mil80, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, B can be identified with an subalgebra of the unital algebra A ⊗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since φ is an isomorphism, it is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, B = Γ(U, A) contains the unit of B ⊗R S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Thus, the argument in the beginning of this proof implies that A is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now h0(A) > 0 follows from the fact that 1 ∈ H0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 24 RASCHID ABEDIN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Rigidity and strongly geometrically admissible algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider the affine variety Alg(d, k) = Hom(kd ⊗ kd, kd) of all possible multiplication maps on kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There is a natural action of the group of invertible d × d-matrices GL(d, k) given by (g · ϑ)(v ⊗ w) = g−1ϑ(gv ⊗ gw) ∀g ∈ GL(d, k), ϑ ∈ Alg(d, k), v, w ∈ kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) The orbit of a multiplication map under this action corresponds to the isomorphism class of the associated algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let M ⊆ Alg(d, k) be a GLn(d, k)-invariant affine subvariety and write A ∈ M for an algebra A = (kd, µ), if µ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A k-algebra A = (kd, µ) is called M-rigid if A ∈ M and the orbit (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) O(A) := {A′ = (kd, µ′) ∈ M | µ′ = gµ for some g ∈ GL(d, k)} contains an open neighbourhood of A in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A sufficient condition for ´etale local triviality is the rigidity of the fiber, as the following result, which is a algebro-geometric version of a generalization of [Kir83, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1], states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0 and M be a GL(d, k)- stable subvariety of Alg(d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, let A be a locally free sheaf of algebras on a reduced k-scheme X such that A|q ∈ M for all q ∈ X closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If A|p is M-rigid for some closed point p ∈ X, A is ´etale A|p-locally free in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, A|q ∼= A|p for all closed points q ∈ X in some neighbourhood of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof is a straight forward adaptation of the proof of [Abe21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11] to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A consequence of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6 is the following important criterion for strong geometric admissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0, M ⊆ Alg(d, k) be a GLn(d, k)-invariant affine subvariety, and (A, β) metric k-algebra in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (A, β) is strongly geometrically admissible if: (1) (A, β) is geometrically admissible in the sense of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) For any ringed A-lattices (O, W) and any maximal ideal m ⊆ O such that Wm is free as Om-module and the pairing Wm × Wm → Om induced by β is perfect the k-algebra W/mW is M-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let ((X, A), (p, c, ζ)) is the geometric datum associated to (O, W) in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Fur- thermore, let U be the set of closed points q ∈ X such that Aq is a free OX,q-module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The restriction ζ(Aq) × ζ(Aq) → c(OX,q) of β from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then A|q is M-rigid for all q ∈ U by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combining Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6 and p ∈ U, U is a non-empty open subset of the set of closed points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, U is connected since X is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, every q ∈ U has an open neighbourhood U ′ ⊆ U such that A|q ∼= A|q′ holds for all q′ ∈ U ′ by virtue of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The connectedness of U therefore implies that A|q ∼= A|p ∼= A for all q ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that (A, β) is strongly geometrically admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Consider M ∈ {Lied, Assd, Jord} where Lied, Assd, Jord ⊆ Alg(d, k) are the varieties of d-dimensional Lie, associative, and Jordan algebras respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1, we discussed that any sim- ple A ∈ M has an, up to multiplication by a scalar, unique algebra metric β and that the metric algebra (A, β) is geometrically admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (O, W) be a ringed A-lattice and m ⊆ O be a maximal ideal such that Wm is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If the restriction Wm × Wm → Om of β from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) is non-degenerate, the k-algebra W/mW ∈ M inherits an algebra metric from β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This algebra metric can be explicitly described using the formula in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 and we see from this description that W/mW is semi-simple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' a direct sum of simple subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If M = Lied we use Cartan’s criterion for semi-simplicity and CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 25 if M ∈ {Assd, Jord} this is a consequence of general results on trace-admissible algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Alb49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since semi-simple algebras in M are rigid (see [HG88] for the case that M ∈ {Lied, Assd} and [Fin90] for the case that M = Jord), we see that (A, β) satisfies the conditions of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Any finite-dimensional simple Lie, associative, or Jordan algebra over an alge- braically closed field of characteristic 0 is strongly geometrically admissible if equipped with its (up to scalar multiple) unique algebra metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Sheaves of algebras on one-dimensional affine algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that over an alge- braically closed field of characteristic 0, a connected affine algebraic group over k of dimension one is either isomorphic to the affine line or the punctured affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us conclude this subsection with a classification of all sheaves of algebras with constant fibers on these schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see [Abe22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1] for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional algebra over an algebraically closed field k of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1) Let B be a k[v, v−1]-algebra satisfying B/(v − λ)B ∼= A for all λ ∈ k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then there exists σ ∈ Autk-alg(A) of order m ∈ N such that B ∼= L(A, σ) := {a ∈ A[�v, �v−1] | a (exp (2πi/m) �v) = σ(a(�v))} as k[v, v−1]-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, the k[v, v−1]-module structure of L(A, σ) is defined by �vm = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) Let B be an k[z]-algebra satisfying B/(z − λ)B ∼= A for all λ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then B ∼= A[z] as k[z]- algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometric A-CYBE datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a triple (X, (A, βA)) geometric A-CYBE datum for a finite-dimensional k-algebra A if: X is an irreducible cubic plane curve over k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A is a coherent sheaf of algebras on X such that: (1) H0(A) = 0 = H1(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) A|p ∼= A for all smooth closed p ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' βA : A × A → OX is a symmetric, perfect, associative OX-bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The name “geometric A-CYBE datum” will become clear in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is well-known that any irreducible plane cubic curve X over k is defined by an equation y2 = x3 + ax + b and precisely one of the following cases occurs: (1) X is smooth if and only if 4b3 + 27a2 ̸= 0, in which case X is an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) X has a unique nodal singularity if 4b3 = −27a2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this case, X \\ {s} is isomorphic to the punctured affine line Spec(k[v, v−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) X has a unique cuspidal singularity if a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this case, X \\ {s} is isomorphic to the affine line Spec(k[z]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us note that, up to isomorphism, irreducible cubic plane curves are precisely irreducible pro- jective curves over k of arithmetic genus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ The following lemma is important for the identification of geometric A-CYBE data below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0, X be an irreducible plane cubic curve over k, and F be coherent sheaf on X satisfying: H1(F) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There exists a non-degenerate symmetric OX-bilinear form βF : F × F → OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then βF is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, βF|p is non-degenerate for all p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since βF is non-degenerate, we have a short exact sequence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) 0 −→ F βa F −→ F∗ −→ C −→ 0, 26 RASCHID ABEDIN where C := Cok(βa F) is a torsion sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We obtain the long exact sequence in cohomology 0 −→ H0(F)−→H0(F∗) −→ H0(C) −→ H1(F)−→H1(F∗) −→ H1(C) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) The dualizing sheaf of any irreducible cubic plane curve is trivial, so Serre duality implies that h0(F∗) = h1(F) = 0 and thus H0(C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since C is a torsion sheaf, we see that C = 0, so βa F is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometric solutions of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this subsection, we will repeat the construction of geometric solutions of the usual CYBE from [BG18] in our general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This will result in a construction of geometric solutions of a geometric A-CYBE from a geometric A-CYBE datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, this explains the name “geometric A-CYBE datum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be a finite-dimensional k-algebra, (X, (A, βA)) be a geometric A-CYBE datum, and C ⊆ X be the set of smooth points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Fix a global section η ∈ H0(ωX) of the dualizing sheaf ωX of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us remark that ωX can be identified with the sheaf of so-called Rosenlicht-regular 1-forms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Con00, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider the diagonal residue sequence 0 −→ OX×C −→ OX×C(∆) resη ∆ −→ δ∗OC −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) Here, ∆ ⊆ X × C is the image of δ: C → X × C defined by p �→ (p, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, resη ∆ is determined by (u1 − u2)−1 �→ µ locally around any closed point p ∈ C, where: u is a local parameter of p defined on an affine open subset U of C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ωC and OX×C(−∆) are locally generated by du and u1 − u2 := u ⊗ 1 − 1 ⊗ u ∈ Γ(U, OX) ⊗ Γ(U, OX) ∼= Γ(U × U, OX×X) respectively, after potentially shrinking U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ηp = µdu holds for some uniquely determined µ ∈ Γ(U, OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The tensor product of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) with A ⊠ A|C := pr∗ 1A ⊗OX×C pr∗ 2A|C for the canonical projections X pr1 ←− X × C pr2 −→ C gives the short exact sequence 0 −→ A ⊠ A|C −→ A ⊠ A|C(∆) −→ δ∗(A|C ⊗OC A|C) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) Using the K¨unneth formula and H0(A) = 0 = H1(A) results in H0(A ⊠ A|C) = H0(A) ⊗ H0(A|C) = 0 and H1(A ⊠ A|C) = � H1(A) ⊗ H0(A|C) � ⊕ � H0(A) ⊗ H1(A|C) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8) Therefore, the long exact sequence in cohomology induced by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) results in an isomorphism R: H0(A ⊠ A|C(∆)) → H0(A|C ⊗ A|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The pairing βA of A induces an isomorphism B : A|C ⊗OC A|C → EndOC(A|C) defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9) a ⊗ b �−→ βA(b, −)a for all U ⊆ C affine open a, b ∈ Γ(U, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with R, we obtain an isomorphism Φ := BR: H0(A ⊠ A|C(∆)) → EndOC(A|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider the section ρ := Φ−1(idA|C) ∈ H0(A ⊠ A|C(∆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then, if U ⊆ C is any affine open subset such that η = µ−1du for µ, u ∈ Γ(U, OX, we can write (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10) ρ|U×U = (1 ⊗ µ)χ u1 − u2 + s for some s ∈ Γ(U × U, A ⊠ A), where χ ∈ Γ(U × U, A ⊠ A) is any preimage of idA|U under the surjective map Γ(U × U, A ⊠ A) −→ Γ(U, A ⊗ A) → EndOU (A|U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' One should think of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10) as an analog of the standard form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) of (n, λ)-type series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By repeating the arguments in [BG18, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3], we can see that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11) ρ13ρ12 − ρ12ρ23 + ρ13ρ23 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 27 Here, the summands on the left-hand side can be understood as a rational section of A⊠ A⊠ A by adapting the notations from Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 to this geometric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, ρ is a solution of a geometric version of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrization of Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that k is an algebraically closed field of char- acteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let (A, β) be a strongly geometrically admissible k-algebra, n ∈ N, λ ∈ k[[z]]×, and ((Dn(A), β(n,λ)), A[[z]], W) be the Manin triple associated to that data in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 that n ∈ {0, 1, 2} and that we may assume λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The goal of this section is to assign a geometric A-CYBE datum to ((Dn(A), β(n,1)), A[[z]], W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrization in case n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There exists a particular O ⊆ M := {f ∈ k((z)) | fW ⊆ W} such that dim(k((z))/(k[[z]] + O)) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, the integral closure N of M either satisfies dim(k((z))/(k[[z]] + N)) = 1 or N = k[u] for some u ∈ z−1 + zk[[z]]×, and then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12) O := � N if dim(k((z))/(k[[z]] + N)) = 1, k[u′, u′u] if N = k[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Applying the geometrization procedure form Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 to (O, W) gives a geometric datum ((X, A), (p, c, ζ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that A satisfies ζ : � Ap ∼ = −→ A[[z]] and H0(A) = 0 = H1(A) (combine A((z)) = A[[z]] ⊕ W with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If X is smooth it is a smooth irreducible cubic plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If O = k[u′, u′u] for u ̸= z−1, we have −u′ = u2 − a for some a ∈ k \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This equation is equivalent to u′,2(a − u′) = u′,2u2, so putting y = u′u and x = u′, we see that X is a nodal irreducible cubic plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Finally, if O = k[z−2, z−3], X is clearly a cuspidal irreducible cubic plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In order to see that (X, A) gives rise to a geometric A-CYBE datum, we have to construct an appropriate OX-bilinear map βA : A × A → OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If X is smooth (which is equivalent to the fact that O is integrally closed), the geometrically admissible metric β defines a pairing βA : A × A → OX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us assume that X is singular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' O = k[u′, u′u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since β is geometrically admissible, it induces a pairing W × W → N = k[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since W is Lagrangian, the image under this pairing lies in the kernel of res0 restricted to k[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that this kernel is equal to O, so the coefficient-wise application of β defines a map W × W → O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is now straight forward to see that for every U ⊆ X, the commutative diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13) Γ(U, A) × Γ(U, A) � ζ×ζ � Γ(U, OX) c � A((z)) × A((z)) β � k((z)) defines a pairing βA : A × A → OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The triple (X, (A, βA)) is a geometric A-CYBE datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, A|q ∼= A for all smooth closed q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since by construction the fiber of βA is β at p ∈ X, the kernel of the canonical morphism A → A∗ is a torsion subsheaf of the torsion free sheaf A, hence zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, βA is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11, βA is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It remains to prove that A|q ∼= A for all smooth closed q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that A|p ∼= A already holds, so we may assume q ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let m ⊆ O be the maximal ideal associated to q ∈ X \\ {p} ∼= Spec(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then βA,q is identified with the restriction Wm ×Wm → Om of β from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) through Aq ∼= Wm and OX,q ∼= Om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since A is strongly geometrically admissible (recall the definition from Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) and βA,q is perfect, we obtain A|q ∼= W/mW ∼= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 28 RASCHID ABEDIN Let us note the following important consequence of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The following results are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1) If X is elliptic, A is non-unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) Let X be singular and ρ be the section constructed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 from (X, (A, βA)) and η := dv ∈ H0(ωX), where v := u/u′ is the local generator of p associated to the representation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then image of ρ under the Taylor expansion H0(A ⊠ A|C(∆)) −→ lim ←− k � Γ(X \\ {p}, A) ⊗ � Ap/mp � Ak p � ζ⊗ζ −→ (A((z)) ⊗ A)[[z]] at X × {p} trivialized by (c, ζ) is equivalent to the solution r of the A-CYBE associated to the Manin triple ((A((z)), β(0,0)), A[[z]], W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume X is elliptic and A is unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' According to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4, A is A-´etale locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5, this contradicts h0(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, A is non-unital if X is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The proof is a straight forward repetition of the the proof of [Abe21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17] (see also [Abe22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrization in case n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let W+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' W−) be the projection of W ⊆ D1(A) = A((z)) × A onto the A((z)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (4), k[z−1]W+ ⊆ W+ and we can consider the geometrization ((Y, W), (p, c, ζ)) of (k[z], W+), where Y = P1 and s− := p = (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let s+ ∈ P1 be the point corresponding to the ideal (z−1) ⊆ k[z−1] via c(Γ(P1 \\ {s−}, OX)) = k[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let the sheaf of algebras V on P1 be defined as the pull-back (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14) V � � W− � W � W|s− ∼= A where A, W−, and W|s− are understood as skyscraper sheaves at s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, V fits into the short exact sequence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) 0 −→ V −→ W ⊕ W− −→ A −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since the morphism W− → A is injective, the morphism V → W is too and we can identify V with a subsheaf of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let βW : W × W → OP1 be the pairing induced by β in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) and βV : V × V → OP1 be the restriction to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The following is true: (1) H0(V) ∼= ι(g[[x]]) ∩ (W+ × W−), H1(V) = 0 and V|P1\\{s−} = W|P1\\{s−};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) There exist canonical surjective morphisms θ± : V|s± → W±/W ⊥ ± such that βV|s±(a, b) = β± (1,1)(θ±(a), θ±(b)) holds for all a, b ∈ V|s±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The restriction of the short exact sequence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) to P1 \\ {s−} results in V|P1\\{s−} = W|P1\\{s−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since the first cohomology group of torsion sheaves vanishes and A[[z]] + W+ = A((z)) implies H1(W) = 0 because of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5), the long exact sequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) in cohomology reads (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16) 0 −→ H0(V) −→ H0(W) ⊕ W− −→ A −→ H1(V) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, H0(W) ∼= A[[z]] ∩ W+ implies H0(V) ∼= A[[z]] ∩ (W+ × W−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 29 The image W + of A[[z]] ∩ W+ → A under the canonical map A[[z]] → A satisfies W + + W− = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, H0(W) ⊕ W− → A in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, H1(V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The algebra W+ is a torsion-free as k[z−1]-module, so it is a free k[z−1]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since β is geometrically admissible, this implies β(W+, W+) ⊆ k[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17) β+ (1,1)(z−1a, b) = res0z−2β(a, b) = 0 for all a, b ∈ W+, so z−1W+ ⊆ W ⊥ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we have a surjective morphism θ+ : V|s+ ∼= W+/z−1W+ −→ W+/W ⊥ + intertwining the corresponding forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' On the other hand, the construction of V as pull-back gives a canonical map V → W− which is surjective since W → W|s− is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This morphism factors through a surjective morphism V|s− → W− which respects the forms and this map induces θ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Let X be an irreducible cubic plane curve with nodal singularity s and chose the normalization ν : P1 → X in such a way that ν−1(s) = {s+, s−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us understand W±/W ⊥ ± as skyscraper sheaf at s± and let θ be the direct image under ν of the morphism (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='18) V −→ V|s+ × V|s− (θ+,θ−) −→ W+/W ⊥ + × W−/W ⊥ − for θ± from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be defined as pull-back (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='19) A � � W/(W ⊥ + × W ⊥ − ) � ν∗V θ � ν∗(W+/W ⊥ + × W−/W ⊥ − ) where W/(W ⊥ + × W ⊥ − ) is viewed as skyscraper sheaves at s ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Again, this is equivalent to the short exact sequence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='20) 0 −→ A −→ ν∗V ⊕ (W/(W ⊥ + × W ⊥ − )) −→ W+/W ⊥ + × W−/W ⊥ − −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, A can be identified with a subsheaf of ν∗V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let βA : A × A → ν∗OP1 be the restriction of ν∗βV to A, where we recall that βV : V × V → OP1 is obtained by restriction from βW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The datum (X, (A, βA)) is a geometric A-CYBE datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, A|q ∼= A for all smooth closed q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The long exact sequence in cohomology of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='20) is given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='21) 0 −→ H0(A) −→ H0(V) ⊕ (W/(W ⊥ + × W ⊥ − )) −→ W+/W ⊥ + × W−/W ⊥ − −→ H1(A) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, we used that the first cohomology group of torsion sheaves vanishes and H1(V) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The canonical map H0(V) → W+/W ⊥ + × W−/W ⊥ − thereby coincides with the inclusion A[[z]] ∩ (W+ × W−) −→ W+/W ⊥ + × W−/W ⊥ − under the identification H0(V) ∼= A[[z]] ∩ (W+ × W−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) implies that the middle arrow in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='21) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, H0(A) = 0 = H1(A), so property (1) of in the definition of a geometric A-CYBE datum in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Next, we want to see that βA actually takes values in OX ⊆ ν∗OP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let a, b ∈ A|s and a±, b± ∈ W± be representatives of the images of a, b under the canonical maps A|s → V|s± → W±/W ⊥ ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='22) βA|s(a, b) = (β+ (1,1)(a+, b+), β− (1,1)(a−, b−)) ∈ k × k ∼= ν∗OP1|s 30 RASCHID ABEDIN holds, since the θ± : V|s± → W±/W ⊥ ± intertwine the forms β± (1,1) and βV|s±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The definition of A implies that (a+, a−), (b+, b−) ∈ W, and the Lagrangian property of W gives (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='23) 0 = β(1,1)((a+, a−), (b+, b−)) = β+ (1,1)(a+, b+) − β− (1,1)(a−, b−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We obtain βA|s(a, b) ∈ {(λ, λ) | λ ∈ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that βA takes values in OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since βW|s− can be identified with the algebra metric β on W|s− ∼= A, βW, and consequently βA, is non-vanishing on an open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6 this implies that there exists a closed point q ∈ P1 \\ {s+, s−} such that A ∼= W|q ∼= A|ν(q) and βA|ν(q) is a non- zero associative bilinear form on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, βA|q is automatically non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, the kernel of the canonical morphism A → A∗ induced by βA is a torsion subsheaf of the torsion free sheaf A, hence zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, βA is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11 now states that βA is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, for all closed q ∈ P1 \\ {s+, s−} the bilinear form βW,q, which can be identified with βA,ν(q) via Wq ∼= Aν(q), is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since (A, β) is strongly geometrically admissible, this implies that W+/(z − a)W+ ∼= W|q ∼= A|ν(q) is isomorphic to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, for a ∈ k× the maximal ideal (z − a) ⊆ k[z, z−1] defines the point q ∈ P1\\{s+, s−} ∼= Spec(k[z, z−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In conclusion, (X, (A, βA)) is a geometric A-CYBE datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There exists a ϕ ∈ Autk[[z]]-alg(A[[z]]), unique {tk,i ∈ A[z] | k ∈ N, i ∈ 1, n}, and N ∈ N such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='24) ϕ(W) = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} and tk,i = 0 for all k ⩾ N, i ∈ 1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, the wk,i were defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15 implies W|q ∼= A|ν(q) ∼= A for all q ∈ P1 \\ {s+, s−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with W|s− ∼= A, this implies that B := ζ(Γ(P1 \\ {s+}, W)) ⊆ A[[z]] is a free k[z] = c(Γ(P1 \\ {s+}, OP1))-algebra satisfying B/(z − λ)B ∼= A for all λ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, B ∼= A[z] by virtue of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Completing said automorphism in the (z)-adic topology yields ϕ ∈ Autk[[x]]-alg(A[[z]]) with the property ϕ(B) = A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since W is a sheaf, we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='25) ϕ(W+) = ϕ(ζ(Γ(P1 \\ {s−}, W)) ⊆ ϕ(ζ(Γ(P1 \\ {s+, s−}, W))) = ϕ(B)[z−1] = A[z, z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This, combined with the fact that W+ is a free k[z−1]-module, implies that ϕ(W+) ⊆ zN−1A[z−1] holds for a sufficiently large integer N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='26) z−NA[z−1] ⊆ ϕ(W+)⊥ ⊆ ϕ(W+) ⊆ zN−1A[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since A[[z]] ⊕ ϕ(W+) = D1(A), we can now write (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='27) ϕ(W) = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} for uniquely determined {tk,i ∈ A[[z]] | k ∈ N, i ∈ 1, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='26) now implies that tk,i ∈ A[z] and tk,i = 0 for all k ⩾ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geometrization in case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Similar to the previous case, W ⊆ D2(A) = A((z)) × A[z]/x2A[z] and we denote by W+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' W−) the projection of W to A((z)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A[z]/z2A[z]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The following facts are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1) W = W+ × W−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) W+ ∩ z2A[[z]] = {0}, so W+ ∩ A[[z]] can be identified with a subalgebra of A[z]/z2A[z];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) (W+ ∩ A[[z]]) ⊕ W− = A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For (1), observe that β(W+, W+) ⊆ k[z−1] holds since (A, β) is geometrically admissible and W+ is a free k[z−1]-algebra by virtue of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, z−2β(W+, W+) ⊆ z−2k[z−1] implies β(2,1)(a, b) = res0z−2β(a, b) = 0 for all a, b ∈ W+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, W+ ⊆ W ⊥ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Together with W ⊥ + ⊆ W+ we arrive at W+ = W ⊥ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) implies W− = W ⊥ − , so W ⊥ + × W ⊥ − ⊆ W ⊆ W+ × W− concludes the proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The identities {0} = (A[[z]] + W+)⊥ = z2A[[z]] ∩ W ⊥ + = z2A[[z]] ∩ W+ imply (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Part (3) now follows from (2) and A[[z]] ⊕ (W+ × W−) = A((z)) × A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Consider the geometrization ((Y, W), (p, c, ζ)) of (k[z−1], W+), where as in the last section we have Y = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let X be an irreducible plane cubic curve with cuspidal singularity s and chose the normalization ν : P1 → X in such a way that ν(p) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The isomorphism ζ : � Wp → A[[z]] implies that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='28) ν∗W|s ∼= ζ(� Wp)/z2ζ(� Wp) = A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This yields a surjective morphism ν∗W → A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let A be the sheaf of algebras defined as the pull-back (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='29) A � � W− � ν∗W � A[z]/z2A[z] where A[z]/z2A[z] and W− are understood as skyscraper sheaves at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Equivalently, A fits into the short exact sequence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='30) 0 −→ A −→ ν∗W ⊕ W− −→ A[x]/z2A[z] −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let βW : W×W → OP1 be the pairing induced by β in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 and let βA : A×A → ν∗OP1 be the the restriction of ν∗βW to A ⊆ ν∗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The datum (X, (A, βA)) is a geometric A-CYBE datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, A|q ∼= A for all smooth closed q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, there exists ϕ ∈ Autk((z))-alg(A((z))) such that the identity ϕ(W+) = A[z−1] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The global section of ν∗W → A[z]/z2A[z] coincides with the canonical morphism A[[z]] ∩ W+ → A[z]/z2A[z] if H0(W) is identified with A[[z]] ∩ W+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, the middle arrow in the long exact sequence in cohomology (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='31) 0 −→ H0(A) −→ H0(W) ⊕ W− −→ A[z]/z2A[z] −→ H1(A) −→ 0 of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='30) is an isomorphism by virtue of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here we used again that: H1(W) = 0 by virtue of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The first cohomology group of torsion sheaves vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, H0(A) = 0 = H1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us now show that βA : A × A → ν∗OX takes values in OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any a, b ∈ A|s we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='32) ν∗βW|s(a, b) = β(a1, b1) + [z](β(a1, b2) + β(a2, b1)) ∈ k[z]/(z2), where a1 + [z]a2 and b1 + [z]b2 ∈ A[z]/z2A[z] are the images of a and b respectively under A|s → ν∗W|s ∼= A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By definition of A, a1 +[z]a2, b1 +[z]b2 ∈ W− and β(a1, b2)+β(a2, b1) = 0 since W− ⊆ A[z]/z2A[z] is Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, βA|s(a, b) = β(a1, b1) ∈ k, implying that βA takes values in OX ⊆ ν∗OP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 32 RASCHID ABEDIN Repeating the arguments in the end of the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15, we can deduce that (X, (A, βA)) is a geometric A-CYBE datum and A|q ∼= A for all smooth closed q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='29) implies that W|q ∼= A|ν(q) ∼= A for all q ∈ P1 \\ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, W+ = ζ(Γ(P1 \\ {p}, W)) ⊆ A((z)) is a free k[z−1] = c(Γ(P1 \\ {p}, OP1))-algebra satisfying W+/(z−1 − λ)W+ ∼= A for all λ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, W+ ∼= A[z−1] by virtue of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This induces the automorphism ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ We can now copy the arguments of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16 to deduce that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There exists a ϕ ∈ Autk[[z]]-alg(A[[z]]), a set {tk,i ∈ A[z] | k ∈ N, i ∈ 1, n}, and a natural number N ∈ N such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='33) ϕ(W) = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} and tk,i = 0 for all k ⩾ N, i ∈ 1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, the wk,i were defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall the notation and statement of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1: k is an algebraically closed field of characteristic 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A, β) is a unital strongly geometrically admissible metric k-algebra and γ ∈ A ⊗ A is its canonical A-invariant element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ((Dn(A), β(n,λ)), A[[z]], W) is a Manin triple of the form 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4 for some n ∈ N and λ ∈ k[[z]]×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' r is the solution of the A-CYBE associated to the Manin triple ((Dn(A), β(n,λ)), A[[z]], W) via Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then precisely one of the following cases occurs: (1) If n = 0, the curve X from the A-CYBE datum (X, A) of ((Dn(A), βn), A[[z]], W) constructed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 is either a nodal or cuspidal irreducible cubic plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore: (a) X is nodal if and only if r is trigonometric in the sense of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (b) X is cuspidal if and only if r is rational in the sense in the sense of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) n = 1 if and only if r is quasi-trigonometric in the sense of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) n = 2 if and only if r is quasi-rational in the sense of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' First of all, since A is unital, X cannot be elliptic by virtue of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, X is either a nodal or a cuspidal irreducible plane cubic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let s ∈ X be the unique singularity in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let η and ρ be as in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) and chose isomorphisms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='34) C := X \\ {s} f −→ � Spec(k[v, v−1]) if X is nodal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Spec(k[z]) if X is cuspidal such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='35) η = � v−1dv if X is nodal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' dz if X if cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In both cases we can chose U = C in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10) in order to obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='36) ρ|C×C = (1 ⊗ µ)χ u1 − u2 + s where u = v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' u = z) and µ = v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' µ = 1) if X is nodal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' if X is cuspidal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that s is some element in H0(A|C ⊠ A|C) = H0(A|C) ⊗ H0(A|C) and χ is some preimage of idA|C under H0(A|C ⊠ A|C) −→ H0(A|C ⊗OC A|C) −→ EndOC(A|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 33 Using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9 we can see that there exists a f ♯-equivariant isomorphism (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='37) H0(A|C) φ1 −→ � L(A, σ) if X is nodal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A[z] if X is cuspidal, where in the nodal case σ ∈ Autk-alg(A) is of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, f ♯ is the map k[v, v−1] → Γ(C, OX) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' k[z] → Γ(C, OX)) defined by f if X is nodal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' cuspidal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us conclude the proof of (1) in a case by case fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Case (a): X is nodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let Aj := {a ∈ A | σ(a) = εja} for the m-th root of unity ε ∈ k from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Note that βA induces an algebra metric L(A, σ) × L(A, σ) → k[v, v−1] defined by the coefficient-wise application of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, since v = �vm and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='38) β(�vka, �vℓ) = β(a, b)�vk+ℓ ∈ k[v, v−1] holds for all a ∈ Ak, b ∈ Aℓ, we have β(Ak, Aℓ) = {0} if k + ℓ /∈ mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, β(σ(a), b) = εkβ(a, b) = εk+ℓ−ℓβ(a, b) = β(a, σ−1(b)) holds for k + ℓ ∈ mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined, we see that σ is orthogonal with respect to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since σ is orthogonal with respect to β, it is easy to see that γ = �m−1 j=0 γj ∈ �m−1 j=0 (Aj ⊗ A−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We can choose χ as the preimage of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='39) m−1 � j=0 � �v �w �j γj ∈ L(A, σ) ⊗ L(A, σ) under the isomorphism φ1 ⊗ φ1 : H0(A|C)⊗ H0(A|C) = H0(A|C ⊠ A|C) → L(A, σ)⊗ L(A, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='40) (φ1 ⊗ φ1)ρ|C×C = 1 (v/w) − 1 � �v �w �j γj + t holds for t := (φ1 × φ1)s ∈ L(A, σ) ⊗ L(A, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let exp be the completion of k[v, v−1] → k[[z]], v �→ exp(z) with respect to the ideal (v − 1) and φ2 ∈ Autk-alg(A[[z]]) be the exp-equivariant isomorphism obtained by completing the map L(A, σ) → A[[z]], f �→ f(exp(z/m)) at the same ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2), we can see that the automorphism φ := φ2φ1ζ−1 ∈ Autk-alg(A[[z]]) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='41) (φ ⊗ φ)r(x, y) = 1 exp (x − y) − 1 m−1 � j=0 exp �x − y m � γj + s � exp � x m � , exp � y m �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This concludes the proof in the nodal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Case (b): X is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We can chose χ ∈ H0(A|C)⊗H0(A|C) as the preimage of γ ∈ (A⊗A)[x, y] under the isomorphism φ1 ⊗ φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='42) (φ1 ⊗ φ1)ρ|C×C = γ x − y + t holds for t := (φ1 ⊗ φ1)s ∈ (A ⊗ A)[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let φ2 ∈ Autk[[z]]-alg(A[[z]]) be the completion of A[z] → A[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2), we can see that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='43) (φ ⊗ φ)r = γ x − y + t holds for φ := φ2φ1ζ−1 ∈ Autk[[z]]-alg(A[[z]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This concludes the proof in the cuspidal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 34 RASCHID ABEDIN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='19 there exist {tk,i ∈ A[z] | k ∈ N, i ∈ 1, n} and N ∈ N such that, up to isomorphism of Manin triples, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='44) W = Spank{wk,i + tk,i | k ∈ N, i ∈ 1, n} and tk,i = 0 for all k ⩾ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, the wk,i ∈ Dn(A) are defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The solution r of the A-CYBE of W can now be determined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='45) r(x, y) = ∞ � k=0 d � i=1 (wk,i + tk,i) ⊗ biyk = ynγ x − y + t(x, y), where t = �N k=0 �d i=1 tk,i(x) ⊗ biyk ∈ (A ⊗ A)[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Classification of associative D-bialgebra structures over series 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Non-triangular topological associative D-bialgebras on series are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The final goal of this paper is the classification of all non-triangular topological associative D- bialgebra structures on A[[z]] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' topological D-bialgebra structures in the category of associative algebras) for any finite-dimensional simple associative algebra A over an algebraically closed field k of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that these are exactly the co-opposites of (non-triangular) topological balanced infinitesimal D-bialgebra structures on A[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, the classification of the latter is equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In order to use Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1, we begin by proving that, as in the case of a simple Lie algebra over k, these are all non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be algebraically closed of characteristic 0 and (A, β) be a finite-dimensional, simple, associative, metric k-algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A ∼= Mn(k) is the space of n × n-matrices with entries in k and β is a scalar multiple of the algebra metric defined by the trace of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Any non-triangular topological associative D-bialgebra structure δ: A[[z]] → (A ⊗ A)[[x, y]] is non-degenerate in the sense of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us begin by proving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be algebraically closed of characteristic 0 and A be a finite dimensional asso- ciative k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Every associative algebra B containing A as subalgebra is isomorphic to A ⊗ R for some unital associative k-algebra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, if B is equipped with an algebra metric �β, then for all a, b ∈ A and r, s ∈ R (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) �β(a ⊗ f, b ⊗ g) = β(a, b)t(rs) for some t: R → k such that the associated pairing (r, s) �→ t(rs) is an algebra metric of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The algebra B splits into a direct sum of irreducible A-bimodules: B = � i∈I AriA, where I := {r ∈ B | ArA is irreducible}/ ∼ for r ∼ s if ArA = AsA and i �→ ri is some choice function I → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The modules AriA are all isomorphic to A itself, so B ∼= A ⊗ R as A-bimodule, for the vector space R over k with basis {ri}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us write the original copy of A in B as A ⊗ 1 for some distinguished element 1 ∈ R and note that (a ⊗ 1)(b ⊗ r) = ab ⊗ r for all a, b ∈ A and r ∈ R by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider (1 ⊗ ri)(1 ⊗ rj) = � k∈I ak ⊗ rk, where only finitely many ak are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now [a ⊗ 1, 1 ⊗ ri] = 0 = [a ⊗ 1, 1 ⊗ rj] implies [a, ak] = 0 for all a ∈ A, k ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, ak ∈ k1 ⊆ A for all k ∈ I, so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) (1 ⊗ ri)(1 ⊗ rj) = 1 ⊗ � k∈I Ck ijrk CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 35 for some {Ck ij}k∈I ⊆ k which are almost all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, R is a k-algebra with multiplication determined by (1 ⊗ r)(1 ⊗ s) = 1 ⊗ rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then 1 ∈ R is a unit and since B is associative, R is too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For the second part of the statement, note that �β(a ⊗ 1, b ⊗ 1) = λβ(a, b) for some λ ∈ k×, so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) t(r) := 1 nλ �β(1 ⊗ 1, 1 ⊗ r) is the desired map t: R → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let R be an alternative algebra over a field of characteristic larger then 3 equipped with a linear map t: R → k such that (r, s) �→ t(rs) is an algebra metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume there exists a reduced, commutative, associative subalgebra S ⊆ R satisfying S⊥ ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The algebra R is commutative and associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let p, q ∈ S and r, s ∈ R be arbitrary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The identities (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) t(p(qr)) = t((pq)r) = t((qp)r) = t(r(qp)) = t((rq)p) = t(p(rq)) show that t(p[q, r]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As a consequence we see that [S, R] ⊆ S⊥ ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, since R is alternative, the subalgebra k[q, r] ⊆ R is associative and we see that 0 = [q, [q, r2]] = [q, [q, r]r + r[q, r]] = [q, [q, r]]r + [q, r]2 + [q, r]2 + r[q, [q, r]] = 2[q, r]2, where we used that [q, r], [q, r2] ∈ S implies [q, [q, r2]] = 0 = [q, [q, r]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since R is reduced, we deduce that [S, R] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5) t(q[r, s]) = t([qr, s]) = 0 so [R, R] ⊆ S⊥ ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Combined with [R, S] = 0, this implies [[r, s], s] = 0 = [[r, sr], s], so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6) [r, s]2 = [[r, s]r, s] = [[r, sr], s] = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since R is reduced, we deduce that [R, R] = 0 and the fact that any unital commutative associative algebra over a field of characteristic larger 3 is associative concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ We can now proof Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2, we have D(A[[z]], δ) ∼= A ⊗ R for some unital associative k-algebra R and ev(a ⊗ r, b ⊗ s) = β(a, b)t(rs) for some t: R → k which defines an algebra metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since A[[z]] ⊆ D(A[[z]], δ) is a Lagrangian subalgebra, k[[z]] ⊆ R is a Lagrangian subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 implies that R is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is now easy to see that (R, t) is a trace extension of k[[z]] in the sense of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Categorization of topological associative D-algebra structures on series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that any finite-dimensional simple associative k- algebra is isomorphic to the algebra A = Mn(C) of n×n-matrices with entries in k and the bilinear form β : A × A → A defined by the trace (a, b) �→ tr(ab) is strongly geometrically admissible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' see Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 states that we have four different types of non-triangular associative topological D-bialgebra structures on A[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, those associated to solutions of the A-CYBE which are either trigonometric, rational, quasi-trigonometric, or quasi-rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3, we will show that there are no trigonometric nor quasi-trigonometric solutions of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' So we are left with two different types of non-triangular associative topological D-bialgebra structures on A[[z]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Namely, those associated to solutions of the A-CYBE which are either rational or quasi-rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In the remainder of this section, we will establish the structure theory of (quasi-)rational solu- tions of the A-CYBE by combining the methods from [Agu01] with the approach of [Sto91] to the structure theory of (quasi-)rational solutions of the sln(C)-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 36 RASCHID ABEDIN 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Absence of (quasi-)trigonometric solutions of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 0, A = Mn(k) be the k-algebra of n × n-matrices, and β be the trace pairing of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this section, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' There are no quasi-trigonometric nor trigonometric solutions of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We will thereby proceed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' First, we show that (quasi-)trigonometric solutions of the A-CYBE define certain subalgebras of A[v, v−1] × A[v, v−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then, using the classification of trigonometric solutions of the sln(k)-CYBE from [BD83] in the formulation of [AM21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' AB21], we prove that these subalgebras cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (Quasi-)trigonometric solutions of the A-CYBE and subalgebras of L := A[v, v−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider L := A[v, v−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us prove that any (quasi-)trigonometric solution r of the A-CYBE defines a subspace Wr ⊆ L × L such that: (1) Wr is a subalgebra complementary to the diagonal D := {(a, a) | a ∈ L}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' L × L = D ⊕ Wr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2) Wr is Lagrangian with respect to the algebra metric �β on L × L defined by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7) �β((a1, a2), (b1, b2)) := res0 1 v (β(a1(v), b1(v)) − β(a2(v), b2(v)) for a1, a2, b1, b2 ∈ L, where β(a(v), b(v)) is the coefficient-wise trace of a(v)b(v) ∈ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (3) Wr is commensurable with V := A[z]×A[z−1] in the sense that dim((Wr +V )/(Wr ∩V )) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Construction of Wr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since all automorphisms of A are inner, L(A, σ) ∼= A[v, v−1] for all σ ∈ Autk-alg(A) of finite order (see [Pia05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, both trigonometric and quasi-trigonometric solutions of the A-CYBE are described by expressions of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8) r(v, w) = wγ v − w + t(v, w) for some t ∈ (A ⊗ A)[v, v−1, w, w−1] such that r(exp(x), exp(y)) is a solution of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We construct a subspace Wr to r in a similar fashion as subalgebras were associated to solutions of the A-CYBE in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Note that the natural embedding L ⊗ L → (L ⊗ A)((w±1)) extends to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9) (L ⊗ L)[(v − w)−1] −→ (L ⊗ A)((w±1)) by interpreting (v − w)−1 as (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10) � k∈N v−k−1wk ∈ k[v, v−1]((w)) and − � k∈N vkw−k−1 ∈ k[v, v−1]((w−1)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' These embeddings can be understood as the Laurent expansions in w = 0 and w = ∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us consider an r of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8), chose an orthonormal basis {bi}d i=1 ⊆ A with respect to the trace pairing β, and let (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='11) � k∈N d � i=1 r+ k,i(v) ⊗ biwk ∈ (L ⊗ A)((w−1)) and � k∈N d � i=1 r− k,i(v) ⊗ biwk ∈ (L ⊗ A)((w)) be the Laurent expansions of r in w = ∞ and w = 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If t = � k∈Z �d i=1 tk,i(v) ⊗ wkbi, where only finitely many tk,i are non-zero, and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12) w− k,i := � biv−k k > 0 0 k ⩽ 0 and w+ k,i := � 0 k > 0 −biv−k k ⩽ 0 we have r± k,i = w± k,i + tk,i for k ∈ Z and i ∈ 1, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us define (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='13) Wr := Spank{(r+ k,i, r− k,i) | k ∈ Z, i ∈ 1, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Clearly, Wr is commensurable with V , so we have to verify that conditions (a) and (b) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 37 Wr satisfies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that L × L = D ⊕ Wr, so we have to show that Wr ⊆ L × L is a subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Similar to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2, we can define for every s ∈ (L ⊗ L)[(v − w)−1] = (A ⊗ A)[v, v−1, w, w−1, (v − w)−1] the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='14) CYB(s) = s13s12 − s12s13 + s23s12 ∈ (L ⊗ L ⊗ L) � 1 (v1 − v2)(v1 − v3)(v2 − v3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' by using the notations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='12) coefficient-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then s satisfies the CYB(s) = 0 if and only if s(exp(x), exp(y)) satisfies the usual A-CYBE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, we can see that CYB(s) = 0 implies already that s is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Similar arguments as in the Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 show that if r is a skew-symmetric, we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='15) CYB(r) ∈ L ⊗ L ⊗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we can rewrite this expression using the Laurent expansions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9) in v3 = ∞ and v3 = 0 to obtain CYB±(r) = � k,ℓ∈Z d � i,j=1 r± ℓ,jr± k,i ⊗ bizk 2 ⊗ bjzℓ 3 − � m∈N d � i=1 r± k,i ⊗ � zk 2b(1) i r(z2, z3) − r(z2, z3)b(2) i zk 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16) If CYB(r) = 0, then CYB+(r) = 0 = CYB−(r) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='16) implies that Wr ⊆ L × L is a subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ Wr satisfies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The fact that r is skew-symmetric is equivalent to t = t − γ, which means (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17) tℓ,j k,i = −tk,i ℓ,j − δijδk0δℓ0 if t = � k,ℓ∈Z tℓ,j k,ibjvℓ ⊗ biwk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, the identities �β((w+ k,i, w− k,i), (w+ ℓ,j, w− ℓ,j)) = δijδk0δℓ0 and �β((w+ k,i, w− k,i), (bjvℓ, bjvℓ)) = −δijδkℓ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='18) are easily verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that, if t is identified with its image under L ⊗ L ∼= D ⊗ D, �β((r+ k,i, r− k,i), (r+ ℓ,j, r− ℓ,j)) = �β((w+ k,i, w− k,i), (w+ ℓ,j, w− ℓ,j)) � �� � =δijδk0δℓ0 + �β((tk,i, tk,i), (tℓ,j, tℓ,j)) � �� � =0 + �β((w+ k,i, w− k,i), (tℓ,j, tℓ,j)) � �� � tk,i ℓ,j + �β((w+ k,i, w− k,i), (tℓ,j, tℓ,j)) � �� � tℓ,j k,i = tℓ,j k,i + tk,i ℓ,j + δijδkℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='19) We conclude that Wr ⊆ W ⊥ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This, L × L = D ⊕ Wr, and D⊥ = D imply Wr = W ⊥ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let π: A → g := sln(k) be the surjective Lie algebra homomorphism defined by a �→ a − tr(a) n and let its coefficient-wise extension L → L := g[v, v−1] be denoted by the same symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that for an r of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8), which defines a trigonometric (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' quasi-trigonometric) solution of the A-CYBE, (π ⊗ π)r defines a trigonometric (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' quasi-trigonometric) solution of the g-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We can assume that in the orthonormal basis {bi}d i=1 of A, we have bd = 1 √n ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Using this choice, it is straight forward to see that our construction of W := Wr is consistent with the construction of the subalgebra W := W(π⊗π)r associated to (π ⊗ π)r in [AM21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' AB21] in the sense 38 RASCHID ABEDIN that W = (π ⊗ π)W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, it is shown there, that using the classification of trigonometric solutions of the g-CYBE from [BD83], there exists g ∈ SLn(k) such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='20) Adg(W±) = N± ⊕ (W± ∩ h) ⊕ (W± ∩ N∓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, we used the following notation: g = n+ ⊕ h ⊕ n− is the triangular decomposition into the subalgebras n+, n−, h ⊆ g of upper, lower, trace-less diagonal matrices and N± := n± ⊕ z±1g[z±1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' W± := pr±(W) for the projections pr± : L × L → L defined by (a+, a−) �→ a±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since Adg leaves k[v, v−1] = Ker(π) ⊆ L point-wise fixed for every g ∈ SLn(k), this implies that the subalgebra W± := pr±(W) ⊆ L satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='21) Adg(W±) = N± ⊕ (W± ∩ H) ⊕ (W± ∩ N∓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, N± := n± ⊕ z±1A[z±1] and H ⊆ A is the subalgebra of diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since W is Lagrangian with respect to �β, we can deduce that W± ⊆ A[v, v−1] is coisotropic with respect to the bilinear form β+ (1,1) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, the subalgebras H± := H ∩ W± ⊆ H satisfy H⊥ ± ⊆ H± with respect to the trace pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, H± are both Artinian k-algebras without nilpotents, hence H± ∼= kℓ± for some ℓ± ⩽ n as algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that any embedding k → H is of the form a �→ ahi1 + · · · + ahik, where hi is the diagonal matrix with 1 at the i-th row and column as only non-zero entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, H = H± ⊕ H⊥ ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This combined with H⊥ ± ⊆ H± implies H± = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now, let us note that we can deduce W+/W ⊥ + × W−/W ⊥ − = {(a, a) | a ∈ L} ⊕ W/(W ⊥ + × W ⊥ − ) by following the same arguments as in the proof of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that W/(W ⊥ + × W ⊥ − ) = {(a, θ(a)) | a ∈ W+/W ⊥ + } holds for some k-algebra isomorphism θ: W+/W ⊥ + → W−/W ⊥ − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consequently, W = {(a, b) ∈ W+ × W− | θ(a) = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, since 1 ∈ H ⊆ W±/W ⊥ ± and θ is unital since it is an isomorphism, we have (1, 1) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' But this contradicts W ∩ D = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In conclusion, W cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Structure theory of rational D-bialgebra structures over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' As in the previous sub- section, let k be an algebraically closed field of characteristic 0, A = Mn(k) be the k-algebra of n × n-matrices, and β be the trace pairing of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The assignment r �→ A(r) defines a bijection between rational solutions of the A-CYBE and Lagrangian subalgebras W ⊆ A((z)) satisfying A[[z]] ⊕ W = A((z)) and z−NA[z−1] ⊆ A(r) ⊆ zNA[z−1] for some sufficiently large N ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, we can apply the associative analog of the maximal order theory developed in [Sto91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Sto95] to study these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' More precisely, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let W ⊆ A((z)) be a subalgebra satisfying z−NA[z−1] ⊆ W ⊆ zNA[z−1] for some N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then there exists g ∈ SL(n, k((z))) such that Ad(g)W ⊆ A[z−1] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' First of all, W ⊆ A[z, z−1] and it suffices to prove that z−NA[z−1] ⊆ W ⊆ zNA[z−1] for some N ∈ N implies the existence of g ∈ SL(n, k[z, z−1]) such that Ad(g)W ⊆ A[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Without loss of generality, we may assume that k[z−1] ⊆ W, since we can pass to the algebra k[z−1]W + k[z−1] which contains W and satisfies z−NA[z−1] ⊆ k[z−1]W + k[z−1] ⊆ zNA[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now, W = π(W) ⊕ k[z−1] as vector spaces, where π: A[z, z−1] → g[z, z−1] is the coefficient-wise application of a �→ a − tr(a) n ∈ g := sln(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The subalgebra π(W) ⊆ g[z, z−1] satisfies z−Ng[z−1] ⊆ π(W) ⊆ zNg[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of [Sto95, Theorem 4’] and the description of maximal orders for g = sln(k) from [Sto91], there exists g ∈ CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 39 SL(n, k[z, z−1]) such that Ad(g)π(W) ⊆ g[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since Ad(g)k[z−1] = k[z−1], this implies that Ad(g)W ⊆ A[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ■ By virtue of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5, for every rational solution r of the A-CYBE exists g ∈ SL(n, k((z))) such that Ad(g)A(r) ⊆ A[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By virtue of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Sto91, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 Sauvage Lemma], there exists d = diag(zd1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zdn) such that g = g−dg+ for g+ ∈ SL(n, k[[z]]) and g− ∈ SL(n, k[z−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, up to equivalence, A(r) ⊆ d−1A[z−1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The fact that A[[z]] + A(r) = A((z)) holds implies that 0 ⩽ di ⩽ 1 for all i ∈ 1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Thus, after reordering the indices, we can assume that d = dk := (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , 1, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=', z), where z appears k-times on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call r rational solution of type k, if A(r) ⊆ Nk := d−1 k A[z−1]dk, where we remark that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='22) Nk := �� A B C D � ∈ L = Mn(k[z, z−1]) ����� A∈Mn−k(k[z−1]), B∈zM(n−k)×k(k[z−1]) C∈z−1Mk×(n−k)(k[z−1]), D∈Mk(k[z−1]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We will now show that these solutions are parametrized by associative versions of Stolin pairs, which parameterize rational solutions of the g-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' To this end, let (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='23) Pk := �� A B 0 D � ∈ A = Mn(k) ����� A ∈ Mn−k(k), B ∈ M(n−k)×k(k), and D ∈ Mk(k) � Then (S, χ) is called associative Stolin pair of type k if S ⊆ A is a subalgebra and χ: S × S → k is a bilinear form such that S + Pk = A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' χ is a Connes 2-cocycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' χ is skew-symmetric and χ(a1a2, a3) + χ(a2a3, a1) + χ(a3a1, a2) = 0 holds for all a1, a2, a3 ∈ S, and χ restricts to a non-degenerate bilinear form on S ∩ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By adjusting the arguments in [Sto91], we will prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Rational solutions of the A-CYBE of type k are in bijection with associative Stolin pairs of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let us note that Stolin pairs of type 0 are simply subalgebras S ⊆ A which admit a non-degenerate Connes 2-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1) states that these are in bijection with rational solutions r of the A-CYBE satisfying A(r) ⊆ A[z−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It is easy to see that r(x, y) = γ x−y + t for a constant tensor t ∈ A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then r is a solution of the A-CYBE if and only if t is a skew-symmetric solution of the A-CYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The fact that these are in bijection with Stolin pairs of type 0 is actually exactly [Agu01, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It suffices to prove that there is a bijection between Lagrangian subalgebras W ⊆ Nk such that A((z)) = A[[z]] ⊕ W and Stolin pairs of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that the image of A[[z]] ∩ Nk in Dǫ := Nk/z−2Nk ∼= A[ǫ]/ǫ2A[ǫ] = A ⊕ ǫA is exactly Pk ⊕ ǫP ⊥ k and Dǫ inherits the algebra metric (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='24) βǫ(a1 + ǫa2, b1 + ǫb2) := β(a1, b2) + β(a2, b1) from A((z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since z−2Nk = N ⊥ k ⊆ W ⊥ = W ⊆ Nk holds, we can see that W �→ W/z−2Nk defines a bijection between Lagrangian subalgebras W ⊆ Nk such that A((z)) = A[[z]]⊕W and Lagrangian subalgebras V ⊆ Dǫ such that Dǫ = (Pk ⊕ ǫP ⊥ k ) ⊕ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, it suffices to establish a bijection between the latter Lagrangian subalgebras and associative Stolin pairs of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let V ⊆ Dǫ be any Lagrangian subspace and S be the image of V under ǫ �→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='25) ǫS⊥ = (S ⊕ ǫA)⊥ ⊆ V ⊥ = V ⊆ S ⊕ ǫA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 40 RASCHID ABEDIN A dimension argument implies that the mapping ǫ �→ 0 defines an isomorphism V/ǫS⊥ → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, there exists a linear map f : S �→ A/S⊥ such that V/ǫS⊥ = {a + ǫf(a) | a ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Consider the bilinear form on S defined by χ(a, b) := β(f(a), b) for a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that, since β pairs S and A/S⊥ non-degenerately, f is completely determined by χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, χ is skew- symmetric since (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='26) 0 = βǫ(a + ǫf(a), b + ǫf(b)) = χ(a, b) + χ(b, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Note that V can be uniquely reconstructed from S and f and hence from the pair (S, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This es- tablishes a bijection between Lagrangian subspaces V ⊆ Dǫ and pairs (S, χ) consisting of subspaces S ⊆ A equipped with a skew-symmetric bilinear form χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' It remains to prove that V ⊆ Dǫ is a subalgebra satisfying Dǫ = (Pk ⊕ ǫPk) ⊕ V if and only if (S, χ) is a Stolin pair of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that V ⊆ Dǫ is a subalgebra if and only if for all a, b ∈ S (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='27) (a + ǫf(a))(b + ǫf(b)) = ab + ǫ(f(a)b + af(b)) ∈ V/ǫS⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' and this is equivalent to f(ab) = f(a)b + af(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now let us note that χ(a1a2, a3) + χ(a2a3, a1) + χ(a3a1, a2) = χ(a1a2, a3) − χ(a1, a2a3) − χ(a2, a3a1) = β(f(a1a2), a3) − β(f(a1), a2a3) − β(f(a2), a3a1) = β(f(a1a2), a3) − β(f(a1)a2, a3) − β(a1f(a2), a3) = β(f(a1a2) − f(a1)a2 − a1f(a2), a3), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='28) where we used the skew-symmetry of χ and the associativity of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since S and A/S⊥ are non- degenerately paired by β, this identity shows that f(ab) = f(a)b+af(b) for all a, b ∈ S is equivalent to the fact that χ is a Connes 2-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' To conclude the proof, we have to show that Dǫ = (Pk ⊕ ǫPk)⊕ V is equivalent to the facts that S + Pk = A holds and χ is non-degenerate on S ∩ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume first that Dǫ = (Pk ⊕ ǫPk) ⊕ V and observe that S + Pk = A immediately follows from Dǫ = (Pk ⊕ ǫP ⊥ k ) + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume that a ∈ S ∩ Pk satisfies χ(a, b) = 0 for all b ∈ S ∩ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In other words, af ∈ (S ∩ Pk)⊥ = S⊥ + P ⊥ k for any representative af of f(a), so af = a1 − a2 for a1 ∈ P ⊥ k and a2 ∈ S⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then a + ǫ(af + a2) ∈ V ∩ (Pk ⊕ ǫP ⊥ k ) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This proves that χ is non-degenerate on S ∩ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Conversely, assume that S + Pk = A and χ is non-degenerate on S ∩ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let a + ε(af + a⊥) = p + ǫp⊥ ∈ (Pk ⊕ ǫPk) ∩ V, for a ∈ S, a⊥ ∈ S⊥, p ∈ Pk, p⊥ ∈ P ⊥ k , and a representative af ∈ A of f(a) ∈ A/S⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then a = p ∈ S∩Pk and af = p⊥−a⊥ ∈ S⊥+P ⊥ k = (S∩Pk)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' This implies that χ(a, b) = βǫ(f(a), b) = 0 for all b ∈ S ∩ Pk, so a = 0 since χ is non-degenerate on S ∩ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Therefore, af + a⊥ = p⊥ ∈ S⊥ ∩ P ⊥ k = (S + Pk)⊥ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Summarized, (Pk ⊕ ǫPk) ∩ V = {0} and by dimension reasoning we see that Dǫ = (Pk ⊕ ǫP ⊥ k ) ⊕ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Structure theory of quasi-rational D-bialgebra structures over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall that k is an algebraically closed field of characteristic 0, A = Mn(k) is the k-algebra of n × n-matrices, and β is the trace pairing of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The assignment r �→ ((D2(A), β(2,1)), A[[z]], A(r)) defines a bijection between quasi-rational solu- tions of the A-CYBE and Manin triples ((D2(A), β(2,1)), A[[z]], W) satisfying z−NA[z−1] ⊆ W+ ⊆ zNA[z−1] for some sufficiently large N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, W+ is the projection of W ⊆ D2(A) = A((z)) × A[z]/z2A[z] onto A((z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Repeating the arguments in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4, we obtain W+ ⊆ Nk = d−1 k A[z−1]dk for some k ∈ 1, n up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Here, Nk is explicitly given in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We call a quasi-rational solution r of the A-CYBE of type k, if A(r) ⊆ Nk × A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' CLASSIFICATION OF D-BIALGEBRA STRUCTURES ON POWER SERIES ALGEBRAS 41 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Quasi-rational solutions of the A-CYBE of type k are in bijection with associative Stolin pairs of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In general, the rational and quasi-rational solution of the A-CYBE associated to the same Stolin pair have no obvious connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' However, if (S, χ) is a Stolin pair of type 0, then the associated rational solution is r(x, y) = γ x−y + t for some t ∈ A ⊗ A and the associated quasi-rational solution is �r(x, y) = xyγ x−y + t = y2γ x−y − xΩ + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, r(x−1, y−1) = �r(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Observe that z �→ z−1 is not an admissible coordinate transformation of A((z)) and D2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ♦ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let r be a quasi-rational solution of the A-CYBE of type k and ((D2(A), β(2,1)), A[[z]], W) be the associated Manin triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Recall from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (1) and its proof that W = W+ ×W− for some Lagrangian subalgebras W+ ⊆ A((z)) and W− ⊆ A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since A[z−1] and consequently Nk is Lagrangian in A((z)), where A((z)) is equipped with β+ (2,1) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='8), W+ ⊆ Nk implies Nk = N ⊥ k ⊆ W ⊥ + = W+ and thus W+ = Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Now Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (2),(3) states that W+ ∩A[[z]] can be embedded into A[z]/z2A[z] in such a way that (W+∩A[[z]])⊕W− = A[z]/z2A[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' But we have seen in the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='6 that this image of W+∩A[[z]] is precisely Pk ⊕[z]P ⊥ k and that the decompositions (Pk ⊕[z]P ⊥ k )⊕W− = A[z]/z2A[z] into Lagrangian subalgebras are in bijection with Stolin pairs of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Since all steps made are invertible, we obtain the desired bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Notations and conventions Throughout this document k denotes the base field we are working over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' From Section 4 onward it will be of characteristic 0 and from Section 6 onward it will be additionally algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' By convention the set of natural numbers N = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='} include 0 and we use the notation m, n = {m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , n} for the set of natural numbers between a number m and larger number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this text, rings are always unital, associative, and commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For a ring R and R-modules M, N, the space of R-linear maps M → N (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' M → M) is denoted by HomR(M, N) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' EndR(M)), while the tensor product of M and N is written as M ⊗R N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For R = k the indices are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The invertible elements of R are denoted by R×, and M ∗ := HomR(M, R) is the dual module of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If R is a domain, Q(R) := (R \\ {0})−1R denotes its quotient field and we write Q(M) := M ⊗R Q(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let f : R → �R be a morphism of rings and � M be an �R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We say that a map g : M → � M is f-equivariant if it is a group homomorphism satisfying g(rm) = f(r)g(m) for all r ∈ R, m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Non-associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In this text, an R-algebra A satisfies no additional assumptions if not mentioned explicitly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A = (A, µA) consists of an R-module A equipped with a multiplication map µA : A ⊗R A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' right) multiplication maps with respect to an element a ∈ A are denoted by La (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Ra), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1) La(b) = ab = Rb(a) for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The group of invertible R-algebra endomorphisms of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' invertible R-linear maps f : A → A satisfying fµA = µA(f ⊗ f), will be denoted by AutR-alg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We note that “ ⊕ ” will always denote the direct sum of modules and not of algebras, while “ × ” is used for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For any a, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , an ∈ A, we write a(i)(a1 ⊗ · · · ⊗ an) = a1 ⊗ · · · ⊗ aai ⊗ · · · ⊗ an (a1 ⊗ · · · ⊗ an)a(i) = a1 ⊗ · · · ⊗ aia ⊗ · · · ⊗ an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2) 42 RASCHID ABEDIN We say that a map β : A×A → R is an algebra metric if it is a non-degenerate symmetric R-bilinear map such that β(ab, c) = β(a, bc) for all a, b, c ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3) In this case, we call the pair (A, β) metric R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Formal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For a module M over a ring R, the module of formal power series in the formal variable z with coefficients in M is denoted by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4) M[[z]] := � m = � k∈N mkzk ����� mk ∈ M for k ∈ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Furthermore, we write M[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]] := M[[z1]] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [[zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The R-module R[[z]] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' R[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]]) is a ring extension of R and M[[z]] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' M[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]]) is an R[[z]]-module (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' R[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]]- module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Then M((z)) := M[[z]][z−1] = Q(M[[z]]) is the module of formal Laurent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' We note that if M is an R-algebra the module M[[z]] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' M((z))) is naturally an R[[z]]-algebra (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' R((z))-algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Elements p in M((z)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' M((z1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ((zk))) will sometimes be denoted with the formal variable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' variables) for convenience, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' p = p(z) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' p = p(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A generic element p ∈ M((z)) is written p(z) = � k∈Z pkzk and p′(z) = � k∈Z kpkzk−1 denotes the formal derivative of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If p(z) ∈ mz−k + z−k+1M[[z]], it is said to be of order k with main part mz−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Finally, M[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]]∨ := {f ∈ M[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]]∗ | f((z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk)mM[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]]) = {0} for some m ∈ N} is the continuous dual of M[[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' , zk]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Let X = (X, OX) be a ringed space and F, G be two OX-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' For a morphism f : X → Y = (Y, OY ) of ringed spaces, we denote the additional structure morphism by f ♭: OY → f∗OX and write f ♯ : f −1OY → OX for the induced morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The set of OX- module homomorphisms F → G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' F → F) is denoted by HomOX(F, G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' EndOX(F)) while its sheaf counterpart is denoted by HomOX(F, G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' EndOX(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, we write F∗ := HomOX(F, OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The tensor product of F and G is written as F ⊗OX G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Assume that X and Y are S-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' The fiber product of X and Y over S is denoted by X ×S Y and F|p is the fiber of F in a point p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' If S = Spec(k), the index S is omitted and Hn(F) denotes the n-th global cohomology group of F, while hn(F) denotes its dimension over k, if said space is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In particular, H0(F) = Γ(X, F) is the space of global sections of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' References [AB21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Abedin and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Burban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Communications in Mathematical Physics (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Abe21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Abedin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Geometrization of solutions of the generalized classical Yang-Baxter equation and a new proof of the Belavin-Drinfeld trichotomy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' prerpint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' arXiv: 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='05678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Abe22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Abedin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Algebraic geometry of the classical Yang-Baxter equation and its generalizations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' url: https://digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='uni-paderborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='de/hs/content/titleinfo/6660394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Agu01] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Aguiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “On the Associative Analog of Lie Bialgebras”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Journal of Algebra 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 (2001), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 492–532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Agu02] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Aguiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Infinitesimal Hopf Algebras and the cd-Index of Polytopes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 3–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Alb49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Albert.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 57–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Kir83] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Kiranagi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Semi-simple Lie algebra bundles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 935–962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [OS08] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Odesskii and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Sokolov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Pairs of Compatible Associative Algebras, Classical Yang-Baxter Equa- tion and Quiver Representations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Commun.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Vanishing of H1 for Dedekind rings and applications to loop algebras”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Comptes Rendus Mathematique 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='9 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 633–638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Pol02] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Polishchuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Classical Yang-Baxter equation and the A∞-constraint”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 56–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Pol09] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Polishchuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the Yang–Baxter Equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Manin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' by Yuri Tschinkel and Yuri Zarhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Birkh¨auser Boston, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Sch55] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Schafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Noncommutative Jordan Algebras of Characteristic 0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='3 (1955), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 472–475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [She71] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Shestakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Certain classes of noncommutative Jordan ring”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='4 (1971), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 407–448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Skr13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Skrypnyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two ap- proaches”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Journal of Mathematical Physics 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='10 (2013), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 103507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Sto91] A.' metadata={'source': 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Algebra And Logic 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='1 (1997), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Zhe98] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Zhelyabin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “Jordan bialgebras of symmetric elements and Lie bialgebras”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: Siberian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='2 (1998), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 261–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' [Zhe99] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Zhelyabin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' “On a class of Jordan D-bialgebras”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' In: St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' Petersburg Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' 11 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content=' ETH Z¨urich, Department of Mathematics, R¨amistrasse 101, 8006 Zurich, Switzerland Email address: raschid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='abedin@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} +page_content='ch' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FPT4oBgHgl3EQfLTSh/content/2301.13022v1.pdf'} diff --git a/oNAzT4oBgHgl3EQf5P6V/content/tmp_files/2301.01857v1.pdf.txt b/oNAzT4oBgHgl3EQf5P6V/content/tmp_files/2301.01857v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78437a6e5b15bfd2649fe97d4e046531f326912b --- /dev/null +++ b/oNAzT4oBgHgl3EQf5P6V/content/tmp_files/2301.01857v1.pdf.txt @@ -0,0 +1,3429 @@ +arXiv:2301.01857v1 [math.AG] 5 Jan 2023 +Geometric G-functions and Atypicality +David Urbanik +January 6, 2023 +Abstract +In a seminal research manuscript, Andr´e showed how the theory of G-functions +could be used to give height bounds on the moduli of smooth projective algebraic +varieties acquiring non-generic algebraic cycles. The method was limited by the lack +of a suitable cohomological interpretation for these G-functions at finite places. In this +paper we use recent developments in p-adic Hodge theory to remove this constraint. +With respect to Andr´e’s strategy for producing height bounds from algebraic rela- +tions on G-function values at special points, we give a general method for producing +relations that hold at all finite places, and show that producing relations at the infinite +places is the essential difficulty. This leads to new cases of the Zilber-Pink conjec- +ture, as well as new height bounds on special moduli, including the first unconditional +finiteness results for CM moduli in non-Shimura settings. +As a more technical contribution, we introduce a refinement of the Pila-Zannier +strategy capable of handling Zilber-Pink-type atypical intersection problems in arbi- +trary dimension and for arbitrary smooth projective families. +Contents +1 +Introduction +2 +1.1 +The G-function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.2 +p-adic Interpretations of G-functions . . . . . . . . . . . . . . . . . . . . . . . +8 +1.3 +Relations on Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +1.4 +Applications to Height Bounds +. . . . . . . . . . . . . . . . . . . . . . . . . . +12 +1.5 +Pila-Zannier for General Atypicality +. . . . . . . . . . . . . . . . . . . . . . . +13 +1.6 +Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1.7 +Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +2 +Cohomological Preliminaries +14 +2.1 +A Model for the Canonical Extension . . . . . . . . . . . . . . . . . . . . . . . +15 +2.2 +ˇCech cohomological recollections +. . . . . . . . . . . . . . . . . . . . . . . . . +17 +2.2.1 +ˇCech cohomology of complexes . . . . . . . . . . . . . . . . . . . . . . +17 +2.2.2 +Cup product in ˇCech cohomology . . . . . . . . . . . . . . . . . . . . . +18 +2.3 +The pro-´etale site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.4 +Coherent Cohomology on Various Sites . . . . . . . . . . . . . . . . . . . . . . +20 +2.5 +The ´etale fundamental group and cohomology . . . . . . . . . . . . . . . . . . +21 +1 + +3 +Cohomological Computations +22 +3.1 +Basic ˇCech Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +3.2 +Evaluation Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.3 +Extending to an ambient variety +. . . . . . . . . . . . . . . . . . . . . . . . . +26 +4 +Realizing G-functions +27 +5 +Algebraic Relations on Functionals +33 +6 +Height Bounds for Families over Curves +36 +6.1 +Setup +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +6.2 +Monodromy-Compatible Frames for H1 . . . . . . . . . . . . . . . . . . . . . . +37 +6.3 +Proof of 1.16 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +6.4 +Proof of 1.17 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +7 +Pila-Zannier for General Intersections +41 +7.1 +Reduction to Height Bounds on Tensors . . . . . . . . . . . . . . . . . . . . . +42 +7.2 +Constraining Heights of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +7.3 +Application in the Abelian Case +. . . . . . . . . . . . . . . . . . . . . . . . . +48 +8 +Applications +50 +8.1 +Families of Curves +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +8.2 +Families of Abelian Varieties +. . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +8.3 +Degenerations to Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +1 +Introduction +To motivate the more technical introduction that follows, we begin with some applications. +Fix some g ě 2, and suppose that Mg is the moduli stack of genus g curves with universal +family C Ñ Mg. Let Mg be its compactification which parameterizes stable curves, as +constructed by Deligne-Mumford in [DM69], and let B Ă MgzMg be the locus of stable +curves Cx “ C1 Y ¨ ¨ ¨ Y Cℓ with smooth components for which +pgpC1q ` ¨ ¨ ¨ ` pgpCℓq ď g ´ 2, +(1) +where pgpCiq denotes the genus of the curve Ci. As explained in [BF22, §1], this condition +is equivalent to δ ´ ℓ ě 1, where δ is the number of nodal singularities of C. +Our first result concerns the Jacobians of points of Mg. We recall that every abelian +variety A over a field k admits a unique isogeny decomposition A « A1 ˆk ¨ ¨ ¨ ˆk Aℓ into +simple factors, where the relation « is given by the existence of an isogeny. Write S Ă MgpCq +for the set of x P MgpCq for which the Jacobian JpCxq admits an isogeny factor with complex +multiplication. +Theorem 1.1. Let S Ă Mg,C with g ě 2 be an irreducible Hodge-generic curve whose +closure S Ă Mg intersects B. Then SpCq X S is finite. +Let us begin by explaining why results of the type described in Theorem 1.1 are difficult. +The set S is infinite, and in particular contains the complex points of infinitely many subva- +rieties of Mg which have dimension linear in g; this can be seen, for instance, by intersecting +2 + +the image of the Torelli map with Hecke translates of loci like Ag´1 ˆ tyu Ă Ag, with y a +point corresponding to a CM elliptic curve. Understanding the geometry of the subvarieties +that give rise to the points of S is a deep problem with numerous arithmetic implications, +and it is difficult to rule out the possibility that infinitely many of these subvarieties might +intersect some curve in Mg. +Because curves in Mg whose Jacobian has a global CM factor exist, some kind of Hodge- +genericity hypothesis on S like the one given in Theorem 1.1 is clearly necessary, but the +role of the hypothesis concerning the intersection of S with the boundary locus B is less +clear. This will turn out to be an artifact of our method. In fact, we will see that the proof +of Theorem 1.1 makes little reference to curves at all, but is instead the consequence of a +general theory of height bounds, in principle effective, for special moduli of one-dimensional +smooth projective families of algebraic varieties. For instance, the same argument will also +give the following: +Definition 1.2. Let V0 be a variety and x P V0 a closed point. We say that pV0, xq is an +order-r normal crossing singularity if it is ´etale locally isomorphic to the locus z1 ¨ ¨ ¨ zr “ 0 +in some neighbourhood of 0 P An for some r ď n. +Definition 1.3. Suppose V0 Ă V is an inclusion of complex analytic varieties whose germ +at some x P V0 is isomorphic to the germ of tx1 ¨ ¨ ¨ xr “ 0u Ă An at zero. Then a degree k +vanishing cycle at pV, xq is a class rγs in HkpV, Zq obtained as the product of k simple loops +around components xi1 “ 0, . . . , xik “ 0 for some subset ti1, . . . , iku Ă t1, . . . , ru. +Definition 1.4. In the setting of an analytic family f : V Ñ Y with special fibre V0 above +0 P Y , we say that two vanishing cycles in V induced by V0 are distinct if for any analytic +neighbourhood B of 0 they are the images of linearly independent cycles inside À +i HipV1, Qq, +where V1 is some fibre of f above B. +Theorem 1.5. Let f : X Ñ S be a Hodge-generic family of abelian varieties of dimension +g ě 3 over an irreducible complex algebraic curve. Suppose that at some point s0 P SpCq the +fibre Xs0 has simple normal crossing singularities which induce at least two distinct degree +one vanishing cycles. Then the set S Ă SpCq of points s P SpCq for which Xs admits an +isogeny factor with complex multiplication is finite. +Remark. A moduli interpretation of Theorem 1.5 in the style of Theorem 1.1 is also possible. +However compactified moduli spaces of abelian varieties are typically constructed using +semiabelian schemes instead of singular normal crossing varieties as degeneration points, +and although one can always construct a normal crossing compactification of a semiabelian +variety, even in families [FC90], there is no canonical way to do so. Thus to understand +which curves in Ag are covered by Theorem 1.5 one has to undertake an analysis of the +different normal crossing compactifications of the fibres of boundary strata in Ag, and as +this will lead us too far astray we do not attempt this here. +The common theme connecting Theorem 1.1 and Theorem 1.5 will turn out to be the +presence of two independent vanishing loops near a degeneration point. These are used in the +following way. Using general moduli-theoretic arguments one can reduce both Theorem 1.1 +and Theorem 1.5 to the case where we have a family f : X Ñ S defined over a number +field K Ă C, S is a smooth curve, and the degeneration point s0 lies in SpKq. We denote +by S1 Ă S the locus over which the family f is smooth, and write f 1 for the map X1 Ñ S1, +where X1 “ f ´1pSq. Replacing S with a finite cover, we may moreover assume that the +monodromy action on the local system V1 “ R1f 1an +˚ Z is unipotent near s0, and extend the +3 + +algebraic de Rham vector bundle H1 “ R1f 1 +˚Ω‚ +X1{S1 canonically to a vector bundle H over +a neighbourhood of s0. Given any section ω of H and vanishing cycle γ as above we may +then consider a period function +hpsq “ +ż +γs +ωs +for s an algebraic local parameter at s0. One checks that this function is given by a K- +algebraic power series in s. +As one ranges over a basis of sections of H and a basis of +vanishing cycles associated to s0, one obtains some finite set G“ thi : 1 ď i ď nu of period +functions. +In his book [And89] on G-functions in geometry, Andr´e observed that one can potentially +use the transcendence theory of such functions to control the heights of points in S at +which the fibres acquire extra algebraic structure. The point is that algebraic cycles induce +Q-algebraic relations between the values of the functions in G, and classical ideas from +transcendence theory can be used to constrain the number of points in S at which such +relations can occur. +In particular Andr´e proves a result he calls a “Hasse principle for +G-functions”, and shows that if one can construct certain “global” relations on the values +of K-algebraic power series which are solutions to a differential system and which have +appropriately bounded coefficients, then effective height bounds on the points at which +those relations occur can be obtained. +The problem is that the “globality” condition requires one to understand the functions +hi, when regarded as power series over K, at each place of K; in particular, one has to be +able to give cohomological interpretations of the functions hi over p-adic fields. With a few +exceptions (which we discuss in §1.6), this has proved elusive. One of our main contributions +will be to give p-adic interpretations of these period functions in full generality, for which +we use recent developments in p-adic Hodge theory due to Scholze [Sch13]. +With a p-adic understanding of period functions in place, we will then turn to techniques +for constructing relations on periods. In addition to the new insights needed to construct +relations at the finite places, we also improve on existing techniques at the infinite places. +In fact, for constructing relations at the finite places we only need one vanishing cycle; the +requirement that we have two will be necessary only at the infinite places. We then prove +the following general result. +Definition 1.6. We say that a Q-Hodge structure V of algebro-geometric origin has complex +multiplication if the algebra of endomorphisms of V generated by algebraic cycles has Q- +vector space dimension equal to dimQ V . +Remark. One can also define what it means for a general Hodge structure to have complex +multiplication in terms of abstract endomorphisms of Hodge structures, with the two defini- +tions coinciding in the geometric setting under the Hodge conjecture. As we are concerned +explicitly with algebraic cycles in this paper we adopt the geometric definition. +Theorem 1.7. Suppose that f : X Ñ S is a projective family of geometrically connected +varieties over the number field K Ă C whose generic fibre is smooth and such that the +primitive local subsystem V1 Ă Rwf 1an +˚ Q is simple, where f 1 is base-change of f to the locus +S1 Ă S above which the fibres are smooth, and S is a curve. Suppose that at some point +s0 P SpKq the fibre Xs0 has simple normal crossing singularities which induce at least two +distinct primitive degree w vanishing cycles. Write S Ă SpCq for the set of points x P SpCq +at which: +- the Hodge conjecture for endomorphisms of the fibre V1 +x holds; and +4 + +- the fibre V1 +x has a Q-Hodge summand with complex multiplication. +Then S Ă SpQq, and for any logarithmic Weil height θ : SpQq Ñ R there exists constants +κ, a P Rą0 such that +θpξq ď κ rKpξq : Ksa +for all ξ P S. +The way that one goes from a result like Theorem 1.7 to a finiteness result for S is by apply- +ing a Pila-Zannier strategy for problems of this type. We note that the usual Pila-Zannier +strategy, used for instance to prove the Andr´e-Oort conjecture [PST`21], is insufficient here. +The reason is that the approach used in Andr´e-Oort-style problems is to produce from S +a large number of Q-algebraic points in a definable period image, but in our setting the +analogous points only have lower transcendence degree than normal. Instead we introduce +a different approach which produces Q-algebraic points in a moduli space for varieties which +intersect the definable period image, and eventually use this to obtain an algebraic inter- +section to which the Ax-Schanuel Theorem applies. This approach is capable of handling +arbitrary smooth projective families and bases S of arbitrary dimension, and contains sev- +eral new ideas. We give more details on our approach in §1.5 below, and a brief comparison +with previous approaches, including the one in [DR18], in §1.6. +Note that, even without the Pila-Zannier strategy, Theorem 1.7 already implies: +Corollary 1.8. In the situation of Theorem 1.7, for any constant d ą 0, the number of +points of S lying inside a number field of degree at most d is finite. +Indeed, Theorem 1.7 gives an absolute height bound in this case, so this is just the Northcott +property. +Let us comment on the fact that Theorem 1.7 produces unconditional results even in +higher weight settings beyond the Shimura case. Indeed, the following is a formal conse- +quence of Theorem 1.7: +Theorem 1.9. Let f, K, V1, f 1, s0, Xs0 and w be as in Theorem 1.7, and write S Ă SpCq +for the set of points x P SpCq for which the Hodge structure V1 +x has complex multiplication. +Then for any logarithmic Weil height θ : SpQq Ñ R there exists constants κ, a P Rą0 such +that +θpξq ď κ rKpξq : Ksa +for all ξ P S. In particular, for each d ą 0, the number of points of S lying a number field +of degree at most d is finite. +Note that the condition that we have two distinct vanishing cycles can be verified in explicit +cases by computing the limiting mixed Hodge structure associated to the degeneration point +s0; for instance, Andr´e computes in [And89, IX, §4.4] the number of such vanishing cycles +for cohomology in middle degree in terms of a cohomological invariant of a dual graph +associated to the singularities in the special fibre. We will use this ourselves when we prove +Theorem 1.1 and Corollary 1.10 below. For instance, one can easily verify the hypotheses +of Theorem 1.9 to show the following: +Corollary 1.10. The conclusion of Theorem 1.9 holds when f 1 is a family of smooth pro- +jective hypersurfaces of degree d in Pn, with d ą n ` 1, and where the fibre Xs0 is a union +of d hyperplanes in general position. +5 + +Remark. The general position assumption is much too strong, one really only needs the +hyperplane arrangement to not be overly degenerate; see the proof in §8.3 for details. +As far as we are aware, results like this beyond the setting of Shimura-type families have not +appeared in the literature before. (Recent work in [Pap22] gives results under additional +arithmetic hypotheses and conjectures near the degeneration point s0; we refer to §1.6 for +more discussion of related work.) +We now give, in more detail, a description of the main technical achievements of this +paper. +1.1 +The G-function Method +Suppose that f : X Ñ S is a projective family of relative dimension n “ ν ´ 1 defined over +a number field K Ă Q Ă C, with X and S both smooth, S a geometrically irreducible curve, +and which has geometrically-connected fibres. Denote by f 1 : X1 Ñ S1 its base-change to +the open locus S1 Ă S above which the fibres are smooth, and fix a degeneration point +s0 P SpKqzS1pKq. For each point ξ P S1pCq we have a smooth projective complex algebraic +variety Xξ. Our goal, loosely phrased, will be as follows: +Goal: Describe the set S Ă S1pCq where the fibre Xξ carries an algebraic sub- +variety Yξ Ă Xξ which does not spread out to a family Y 1 Ă X1 lying over the +generic point of S1. +More generally, one can also formulate this goal with Xξ replaced by all its self-products +Xn +ξ “ Xξ ˆ ¨ ¨ ¨ ˆ Xξ +looooooomooooooon +n times +. +From the theory of relative Hilbert schemes one can show that S Ă SpQq, and so after +choosing a (logarithmic) Weil height θ : SpQq Ñ R one obtains a function S Ñ R which we +also denote by θ. Our goal leads to the following natural question +Question: +How can one bound the heights θpξq for ξ P S? +In his research monograph [And89], Andr´e gave a method for producing such bounds, at +least under certain assumptions on the degeneration point s0. He considers the case where +monodromy around s0 acts via a unipotent linear transformation, and for which the fibre +X0 Ă X at s0 degenerates via a reduced normal crossing; this latter condition means that: +- there is an affine open subset U Ă X and functions z1, . . . , zν on U whose differentials +trivialize Ω1 +U; and +- the equation ze1 +1 ¨ ¨ ¨ zeν +ν “ s defines the graph of f +ˇˇ +U near s0, where s is a uniformizing +function on S at s0, and ej P t0, 1u for all 1 ď j ď ν. +After reordering we obtain an integer µ such that that ej “ 0 for j ą µ and ej “ 1 otherwise. +On U one can then fix a point q in the locus z1 “ ¨ ¨ ¨ “ zµ “ 0 and consider, in an analytic +neighbourhood of s0, complex analytic functions of s given by +Ppsq “ +1 +p2πiqµ´1 +ż +γs +ι˚ +s +ˆ +hq dz2 ¨ ¨ ¨ dzµ +z2 ¨ ¨ ¨ zµ +˙ +, +(2) +where +6 + +- ιs : Xs X U ãÑ U is the inclusion of the fibre above s; +- γs a “vanishing cycle” in the fibre Us obtained as a product of µ ´ 1 simple loops near +q around the divisors zj “ 0 for j “ 2, . . . , µ; and +- hq is a holomorphic function chosen so that hq dz2¨¨¨dzµ +z2¨¨¨zµ +represents the restriction of +a relative class in the algebraic de Rham cohomology of X over S and whose power +series representation in the coordinates z1, . . . , zµ at q has coefficients in K. +As explained by Andr´e in [And89, IX,§4], the functions Ppsq are described by power +series in s with coefficients in K when expanded around s0, and give, in a punctured +neighbourhood around s0, a relative period of the Betti-algebraic de Rham comparison +Rµ´1f 1 +˚Zpµ´1qbOS1an » pRµ´1f 1 +˚Ω‚ +X1{S1qan +C when restricted to S1. In particular, this power +series representation of P is a G-function in the sense of [And89, I]. +Remark. Andr´e actually assumes that µ “ ν, and assumes that X is covered by neighbour- +hoods U of the above type. As it is not substantially more difficult, we will work in greater +generality. +To explicate the relationship between these G-functions and the projective family f +Andr´e classifies, at least in degree n, the period functions P of this form in terms of the +monodromy around the degeneration point. To explain what we mean, let us fix an in- +teger w and denote by V1 the variation of Hodge structure with underlying local system +Rwf 1 +˚Zpwq{tor. modulo torsion, and let H1 “ Rwf 1 +˚Ω‚ +X1{S1 be the associated algebraic de +Rham cohomology vector bundle. The vector bundle H1 has a so-called canonical extension +H to a vector bundle over S which we recall in §2.1. The sections of V1 that extend to sec- +tions of Han +C +under the comparison isomorphism define a local subsystem M Ă V1ˇˇ +B, where +B Ă SpCq is a small analytic ball centered at s0. Poincar´e duality defines a natural pairing +H1anˇˇ +B b M Ñ OB, +(3) +and sections in the image of this pairing we refer to as non-degenerating (relative) periods +at s0; the functions Ppsq described above were of this type. Andr´e gives a description of the +image in the case when w “ ν ´ 1, and for this w shows that when Xs0 has simple normal +crossings the image of (3) is spanned by G-functions of the form (2). +In what follows we write MmˆnpAq for the space of m ˆ n matrices with values in the +ring A; we will also write MmpAq for MmˆmpAq. Andr´e then uses the non-degenerating +periods at s0 to give a method for bounding the heights of the points in the set S, based +on the so-called Hasse principle for G-functions [And89, VII, §5], which may be stated as +follows: +Notation. Given a number field L, we write ΣL for the set of places of L. +Theorem 1.11 (Hasse Principle). Suppose that G “ pG1, . . . , Gmqt P Mmˆ1pKrrxssq sat- +isfies the differential system +d +dxG “ ΓG for some Γ P MmpKpxqq, write Gi “ ř8 +j“0 Gijxj, +and suppose that +lim sup +nÑ8 +˜ +1 +n +ÿ +vPΣK +max +iďm,jďn log` |Gij|v +¸ +ă 8 +where log`ptq “ log maxt1, tu. Let ΞpG, δq denote the set of ξ P Q satisfying the following +property: there exists a homogeneous polynomial P P Kry1, . . . , yms of degree at most δ such +that: +7 + +(1) the relation PpG1, . . . , Gmq does not hold on the level of formal power series; and +(2) for all v P ΣKpξq for which |ξ|v ă 1, either +(i) at least one of the series Gi does not have v-adic convergence radius greater than +|ξ|v; or +(ii) the relation PpG1pξq, . . . , Gmpξqq “ 0 holds v-adically at ξ. +Then there exists an exponent e P Zě0 such that +sup +ξPΞpG,δq +θpξq “ Opδeq, +where e and the implied constant depend only on G, and θ is the standard logarithmic Weil +height function. +Definition 1.12. In the context of an application of the Hasse principle, a point ξ P Q and +a place v of Kpξq, we will say that v is relevant for ξ if |ξ|v ă 1 and the series G1, . . . , Gm +all have v-adic convergence radius greater than |ξ|v; in particular, a place being relevant +means that one has to demonstrate condition (2)(ii) holds if one wants to apply the Hasse +principle. +Remark. The constants implicit in the term Opδeq can be made explicit, and even (in +principle) effective, see the footnote in [And89, pg.129] and the corresponding discussion. +By taking G to be a vector consisting of G-functions at s0 arising from a basis of H1 and +sections of M Andr´e uses this principle to bound the height of elements in certain subsets +of S; here the polynomials P are relations on periods coming from algebraic (or absolute +Hodge) cycles associated to the cohomology groups of the fibre Xξ. However in the absence +of a p-adic interpretation of these power series, he is only able to bound those points ξ +for which he can show (2.i) holds at all finite places, which greatly restricts the method. +The possibility that p-adic cohomological input might remedy the problem is discussed in +[And89, pg.8-10] and [And89, pg.194, Rem.1], but at the time of writing the availability of +such techniques to Andr´e was limited. +The substantial growth in p-adic Hodge theory, and in particular the recent developments +in p-adic Hodge theory due to Scholze [Sch13], provides an opportunity to revisit these ideas. +Our first main technical contribution will be to give a p-adic interpretation of Andr´e’s +G-functions, and show how this greatly expands their applicability to arithmo-geometric +problems. +1.2 +p-adic Interpretations of G-functions +Our analysis will start by giving a purely algebraic-de-Rham description of Andr´e’s G- +functions; a similar description already appears in the proof of [And89, IX, §4, Theorem +2]. To do this, we once again fix an affine open subset U Ă X, with coordinates z1, . . . , zν +inducing an ´etale map U Ñ Aν, and such that the map to S is given by s ÞÑ z1 . . . zµ for +some 1 ď µ ď ν; we also set w “ µ ´ 1. The vector bundle H1 extends canonically to a +K-algebraic vector bundle H over S, as we review in §2.1. From our description of H we +will see that any section ω of H then admits a restriction ωU to a relative de Rham sheaf on +8 + +U. Fixing a point q P UpKq in the locus z1 “ ¨ ¨ ¨ “ zµ “ 0 mapping to s0, we will further +restrict ωU to a formal neighbourhood of q to obtain a unique representation +i˚ +q ωU “ hq dz2 ¨ ¨ ¨ dzµ +z2 ¨ ¨ ¨ zµ +, +in a formal de Rham complex at q, with hq P Krrsss. This K-algebraic power series hq is +obtained without leaving the algebro-geometric category, and its analytification agrees with +the function P in (2) above. Our goal is to provide a p-adic interpretation of the same +object. +The basic difficulty is that there is no robust analogue of homology in the rigid-analytic +setting, making it difficult to find a proper analogue of the integration pairing. However, +because the cycles in question are of a particularly simple form, one can make do with less, +as we now explain. +First, for a connected adic space Y defined over SpapCp, OCpq, we recall in §2.5 the +definition of the ´etale fundamental group π1 +´etpY, yq at a closed point y P Y . We consider +the case where Y “ ∆˝ “ SpapCpxT, T ´1y, OCpxT, T ´1yq is the rigid-analytic torus, and +try to describe an element in π1 +´etp∆˝, yq giving a pro-p analogue of a rigid-analytic “loop” +around the puncture. Unfortunately, the space ∆˝ admits more rigid-analytic coverings +than just those of Kummer type, making a na¨ıve approach difficult. Our idea is basically to +define this “loop” on just those coverings of Kummer type, and then to choose an extension +to π1p∆˝, yq which will be compatible with the formalism of p-adic Hodge theory. This +is not so easy to do in the (possibly non-abelian) setting of fundamental groups, so we +instead work dually, viewing each element of π1p∆˝, yq through its induced functional on +first-degree cohomology via the isomorphism H1p∆˝ +p´et, ˆZpp1qq » Homcontpπ1p∆˝, yq, Zpp1qq. +(The notation p´qp´et denotes the pro-´etale site introduced by Scholze in [Sch13], which we +review in §2.3.) +After fixing a compatible system of p’th roots of unity we obtain a functional α˚ : +H1pGm,p´et, ˆZpp1qq Ñ Zpp1q, where Gm is the multiplicative group, which we then try to +extend to a functional ˆα˚ : H1p∆˝ +p´et, ˆZpp1qq Ñ Zpp1q compatible with pullback by the map +H1pGm,p´et, ˆZpp1qq Ñ H1p∆˝ +p´et, ˆZpp1qq induced by the embedding ∆˝ ãÑ Gm of adic spaces. +Of the possible extensions we could choose, we arrange for one satisfying the property that +ˆα˚ is zero on the kernel of the natural map H1p∆˝ +p´et, ˆZpp1qq Ñ H1p∆˝ +p´et, BdRq, where BdR is +the period sheaf induced by Scholze. In the case of a more general space ∆a,b “ p∆˝qa ˆ ∆b +embedding into Ga +m ˆ Ab, where ∆ “ SpapCpxT y, OCpxT yq, one extends both α˚ and ˆα˚ to +functionals α˚ +a,b and ˆα˚ +a,b on the cohomology in degree a using the Kunneth formula. +To apply this to the study of G-functions, we then consider the localized situation that +arises from our choice of U Ă X and coordinates z1, . . . , zν. Fixing again a point q in the lo- +cus z1 “ ¨ ¨ ¨ “ zµ “ 0 mapping to s0 and a small rigid-analytic disk D “ SpapCpxsy, OCpxsyq +around s0, we may consider the neighbourhood U “ f ´1pDq X U of q. Using the coordi- +nates on U induced by z1, . . . , zν one obtains a neighbourhood C Ă U near q of the form +p∆˝qµ ˆ ∆ν´µ. A similar local description applies to the fibres Us above each point s P D, +giving neighbourhoods ∆w,ν´µ +s +Ă Us isomorphic to p∆˝qw ˆ ∆ν´µ. For each closed point +s P D one can then consider an evaluation functional ˆγ˚ +s : Hwp∆w,ν´µ +s +, ˆZppwqq Ñ Zppwq +obtained by pulling back ˆα˚ +w,ν´µ along an isomorphism ∆w,ν´µ +s +» ∆w,ν´µ induced by the +coordinates z2, . . . , zµ. As the point s varies, the functionals ˆγ˚ +s give a p-adic analogue of +the family of vanishing cycles in the complex analytic setting. We note, however, that these +cycles are defined one fibre at a time. +9 + +Using our fixed compatible system of p’th roots of unity, one obtains a fundamental +p-adic period t P BdR. We then extend ˆγ˚ +s to a map +ˆγ˚ +s,BdR : Hwp∆w,ν´µ +s,p´et , ˆZppwqq b BdR Ñ BdR +and obtain the following p-adic interpretation of Andr´e’s G-functions: +Theorem 1.13. For each closed point s P D one has that +hqpsq “ 1 +tw ˆγ˚ +s,BdR +´ +ρ´1 ´ +ω +ˇˇ +∆w,ν´µ +s +¯¯ +, +(4) +where +ρ´1 : Hw +dRp∆w,ν´µ +s +q b BdR Ñ Hwp∆w,ν´µ +s,p´et , ˆZppwqq b BdR +is a p-adic period map. +Remark. The notation “ρ´1” is formal, used for compatibility with other notation we will +use later, and the map is constructed directly instead of as the inverse of a map “ρ”. +Remark. To make sense of the equality (4) we have fixed an embedding K ãÑ Cp, and the +result holds for all such choices. +We note that period maps like ρ´1 are typically constructed to compare the cohomology of +complete — or at least “nicely” completable — algebraic or rigid-analytic varieties, which +is far from the case here. But the product of disks ∆w,ν´µ is simple enough that one can +describe the map ρ´1 by an explicit ˇCech calculation. +Showing that ρ´1 is sufficiently +compatible with other p-adic period isomorphisms — in particular, enough to consistently +pull back ˆγ˚ +s to the cohomology of the ambient variety Xs — then amounts to us having +chosen our extension of α˚ to not interact with the kernel of the map H1p∆˝ +p´et, ˆZpp1qq Ñ +H1p∆˝ +p´et, BdRq. +For applications one actually needs something more precise than Theorem 1.13, which +we show in Theorem 4.3. The point is that one doesn’t just want an algebro-geometric +interpretation of the function hq that holds in some unspecified neighbourhood D of the +point s0, but instead an interpretation for hq that holds inside its entire radius of convergence +after choosing an embedding of K into C or Cp (or at least when |s|v ă 1, in light of +the condition in Theorem 1.11). This requires additional arguments in both the complex +analytic and rigid analytic settings. In particular, to make this work in the rigid analytic +setting we will choose the coordinates z1, . . . , zν carefully so that neighbourhoods like the +neighbourhood C mentioned above are sufficiently large; see the discussion in §4 and §6.1. +1.3 +Relations on Periods +The above considerations, of course, are purely local, and to produce algebraic relations +between the values taken by Andr´e’s G-functions at a point s for the purpose of applying +the Hasse principle one ultimately needs to relate these values to the cohomology of the +projective fibre Xs. Note that we have an induced natural map +HwpXs,p´et, ˆZppwqq Ñ Hwp∆w,ν´µ +s,p´et , ˆZppwqq +ˆγ˚ +s +ÝÝÑ Zppwq +which by abuse of notation we also denote by ˆγ˚ +s . If one additionally considers the compar- +ison isomorphism of p-adic Hodge theory, one obtains a diagram +10 + +Hw +dRpXsq b BdR +Hw +dRp∆w,ν´µ +s +q b BdR +HwpXs,p´et, ˆZppwqq b BdR +Hwp∆w,ν´µ +s,p´et , ˆZppwqq b BdR +„ +ρ´1 +, +using which one may extend ˆγ˚ +s,BdR consistently along both paths in the diagram to take +values on Hw +dRpXsq, as mentioned above. +After fixing a frame ω1, . . . , ωm of H1ˇˇ +D with corresponding representations +i˚ +q ωj “ hj +dz2 ¨ ¨ ¨ dzµ +z2 ¨ ¨ ¨ zµ +, +our result Theorem 1.13 can be interpreted as follows: +Theorem 1.14. The values h1psq, . . . , hmpsq give the vector +1 +tw ˆγ˚ +s,BdR P HompHµ´1 +dR pXsq b BdR, BdRq +in the dual coordinates induced by ω1,s, . . . , ωm,s. +To apply this fact we now consider the situation where Xs admits a non-trivial algebra E +of algebraic self-correspondences. If our point s P D comes from a point ξ of S defined over a +finite extension L of the fixed number field K, our goal is to produce a K-algebraic relation +on the coordinates h1psq, . . . , hmpsq so that we can apply the Hasse principle for G-functions. +But the strategy employed by Andr´e in [And89, X, §3] at the infinite places doesn’t work, as +it involves the underlying Q-structure of the Betti cohomology of Xξ. However as the p-adic +case is enriched by the presence of a Galois action compatible with the period isomorphism, +we may obtain relations using an entirely new method. +First, we prove the following, which is almost immediate from the construction of ˆγ˚ +s : +Proposition 1.15. Let v be the place of K at which the functional ˆγ˚ +s is defined, and +GKv “ GalpKv{Kvq the associated local Galois group. Then GKv acts on ˆγ˚ +s through the +character χ´w +cycl, where χcycl : GKv Ñ Z˚ +p is the usual cyclotomic character. +The point is that the only choice not invariant under the Galois action made in the con- +struction of ˆγ˚ +s is our choice of a non-trivial compatible system of p-power roots of unity +corresponding to a p-adic “loop” inside the torus ∆˝, and if we “integrate” around w such +loops then GK acts on the “integrals” through the w’th power of χcycl. (There is also a +choice of splitting of the ´etale cohomology of ∆˝, but all such choices lead to the same +functional on the cohomology of Xs.) We observe that this simple fact is already enough to +produce non-trivial algebraic relations on the de Rham coordinates of ˆγ˚ +s in the presence of +an L-algebraic correspondence τ : Xs ��� Xs. Indeed, the cohomology class rτs in both de +Rham and ´etale cohomology is fixed by a finite index subgroup of GKv, hence the functionals +ˆγ˚ +s , ˆγ˚ +s ˝ rτs, ˆγ˚ +s ˝ rτs2, ˆγ˚ +s ˝ rτs3 . . . +(5) +all lie in a subspace of the dual of HwpXs,p´et, ˆZppwqq on which a finite index subgroup of +GK acts by χ´w +cycl. If one expresses the functionals in (5) in a fixed L-algebraic de Rham +basis, then the coordinates of each element of the sequence (5) are L-linear combinations of +the coordinates of ˆγ˚ +s . The Hodge-Tate comparison provides a simple way of bounding the +11 + +dimension of the χ´w +cycl-character space in terms of the de Rham Hodge numbers, and hence +one obtains an L-algebraic relation on the coordinates of ˆγ˚ +s simply by taking determinants +of minors of matrices constructed from the vectors in the sequence (5). To make the relation +K-algebraic one takes the product of this relation with all its Galois conjugates. +Let us remark that, in the cases we consider, we often don’t expect relations like the ones +we construct on the coordinates of ˆγs to actually exist for points s which are p-adically close +to s0. The reason is that one generally expects special moduli to be in some sense bounded +away from singularities in p-adic metrics: this is the case for curves and abelian varieties +with complex multiplication, all of which have potentially good reduction. So instead our +construction of non-trivial relations on the coordinates of ˆγ˚ +s under the (often counterfactual) +assumption that s is a special modulus p-adically close to s0 is to be interpreted as a sort +of integrality constraint on s. +1.4 +Applications to Height Bounds +The generality in which one can now construct algebraic relations on Andr´e’s G-functions at +finite places eliminates a broad class of obstructions to applying the G-function method to +problems in arithmetic geometry. Indeed one can now show, in quite general settings, that +polynomial height bounds on special moduli follow as soon as one can establish K-algebraic +constraints at the infinite places. +We now give a sample result of this type. +Recall the variation of Hodge structure +V1 “ Rwf an +˚ Z{tor. whose fibres have dimension m. In what follows we write S Ă SpQq for +the set of ξ P SpCq such that there exists a simple Hodge summand W Ă V1 +Q,ξ of CM type. +Proposition 1.16. Suppose that V1 is simple and that the Hodge conjecture holds for en- +domorphisms appearing in the fibres of V1. Let s0 be a point in SzS1 at which the fibre Xs0 +acquires a normal crossing singularity of order w at the point q P Xs0, and for which the +associated tuple ph1, . . . , hmq of G-functions is not constant. Then after replacing K with a +finite extension, there exists +(i) a finite covering c : C Ñ S, and a parameter s on C with simple zeros and vanishing +exactly on the set c´1ps0q; and +(ii) for all but finitely many ξ P c´1pSq a Kpξq-algebraic relation on the values +h1pspξqq, . . . , hmpspξqq, +not induced by a functional relation on h1, . . . , hm, and which holds at all finite places +relevant for spξq. +Moreover, the degree of the relation in (ii) may be bounded independently of ξ. +Remark. In Proposition 1.16(ii) we actually mean to replace the original G-functions with +the ones computed in terms of the parameter s; we give a more precise description in §6. +This leads to the following theorem, which reinterprets Theorem 1.7 above: +Theorem 1.17. Suppose that in the setting of (1.16) there exists an additional order w +normal crossing point q1 P Xs0 with an associated non-constant tuple ph1 +1, . . . , h1 +mq linearly +independent from ph1, . . . , hmq. Then for any logarithmic Weil height θ : SpQq Ñ Rą0 there +exists constants κ, a P Rą0 such that +θpξq ď κ rKpξq : Ksa +for all ξ P S. +12 + +1.5 +Pila-Zannier for General Atypicality +To get from Theorem 1.17 to the finiteness results in Theorem 1.1 and Theorem 1.5, one +applies the Pila-Zannier strategy for obtaining finiteness results from lower bounds on Galois +orbits. However the usual Pila-Zannier strategy, for instance used to prove the Andr´e-Oort +conjecture, is insufficient here. In that setting one uses the fact that points ξ P S Ă SpCq +above which the fibre Xξ has complex multiplication produce Q-algebraic points rt inside +a definable period image I Ă D, where D is a period domain for the polarized Hodge +structures appearing in the fibres of V. By producing lots of Q-algebraic points in I of +bounded height and over number fields of bounded degree one can apply a theorem of Pila- +Wilkie to obtain an algebraic curve inside I, and from this functional transcendence results +can be used to relate this to Hodge loci in S. The notion of algebraicity here comes from +the open embedding D Ă qD into a natural ambient flag variety. +If the points in S one is studying are not CM points, the situation becomes more compli- +cated. What happens in this case is that the points rt are no longer Q-algebraic, but merely +have lower-than-expected transcendence degree on account of an intersection between Iand +a Q-algebraic flag subvariety qDM Ă qD determined by the Mumford-Tate M of the Hodge +structure rt. Our observation, which is related to ideas appearing in [DR18], is that one +can obtain results in this more general setting by applying Pila-Zannier-type reasoning to +the moduli of the varieties qDM. More specifically, one can reduce to the case where one +considers only varieties qDM for which the associated Mumford-Tate groups M lie inside a +single GSpCq-orbit for the generic Mumford-Tate group GS of the variation V, where the +action on Mumford-Tate groups is by conjugacy. The situation one is then tasked with deal- +ing with is the situation where there are many Q-algebraic translates g ¨ qDM of the variety +qDM which intersect I atypically. One can understand the elements g that arise in terms +of heights of Hodge tensors defining the associated Mumford-Tate groups gMg´1, and use +this to bound both the heights of such g and the degree of their field of definition. The +Pila-Wilkie theorem then produces, under appropriate bounds on the heights of some Hodge +tensors associated to points of I, an algebraic family of subvarieties of qD which intersect +I atypically, and from this one can run the usual functional transcendence arguments. We +do not need any constraints on S; in particular, we do not use that S is a curve. +As an application of this, we prove the following general result, which we state here +somewhat informally (see §7.3 for the relevant definitions and precise statements). +Theorem 1.18. Suppose that f : X Ñ S is a family of abelian varieties whose algebraic +monodromy agrees with its derived Mumford-Tate group, and S Ă SpCq is the subset of points +in the zero-dimensional Hodge locus which are defined by, and atypical for, the property of +acquiring a non-generic isogeny summand. Then if there exists constants κ, a P Rą0 such +that +θpξq ď κ rKpξq : Ksa +for all ξ P S, with θ some logarithmic Weil height, then S is finite. +1.6 +Related Work +As we have discussed in great detail, the G-function method for bounding heights on special +moduli was introduced in Andr´e’s book [And89], but was limited by the lack of p-adic +interpretations of these functions. Since then, two works of which we are aware of have +managed to apply this method in a way which produces cohomological relations on values of +G-functions at finite primes. The first is follow-up work of Andr´e [And95], and in particular +13 + +[And95, Theorem 1]. Here Andr´e works in the special setting of families of abelian varieties +and gives a p-adic interpretation of values of G-functions using crystalline cohomological +period matrices. However to obtain bounds on heights of special moduli, he is forced to +restrict to moduli satisfying an integrality condition at all but finitely many primes, as the +relations he obtains at the finite places do not hold for all finite places at once. +A second more recent approach is given by Daw and Orr in [DO22]. Here they are able +to use a concrete p-adic interpretation of certain G-functions coming from Tate’s p-adic +uniformization of elliptic curves. They are able to control all finite places at once, and prove +results similar to us in the special case of curves in Y p1qn (the n-fold product of the moduli +space of elliptic curves) intersecting a certain “very degenerate” point lying on the boundary +of the compactified moduli space. It seems to us difficult to extend this method to more +general types of special moduli. +With respect to obtaining height bounds on special moduli in more general settings, +specifically beyond the case of abelian-type families, recent work of Papas [Pap22] gives +general results under algebraic-cycle and arithmeticity conjectures. We have found his work +useful for giving an outline of the general strategy of obtaining height bounds in settings +beyond those considered by Andr´e, Daw and Orr. In particular, he gives in [Pap22, Theorem +1.1] a general theorem for producing height bounds of the kind we are interested in under +the Hodge conjecture and a conjecture on the existence of certain “good” arithmetic models +for smooth projective families. The arithmetic models assumption has the effect of showing +that the special points he studies do not lie in sufficiently small v-adic neighbourhoods near +the degeneration point, and so is in effect a kind of integrality assumption. Our approach +gives a way of removing this hypothesis. +Finally, with respect to the Pila-Zannier strategy for Zilber-Pink-type atypical inter- +sections, Daw and Ren in [DR18] give an approach for the special case of subvarieties of +Shimura varieties. The basic idea is in some sense similar in that one tries to argue that +having “many” special points in S will allow one to produce some low-dimension algebraic +variety which interacts exceptionally with an analytic period image in order to contradict +an Ax-Schanuel principle. To do this they introduce and assume several conjectures relating +to various notions of “complexity” of special subvarieties, and establish a Zilber-Pink-type +theorem for point-like atypical intersections inside Shimura varieties under these assump- +tions. Our results are similar, except that we are able to work in a general algebro-geometric +setting beyond the case of Shimura varieties, and various aspects of our approach seem sim- +pler to us. We note that aside from Galois-orbit lower bounds and that special varieties in +the general-algebraic-family setting behave well under Galois-actions (a basic requirement +fundamental to the study of atypical intersections), we do not need other conjectures; see +for instance Corollary 7.7 and Corollary 7.14. +1.7 +Acknowledgements +We thank Jacob Tsimerman, Chris Daw, Martin Orr, and Georgios Papas for comments on a +draft of this manuscript. We also thank Donu Arapura for a MathOverflow comment which +suggested to the author the idea of considering hypersurface degenerations to hyperplanes +as an application of Theorem 1.7. +2 +Cohomological Preliminaries +We continue with the notation and setup established in the introduction. +14 + +2.1 +A Model for the Canonical Extension +We begin by describing an explicit model for the canonical extension H of H1 referenced +in the introduction, following Steenbrink [Ste76]. As before we assume that E “ XzX1 “ +f ´1ps0q, where ts0u “ SzS1 and E is a divisor with normal crossings, and define the de Rham +complex Ω‚ +Xplog Eq of algebraic differentials with logarithmic poles along E as follows: for an +open set U Ă X the sections of Ωp +Xplog Eq over U are the algebraic forms ω on UzE such that +ω and dω have at most a simple pole along E. If one chooses local coordinates pz1, . . . , zνq +around a point q P E so that E is defined by z1 ¨ ¨ ¨ zµ “ 0 for some 1 ď µ ď ν, then the stalk +Ω1 +Xplog Eqpqq is a free module over OX,pqq with generators dz1{z1, . . . , dzµ{zµ, zµ`1, . . . , zν +and Ωp +Xplog Eq “ Źp Ω1 +Xplog Eq. We further define Ωp +X{Splog Eq as the pth exterior power +of the quotient Ω1 +Xplog Eq{f ˚Ω1 +Splogts0uq, with Ω1 +Splogts0uq defined analogously via differ- +entials with at most a logarithmic pole at s0. +The following is proven in [Ste76, 2.18] (note that it makes no difference whether one +uses the algebraic or analytic site, c.f. §2.4 below): +Proposition 2.1. For all w ě 0, the sheaf Rwf˚pΩ‚ +X{Splog Eqq is locally free on S and for +all s P S the canonical map +Rwf˚pΩ‚ +X{Splog Eqq bOS pOS,s{mS,sq Ñ HwpXs, Ω‚ +X{Splog Eq bOX OXsq +is an isomorphism. +We may therefore take H “ Rwf˚Ω‚ +X{Splog Eq. Let us now consider the setup in the +introduction, where U Ă X was a fixed affine Zariski open subset with coordinates z1, . . . , zν +trivializing Ω1 +U and such that the map U Ñ S takes the form s ÞÑ z1 ¨ ¨ ¨ zµ for some +1 ď µ ď ν. As before, we set w “ µ ´ 1. We have a natural map Rwf˚Ω‚ +X{Splog Eq Ñ +Rwf˚Ω‚ +U{SplogpE X Uqq. For S affine, the cohomology module Rwf˚Ω‚ +U{SplogpE X Uqq may +be identified with RwΓ Ω‚ +U{SplogpE X Uqq, as follows from the Leray spectral sequence. +Moreover, because U is affine, this can in turn be identified with the cohomology in degree +w of the complex +0 Ñ OU Ñ Ω1 +U{SplogpE X Uqq Ñ ¨ ¨ ¨ Ñ Ωn +U{SplogpU X Eqq, +viewed as a module over OS. Restricting to the completed stalk at a point q in the locus +z1 “ ¨ ¨ ¨ “ zµ “ 0 mapping to s0 one obtains a complex of pOS,ps0q-modules +0 Ñ pOU,pqq Ñ pΩ1 +U{SplogpE X Uqqpqq Ñ ¨ ¨ ¨ Ñ pΩn +U{SplogpE X Uqqpqq, +and by composition a restriction map +η : ΓpRwf˚Ω‚ +X{Splog Eq, Sq Ñ Cohomw ” +pΩ‚ +U{SplogpE X Uqqpqq +ı +, +where we denote by Cohomw the na¨ıve cohomology in degree w. This map will be used in +the proof of Theorem 4.3 to construct G-functions. +The following Lemma describes the form of the elements in Cohomw ” +pΩ‚ +U{SplogpE X Uqqpqq +ı +, +and also the analogous modules obtained by considering convergent power series in the com- +plex and rigid-analytic topologies. +15 + +Lemma 2.2. Suppose that A is any of the rings +tKrrz1, . . . , zνss, Ctz1, . . . , zνu, kxz1, . . . , zνy, kxz1, . . . , zνyR1{µu +which are (in order) formal power series over the characteristic zero field K, germs of +complex analytic power series, germs of k-analytic power series over the non-archimedian +valued field k, and power series convergent in the (closed or open) non-archimedian ball of +radius R1{µ, respectively. Let B be the corresponding ring in tKrrsss, Ctsu, kxsy, kxsyRu and +consider the map B Ñ A given by s ÞÑ z1 ¨ ¨ ¨ zµ. Consider the complex +0 Ñ A Ñ Ω1 +A{Bplog Eq Ñ ¨ ¨ ¨ Ñ +ν +ľ +Ω1 +A{Bplog Eq Ñ 0, +(6) +where Ω1 +A{Bplog Eq is the quotient of Ω1 +Aplog Eq and A bB Ω1 +Bplogt0uq, with +Ω1 +Aplog Eq “ Adz1 +z1 +‘ ¨ ¨ ¨ ‘ Adzµ +zµ +‘ Adzµ`1 ‘ ¨ ¨ ¨ ‘ Adzν, +and Ω1 +Bplogt0uq “ B ds +s . Then every element α of Cohomw ” +Ω‚ +A{Bplog Eq +ı +admits a unique +representation of the form +hdz2 ¨ ¨ ¨ dzµ +z2 ¨ ¨ ¨ zµ +where h is a uniquely determined element of B. +Proof. In the complex analytic setting the entire cohomology of the complex (6) is described +in [Ste76, 1.13], and the same proof works in general. For the convenience of the reader +we give some details. The relation s “ z1 ¨ ¨ ¨ zµ induces the relation dz1 +z1 ` ¨ ¨ ¨ ` dzµ +zµ “ 0 in +the complex (6), which gives a natural presentation of the complex in terms of the forms +dz2 +z2 , . . . , dzµ +zµ , dzµ`1, . . . , dzν only. The complex then reduces to a Kozul-type complex L‚ +generated by +Adz2 +z2 +‘ ¨ ¨ ¨ ‘ Adzµ +zµ +‘ A dzµ`1 ‘ ¨ ¨ ¨ ‘ A dzν, +where the differential operators for L‚ are given by Di “ ziBi ´ z1B1 for 2 ď i ď µ and +Di “ Bi for µ ` 1 ď i ď ν. +Suppose that an element +β “ g dzi1 +z +ei1 +i1 +^ ¨ ¨ ¨ ^ dzir +zeir +i1 +in the complex L‚ lies in the kernel of the differential, with g P A a monomial, and where +the exponents eij are chosen so that each factor in the wedge product is one of the specified +generators for the complex. Then from the construction of the Kozul complex we must have +that Djpgq “ 0 for each j not appearing in the set ti1, . . . , iru. If we have ik ą µ for some +k (and hence eik “ 0), and the variable zik occurs with exponent a ě 0 in the monomial g, +we compute that +d +˜ +p´1qik zik +a ` 1g dzi1 +z +ei1 +i1 +^ ¨ ¨ ¨ ^ y +dzik ^ ¨ ¨ ¨ ^ dzir +zeir +i1 +¸ +“ β ` p´1qik zik +a ` 1 +ÿ +j +Djpgq dzj ^ +˜ +dzi1 +z +ei1 +i1 +^ ¨ ¨ ¨ ^ dzir +zeir +i1 +¸ +loooooooooooooooooooooooomoooooooooooooooooooooooon +“0 +16 + +which shows that β can be integrated. Thus, non-trivial contributions to cohomology appear +only when ti1, . . . , iru Ă t2, . . . , µu. In degree r “ w this means that ti1, . . . , iwu “ t2, . . . , µu +and that Djpgq “ 0 for j ą µ, meaning we can assume our class α is of the form h dz2¨¨¨dzµ +z2¨¨¨zµ +with h depending on z1, . . . , zµ only. To ensure that a monomial m in h cannot be integrated, +one checks that we must have additionally that Djpmq “ 0 for 2 ď j ď µ. But this means +that h consists only of terms like c ¨ pz1 ¨ ¨ ¨ zµqe, hence g lies in the image of B. +The uniqueness claim is checked directly from the construction of the complex, as no +non-zero elements of the specified form lie in the image of the differential. +2.2 +ˇCech cohomological recollections +2.2.1 +ˇCech cohomology of complexes +We now develop the general formalism of the ˇCech double complex associated to a complex +pF‚, dFq of sheaves of abelian groups on a site C, generalizing the case of sheaves on a +topological space which appears in [Sta20, Section 01ED] and [Sta20, Section 01FP]. +We assume that C has a final object X, and we let U “ tci : Ui Ñ XuiPI be a covering +of X. We first consider the case where F‚ “ F consists of a single sheaf. We define +CppU, Fq “ +ź +pi0,...,ipqPIp`1 +FpUi0 ˆX ¨ ¨ ¨ ˆX Uipq. +Given s P CppU, Fq we will write si0...ip its value in the factor FpUi0 ˆX . . . ˆX Uipq, and +we define the differential +δ : CppU, Fq Ñ Cp`1pU, Fq +by the formula +δpsqi0...ip`1 “ +p`1 +ÿ +j“0 +p´1qjsi0... pij...ip`1 +ˇˇˇ +Ui0 ˆX¨¨¨ˆXUip`1 +where restriction comes from the natural fibre product projection. One checks that pC‚pU, Fq, δq +is a complex. +The formation of the complex C‚pU, Fq is functorial in F, so given a complex pF‚, dFq +one naturally obtains a double complex C‚pU, F‚q. We write pL‚pU, F‚q, dq for the associ- +ated total complex, with terms +LnpU, F‚q “ +à +p`q“n +ź +i0...ip +FqpUi0...ipq +and with differential of an element α of degree n given by d “ δ ` p´1qp`1dF. Finally, we +write qHppU, F‚q for the cohomology groups of this complex. +We now compare the ˇCech cohomology to sheaf cohomology. We denote the cohomology +of a complex of sheaves F‚ computed on an object V of C using covers of C by H‚pVC, F‚q. +Then we have the following generalization of [Gro60, III, Ch.0, 12.4.6]: +Proposition 2.3. Let F‚ be a bounded below complex of abelian groups on C, and let +U “ tci : Ui Ñ XuiPI be a covering of X. Then there exists a spectral sequence abutting to +H‚pXC, F‚q whose second page is given by +Epq +2 “ CohomppL‚pU, JqpF‚qqq, +where JqpF‚q denotes the complex of presheaves whose j’th term is given by rV ÞÑ HqpVC, Fjqs. +17 + +Proof. One just has to check that all the steps in the argument in [Gro60, III, Ch.0, 12.4.6] +generalize to this situation (c.f. [Sta20, Lemma 08BN]). We consider a Cartan-Eilenberg +resolution L‚‚ of F‚ by injective sheaves, constructed as in [Sta20, Lemma 015I]. From the +functorality of ˇCech cohomology we obtain a tricomplex C‚pU, L‚‚q “ rCipU, Ljkqs which +we may regard as a bicomplex in degrees i and j ` k. Because the sheaves Ljk, and hence +the terms in the total complex of L‚‚, are all injective sheaves of abelian groups, the ˇCech +complex C‚pU, L‚‚q computes the cohomology of L‚‚ regarded as a single complex: this +follows by combining [Sta20, Lemma 03AW], which shows that the positive degree ˇCech +cohomology on U of each injective sheaf is zero, with [Sta20, Lemma 0133] applied to the +map L‚‚ Ñ C‚pU, L‚‚q, where we regard L‚‚ as a single complex, as stated. Because +the total complex of L‚‚ computes the cohomology of F‚, it follows that the map F‚ Ñ +C‚pU, L‚‚q induces an isomorphism on cohomology. +We now consider the tricomplex C‚pU, L‚‚q as a bicomplex in degrees i`j and k. Then +because Lj,‚ is an injective resolution of Fj for all j, the degree q cohomology of the complex +CipU, Lj,‚q is then given by the ˇCech complex CipU, JqpFjqq. Computing the second page +then gives the result. +Corollary 2.4. Suppose that for each U 1 obtained as a fibre product of objects in the cover +U and for each k one has that HqpU 1 +C, Fkq “ 0 for all q ą 0. Then L‚pU, F‚q computes +the cohomology of the complex F‚. +Proof. The assumption ensures that the spectral sequence of Proposition 2.3 degenerates at +the second page, hence the cohomology is computed by the first page, which is naturally +identified with the cohomology of the total ˇCech complex L‚pU, F‚q. +2.2.2 +Cup product in ˇCech cohomology +We also recall how to define the cup product on the ˇCech complex, following [Sta20, +Section 01FP] in the setting of complexes of sheaves on topological spaces. +Given two +complexes of sheaves F‚ and G‚ of abelian groups on the site C, we write TotpF‚ b G‚q for +the complex with terms À +p`q“n Fp b Gq and where the differential is given by +dpα b βq “ dpαq b β ` p´1qdegpαqα b dpβq. +Given a covering U “ tci : Ui Ñ XuiPI, our cup product is then a map +Y : Tot pTotpC‚pU, F‚qq b TotpC‚pU, G‚qqq Ñ TotpC‚pU, TotpF‚ b G‚qqq. +It is given by the rule +pα Y βqi0...ip “ +pÿ +r“0 +εpdeg α, deg β, p, rqαi0...ir b βir...ip, +where εpn, m, p, rq “ p´1qpp`rqn`rp`r. The associativity of the cup product as well as the +identity +dpα Y βq “ dpαq Y β ` p´1qdegpαqα Y dpβq +may be proved by explicit calculation, exactly as is done in [Sta20, Section 01FP] in the +setting of complexes on topological spaces. Moreover, the cup product is compatible with +a graded commutative structure on the complex F‚, as we now explain, following [Sta20, +Section 01FP]. +18 + +Suppose that we have a graded commutative multiplication map +^‚ : TotpF‚ b F‚q Ñ F‚. +This is defined to mean that given sections s of Fa and t of Fb we obtain a section s ^ t of +Fa`b in such a way that s^t “ p´1qabt^s, and that we have dps^tq “ dpsq^t`p´1qas^dptq. +We may then consider the composition +Tot pTotpC‚pU, F‚qq b TotpC‚pU, F‚qqq Y +ÝÑ TotpC‚pU, TotpF‚ b F‚qqq +^ +ÝÑ TotpC‚pU, F‚qq. +It may be checked as in [Sta20, Section 01FP] that this induces a map on ˇCech cohomology +HnpTotpC‚pU, F‚qqq ˆ HmpTotpC‚pU, F‚qqq Ñ Hn`mpTotpC‚pU, F‚qqq. +In our situation of interest, this will reproduce the cup product on both ´etale cohomology +and (algebraic) de Rham cohomology. +2.3 +The pro-´etale site +Let us fix an adic space X over Spapk, Okq, with k a perfectoid field. We will assume that +X is locally noetherian. (This assumption will also continue to be in force in subsequent +sections without further comment.) We will begin by defining some categories (and sites) +associated to X. +First, one has the ´etale site X´et, whose objects consist of ´etale maps +U Ñ X of adic spaces a morphisms between them. Next we consider the category propX´etq: +its objects consist of projective limits lim +ÐÝiPI Ui of objects of X´et and its morphisms are the +natural morphisms of limit diagrams. A map of objects U Ñ V in the category propX´etq +is called ´etale if it is induced by an ´etale morphism of objects U0 Ñ V0 in X´et. A map of +objects U Ñ V is called pro-´etale if we have U “ limi Ui in such a way so that U Ñ V +is given by an inverse limit Ui Ñ V of objects ´etale over V , and such that Ui Ñ Uj is +finite ´etale and surjective for large i ą j. The category Xp´et is then defined to be the full +subcategory of propX´etq consisting of objects which are pro-´etale over X. A covering U in +propX´etq of an object U is given by a family of pro-´etale morphisms U “ tfi : Ui Ñ Uu such +that |U| “ Ť +i fip|Ui|q, where we give pro-objects the limit topology. By [Sch13, Lemma +3.10], this defines a site. +If one instead starts with the category Xf´et of objects finite ´etale over X, one may +carry out the analogous procedure to define a category Xpf´et, which we call the “pro-finite +finite ´etale site”. It is naturally a subcategory of Xp´et, and we have a natural map of sites +Xp´et Ñ Xpf´et. +The pro-finite ´etale site can be used to compute ´etale cohomology with coefficients in +Zp as the cohomology of the sheaf ˆZppUq “ Homcontp|U|, Zpq, where we consider continuous +morphisms of the underlying topological spaces, and Zp has the usual p-adic topology. +We now introduce some important sheaves on Xp´et, following [Sch13]. The first is OX, +the “uncompleted structure sheaf”, which is the pullback γ˚OX´et under the natural map +γ : Xp´et Ñ X´et of sites. Likewise we have the subring of integral elements O` +X “ γ˚O` +X´et. +These sheaves can then be completed to obtain ˆO` +X “ lim +ÐÝ O` +X{pn and ˆOX “ ˆO` +X +” +1 +p +ı +. Next +we have the tilted integral structure sheaf, defined as ˆO` +X5 “ lim +ÐÝΦ O` +X{p, with the inverse +limit over Frobenius; here we use the notion of the tilt X5 of X comes from Scholze’s theory +19 + +of perfectoid spaces [Sch11]. We set ˆOX5 “ ˆO` +X5 bk5` k5. We then define Ainf “ WpˆO` +X5q and +Binf “ Ainf +” +1 +p +ı +. +We have a natural map θ : Ainf Ñ pO` +X which extends to a map Binf Ñ ˆOX. To define +it, we work locally, where the sheaf Ainf is represented by a ring WpA5q, and we wish to +construct a map WpA5q Ñ A. We may represent an element x P WpA5q via its Witt vector +components as a sum ř +i pirxis. We then define +θ +˜ÿ +i +pirxis +¸ +“ +ÿ +i +pix7 +i, +where the operation p´q7 is defined on y P A5, represented by the sequence py1, y2, . . .q, by +choosing lifts ˆyj for all j and setting +y7 “ lim +jÑ8 ˆyj +pj. +We then define B` +dR “ lim +ÐÝ Binf{pker θqn and BdR “ B` +dRrt´1s, where t is any element gener- +ating the kernel of θ. +Finally we define OBinf “ OX bWpκq Binf. +The map θ on Binf extends to a map θ : +OBinf Ñ ˆOX. One then defines OB` +dR “ lim +ÐÝ OBinf{pker θqn and OBdR “ OB` +dRrt´1s, where t is +a generator of ker θ (this makes sense locally, as is checked in [Sch13, §6]). Lastly we define +Ωi +X “ OBdR bOX Ωi +X as sheaves on Xp´et. +2.4 +Coherent Cohomology on Various Sites +An important sort of fact that we will use (often implicitly) throughout the paper is that it +“doesn’t matter” on which site one computes the cohomology of coherent objects associated +to a space X. What is meant by this is is that one has two sites associated to X, say X1 +and X2, with a natural map of ringed sites τ : X2 Ñ X1, and given a complex of coherent +sheaves F‚ on X1 the natural map HipX1, F‚q Ñ HipX2, τ ˚F‚q is an isomorphism for all +i. Note that one typically only needs to check this when F‚ “ F is a single sheaf rather +than a complex of such. The reason is that, in the situations of interest, the sites X1 and +X2 will contain certain types of objects on which the coherent cohomology of any individual +coherent sheaf vanishes (e.g., affine, affinoid, Stein, etc.), and using this fact for each Fi +in the complex F‚ and an appropriate cover one learns that the “same” ˇCech complex +computes cohomology on both X1 and X2, and the resulting fact is formal. +These facts we will only need for complexes of differentials (possibly relative, possibly +logarithmic) and for sufficiently nice spaces X. (Our spaces or maps of spaces will also often +be proper, which makes things even easier, see the remark below.) Nevertheless, we give +some of the required facts in greater generality. +Proposition 2.5. For F‚ a complex of coherent sheaves on X1 and τ : X2 Ñ X1 a map of +sites, the natural map HipX1, F‚q Ñ HipX2, τ ˚F‚q is an isomorphism when +(i) X is a scheme, X1 “ XZar, X2 “ X´et; +(ii) X is a proper C-scheme, X1 “ XZar, X2 “ Xan, and F‚ “ Ω‚ +X; +(iii) X is a Cp-scheme, X1 “ XZar, X2 “ Xad, and F‚ “ Ω‚ +X; +(iv) X is a rigid space, X1 “ Xad, X2 “ X´et; +20 + +(v) X is a adic space, X1 “ X´et, X2 “ Xpro´et. +Proof. For (i) see [Sta20, Proposition 03DW]; for (ii) see the introduction to [Gro66]; for (iii) +see [GK04, Theorem 2.3]; for (iv) see [CT09, Example 2.1.3]; for (v) see [Sch11, Corollary +3.17]. +Remark. Note that one obtains (ii) and (iii) for any coherent sheaf F‚ when X is proper by +the relevant GAGA result. The same is also true if one considers the derived pushforward +in the relative setting of a proper morphism f : X Ñ S of schemes; see for instance [Ray71, +Expose XII] and the appendix to [Con06]. +2.5 +The ´etale fundamental group and cohomology +We now describe the ´etale fundamental group of X and its relation to the cohomology of +X. We write Xf´et for the category of adic spaces Y which are finite ´etale over X. Fixing a +geometric point x of X, we obtain a natural fibre functor FX,x : Xf´et Ñ Set, and as usual +the group π´et +1 pX, xq is defined as the group of automorphisms of this functor. For any finite +abelian group Λ, we now describe a natural isomorphism +Hompπ´et +1 pX, xq, Λq „ +ÝÑ H1pXp´et, Λq. +We note that to compute H1pXp´et, Λq for a finite abelian group Λ it suffices to use the usual +´etale site X´et, since the natural map H1pXp´et, Λq Ñ H1pX´et, Λq induced by the map of sites +X´et Ñ Xp´et is an isomorphism; this is due to [Sch11, Corollary 3.17], as mentioned above. +The description of this isomorphism is essentially identical to the case of schemes, for +which [Mil13, I.§11] is a reference. We will give some details. In what follows we also denote +by Λ the constant sheaf on X´et it defines, and we use multiplicative notation for group +multiplication. A sheaf L of sets on X´et on which Λ acts is called a torsor for Λ if: +(i) there exists a covering U “ tci : Ui Ñ XuiPI in X´et such that LpUiq ‰ ∅ for all i; +and +(ii) for every object U Ñ X in X´et and s P LpUq the map Λ +ˇˇ +U Ñ L +ˇˇ +U given by g ÞÑ gs is +an isomorphism of sheaves over U´et. +A covering U “ tci : Ui Ñ XuiPI for which (i) holds is said to split L. Supposing we +have such a covering, we construct a ˇCech cocycle rLs P H1pU, Λq as follows. Choose some +sections si P LpUiq for each i. By (ii), on each “intersection” Uij arising from the cover U +there exists a unique element λij P ΛpUijq such that λij ¨ si +ˇˇ +Uij “ sj +ˇˇ +Uij. Then pgijqIˆI is a +cocycle, and defines a class rLs in H1pU, Λq. Moreover we have: +Lemma 2.6. The map L ÞÑ rLs defines a bijection from the set of isomorphism classes of +torsors for Λ split by U to H1pU, Λq. +Proof. In the case of the ´etale site of a scheme this is [Mil13, I. Prop 11.1], and the proof is +identical in our case. +We now use the fact that there is a further bijection +tisom. classes of Λ-torsorsu ÐÑ Hompπ1pX, xq, Λq. +(7) +21 + +This is true in a great deal of generality by the work of [AM69] (c.f. +the discussion in +[H¨ub18, §9]). We describe this correspondence in the special case where the Λ-torsor L is +representable by a Galois covering Y Ñ X. More precisely, we assume that Λ “ AutXpY q +and that LpUq “ HomXpU, Y q for every U P Xf´et, with the natural action of Λ on L. Using +the fact π1pX, xq “ AutpFX,xq, we may define the map π1pX, xq Ñ AutXpY q by sending +η P AutpFX,xq to the automorphism α P AutXpY q for which ηpyq “ αpyq for all y P FX,xpY q; +that such an element exists follows from the assumption that Y Ñ X be a Galois cover. +In the situation where Λ “ lim +ÐÝ Λn is a pro-finite group, one can take the limit of both +sides of (8) to obtain a bijection +tisom. classes of Λ-torsorsu ÐÑ Homcontpπ1pX, xq, Λq, +(8) +with a similar explicit description in the case of a torsor coming from a limit of Galois +coverings. +3 +Cohomological Computations +3.1 +Basic ˇCech Computations +Let k be a complete algebraically closed non-archimedian local field with ring of integers Ok +and residue field κ. Write ∆˝ for the adic space given by SpapkxT ˘1y, OkxT ˘1yq, which can +be thought of as a rigid-analytic annulus. We consider the natural cover of ∆˝ inside ∆˝ +pro´et +with the covering space modelled by the infinite tower +r∆˝ “ lim +ÐÝ +T ÞÑT p +∆˝. +The space r∆˝ is then perfectoid of the form +SpapkxT ˘1{p8y, OkxT ˘1{p8yq “ lim +ÐÝ SpapkxT ˘1{pjy, OkxT ˘1{pjyq, +and the covering map c : r∆˝ Ñ ∆˝ with respect to these presentations is simply given by +T ÞÑ T . +Define Zpp1q “ lim +ÐÝj µpj, where the transition maps are given by x ÞÑ xp. We consider +the self-product r∆˝2 “ r∆˝ˆ∆˝ r∆˝, and observe that its connected components are naturally +indexed by Zpp1q. Indeed, one has that +r∆˝2 “ +lim +T ÞÑT pj ∆˝ ˆ∆˝ ∆˝ +looooomooooon +∆˝ +j +. +Where the fibre product ∆˝ +j may be modelled as +∆˝ +j “ SpapkxT ˘1 +1 , T ˘1 +2 y{pT pj +1 ´ T pj +2 q, OkxT ˘1 +1 , T ˘1 +2 y{pT pj +1 ´ T pj +2 qq, +and the transition maps are given by pT1, T2q ÞÑ pT p +1 , T p +2 q. It is clear that the components of +∆˝ +j are naturally identified with the group µpj of pj’th roots of unity, with ζj P µpj identified +with the component on which T1 ´ ζjT2 “ 0, and hence the idempotents of the coordinate +ring of r∆˝2 are identified with a compatible system of such roots and hence with Zpp1q. +22 + +Let F‚ “ Ω‚ +∆˝ be the sheaf of BdR-differentials on the pro-´etale site, as defined in §2.3. +We let U “ tcu be our cover, and form the ˇCech complex C‚pU, F‚q and the associated +total complex L‚ “ L‚pF‚q. We will consider the cocycle sc “ logpT {rT 5sq. To be more +precise, [Sch13, Cor. 6.13] shows that the sequence +0 Ñ BdR Ñ OBdR +dÝÑ Ω1 +∆˝ Ñ 0 +(9) +is exact, and does so by showing that the map OB` +dR +ˇˇ r∆˝ +„ +ÝÑ B` +dR +ˇˇ r∆˝rXs defined by X ÞÑ +T b 1 ´ 1 b rT 5s is an isomorphism; here T 5 is defined as in [Sch13, §6]. One does this by +showing that B` +dR +ˇˇ r∆˝rXs admits the structure of an O∆˝ +ˇˇ r∆˝-algebra, satisfying T ÞÑ rT 5s`X +and compatible with the one on the quotient +B` +dRrrXss{pkerθq “ ˆO∆˝. +This then gives a natural map +´ +O∆˝ bWpκq WpˆO` +∆˝5q +¯ ˇˇˇ r∆˝ +„ +ÝÑ B` +dR +ˇˇ r∆˝rrXss +inducing the inverse of the map X ÞÑ T b 1 ´ 1 b rT 5s. +Using this description, one can define logpT {rT 5sq by applying the inverse of the isomor- +phism, computing logp1 ` X{rT 5sq (using the power series expansion) and then using the +isomorphism to translate the resulting expression back. The resulting function satisfies the +property that dplogpT {rT 5sqq “ dT {T , and that logpapT {rT 5sqq “ logpaq ` logpT {rT 5sq for +any non-zero a P BdR. +We note that we have two natural maps pi : r∆˝2 Ñ r∆˝ with i P t1, 2u. If one models +r∆˝2 as the space +r∆˝2 “ SpapkxT ˘p8 +1 +, T ˘p8 +2 +y{pT1 ´ T2q, OkxT ˘p8 +1 +, T ˘p8 +2 +y{pT1 ´ T2qq, +then these maps are given by T ÞÑ Ti. The component r∆˝2 +ζ‚ Ă r∆˝2 corresponding to the +sequence ζ‚ “ pζ1, ζ2, . . .q is then given by imposing the infinitely many relations +T 1{p +1 +“ ζ1T 1{p +2 +, +T 1{p2 +1 +“ ζ2T 1{p2 +2 +, +. . . +We we consider the restrictions (isomorphisms) pi,ζ‚ : r∆˝2 +ζ‚ +„ +ÝÑ r∆˝ associated to this compo- +nent, they are, on the level of the ring maps ri,ζ‚, related by the fact that r1,ζ‚pT 1{pkq “ +ζkr2,ζ‚pT 1{pkq for all k ě 1. +We now apply the differential d ` δ to the cocycle sc. The result is the direct sum of +dT {T , regarded as a differential form on r∆˝, and the difference logpT2{rT 5 +2sq ´ logpT1{rT 5 +1sq. +If we compute this latter difference on the component r∆˝2 +ζ‚ we obtain ´ logprζ‚sq, coming +from the fact that rT 5 +1s “ rζ‚srT 5 +2s. We thus have that pd`δqpscq “ dT {T ´t, where t is the +section of BdR +ˇˇ r∆˝2 whose value on each component r∆˝2 +ζ‚ is logprζ‚sq. Our conclusion is that +Proposition 3.1. In the ˇCech complex associated to the sheaf Ω‚ +∆˝ and the cover U “ tcu +the cycles dT {T and t are cohomologous. +Corollary 3.2. The class of dT {T is non-zero in the cohomology of the complex Ω‚ +∆˝. +23 + +Proof. Using the exactness of (9), it suffices to check the class of t is non-zero regarded as an +element of H1p∆˝ +p´et, BdRq. Note that because our covering c is perfectoid, the cohomology +of BdR in positive degree vanishes on this cover (see [Sch13, Theorem 6.5]), which means we +can compute these groups using the above ˇCech complex as a consequence of Corollary 2.4. +Thus we are asking whether there is an element s of BdR +ˇˇ r∆˝ such that s +ˇˇ +p1,ζ‚ ´ s +ˇˇ +p1,ζ‚ is +a constant in Fil1B` +dRzFil2B` +dR for each choice of ζ‚, where the filtration is defined as in +[Sch13, Definition 6.1]. Because the restriction maps preserve the filtration it suffices to +assume s P Fil1B` +dR, and then to check that this is not even possible after passing to the +quotient Fil1B` +dR{Fil2B` +dR – ˆO∆˝. The two restrictions are related by an automorphism of +r∆˝2 +ζ‚ inducing an automorphism of BdR +ˇˇ r∆˝2 +ζ‚ and hence of pOr∆˝2 +ζ‚ . But one easily checks that +this automorphism, which is induced by scaling p-powers of T by the corresponding elements +of ζ‚, cannot shift any function by a non-zero constant. +3.2 +Evaluation Functionals +We now consider the more general setting where we have a product of annuli ∆a,b “ p∆˝qaˆ +∆b´a, corresponding to the adic space +SpapkxT ˘1 +1 , . . . , T ˘1 +a , Ta`1, . . . , Tby, OkxT ˘1 +1 , . . . , T ˘1 +a , Ta`1, . . . , Tbyq. +In this section we will view all spaces, including algebraic varieties, as adic spaces over +Spapk, Okq; in particular we consider the multiplicative group Gm and the affine line A1 as +adic spaces. We wish to define certain “evaluation functionals” ˆα˚ +a,b : Hap∆a,b +p´et, ˆZppaqq Ñ +Zppaq and study their relationship with the p-adic Hodge comparisons and our calculation +in the previous section. We will actually have some freedom in the definition, as we will only +require that ˆα˚ +a,b has the “right” behaviour on Kummer-type covers, as we now explain. +By a Kummer-type cover of ∆a,b we mean a cover obtained by pullback from a finite +´etale cover of V a,b “ Ga +m ˆ Ab under the natural map ∆a,b Ñ V a,b. We have a natural +induced map HapV a,b +pf´et, ˆZppaqq Ñ Hap∆a,b +p´et, ˆZppaqq induced by the map ∆a,b +p´et Ñ V a,b +pf´et of sites, +and we will denote by Ia,b its image. We note that the etale cohomology of V a,b may be +computed with only finite ´etale covers (products of tori are Kpπ, 1q spaces), so this agrees +in particular with the image of HapV a,b +p´et , ˆZppaqq. +Lemma 3.3. For all k, the map HkpV a,b +pf´et, ˆZpq Ñ Hkp∆a,b +p´et, ˆZpq is injective. +Proof. Both the cohomology of V a,b and the cohomology of ∆a,b are generated by the +cohomology in degree one by the rigid-analytic K¨unneth formula (see [Ber93, 7.7.3] for +a reference in the equivalent Berkovich setting). +As the K¨unneth maps are natural in +the underlying site, it suffices to show that the maps H1pV a,b +pf´et, ˆZpq Ñ H1p∆a,b +p´et, ˆZpq in +degree one are injective. From our discussion in §2.5 we may identify these maps with the +natural maps Hompπ1 +´etpV a,bq, Zpq Ñ Hompπ1 +´etp∆a,bq, Zpq, so we are reduced to showing that +π1 +´etp∆a,bq Ñ π1 +´etpV a,bq is surjective. Using the natural factorization, this reduces to the +same statement for π1 +´etp∆˝q Ñ π1 +´etpGmq. By [Sta20, Lemma 0BN6], we reduce to showing +that every connected finite ´etale cover of Gm pulls back to a connected finite ´etale cover +of ∆˝, which is obvious. (Note that the rigid analytic finite ´etale coverings of Gm are just +those of Kummer type as a consequence of the rigid analytic Riemann existence theorem +[L¨ut93].) +24 + +Using the Lemma, we will define ˆα˚ +a,b as follows. We will first define a functional α˚ +a,b +on HapV a,b +pf´et, ˆZppaqq, which induces a functional on Ia,b. We will then choose a splitting +Hap∆a,b +p´et, ˆZppaqq “ Ia,b ‘ Ja,b, and define ˆα˚ +a,b by extending by zero. We begin by fixing +a distinguished system tζ‹ +i uiě1 of p-power roots of unity of k. This induces the following +data: +- A p-adic period t “ logprζ‹ +‚sq P BdR. +- Via the automorphisms T ÞÑ ζ˚ +i T , an element, denoted α, of the pro-p fundamental +group π1 +´etpGmqppq; note that by the rigid analytic Riemann existence theorem [L¨ut93] +the covers T ÞÑ T pi described above exhaust the connected finite ´etale coverings of +Gm with degree a power of p, even on the adic finite ´etale site. +- A map α˚ : H1pGm,p´et, Zpp1qq Ñ Zpp1q. This uses the identification H1pGm,p´et, Zpp1qq » +Homcontpπ1 +´etpGmq, Zpp1qq and is defined by evaluation on α. +- Using the identification H1pGm,p´et, Zpp1qq » H1pGm,fp´et, Zpp1qq, a map, also denoted +α˚, on the latter cohomology group. +- A map ˆα˚ : I1,0 Ñ Zpp1q, obtained by pulling back α˚ along the map I1,0 Ñ +H1pGm,fp´et, Zpp1qq. +- A linear functional +ˆα˚ +BdR : I1,0 b BdR Ñ BdR, +which is defined by evaluating on α and extending scalars along the map Zpp1q ãÑ BdR +given by logpr´sq. +From the coverings T ÞÑ T pj of ∆˝ in the previous section we obtain torsors Lj on ∆˝ +p´et +and a class rL8s “ plimjrLjsq P I1,0. To complete our definition we will need the following +facts, both of which are formal verifications. +Lemma 3.4. Fix a, b ě 0, and let ˆα˚ +i : H1pV a,b +pf´et, Zpp1qq Ñ Zpp1q be the pullback of ˆα˚ along +the i’th projection H1pV a,b +pf´et, Zp1qq Ñ H1pGm,pf´et, Zp1qq induced by inclusion of factors. Then +the map +ˆα˚ +a,b :“ ˆα˚ +1 b ¨ ¨ ¨ b ˆα˚ +a : H1pV a,b +pf´et, Zpp1qq b ¨ ¨ ¨ b H1pV a,b +pf´et, Zpp1qq +loooooooooooooooooooooooooomoooooooooooooooooooooooooon +“HapV a,b +pf´et ,Zppaqq +Ñ Zppaq +takes the value tζ‹ +‚ub¨ ¨ ¨btζ‹ +‚u on the element rL8,1sb¨ ¨ ¨brL8,as, where rL8,is is induced +from rL8s by the inclusion H1pGm,pf´et, Zp1qq Ñ H1pV a,b +pf´et, Zp1qq coming from the projection +onto the i’th factor. +Proof. Immediate from the definitions and functorality of cohomology. +Lemma 3.5. Denote by ε the natural comparison map +ε : Hap∆a,b +p´et, Zppaqq b BdR Ñ Hap∆a,b +p´et, Ω‚ +∆a,bq. +Then ε maps the element prL1,8s Y ¨ ¨ ¨ Y rLa,8sq to the element dT1{T1 ^¨ ¨ ¨^dTa{Ta, and +this element is non-zero in Hap∆a,b +p´et, Ω‚ +∆a,bq. +25 + +Proof. If we define cup product using ˇCech cohomology on both sides and also on the +cohomology of the sheaf BdR, the first part of the result (i.e., ignoring the non-zeroness of +dT1{T1^¨ ¨ ¨^dTa{Ta) will follow from the compatibility of cup product with the differential +graded structure on Ω‚ +∆a,b (as discussed in §2.2.2), as well as our result Lemma 3.4 above, +as long as we can show that rL1,8s maps to t1, where t1 is the class in H1p∆a,b +p´et, BdRq +obtained as the image of t under the map H1p∆˝ +p´et, BdRq Ñ H1p∆a,b +p´et, BdRq coming from +projection. From functoriality it suffices to show that rL8s maps to t under the natural map +H1p∆˝ +p´et, ˆZpp1qq Ñ H1p∆˝ +p´et, BdRq. Working on the level of ˇCech complexes with respect +to the perfectoid cover c as in §3.1, the gluing data for the torsor rL8s assigns the system +of compatible roots ζ‚ to the component r∆˝2 +ζ‚. As the map Zpp1q Ñ BdR is induced by +pζiqiě1 ÞÑ logprζ‚sq, one obtains the cycle t as desired. +For the non-zeroness of dT1{T1 ^ ¨ ¨ ¨ ^ dTa{Ta we may argue as follows. We may first +reduce to the case of b “ 0 by using the natural map Hap∆a,b +p´et, BdRq Ñ Hap∆a,0 +p´et, BdRq +coming from inclusion. Because Proposition 3.1 and the cup-product compatibility implies +that the class of dT1{T1 ^ ¨ ¨ ¨ ^ dTa{Ta is cohomologous to the class of t1 Y ¨ ¨ ¨ Y ta, we +may reduce to the same statement for the latter. Using the Kunneth formula to compute +the ´etale cohomology of ∆a,0 this then reduces to showing that each ti is non-zero, which +is what we showed in the argument of Corollary 3.2. +From the fact that ε maps a class generating Ia,b to a non-zero element, it follows that +we may choose a splitting Hap∆a,b +p´et, ˆZppaqq “ Ia,b ‘ Ja,b such that Ja,b contains the kernel +of Hap∆a,b +p´et, ˆZppaqq Ñ Hap∆a,b +p´et, BdRq. We then define ˆα˚ +a,b on all of Hap∆a,b +p´et, ˆZppaqq by +extending by zero. Finally, we define: +Definition 3.6. The map ρ´1 : Hap∆a,b, Ω‚ +∆a,bq Ñ Ia,b b BdR is the unique BdR-linear +map which sends the class of dT1{T1 ^ ¨ ¨ ¨ ^ dTa{Ta to the class rL1,8s Y ¨ ¨ ¨ Y rLa,8s. +3.3 +Extending to an ambient variety +We now suppose that Y is a smooth proper algebraic variety over Cp, and that we have an +adic neighbourhood ∆a,b Ă Y ad. There are two ways in which one could imagine extending +the functional ˆα˚ +a,b to the cohomology of Y : by pulling back along HapY ad +p´et, Zppaqq Ñ +Hap∆a,b +p´et, Zppaqq, and by pulling back along ρ´1 and HapY ad, Ω‚ +Y adq Ñ Hap∆a,b, Ω‚ +∆a,bq. We +now check that these two extensions are compatible with the p-adic period isomorphism, +meaning that we can consistently identify the extensions without issues. We note that by +Proposition 2.5 we may compute coherent cohomology on the pro-´etale site. +From the natural morphisms ˆZppaq Ñ BdR, BdR Ñ Ω‚ +p´q and Ω‚ +p´q Ñ Ω‚ +p´q of sheaves on +the pro-´etale site, one obtains the following diagram: +HapY ad +p´et, ˆZppaqq b BdR +HapY ad +p´et, BdRq +HapY ad +p´et, Ω‚ +Y adq +HapY ad +p´et, Ω‚ +Y adq b BdR +Hap∆a,b +p´et, ˆZppaqq b BdR +Hap∆a,b +p´et, BdRq +Hap∆a,b +p´et, Ω‚ +∆a,bq +Hap∆a,b +p´et, Ω‚ +∆a,bq b BdR +„ +„ +„ +σ +„ +ρ´1 +26 + +All the squares in the diagram are commutative by general cohomological principles ex- +cept (a priori) possibly the ones involving ρ´1. That the middle horizontal rightward arrows +are isomorphisms is [Sch13, 6.13]. That the upper left horizontal arrow is an isomorphism +is [Sch13, 8.4], and that the upper right horizontal arrow is shown in the proof of [Sch13, +7.11]. +Lemma 3.7. After inverting the isomorphisms in the above diagram, the functional ˆα˚ +a,b +admits a pullback ˆγ˚ to any object in the top row of the diagram, and the resulting functional +is independent of the path along which one takes the pullback. +Proof. We check the consistency of the pullback to HapY ad +p´et, Ω‚ +Y adqbBdR, as this contains all +the essential difficulties. One may easily check that if x P HapY ad +p´et, Ω‚ +Y adqbBdR is an element +in the top right group, then it admits a unique image in every object in the diagram except +possibly the group Hap∆a,b +p´et, ZppaqqbBdR in the bottom left. This in particular means that +the difference between any two images of x inside Hap∆a,b +p´et, Zppaqq b BdR lies in the kernel +of σ. It then suffices to check that the kernel of σ lies in the kernel of ˆα˚ +a,b, but this is by +construction since ker σ Ă Ja,b b BdR. +Corollary 3.8. The induced functional ˆγ˚ on the cohomology of Y is independent of the +summand Ja,b containing the kernel of σ chosen. +Proof. Since, as a consequence of Lemma 3.7, the functional may be defined by pulling back +first along ρ´1 and then the right vertical arrow, it only depends on the the restriction of +ˆα˚ +a,b to Ia,b Ă impρ´1q, and hence is independent of the choice of summand Ja,b whose +intersection with Ia,b is zero. +4 +Realizing G-functions +We now give our main technical result, which will give a cohomological interpretation of +Andr´e’s G-functions at all places of our fixed number field K. We recall the setup. We +have a projective family f : X Ñ S over K of relative dimension ν ´ 1, with S an algebraic +curve, and which has geometrically connected fibres. There is a fixed K-point s0 P S such +that the family f 1 : X1 Ñ S1 with S1 “ Szts0u is smooth. The fibre E Ă X above s0 is +assumed to have simple normal crossings. We have an affine Zariski open subset U Ă X with +coordinates z1, . . . , zν trivializing Ω1 +U, a uniformizing parameter s at s0, all such that the +map fU “ f +ˇˇ +U is surjective and s ÞÑ z1 ¨ ¨ ¨ zµ for some µ ě 2. We let H “ Rwf˚Ω‚ +X{Splog Eq +where w “ µ ´ 1 and set H1 “ H +ˇˇ +S1. We additionally assume for simplicity that s is defined +on all of S and that ds trivializes Ω1 +S, which becomes true after removing finitely many +points from S. This in particular implies that S is affine and H may be identified with its +module of global sections. Finally, write HU “ Rwf˚Ω‚ +U{SplogpE X Uqq, and note that there +is a natural restriction map H Ñ HU. +We have a commuting diagram +U +Spec Krx1, . . . , xνs +S +Spec Krts +px1,...,xνqÞÑpz1,...,zνq +f +ˇˇˇ +U +tÞÑx1¨¨¨xµ +tÞÑs +. +(10) +27 + +We will denote the top arrow by g, the bottom arrow by u, and write j for the ar- +row on the right. +Because g is ´etale, its image is an open K-algebraic subvariety V Ă +Spec Krx1, . . . , xνs. Write T Ă Spec Krts for the image of V . +By a scaling of the coordinates pz1, . . . , zνq we mean coordinates pλz1, . . . , λzνq for some +λ P Kˆ. If one replaces the coordinates pz1, . . . , zνq with a scaling pλz1, . . . , λzνq and the +coordinate s with λµs the diagram (10) continues to commute; when say “replace pz1, . . . , zνq +with a scaling” we mean to make such a change of coordinates. +Lemma 4.1. Choose a K-point q P g´1p0q. After replacing pz1, . . . , zνq with a scaling by +N ´1, where N P Z, the following property holds: for any embedding ι : K ãÑ Kv with v +a finite place of K, the map g is invertible in the open ball of v-adic radius 1 around 0 P +Spec Krx1, . . . , xνs onto a neighbourhood containing q. (In particular, this ball is contained +inside V .) This property continues to hold if N is replaced by some N 1|N. +Proof. The idea is that one can write down a formal inverse to the map of germs pU, qq Ñ +pV, 0q and have this inverse converge at each finite place in the desired neighbourhood after +scaling coordinates. More explicitly, let us begin by embedding the affine variety U as a +closed subvariety of SpecKry1, . . . , yσs defined by polynomials p1, . . . , pℓ P Kry1, . . . , yσs. +After translation we may identify q with the origin in Spec Kry1, . . . , yσs. The map g is then +given by component polynomials g1, . . . , gν P Kry1, . . . , yσs with no constant terms. The +formal inverse A we wish to compute is then given by power series +Aipx1, . . . , xνq “ +ÿ +J +Ai,JxJ +(11) +for 1 ď i ď σ, where J ranges over the set C of all appropriate compositions of integers ě 0 +and we use multi-index notation to exponentiate the vector x “ px1, . . . , xνq. +The fact that g ˝ A “ id and p ˝ A “ 0 gives a system of linear equations for each +coefficient appearing in each Ai in terms of coefficients of the polynomials gj and pk. Let +us suppose we can solve this system for a formal function A, and that this formal function +converges in the open ball of v-adic radius 1 around 0 for all finite places of K outside of +a finite set Σ. Then it will suffice to scale the coordinates px1, . . . , xνq by multiplying each +xi by a sufficiently large integer N whose prime factors all lie above places of Σ: indeed, +doing so does not affect the radius of convergence for finite places outside of Σ, and the +radius of convergence of the resulting power series at a place v P Σ will increase by a factor +of 1{|N|v and hence be greater than 1 as soon as |N|v is small enough. We are reduced to +the following more formal fact: +Lemma 4.2. Suppose that we have formal power series (11) with coefficients in a number +field K that are defined by the property that Bi ˝ A “ Ci, where B1, . . . , Bc, C1, . . . , Cc P +Kry1, . . . , yσs are finitely many polynomials with coefficients in K. Then A converges in the +open v-adic ball of radius 1 away from finitely many places v of K. +Proof. To understand the system of equations defined by these polynomials we recall the +multivariate Fa`a di Bruno formula [CS96], which says that the derivatives of a composition +C “ B ˝ A of functions given by power series centred at zero are given by +pBJCqp0q “ +ÿ +1ď|λ|ď|J| +pBλBqp0q +|J| +ÿ +s“1 +ÿ +CspJ,λq +J! +s +ź +j“1 +rAℓjp0qskj +pkj!qrℓj!s|kj| , +(12) +where we have made use of the following notation: +28 + +- the vectors λ and kj come from Zσ +ě0 and the vectors J and ℓj come from Zν +ě0; +- for any vector u “ pu1, . . . , urq P pZě0qr we have |u| “ u1 ` ¨ ¨ ¨ ` ur; +- we have +CspJ, λq “ +! +pk1, . . . , ks; ℓ1, . . . , ℓsq : +|ki|ą0, +0ăℓ1㨨¨ăℓs +řs +i“1 ki“λ and řs +i“1 |ki|ℓi“J +) +, +- for vectors u “ pu1, . . . , urq and u1 “ pu1 +1, . . . , u1 +rq, the symbol u ă u1 means that one +of the following conditions holds: +(i) |u| ă |u1|; +(ii) |u| “ |u1| and u1 ă u1 +1; or +(iii) |u| “ |u1|, u1 “ u1 +1, . . . , uk “ u1 +k and uk`1 ă u1 +k`1 for some 1 ď k ă r; +- the notation Aℓ for ℓ “ pℓ1, . . . , ℓνq means pBℓA1, . . . , BℓAνq; and +- for a vector u “ pu1, . . . , urq, we have u! “ u1! ¨ ¨ ¨ ur!. +The terms on the right-hand side of the equation (12) involving the components of AJp0q +are then +J! +σÿ +i“1 +pBiBqp0qAi,J +J! . +(13) +Now let us suppose that B is a polynomial over K, and that pBJCqp0q “ 0; this is the +case for B P tB1, . . . , Bcu and for |J| sufficiently large. Then at a finite place v, for all but +finitely many v, the norms |pBλBqp0q|v are ď 1 if |λ| ď deg B, and equal to 0 if |λ| ą deg B. +In particular, the equation (12) induces the following linear equation for AJ{J!, at least +when |J| ą deg B: +σÿ +i“1 +pBiBqp0qAi,J +J! +“ ´ +ÿ +2ď|λ|ďdeg B +pBλBqp0q +|J| +ÿ +s“1 +ÿ +CspJ,λq +˜ s +ź +j“1 +1 +pkj!q +¸ ˜ s +ź +j“1 +rAℓjp0qskj +rℓj!s|kj| +¸ +. (14) +We note that there are only finitely many possibilities for the coefficients śs +j“1 +1 +pkj!q which +appear in (14), and these possibilities are independent of J and depend only on deg B. +Indeed, because only finitely many λ ever occur in all such terms, the equation řs +i“1 ki “ λ +together with the condition |ki| ą 0 for all i ensures that only finitely values for the tuple +ps, k1, . . . , ksq ever appear, and hence there are only finitely many śs +j“1 +1 +pkj!q which appear. +After excluding a further finite set of norms | ¨ |v, we may assume that all these coefficients +have norm 1, and the non-zero coefficients of B and its derivatives also have norm 1. Letting +B range over the finitely many polynomials in the set B P tB1, . . . , Bcu, we have proven +that: +For |J| ą deg B, and for all but finitely many places v, the vector +1 +J!AJ is the +solution to a system of linear equations +M AJ +J! “ N, +(15) +29 + +where M is a c ˆ σ matrix, independent of J, whose non-zero entries all have +unit v-norm, and N is a vector with c entries whose norms are at most +max +s,CpJ,λq +››››› +s +ź +j“1 +rAℓjp0qskj +rℓj!s|kj| +››››› +v +. +(16) +We may assume c ě σ or else the solution is not uniquely determined, and then c “ σ by +choosing a linearly-independent subset of the rows of M. We now use this to show that, after +possibly throwing out a further finite set of places v, one has }Ai,J{J!}v ď 1 for all i and all +J. We prove this by induction, starting from the case where |J| “ maxtdeg B1, . . . , deg Bcu; +we note that the base cases with smaller |J| can be assumed after removing a further finite +set of places. Removing a further finite set of places to ensure that } detpMq}v “ 1, we +may use Cramer’s rule and the equation (15) to compute the entries of Ai,J{J! as quotients +detpM 1q{ detpMq, where M 1 is a matrix obtained from M by replacing a column with the +vector N. By induction the bound (16) is at most 1, so it follows that detpM 1q{ detpMq, +and hence the entries of Ai,J{J!, have v-adic norm at most 1. +Theorem 4.3. Scale coordinates as in Lemma 4.1, and fix a point q in the common vanish- +ing locus z1 “ ¨ ¨ ¨ “ zµ “ 0 with image s0 P S. Then q induces a K-linear map, compatible +with base change along a finite extension L{K, +Γ : HpSq Ñ Krrtss, +whose image consists of G-functions. These G-functions satisfy the following two “realiza- +tion” properties: +(i) Fix an embedding ι : K ãÑ C, suppose that tωiuiPI is a subset of HpSq, and that +R ą 0 is a real number such that Γpωiqι has radius of convergence at least R for all +i P I. Denote by DR Ă San +ι +the component containing s0 of the complex analytic +neighbourhood defined by |s| ă R. Then for each point s1 P DRzts0u, there exists a +linear functional +γ˚ +1 : HwpXan +s1 , Zpwqq Ñ Zpwq, +such that if ρ is the natural isomorphism +HwpXan +s1 , Zpwqq b C „ +ÝÑ Hw +dRpXs1q, +then we have +1 +p2πiqw pγ˚ +1,C ˝ ρ´1qpωi,s1q “ pΓpωiqιqpups1qq +(17) +for all i P I. +(ii) Fix an embedding ι : K ãÑ Cp for some prime p, suppose that tωiuiPI is a subset of +HpSq, and that 1 ě R ą 0 is a real number such that Γpωiqι has radius of convergence +at least R for all i P I. Denote by DR Ă Sad +ι +the component containing s0 of the +adic neighbourhood defined by |s| ă R. Then for each closed point s1 P DRzts0u there +exists a neighbourhood ∆w,µ´ν +s1 +Ă Xad +s1 such that the linear functional +ˆγ˚ +1 : HwpXad +s1,p´et, ˆZppwqq Ñ Zppwq, +30 + +constructed from this neighbourhood as in §3.3 satisfies the property that +1 +tw pˆγ˚ +1,BdR ˝ ρ´1qpωi,s1q “ pΓpωiqιqpups1qq +(18) +for all i P I, where ρ is the natural isomorphism +HwpXad +s1,p´et, ˆZppwqq b BdR +„ +ÝÑ Hw +dRpXs1q b BdR. +Remark. The p-adic period t in the statement of (ii) is not to be confused with the coordinate +t in the diagram (10). +The remainder of this section is devoted to the proof of Theorem 4.3. +We begin by +constructing the map Γ. From functoriality, the sections tωiuiPI all have restrictions to the +sheaf RwfU,˚Ω‚ +U{S, which as we explained in §2.1 is represented by the cohomology in degree +w of the complex +0 Ñ OU Ñ Ω1 +U{SplogpE X Uqq Ñ ¨ ¨ ¨ Ñ Ωn +U{SplogpU X Eqq. +(19) +We may then further restrict to a formal neighbourhood of q and consider the complex +0 Ñ pOU,pqq Ñ pΩ1 +U{SplogpE X Uqqpqq Ñ ¨ ¨ ¨ Ñ pΩn +U{SplogpE X Uqqpqq, +(20) +and obtain a K-linear map +η : HpSq Ñ Cohomw ” +pΩ‚ +U{SplogpE X Uqqpqq +ı +. +As we saw in Lemma 2.2, the target of η is naturally a 1-dimensional free module over +pOS,ps0q, and so we may define Γ as the composition +HpSq +ηÝÑ Cohomw ” +pΩ‚ +U{SplogpE X Uqqpqq +ı +„ +ÝÑ pOS,s0 +uÝÑ pOSpec Krts,0 “ Krrtss. +Before turning to the proof of (i), we briefly explain why the image consists of G- +functions. The point is that, within the degree-w cohomology of the complex (19), the each +relative form ω is represented by h dz2¨¨¨dzµ +z2¨¨¨zµ , where h is a function algebraic over a rational +function field; in particular, the power series hq in the coordinates z1, . . . , zµ representing h +at q is algebraic over a rational function field. The calculation of Lemma 2.2, which we will +see again in the proof of (i), computes Γpωq as the µ-diagonal of hq, i.e., the power series in +one variable t “ z1 ¨ ¨ ¨ zµ obtained by keeping all terms a ze1 +1 ¨ ¨ ¨ zeµ +µ +with e1 “ ¨ ¨ ¨ “ eµ and +discarding the others. It is known (c.f. [And89, I, §4.2] and the discussion on pg. 965 of +[AB13]) that any function obtained in this way is a G-function, which Andr´e himself uses +in his proof in [And89, IX, §4.4]. +Proof of (i): We work entirely in the complex analytic category, and view the diagram (10) +in the complex analytic category using base-change along ι : K ãÑ C and analytifying. +Denoting by UR the fibre of f over DR, we obtain a complex: +0 Ñ OUR Ñ Ω1 +UR{DRplogpE X URqq Ñ ¨ ¨ ¨ Ñ Ωn +UR{DRplogpUR X Eqq. +(21) +If one restricts (21) to the stalk at q, one obtains a complex pΩ‚ +UR{DRqpqq whose formal +completion is naturally identified with the complexification of (20). Applying Lemma 2.2 to +31 + +this complex, one obtains, for each i, analytic functions Pi such that Pi +dz2¨¨¨dzµ +z2¨¨¨zµ +represents +the image of ωi,C in the cohomology Cohomw ” +pΩ‚ +UR{DRqpqq +ı +. As the Pi are obtained using +the same calculations that produced Γpωiqι, these functions agree at the formal level. +We now give a geometric interpretation of the Pi following [And89, IX, §4.4], as follows. +Fix a sufficiently small disk D Ă DR centred at s0 and a neighbourhood U Ă f ´1pDq +above D containing q such that the map g maps U isomorphically to a product of complex +analytic disks ∆ν, where xi is the coordinate on the i’th factor. The map f is then locally +identified with the map ∆ν Ñ ∆ given by t ÞÑ x1 ¨ ¨ ¨ xµ, and the special fibre with the locus +x1 ¨ ¨ ¨ xµ “ 0. We can choose a family ε2,t, . . . , εµ,t of small loops inside ∆ν, with εi,t a loop +around the divisor xi “ 0 inside the fibre ∆ν +t defined by x1 ¨ ¨ ¨ xµ “ t, and consider the +family of integrals +P 1 +iptq “ +1 +p2πiqw +ż +εt +ωi +ˇˇ +∆ν +t +where εt “ ε2,t ˆ ¨ ¨ ¨ ˆ εµ,t. Representing the restriction ωi +ˇˇ +∆ν +t as a function of the form +hi +dx2¨¨¨dxµ +x2¨¨¨xµ +with h a power series in x1, x2, . . . , xµ, one computes using the residue formula +that P 1 +iptq is the µ-diagonal of h, i.e., the function whose power series is obtained from that +of h by substituting in a te1 for all terms a xe1 +1 ¨ ¨ ¨ xeµ +µ with e1 “ ¨ ¨ ¨ “ eµ, and ignoring all +other terms. This is compatible with the calculation in the proof of Lemma 2.2, and we +have that P 1 +i “ Pi. +By taking the image of the cycles εt inside the fibres Xt, this calculation realizes the +functions Pi (and hence the functions Γpωiqι) as functions inside the image of the integration +pairing +Rwf˚Zp´wq b Rwf˚Ω‚ +X1{S1 Ñ OS +restricted to the neighbourhood D. By analytic continuation, the cycles εt extend to a +(possibly multi-valued) section rε of Rwf˚Z +ˇˇ +DR, and hence produce a (a priori possibly +multi-valued) function rPi inside OS +ˇˇ +DR after pairing with ωi. +But since rPi agrees with +the analytification of Γpωiqι near s0 and its power series representation converges on DR, +the analytic function rPi is single-valued, and gives an analytic realization of Γpωiqι. (In +the case where the ωi’s give a frame of the de Rham cohomology over DR, one may also +conclude that rεs is single-valued.) +To complete the proof of (i), it suffices to define, for each s1 P DR, the functional γ˚ +1 . +This we define as the evaluation functional +γ˚ +1,C : HwpXan +s1 , Cq Ñ C, +«ż +p´q +ω ÞÑ +ż +rεs1 +ω +ff +. +To make sense of this definition, we are using the canonical isomorphism HwpXan +s1 , Cq » +Hw +top-dRpXan +s1 q b C (with topological de Rham cohomology) to represent each element of +HwpXan +s1 , Cq as an integration functional, and then defining γ˚ +1,C by evaluating this functional +on rεs1. Note that because the section rε may in principle by multi-valued, this is also true of +the function γ˚ +1,C. However the equality (17), which amounts to the above observation that +the relative period with ωi is given by Γpωiq, holds regardless of which choice we make. To +complete the proof we observe that this functional descends to Zpwq. +32 + +Proof of (ii): As before, we will regard all spaces as analytic spaces over Cp using the fixed +embedding ι : K ãÑ Cp. +We wish to mirror the argument we made in the complex analytic case. The argument is +in many respects simplified by our choice of coordinates, which ensure that the property in +Lemma 4.1 holds over the neighbourhood DR since R ď 1. In particular if one writes D Ă S +for the component of the neighbourhood |s| ă 1 containing s0, and one writes U Ă f ´1pDq +for the neighbourhood around q defined by |pz1, . . . , zνq|ι ă 1, then we are guaranteed that +the family U Ñ D is isomorphic to the family V Ñ E obtained by restricting j to the open +ball of radius 1 around 0. +Now let us construct a subspace isomorphic to ∆w,µ´ν inside Xs1X U. Letting t1 “ ups1q, +we may instead construct such a subspace inside Vt1. Write R1 “ |t1|v ă R ď 1. The fibre +Vt1 is defined by the equation x1 ¨ ¨ ¨ xµ “ t1 inside the open ball defined by maxi|xi| ă 1, +and we may embed ∆w,µ´ν inside this neighbourhood via the map xi ÞÑ R1{µ +1 +Ti´1 for +2 ď i ď ν, and x1 ÞÑ R´w{µ +1 +t1{pT1 ¨ ¨ ¨ Twq. +This embedding identifies ∆w,µ´ν with the +closed neighbourhood Vw,ν´µ +t1 +inside Vt1 defined by +|xi| “ R1{µ +1 +for 1 ď i ď µ, +and +|xi| ď R1{µ +1 +for i ą µ. +We may then define ˆγ˚ +1 by pulling back the functional ˆα˚ +w,ν´µ on the cohomology of ∆w,ν´µ +constructed in §3.2, and then extend this to the entire fibre Xs1 as in §3.3. +Note that +Corollary 3.8 ensures that the extension is uniquely determined by the neighbourhood +Uw,ν´µ +s1 +isomorphic to Vw,ν´µ +t1 +» ∆w,µ´ν, and furthermore Lemma 3.7 ensures that the +resulting map is compatible with the p-adic comparison isomorphism associated to the fibre +Xs1. +It now suffices to verify the desired equality; let UR be the fibre of U Ñ D above DR. +As before, we may restrict each ωi to the cohomology in degree w of the complex Ω‚ +UR{DR +to obtain, via Lemma 2.2, sections hi +dz2¨¨¨dzµ +z2¨¨¨zµ , with hi a function on DR. We first observe +that the hi agree with the functions Γpωiqι ˝ u on DR: this is true by construction at the +formal level, and that implies that these functions agree in a small enough ball around s0, +so this follows by uniqueness of analytic continuation of rigid-analytic functions on affinoid +balls (c.f. [Meh19, Prop. 1.6.24]). Now the coherent cohomology group HwpUs1,p´et, Ω‚ +Us1 q is +computed by the cohomology of the na¨ıve de Rham complex Ω‚ +Us1 . The natural restriction +map +HwpUp´et, Ω‚ +U{DplogpE X Uqqq Ñ HwpUs1,p´et, Ω‚ +Us1q +is then represented by the natural map +Cohomw ” +Ω‚ +U{DplogpE X Uqq +ı +Ñ Cohomw ” +Ω‚ +Us1 +ı +, +and this map sends hi +dz2¨¨¨dzµ +z2¨¨¨zµ +to hips1q dz2¨¨¨dzµ +z2¨¨¨zµ . If we then evaluate this element with γ˚ +1,BdR, +one sees by combining the definition (3.6) of ρ´1 and the computation Lemma 3.4 that the +result is simply twhips1q, which completes the proof. +5 +Algebraic Relations on Functionals +We now introduce a new way to obtain relations on Andr´e’s G-functions at finite places, +which uses the explicitly p-adic nature of our construction. The first key observation is the +following: +33 + +Lemma 5.1. Let Y be a proper algebraic variety over the finite extension Kv of Qp, and +let ˆγ˚ : HapYCp,p´et, ˆZppaqq Ñ Zppaq be constructed as in §3.3. +Then the Galois group +GKv “ GalpKv{Kvq acts on ˆγ˚ through χ´a +cycl, where χcycl is the cyclotomic character. +Proof. Write η for the map HapYp´et, ˆZppaqq Ñ Hap∆a,b +p´et, ˆZppaqq. +We recall that ˆγ˚ was +constructed from a functional ˆα˚ +a,b on the GKv-invariant subspace Ia,b Ă Hap∆a,b +p´et, ˆZpq +after choosing an appropriate splitting Hap∆a,b +p´et, ˆZppaqq “ Ia,b ‘ Ja,b. Write ˆβ˚ +J for the +extension-by-zero of ˆα˚ +a,b with respect to this splitting, so that ˆγ˚ “ ˆβ˚ +J ˝ η. Note that by +Corollary 3.8, this is the same as ˆβ˚ +g¨J ˝η for any g P GK, where ˆβ˚ +g¨J is the extension-by-zero +of ˆα˚ +a,b with respect to the splitting Ia,b ‘ pg ¨ Ja,bq; here we use the fact that the kernel of +σ : Hap∆a,b +p´et, ˆZppaqq Ñ Hap∆a,b +p´et, BdRq is GKv-invariant. +Now given g P GK, the functional g ¨ ˆγ˚ is equal to pg ¨ ˆβ˚ +Jq˝η “ pg ¨ ˆβ˚ +g¨Jq˝η as the map η +is Galois-equivariant. The map g ¨ ˆβ˚ +g¨J is the extension-by-zero of g ¨ ˆα˚ +a,b with respect to the +splitting Ia,b ‘Ja,b. Given a vector v P HapYp´et, ˆZppaqq, with decomposition ηpvq “ wI `wJ +with respect to the splitting Ia,b ‘ Ja,b, we therefore have that +pg ¨ ˆγ˚qpvq “ pˆβ˚ +g¨J ˝ ηqpvq +“ pˆβ˚ +g¨JqpwIq ` 0 +“ pg ¨ ˆα˚ +a,bqpwIq +“ ˆα˚ +a,bpwIqχ´a +cylcpgq +“ ˆγ˚pvqχ´a +cylcpgq. +Note that to show that g acts on ˆα˚ +a,b via χ´a +cycl we have used the definition of ˆα˚ as a product +of individual maps ˆα˚ +i in Lemma 3.4, as well as the fact that as Galois representations we +have HompIa,b, Zppaqq » Zpp´aq; this is true since Ia,b » HaprGa +m ˆ Abs´et, Zppaqq » Zpp2aq +as Galois representations. +We now adopt a slightly more abstract perspective to produce relations on de Rham +coordinates of ˆγ˚. In the proposition below, the most typical application will be with Y +a variety defined over a number field K, and V´et “ HapYK,´et, Qppaqq, VdR “ HwpY, Ω‚ +Y q +its ´etale and de Rham cohomology groups, although the extra generality will be useful to +handle summands appearing in such cohomology groups as well. +Proposition 5.2. Let V´et be a BdR-admissible GKv-representation of weight 2a such that +VdR,Kv “ pV´et bQp BdRqGKv admits a K-structure VdR bK Kv » VdR,Kv, where K Ă Kv is +a finite extension of Q. Suppose that +- we have a GKv-invariant endomorphism τ : V´et Ñ V´et, whose de Rham realization is +defined over K; +- a functional ˆγ˚ : V´et Ñ Qppaq on which GKv acts through the ´a’th power of χcycl; +and +- the dimension k of the GKv-invariant subspace of +HompV´et, Qpp2aqq » HompV´et, Qppaqq bQp Qpp´aq +is less than the degree of the minimal polynomial of τ. +34 + +Then with respect to any choice of basis ω1, . . . , ωm for VdR, there is a K-algebraic relation +on the dual coordinates of ˆγ˚ +BdR of degree equal to k`1, which does not hold for the coordinates +of a generic functional on V´et. The relation depends only on the coordinates of τ in the basis +ω1, . . . , ωm. +Proof. It suffices to show that the set +ˆγ˚ +BdR, ˆγ˚ +BdR ˝ τ, . . . , ˆγ˚ +BdR ˝ τ k +(22) +is linearly dependent. Indeed, if we evaluate the vectors in this set on the basis ω1, . . . , ωm +we will obtain a sequence of vectors v1, . . . , vk`1 P Bm +dR such that vi is obtained by applying +a non-zero linear transformation to v1 whose matrix entries in the basis ω1, . . . , ωm lie in +K. But we may then get an K-algebraic relation on the coordinates of v1 by taking the +determinant of a square submatrix of rv1| ¨ ¨ ¨ |vk`1s. This relation has degree k ` 1 and +depends only on the coordinates of τ. That this relation does not hold generically follows +from the definition of minimal polynomial. +To show this linear dependence, we observe that τ is invariant under GKv, and therefore +the vectors (22) are all obtained from scalar extension of vectors in HompV´et, Qppaqq, and +these vectors are all invariant under the action of GKv on HompV´et, Qpp2aqq. But the space +of GKv-invariants on HompV´et, Qpp2aqq has dimension at most k by assumption. +Corollary 5.3. If in the setting of Proposition 5.2 the characteristic polynomial P of τ is +equal to its minimal polynomial, then one may assume that the relation is a product of linear +relations, with each linear factor defined over KF, with F the splitting field of P. +Proof. That the characteristic polynomial of τ is equal to its minimal polynomial means +that V ˚ +dR admits only finitely many τ-invariant subspaces, each of which is defined over the +splitting field of P. The proof of Proposition 5.2 shows that ˆγ˚ +BdR lies inside one of these +subspaces, so one may take a product of linear relations associated to each subspace. +Lemma 5.4. In the setting of Proposition 5.2, the dimension of the GKv-invariant subspace +of HompV´et, Qpp2aqq is at most the size of the first Hodge number of V ˚ +dR. +Proof. Letting Cv be the completion of Kv, the module HompV´et, Qpp2aqq becomes isomor- +phic to HompV´et b Cv, Cvq b Cvp´2aq after scalar extension, which the Hodge-Tate com- +parison shows is isomorphic to a sum À +i`j“´2a griV ˚ +dR bK Cvpj ´ 2aq. The GK-invariant +subspace necessarily maps to the summand with indices pi, jq “ p0, 2aq, and this summand +has dimension at most dimK gr0V ˚ +dR. +In the situation where the endomorphism τ appearing in Proposition 5.2 is defined over +a finite extension L of K, we will also want to control the degree of the extension rL : Ks, +for which the following fact, proven in [Pap22], will be useful: +Proposition 5.5. Suppose that Y is an algebraic variety defined over a number field K, +and that the absolute Hodge conjecture holds in degree a for cohomological endomorphisms +associated to Y . Then the algebra of absolute Hodge endomorphisms may be identified with a +subalgebra of EndpHapY, Ω‚ +Y qqL, where L{K is a finite Galois extension with degree bounded +only in terms of m “ dimK HapY, Ω‚ +Y q. +Proof. This is [Pap22, Prop. 5.1] and [Pap22, Prop. 5.2]. Note that the absolute Hodge +conjecture is only assumed for endomorphisms associated to Y in the proof. +35 + +6 +Height Bounds for Families over Curves +In this section we prove Proposition 1.16 and Theorem 1.17. This will require some addi- +tional setup due to an important subtly that occurs in applications of the G-function method +to bound heights on curves. To understand why, suppose that s is some uniformizing pa- +rameter on our curve S at s0 and our G-functions are, as in Theorem 4.3, obtained from +expanding periods near s0 in terms of s. Then if ξ P S is a special point and v is a place of +Kpξq which is relevant for spξq, it could be that ξ does not lie sufficiently close to s0; more +precisely, if D Ă A1 is a v-adic disk on which the G-functions are defined, ξ and s0 could lie +in different components of s´1pDq. What we need instead is to be in the situation where +every component of s´1pDq contains an appropriate degeneration point. This is the reason +for passing to the finite cover in Proposition 1.16(i). +6.1 +Setup +To give the covering we use the following lemma, proven in [DO22, Lemma 5.1]: +Lemma 6.1. Let C1 be an irreducible projective curve over a characteristic zero field K +with s0 P C1pKq a point. Then there exists a finite extension L{K, a smooth projective +curve C over L, a finite map c : C Ñ C1 +L, and a rational function s on C such that +- every zero ξ0 P CpLq of s is simple; +- every zero ξ0 P CpLq of s maps under c to s0; +- s : C Ñ P1 +L is a finite Galois covering (not necessarily ´etale). +In our setting we may complete S a smooth projective curve S, and apply the Lemma to +obtain a covering c : C Ñ SL. By pullback we obtain a family Xc´1pSq Ñ c´1pSq and simi- +larly for S1. After replacing K with L and S with c´1pSq we have reduced Proposition 1.16 +to the setting where: +(A) we have a parameter s : S Ñ A1 with only simple zeros; +(B) for each zero ξ0 of s the fibre Xξ0 is isomorphic to Xs0; and +(C) for any extension L of K, any place v of L, and any R ą 0, each connected component +of the analytic neighbourhood |s|v ă R in the associated analytic space SL,v (complex +or rigid) contains a zero of s. +To see the last property note that the number of components of |s|v ă R is bounded by the +size of a generic fibre of s, and the automorphism group of the covering s acts transitively +on such components because it acts transitively on fibres. +We now construct the set of G-functions Gwe will be working with. With an eye toward +future applications, and so that our setup here is easily reused, we will do it in the general +setting where we start with ℓ different order-w normal crossing singularities in the fibre Xs0, +where ℓ is arbitrary. We then construct = G“ tG1, . . . , GMu, which we regard as elements +of the formal power series ring Krrtss, as a union G “ Gξ1 \ ¨ ¨ ¨ \ Gξk of G-functions +associated to the elements of the fibre s´1p0q “ tξ1, . . . , ξku; we may assume these points +are defined over K after passing to a finite extension. Fixing a point s0 P s´1p0q, we will +36 + +further sub-divide the set Gs0 as a union Gs0 “ Gs0,q1 \ ¨ ¨ ¨ \ Gs0,qℓ, where q1, . . . , qℓ are +the points of the fibre Xs0 where the normal crossings occur; again we can assume these are +defined over K after passing to a finite extension. +Choosing a sufficiently small affine neighbourhood U qi Ă X containing qi we may find +functions zi1, . . . , ziν on U qi such that: +- the equation zi1 ¨ ¨ ¨ ziµ “ 0 defines Xs0 X U qi where µ “ w ` 1; +- the point qi lies in the locus zi1 “ ¨ ¨ ¨ “ ziν “ 0; +- the function s maps to z1i ¨ ¨ ¨ ziµ; and +- the map U qi Ñ Aν +K defined by pzi1, . . . , ziνq is ´etale. +Indeed, the image of s inside U qi necessarily vanishes on Xs0, hence lies in the ideal defining +Xs0 X U qi, which is necessarily locally principal generated near qi by a function zi1 ¨ ¨ ¨ ziµ +such that the locus defined by zi “ 0 is smooth near qi for 1 ď i ď µ and and the differentials +dzi1, . . . , dziµ are independent in a neighbourhood of qi. After shrinking U qi we may then +extend to a local ´etale coordinate system zi1, . . . , ziν at qi. With this setup we are now, +after removing finitely many points from S so that ds trivializes Ω1 +S, in the situation of the +setup in §4; we define Gs0,qi “ tΓpωi1q, . . . , Γpωimqu, where ωi1, . . . , ωim is a frame of HpSq +over S and Γ is as in Theorem 4.3. In general we will actually want to choose the frames +ωi1, . . . , ωim to be compatible with certain monodromy data, as this will make it easier to +analyze functional relations on the associated G-functions; we give more details in the next +section. +In general an application of Theorem 4.3 results in functions in a scaled parameter λs0is, +where λs0i “ N ´1 +s0i for Ns0i P Z. However because, by Lemma 4.1, any N 1 with N 1|Ns0i will +suffice, we may arrange for there to be some common N for all elements of G by taking +a product (or greatest common multiple) of the individual Ns0i’s. We then replace s with +N ´1s. +6.2 +Monodromy-Compatible Frames for H1 +Recall that H1 carries a K-algebraic Gauss-Manin connection ∇1 such that the (analytic) +flat sections of ∇1 may be identified with the sections of V1 +C under the Betti-de Rham +comparison. We now explain how to use ∇1 to choose frames for H1 which make it easy +to analyze functional relations on period matrices. Write H1a,b “ pH1baq b pH1˚qbb with +a, b, ě 0 integers for the associated tensor vector bundles. Write T Ă À +a,b H1a,b for the +subbundle of ∇1-flat tensors. We write AutpH1q for the bundle whose fibre above s P S1 is +GLpH1 +sq; note that AutpH1q acts naturally on each vector bundle H1a,b. We then consider +the subbundle M Ă AutpH1q which is defined fibrewise as the stabilizer of T. Fix a K-point +x P S1pKq. Then M is a Mx-torsor with a canonical section 1 : S1 Ñ M given by the +identity, so we obtain a trivialization M „ +ÝÑ Mx ˆ S1 over S1. (Note that it is clear from the +local system picture that M is an Mx-torsor complex analytically, and one can descend the +complex analytic torsor structure and hence the trivialization, c.f. [BT22, §2].) +Write FrpH1q for the frame bundle of H1, which is an algebraic vector bundle whose fibres +FrpH1qs for s P S1 may be identified with the invertible maps in the set HomκpsqpH1 +s, κpsqmq. +The bundle AutpH1q acts on FrpH1q fibrewise by pullback. Given any section F of FrpH1q +and any section t of H1a,b one obtains an element Fptq of pOm +S1qa,b “ pOS1qbm b pO˚ +S1qbb by +evaluation. +37 + +Definition 6.2. A monodromy-compatible frame of H1 is a frame ω1, . . . , ωm for which the +associated section F of FrpH1q satisfies the following property: for every a, b ě 0 and every +global flat tensor t P H1a,b the image Fptq in pOm +S1qa,b “ pOS1qbm b pO˚ +S1qbb is a constant +function. +Lemma 6.3. Monodromy-compatible frames exist locally in the ´etale topology on S1, and +given such a frame with associated section F and a section M of M the product M ¨ F is a +monodromy-compatible frame. +Proof. The second property is clear. For the first property we may start by fixing a point Fx +of FrpH1qx. We then consider the algebraic locus F Ă FrpH1q defined by the property that +for each section t of T and each point Fy P F above some y P S1 we have Fyptyq “ Fxptxq. +Then the fibrewise action of M on FrpH1q preserves F. +We now show that the natural map F Ñ S1 is flat. Indeed, we can replace K with C +and work locally on an analytic ball B inside S1, in which case we know that a monodromy- +compatible frame F which agrees with Fx on global flat tensors exists. The restriction F +ˇˇ +B +is then isomorphic to M +ˇˇ +B via the map pg, sq ÞÑ pg ¨ F, sq, and the map Mx ˆ B Ñ B is +clearly flat. Knowing that FÑ S1 is flat we may consider FÑ S1 to be an fppf covering of +S1, and observe that FˆS1 F is isomorphic to the trivial Mx-torsor MˆS1 F via the map +pg, Fq ÞÑ pF, g ¨Fq; in particular, Fis an fppf Mx-torsor. For smooth algebraic groups every +fppf torsor is an ´etale torsor by [hmb], so the result follows. +Remark. Nothing in this subsection relied on the fact that S1 is a curve. +6.3 +Proof of 1.16 +From the results of the previous section we will assume that the frames of H used to define +the functions in G restrict to monodromy-compatible frames of H1. There is no harm in +this, as we are allowed to remove finitely many points from S1, and we could have started +the argument by replacing the family X Ñ S with an ´etale base-change. +Choose ξ P S and a finite place v of Kpξq which is relevant for ξ. Let R be the minimum +v-adic convergence radius of the functions in G. Then in particular ξ lies in some component +DR of the neighbourhood of Sad +Kpξqv defined by |s|v ă R. By (C) above, necessarily such a +component must contain a point s0 in the fibre s´1p0q. We may then identify our neighbour- +hood DR with the one in the statement of Theorem 4.3, and work with the G-functions in +the set Gs0 “ Gs0,q corresponding to s0; we note that any relation satisfied by G-functions +in this set can be interpreted as a relation satisfied by the G-functions in G. +Note that, by assumption, the G-functions in the set Gs0 “ th1, . . . , hmu are not all +constant; here hi “ Γpωiq. +(This was assumed before we passed to the finite covering, +but can be reinterpreted as a claim about the functionals γ˚ +1 and ˆγ˚ +1 of Theorem 4.3 in a +neighbourhood of s0 and so continues to hold after passing to the covering.) +Write π : V1 +ξ Ñ W for the Hodge-theoretic projection. By assumption, this projection is +induced by an algebraic cycle class, and so has cohomological realizations in both ´etale and +algebraic de Rham cohomology, compatibly with the comparison isomorphisms. In partic- +ular, the image of π corresponds to a summand W´et Ă HwpXξ,´et, Qppwqq and a summand +WdR Ă HwpXξ, Ω‚ +Xq which correspond under the p-adic Hodge comparison maps. +Possible vanishing of ˆγ˚ on W´et: We first consider the situation where the restriction +ˆγ˚ˇˇ +W´et vanishes. From the compatibility with the p-adic Hodge comparison, this means that +38 + +ˆγ˚ +BdR vanishes when restricted to WdR. The subspace WdR Ă HwpXξ, Ω‚ +Xξq is defined over +an extension L of Kpξq which, by Proposition 5.5, has degree bounded independently of ξ. +Let ω P WdR be a vector expressed as an L-linear combination ω “ ř +i aiωi,ξ. Applying +Theorem 4.3(ii) we find that ř +i aihipspξqq “ 0. Thus, we will be in the situation where we +can apply the Hasse principle for G-functions if we can show that the functions h1, . . . , hm +do not satisfy any non-trivial linear relations at the functional level. +Ruling out Functional Relations: The statement that h1, . . . , hm are linearly indepen- +dent as functions can be interpreted as a statement about the associated formal power series +at s0, and deduced from the same statement about the associated complex analytic power +series where we view K as a subfield of C. In this setting the functionals γ˚ +1 of Theorem 4.3(i) +glue into a section (denoted by the same notation) of the degree w homology local system +over a punctured neighbourhood of s0 in SpCq, and it will suffice to show that the germs of +h1, . . . , hm at some point s1 sufficiently close to s0 do not satisfy a non-zero linear relation. +After some setup, we will see that this is a consequence of the Ax-Schanuel theorem for +periods, recently proved in [BT22] as an application of a general theorem of [BSCFN21]. +To set things up, fix a small ball B around s1 and extend γ˚ +1 to a global frame γ˚ +1 , . . . , γ˚ +m +of the dual local system rV1ˇˇ +Bs˚. We write γ1, . . . , γm for the basis of V1ˇˇ +B dual to γ˚ +1 , . . . , γ˚ +m. +We consider the analytically-varying period matrix M “ Mij on B defined by the formula +Mij “ γ˚ +i pωjq; note that pM11, . . . , M1mq “ ph1, . . . , hmq. We now give a description of +the Zariski closure of MpBq Ă GLmpCq using [BT22, Theorem 1.1]. Because the frame +ω1, . . . , ωm is monodromy-compatible, flat sections of tensor powers of H1 are constant in +this frame, and hence MpBq lies inside the Ms1-torsor in GLmpCq consisting of the set of +matrices which send flat tensors of H1 (in the ω1, . . . , ωm coordinates) to flat tensors of +V1 (in the γ1, . . . , γm coordinates). Let P Ă GLmpCq be the connected component of this +torsor containing the image of M, which is a torsor for the identity compoent M˝ +s1. The +group M˝ +s1 is naturally identified with the algebraic monodromy-group of the variation of +Hodge structures V1, and it is then a consequence of [BT22, Theorem 1.1] that +MpBq +Zar “ P. +Now suppose the functions h1, . . . , hm satisfy a linear relation ř +i aihi “ 0 in a neighbour- +hood of s1, where ai P C are constants. This relation can be interpreted as a relation on the +first row of points in GLmpCq, and from the above one learns that it necessarily vanishes on +all of P. In particular, one learns that if g P M˝ +s1 is any element then ř +i aiγ˚ +1 pg ¨ ωi,s1q “ 0. +Interpreting this dually via the action of the (the Betti realization of) the group M˝ +s1 on the +local system V1˚, one learns that the vector γ˚ +1,s1 lies inside a proper monodromy-invariant +subspace. Since γ˚ +1,s1 is rational, this means that V1˚ admits a non-trivial rational summand. +Via a choice of polarizing form one knows that V1˚ is abstractly isomorphic to V1, hence V1 +also admits a non-trivial rational summand. Note that any summand of the underlying local +system of V1 is automatically a summand on the level of variations of Hodge structures as +a consequence of the Theorem of the Fixed Part. This contradicts the assumed simplicity +of V1, so we conclude that no functional linear relations on the period functions h1, . . . , hm +exist. +Relations in the non-vanishing case: As W is a CM Hodge structure, its endomor- +phism algebra is a field, hence has a primitive element ϕ. In particular, the characteristic +polynomial P of ϕ is equal to its minimal polynomial, and this is true regardless of the +cohomological realization of ϕ chosen. Combining Corollary 5.3 and Lemma 5.4, we obtain +a linear algebraic relation R on the dual coordinates of the restriction ˆγ˚ˇˇ +W´et with respect +39 + +to a basis of WdR; note here that W being a CM Hodge structure implies that it has more +than one non-zero Hodge number. If one expresses this basis as a linear combination of the +basis vectors ω1, . . . , ωm, one then obtains a linear relation on the dual coordinates of ˆγ˚ in +the basis ω1, . . . , ωm. This relation is defined over KFL, where F is the splitting field of P +and L is the field of definition of the de Rham realization of π. Moreover if one includes all +factors of the relation given to us by the statement of Corollary 5.3, the resulting relation is +defined over KL and holds at all finite places as it depends only the de Rham coordinates +of ϕ. +Conclusion: We have produced, by the arguments in the vanishing and non-vanishing cases, +linear relations Rvan,1, . . . Rvan,j and Rnv on the coordinates of ˆγ˚ +1 in the basis ω1,ξ, . . . , ω1,ξ +determined by π and ϕ and defined over a field KLF. To obtain the conclusion we just have +to argue that the relation obtained as the product of all Galois conjugates of Rvan,1, . . . Rvan,j +and Rnv over Kpξq has degree bounded independently of ξ. (Note that this replacement +does not affect whether the G-functions satisfy this relation on the functional level; if the +G-functions satisfied a polynomial relation with linear factors they would satisfy a linear +relation, and our above argument rules out this possibility.) This is a matter of bounding +rL : Kpξqs and rF : Qs independently of ξ. But the field F is a splitting field of a polynomial +of bounded degree, and the degree of rL : Kpξqs is bounded by Proposition 5.5. +6.4 +Proof of 1.17 +Repeat the setup in §6.1 using monodromy-compatible frames, except now with our set Gof +G-functions also containing those G-functions associated to the normal crossing intersection +at q1. Repeat the argument of Proposition 1.16 to establish relations exist at relevant finite +places. +We consider a point ξ P S and a relevant infinite place v for spξq. Let R be the minimum +of the convergence radii of the elements in G. As before, using property (C), we can find a +component DR of the complex analytic neighbourhood defined by |s|v ă R which contains +both ξ and a degeneration point s0. We write ph1, . . . , hmq and ph1 +1, . . . , h1 +mq for the G- +functions associated to the two normal crossing singularities in the fibres Xs0, which we +regard as complex analytic power series using the embedding v. +We now construct a Kpξq-algebraic relation on the values +h1pspξqq, . . . , hmpspξqq, h1 +1pspξqq, . . . , h1 +mpspξqq +(23) +We denote by γ˚ +1 and γ1˚ +1 the functionals given to us by Theorem 4.3(i). We consider two +cases: +Dependent Case: Suppose that γ˚ +1 +ˇˇ +W and γ1˚ +1 +ˇˇ +W are linearly dependent. Then we can +construct a linear relation among the values in (23) which, in the coordinates induced by +ω1,ξ, . . . , ωm,ξ, is defined over the field of definition L of the de Rham realization of W. (It +is defined over Q in the Betti coordinates.) One shows that such a linear relation does not +extend to the functional level by arguing as in the proof of Proposition 1.16 above, noting +that if γ˚ +1 ´ λγ1˚ +1 +vanishes on W for some scalar λ P Q, then this can be interpreted as a +linear relation on the coordinates of γ˚ +1 ´ λγ1˚ +1 . +Independent Case: We now suppose that γ˚ +1 +ˇˇ +W and γ1˚ +1 +ˇˇ +W are linearly independent. We +write γ˚ +2 “ γ1˚ +1 , and extend this to a basis γ˚ +1 , . . . , γ˚ +m with dual basis γ1, . . . , γm. Let E be the +endomorphism algebra of W, which is a field of dimension dimQ W, and let F be its Galois +40 + +closure. Then W bQ F decomposes as a direct sum À +σ:EãÑC Wσ of one-dimensional weight +spaces for the action of E, so necessarily there is some F-linear combination γ “ λ1γ1`λ2γ2 +and some τ : E ãÑ C such that γ lies inside À +σ‰τ Wσ. +On the de Rham side, the subspace W corresponds to a summand WdR of HwpXξ, Ω‚ +Xξq +defined over some number field L in the de Rham coordinates. The weight space decompo- +sition of W also admits a de Rham counterpart WdR “ À +σ WdR,σ defined over LF in the +de Rham coordinates, and choosing an LF-algebraic vector ωτ P WdR,τ one has γ˚pωτq “ 0 +by construction. Expressing ωτ in terms of the monodromy-compatible frame ω1,ξ, . . . , ωm,ξ +one thus obtains a linear relation on the values in (23) defined over LF. +Functional Relations in the Independent Case: That the linear relation R obtained +in the independent case does not hold on the functional level can be argued as in the proof +of Proposition 1.16 by working with the vector γ˚, however there is one additional subtly +to consider. After showing that the relation R, if it were to hold at the functional level, +would define a monodromy-invariant subspace of the monodromy-representation associated +to V1˚, one cannot immediately conclude that V1˚ has a non-trivial Q summand because the +vector γ˚ is defined over a possibly non-trivial extension F of Q. +However, the argument shows that one does obtain a monodromy-invariant subsystem +M Ă V1˚ +C defined by R in which γ˚ lies. Letting L Ă V1˚ +C be the simple summand containing +γ˚, one may consider some non-trivial conjugate Lc of L. Because γ˚ +1 and γ˚ +2 are Q-vectors +one necessarily learns that L‘Lc contains spanCtγ˚ +1 , γ˚ +2 u. Then because this span is defined +over Q and V1˚ is Q-simple one necessarily finds that V1˚ “ L ‘ Lc. As the sub-local system +M contained L one then finds L “ M, and finally because M was defined by a single linear +relation one finds that rank V1 “ rankV1˚ “ 2. But a rank two variation of Hodge structure +with non-trivial monodromy is necessarily isomorphic to a representation coming from an +elliptic family, and such variations have absolutely irreducible monodromy representations. +Conclusion: The above reasoning shows that at each relevant place v for spξq we may +find a linear KLF-algebraic relation on the values of G at spξq which does not hold at the +functional level, and such that the degrees rL : Kpξqs and rF : Qs are bounded independently +of ξ; note that to obtain the bound on rL : Kpξqs we are using Proposition 5.5. Taking the +product of conjugates of these relations we obtain Kpξq-algebraic relations Rv of degree +bounded by a constant κ independent of ξ and the place v; as we saw in the proof of +Proposition 1.16 the relations Rv may be assumed to be the same for all finite places. The +product R “ ś +v Rv taken over the relevant infinite places and a relevant finite place (if it +exists) is then a Kpξq-algebraic relation of degree at most κ prKpξq : Qs ` 1q which holds +at all relevant places for spξq. Using the Weil height θ induced by the parameter s one +concludes from Theorem 1.11 that for all ξ P S +θpξq ď κ1 rKpξq : Qsa, +for some constants κ1, a P Rą0. +7 +Pila-Zannier for General Intersections +In this sections we drop all “primed” superscripts; i.e., write S instead of S1, etc., as com- +pactifications and degeneration points will not be relevant. For a general overview of the +ideas in this section, we refer the reader to §1.5. +We make an additional comment about our presentation. The Zilber-Pink conjecture has +a “geometric” part and an “arithmetic” part. With reference to the formulation appearing +41 + +in [BKU21], which works in the setting of a general polarizable integral variation of Hodge +structure V on a complex algebraic variety S, the difference between the two is whether +the “atypical” Hodge loci one considers in the base of the variation of Hodge structures are +positive or zero dimensional (in the sense of period dimension, see [BKU21, Def. 1.2]). It is +known as a consequence of [BKU21, Theorem 3.1] that geometric Zilber-Pink statements — +statements about non-Zarski density of positive-dimensional atypical Hodge loci — can be +be proven without any arithmetic input, using only general Hodge-theoretic facts. (There is +an exception in that one cannot resolve Zilber-Pink-type statements for positive-dimensional +Hodge loci coming from splittings of the generic Hodge datum associated to S, but this can +be reinterpreted as a failure to resolve the conjecture for zero-dimensional Hodge loci for +an auxiliary variation constructed from this datum.) We therefore focus exclusively on the +zero-dimensional, i.e. arithmetic, part of Zilber-Pink here. +7.1 +Reduction to Height Bounds on Tensors +To begin this section we assume we have a smooth projective C-algebraic family f : X Ñ S +over a smooth base S; we note in particular we do not assume S is a curve. We let π : rS Ñ S +be the universal covering. We fix a frame γ1, . . . , γm of rV “ π˚ pRwf an +˚ Zq, using which we +make the identification of local systems rV » Zm. This give us a natural height function rθ +on all tensor spaces associated to any fibre of rV induced by the standard Weil height on the +affine space associated to Zm. Fixing a polarization Q : Zm b Zm Ñ Z, we then consider +the space D of all polarized Hodge structures on Zm with the same Hodge numbers as the +variation V. We assume that map rϕ : rS Ñ D given by sending rs to its associated Hodge +flag F ‚ +rs has discrete fibres; this is the same as saying that, if we set Γ “ AutpZm, QqpZq, the +associated period map ϕ : S Ñ ΓzD is quasi-finite. We fix a fundamental domain F Ă D +for the Γ-action which intersects the image of rϕ, and we write I Ă F for the intersection +rϕprSqXF. By [BKT20, Theorem 1.3], this set is definable in the o-minimal structure Ran,exp. +In what follows we will freely use the notion of special subvariety of S for the variation +V, defined for instance as in [BKU21, Def. 4.4]. A special point is a zero dimensional special +subvariety. We will also refer to points in I which are images of lifts of special points as +special points. +Definition 7.1. For an irreducible subvariety Y Ă S, the Mumford-Tate group of GY is +the Mumford-Tate group of Vs above a generic point s P Y (i.e., a point outside the Hodge +locus of V +ˇˇ +Y ). +Notation. We write HS Ă GS for the algebraic monodromy group of S; the identity com- +ponent of the Zariski closure of the image of the monodromy representation. +Definition 7.2. Given a special subvariety Y Ă S, we say that a Mumford-Tate group +GY Ă M Ă GS defines Y if there does not exist Y Ĺ Y 1 Ă S such that GY 1 Ă M. +Lemma 7.3. There are finitely many Mumford-Tate subgroups M Ă GS up to GSpCq- +conjugacy. +Proof. For GLmpCq-conjugacy this is [Voi10, Theorem 4.14]; the same proof works in general. +Lemma 7.4. Let M Ă GS be a Mumford-Tate group. Then the number of special points +in I defined by M is bounded by a constant κ independent of M. +42 + +Remark. We emphasize here that we are using the basis γ1, . . . , γm to identify the fibres of +the local system on rS, and so we understand Mumford-Tate groups as subgroups of GLm,Q +with respect to this choice of coordinates. +Proof. Because Mumford-Tate subgroups of GS lie in finitely many GSpCq-conjugacy classes, +we may reduce to the case where we just consider special points x P I defined by Mumford- +Tate groups M in a fixed GSpCq-conjugacy class. These points satisfy the property that +they are isolated in the subset IM Ă I of points whose Mumford-Tate group is contained +in M. +Each Mumford-Tate group M defines a “Noether-Lefschetz” locus | +NLM Ă qD consisting +of all Hodge flags with Mumford-Tate group contained in M in the sense of [GGK12, pg.49]. +These loci can be extended to an algebraic family of subvarieties of qD over a Noetherian +base, all of whose fibres are GSpCq-translates of | +NLM. If we consider the collection of points +x P I which are isolated in IM1 for some M 1 “ gMg´1, where g P GSpCq, then all such +points are isolated points in definable intersections of the form I X pg ¨ | +NLMq. But the +number of isolated points in any fibre of the definable family tI X pg ¨ | +NLMqugPGSpCq is +bounded by a uniform constant κ as a consequence of definability. +Given a Mumford-Tate group M Ă AutpZm, Qq defined as the stabilizer of a set T Ă +À +aě1pQmqba of tensors, we say that T is a set of minimal type defining M if: +(i) T is linearly independent. +(ii) Suppose T is partitioned as T “ T1 \ T2 \ ¨ ¨ ¨ by tensor degree. Then subject to +(i), the sequence p|T1|, |T2|, . . .q is maximized with respect to the natural lexicographic +total order on elements of NN. (We have as many low-order tensors as possible.) +We additionally say that T is a minimal set defining M if in addition: +(iii) Subject to (i) and (ii), the set T is chosen to have minimal total height, where the +height rθpT q is the maximum of the heights of all its (necessarily finitely many) ele- +ments. +Proposition 7.5. Fix the data of: +- a Mumford-Tate subgroup M Ă GS and its GSpCq-conjugacy class C; +- a set S Ă SpCq of special points defined by Mumford-Tate groups in C; +- for each ξ P S, a lift rξ of ξ mapping to I and a Mumford-Tate group Mrξ Ă GLprVrξq » +GLpZmq which defines ξ and belongs to C. +Let | +NLM Ă qD be the locus consisting of all Hodge flags with Mumford-Tate group contained +in M. Then if P1, . . . , Pr P | +NLMpCq are representatives of the components of | +NLM, there +exists a map gp´q : SÑ GSpQq, denoted ξ ÞÑ gξ, with the following properties: +(1) there exists a constant d such that each point in the image of gp´q lies inside a number +field of degree at most d; +(2) there exists a constant c such that each fibre of gp´q has cardinality at most c; +43 + +(3) the height rθpgξq is bounded by a polynomial in rθpTrξq, where Trξ is the minimal set of +definition for Mrξ; and +(4) for each ξ P S we have gξ ¨ Mrξ ¨ g´1 +ξ +“ M and that rϕprξq P pg´1 +ξ +¨ MpCq ¨ Pjq for some j. +Proof. We note that, because all Mumford-Tate are defined by the tensor invariants they +stabilize and the spaces of these tensor invariants are defined over Q, any two minimal +sets defining Mumford-Tate groups in C have the same number of tensors in each degree. +Denoting by d1, d2, . . . , dk the dimensions of the associated subspaces of tensors, where k is +the largest k for which dk “ dimQpT XpQmqbkq is positive for such a minimal set T , we may +the consider an affine parameter space +T“ Aff rpQmqsd1 ˆ Aff +“ +pQmqb2‰d2 ˆ ¨ ¨ ¨ ˆ Aff +“ +pQmqbk‰dk , +where for a vector space V we write AffrV s for the associated affine space. To any point +T P T we may associate a group MT defined as the stabilizer of T . We then define T0 Ă T +as the locus of T for which MT is conjugate to M, and consider the Q-algebraic family +y : G Ñ T0 whose fibre above T consists of those g P GSpCq satisfying the property that +g ¨ MT ¨ g´1 “ M. +It will suffice to show that, for any T P T0pQq, we can construct gpT q P y´1pT qpQq +defined over a number field of uniformly bounded degree and whose height is bounded by +a polynomial in rθpT q. Indeed, suppose we can do this, and define gξ “ gpTrξq where Trξ is +a minimal set of definition for Mrξ. We may then see that (1) is true by assumption. For +(2) we note that if gpTrξ1q “ gpTrξ2q then Mrξ1 “ Mrξ2, and the number of lifts rξ mapping to +I of points in S for which this can occur is bounded by Lemma 7.4. Then (3) is true by +assumption, and as | +NLM rξ “ g´1 +ξ +¨ | +NLM statement (4) reduces to the claim that rϕprξq has +Mumford-Tate group contained in Mrξ, which is true because Mrξ defines ξ. +Continuing now with our claim about the family y, we may consider a stratification +T0 “ T1 \¨ ¨ ¨\ Tℓ such that for each i the base-change yi : Gi Ñ Ti along Ti ãÑ Tof y is flat +and each Ti is smooth; base-changing to one of these strata, we may assume that y : GÑ T0 +is both smooth and flat. Because y is flat with smooth fibres, it is then necessarily a smooth +morphism, and then this necessarily implies Gis a smooth variety. Each fibre of y is naturally +a torsor for the normalizer N of M in GS, and one has a natural map N ˆ G Ñ GˆT G +bijective on complex points; since this is a map of two smooth varieties, this means it is +necessarily an isomorphism. We conclude that y is an fppf N-torsor, and then because N +is a smooth algebraic group, an ´etale N-torsor by [hmb]. +Choose an ´etale covering e : U Ñ T0 and a trivialization σ : N ˆ U +„ +ÝÑ G ˆT0 U. +For T P T0, construct gpT q by choosing any element ζ of the fibre e´1pT q and then defining +gpT q “ σp1, ζq. That the degree of the resulting field of definition is bounded is a consequence +of the fact that the degree of e is bounded, and that the resulting height is polynomial in +the height of T is an easy consequence of how heights behave under polynomial maps. +To give a finiteness criterion for special points, we now need to introduce some language +to talk about atypical intersections. Recall that a Hodge structure h on a Q-vector space V +can be thought of as a map h : S Ñ GLpV qR, where S “ ResC{RGm,R is the Deligne torus. +Given a Q-subgroup M Ă GLm whose real points contain the image of S, one obtains an +induced Hodge structure on the Lie algebra m of M through the adjoint action; in particular +this is true for the Mumford-Tate group of the Hodge structure. +44 + +Notation. Given a Hodge structure h : S Ñ GLpV qR factoring through a Mumford-Tate +group M with Lie algebra m, we write δpM, hq for the sum of the positive Hodge numbers +associated to the induced Hodge structure on m. +Notation. Given a point s P S, write hs (resp. Gs) for the Hodge structure (resp. Mumford- +Tate group) at s. +Notation. Given a complex subvariety Y Ă S and a Mumford-Tate group GY Ă M Ă GS, +write δpM, Y q “ δpM, hsq for s P Y (the quantity is independent of the point chosen; here +we understand M up to conjugacy by the monodromy on Y ). +Definition 7.6. Given some Mumford-Tate group GY Ă M Ă GS and a special subvariety +Y Ă S, we say that Y is atypical for M if M defines Y and dim S ´ dim Y ă δpS, hsq ´ +δpM, hsq for s P Y . We say that Y is atypical if Y is atypical for GY . +We now restrict to the case where f : X Ñ S is defined over a number field K Ă C, and +let SHg,` Ă SpCq be the positive dimensional Hodge locus: the collection of points which +lie inside a special subvariety of positive dimension. +Corollary 7.7. Fix a Mumford-Tate group M Ă GS, and let SĂ SpCqzSHg,` be a collection +of special points which are defined by, and atypical for, some GSpCq-conjugate of M. Suppose +that +(i) the group HS is equal to the derived subgroup of GS; +(ii) each point of S is defined over Q, and the GalpQ{Kq-action preserves S; and +(iii) that there exists constants a, b P Rą0 independent of ξ P S, and for each ξ P S a lift rξ +mapping to I for which +rθpTrξq ď a rKpξq : Ksb, +where Trξ is a set of definition of minimal type for some Mumford-Tate group Mrξ which +defines ξ atypically and is GSpCq-conjugate to M. +Then S is finite. +Proof. We apply Proposition 7.5. In particular, we consider the definable locus +G:“ +rď +j“1 +tg P GSpCq : pg´1 ¨ MpCq ¨ Pjq X I ‰ ∅u +looooooooooooooooooooooooooomooooooooooooooooooooooooooon +Gj +with P1, . . . , Pj as in Proposition 7.5 and observe that the construction of Proposition 7.5 +produces a point gξ of Gfor each ξ P S. These points are all defined over Q, and in particular +over a number field of degree at most some fixed constant d by Proposition 7.5(1). Applying +Proposition 7.5(3) and our assumption (iii), there exists constants a1, b1 ą 0 such that +rθpgξq ď a1 rKpξq : Ksb1 +for all ξ P S. Assume for contradiction that S is infinite. Using Proposition 7.5(2) and (ii), +one has for infinitely many positive integers N that +ˇˇˇ +! +gξ : rθpgξq ď a1N b1)ˇˇˇ ě 1 +cN. +45 + +Applying the Pila-Wilkie theorem [Pil09, Theorem 1.6] in the case of S infinite one finds +that there exists an index j, an irreducible complex algebraic subvariety V Ă GSpCq, and an +analytic open neighbourhood U1 intersecting V such that V X U1 Ă Gj. (A na¨ıve application +of [Pil09, Theorem 1.6] gives only a real-algebraic such V , but one can use [PT13, Lemma +4.1] to ensure this is a complex algebraic subvariety.) Morever V contains infinitely many +elements gξ for ξ P S, hence V ´1 ¨ MpCq ¨ Pj intersects I in a locus of positive dimension +(the intersection contains infinitely many points rϕprξq and is definable). +We now choose an irreducible algebraic curve C Ă V containing a point c “ gξ associated +to some ξ P S subject to the conditions: +- there exists an analytic locus F Ă pC´1 ¨ MpCq ¨ Pjq X I such that dim rϕprξq F ą 0; and +- F does not lie inside any complex orbit in qD of a proper Q-normal factor of HS. +Both of these conditions may be understood in terms of tangency: because for any g P V X U1 +near c the intersections pg´1 ¨ MpCq ¨ Pjq X I are non-empty, we may always choose C along +a direction in V for which the intersections pg´1 ¨ MpCq ¨ Pjq X I are both non-empty and +varying. Similarly, because no germ of I lies inside any orbit of a proper normal factor of +HS, and the orbits of such a normal factor foliate qD, one can even choose C so that the +intersections C´1 ¨ MpCq ¨ Pj X I vary in a direction transverse to each such foliation. +Now in particular by Proposition 7.5(4), one has gξMrξg´1 +ξ +“ M and rϕprξq is a point of +O “ c´1¨M ¨Pj. The fact that ξ is atypically defined by Mrξ means that dim O “ δpMrξ, rϕprξqq +satisfies +dim S ` dim O ă δpSq “ dim HSpCq ¨ ϕprξq, +(24) +where for the equality we use the fact that the derived subgroup of GS is HS. By construc- +tion, the (constructible) variety E “ C´1MpCq ¨ Pj has dimension dim O ` 1 and intersects +I in a locus of dimension at least 1 at F. Letting qT “ HSpCq ¨ ϕprξq one has that +codim qT F ă codim qT I` codim qT E +as a consequence of (24). From the Ax-Schanuel Theorem for period mappings [BT17], one +the learns that ξ, which lies inside ϕ´1pπDpFqq with πD : D Ñ ΓzD the natural projection, +lies inside a (necessarily positive dimensional) weakly special subvariety W Ă S. (We note +that in [BT17], the Ax-Schanuel theorem is formulated for intersections with graphs of period +mappings, but one can recover our formulation by pulling back an algebraic intersection with +the image to an algebraic intersection with the graph.) +By assumption the point ξ does not lie inside SHg,`, so the weakly special subvariety W +containing ξ does not either; in particular, the Mumford-Tate group of W is GS. On the +other hand, because W contains the image of F, it is not defined by an orbit of a proper +normal factor of HS; in particular, its algebraic monodromy group is not a normal factor +of GS. This contradicts [Yve92, §5], which ensures that algebraic monodromy groups of +varieties are always Q-algebraic normal subgroups of their Mumford-Tate groups. +7.2 +Constraining Heights of Tensors +We will now try to put ourselves in a situation where we can apply Corollary 7.7, and in +particular verify hypotheses (ii) and (iii). We begin by giving a way to estimate the heights +rθpTrξq that appear in Corollary 7.7 using the theory of Siegel sets. We set G “ AutpZ, Qq +46 + +and recall that D is naturally a homogeneous space for GpRq. If one fixes a reference Hodge +structure x0 P FĂ D we obtain an identification q : GpRq{H +„ +ÝÑ D, where H Ă GpRq is the +centralizer of the map x0 : S Ñ GR which defines the Hodge structure x0. From the theory +in [Bor69], there exists a Siegel set S Ă GpRq whose image is the fundamental domain Ffor +the Γ “ GpZq-action fixed above; in particular, every fibre of the map q above Fis contained +in S. (We refer to [Orr18, §2] for the definition of Siegel set, as we will only need their +abstract properties here; in particular, we will just use the fact that they are fundamental +sets for the Γ-action compatibly with the map q.) +Consider now a rational element ϕ P GpQq which induces an endomorphism of a Hodge +structure x P F. Then if gh P G is chosen such that x “ gx0, then g´1ϕg is an endomorphism +of x0, and hence lies in H. Writing h “ g´1ϕg, one then has ϕ “ ghg´1, which is a rational +point contained in GG´1. +Orr [Orr18, Theorem 1.1] has shown the following quantitative result about Siegel sets: +Theorem 7.8 (Orr). There exists a constant C1 (depending on a choice of reductive Q- +algebraic subgroup of G Ă GLm and Siegel set in G) such that for all ϕ P GG´1 X GpQq one +has +rθpϕq ď maxtC1ndm, du, +where n “ |det ϕ|, and d is the maximum of the denominators in the entries of ϕ (regarded +as a point in GLmpQq). +As shown in [Orr18, Theorem 1.2], one can also construct a Siegel set Gm for GLm,Q con- +taining the centralizer Hm of x0 such that finitely many rational translates of Gm contain +G, so one can even use this theorem to deal with endomorphisms ϕ that don’t preserve the +polarization at the cost of changing the constant C1; with respect to the above analysis all +that changes is that H is replaced by Hm. +Using this, we now bound heights of endomorphisms of Hodge structures in F in terms +of more geometric notions. +Definition 7.9. Given any rational endomorphism ϕ of a Hodge structure x, the map +ϕ is diagonalizable, hence is invertible when restricted to its image impϕq. We define its +restricted determinant detrpϕq to be the determinant of this restriction. +Definition 7.10. We call a map y P Hompx, x1q between Hodge structures x and x1 on +Zm an isogeny if the scalar extension yQ is invertible. Its dual y_ is the map in Hompx1, xq +induced from the natural map x1_ Ñ x_ by using the isomorphisms x1 » x1_ and x » x_ +induced by the fixed polarization. +Definition 7.11. Given an integral Hodge structure x on the lattice Zm, a Hodge en- +domorphism ϕ P EndpxqQ, and the Hodge idempotent πϕ defining the Hodge summand +impϕq, we define the isogeny-height θisopϕq to be the minimum degree of an integral isogeny +y P Hompx, x1q to some integral Hodge structure x1 on Zm such that both yQ ˝ ϕ ˝ y´1 +Q +and +yQ ˝ πϕ ˝ y´1 +Q +are Q-scalar extensions of elements of Endpx1q. +Lemma 7.12. The height of any rational endomorphism ϕ associated to some Hodge struc- +ture x P F is bounded by a polynomial in its isogeny-height and the absolute value of its +restricted determinant. +Proof. Let us begin by reducing to the case where ϕ and its associated idempotent πϕ are +integral. Let y P Hompx, x1q be an integral isogeny to some x1 P Fsuch that ϕ1 +Q “ yQ˝ϕ˝y´1 +Q +47 + +and πϕ1,Q “ yQ ˝ πϕ ˝ y´1 +Q +are the Q-scalar extensions of integral endomorphisms ϕ1 and πϕ1. +As y is a homomorphism, it conjugates the actions of the tori associated to x and x1. If +one chooses gx and gx1 such that x “ gxx0 and x1 “ gx1x0, where x0 is the reference Hodge +structure fixed above, then g´1 +x1 ygx centralizes x0 and hence lies in Hm, the centralizer of +x0 in GLmpRq. It follows that y lies in Gm ¨ G´1 +m . The element y is assumed integral, so its +height is bounded as a consequence of Theorem 7.8 by a linear multiple of its determinant. +The heights of ϕ is then bounded by a polynomial in its isogeny height (i.e., a polynomial +in the minimum determinant of such a y) and the height of an integral endomorphism ϕ1 +with the same restricted determinant and whose associated idempotent πϕ1 is integral. +Now assuming ϕ and πϕ are integral, we observe that, again by applying Theorem 7.8 +and the reasoning at the beginning of this section, we are done in the case where ϕ has +trivial kernel (and hence πϕ “ id). In the cases where ϕ has non-trivial kernel we have +ϕ “ pϕ ` p1 ´ πϕqq ´ p1 ´ πϕq, where we observe that ϕ ` p1 ´ πϕq has trivial kernel and +an isogeny height of 1. A polynomial height bound for the sum follows from a polynomial +height bound for its summands, so we are reduced to the case of integral idempotents, and +in particular the case where ϕ “ πϕ. +It now suffices to show that there are only finitely many ϕ which are integral polarization- +preserving idempotents for Hodge structures x P F. Indeed, because projections preserving a +non-degenerate bilinear form are uniquely determined by their image, it suffices to show that +only finitely many integral summands of Zm occur for Hodge structures in F. Because Γ “ +AutpZm, QqpZq is the discrete group defining the equivalence for which F is a fundamental +set, the set F has only one Hodge structure from each integral isomorphism class. Given +two distinct direct summand decompositions Zm “ V ‘ W and Zm “ V 1 ‘ W 1 where +a “ dim V “ dim V 1 and b “ dim W “ dim W 1 respectively, any Hodge structure compatible +with the first decomposition is isomorphic to one compatible with the second. One may then +see that in fact only one such decomposition is associated to Hodge structures in F from +the fact that, after fixing such a decomposition Zm “ V ‘ W, one has a natural map +FV ˆ FW Ñ Fof fundamental sets sending a pair of polarized Hodge structures to its direct +sum. +7.3 +Application in the Abelian Case +The previous proposition allows one to bound the heights rθpTrξq appearing in Corollary 7.7, +at least in the case where Trξ consists of endomorphisms, in terms of quantities of a geometric +nature. We now consider the more concrete case of Hodge structures arising from abelian +varieties where this is easier to analyze; in particular, we start by verifying hypothesis (ii) +of Corollary 7.7 in this case. +Lemma 7.13. Suppose in the situation of Corollary 7.7 that f is an abelian family. Then +after replacing K with a finite extension, hypothesis (ii) holds for the set S consisting of +all special points not lying in SHg,` which are defined by a Mumford-Tate group which is +GSpCq-conjugate to M. +Proof. This is an easy consequence of Deligne’s verification [Del82] of the absolute Hodge +conjecture for abelian families. First, we note that the fact that the action of GalpQ{Kq +preserves SHg,` is known, at least in the case where HS is Q-simple, in much greater gener- +ality, see [KOU20, Corollary 1.13]. To see that GalpQ{Kq will also preserve zero-dimensional +Hodge loci outside of SHg,` it then suffices to check that it preserves Hodge loci, and that +this is a consequence of the absolute Hodge conjecture is explained in [Voi10]. +48 + +We must also check that if Mξ Ă GLpVξq is a Mumford-Tate group in the GSpCq- +conjugacy class of M defining ξ atypically, and if σ P GalpQ{Kq is an automorphism, then +ξσ admits a Mumford-Tate group M σ +ξ Ă GLpVξσq which defines ξσ atypically and is GSpCq- +conjugate to M. For this consider the composition +Vξ,C +„ +ÝÑ Hξ,C +σÝÑ Hξσ,C +„ +ÝÑ Vξσ,C. +The absolute Hodge conjecture is the statement that the induced map on tensor spaces +sends Hodge tensors to Hodge tensors, so in particular the induced map on spaces of linear +transformations sends the Mumford-Tate group Mξ to some Mumford-Tate group M σ +ξ Ă +GLpVξσq which contains the Mumford-Tate group of Mξσ. Moreover, this induced map also +sends the realization of GS in GLpVξq to the realization of GS in GLpVξσq, hence there is an +induced outer automorphism η of the abstract group GSpCq which sends Mξ (regarded as +a subgroup through the realization of GS in the fibre at ξ) to M σ +ξ (regarded as a subgroup +through the realization of GS in the fibre at ξσ). Now if gMξg´1 “ M for some g P GSpCq, +one finds that ηpgqM σ +ξ ηpgq´1 “ ηpMq. +We claim that, after possibly replacing K with a finite extension, the group ηpMq is +GSpCq-conjugate to M. The point is that one can work entirely with the algebraic-de Rham +realizations of these groups, in particular, we may denote by C1, . . . , Cℓ the GS-conjugacy +classes of Mumford-Tate subgroup of GS and consider the algebraic vector bundles yi : +Gi Ñ S whose fibres are the moduli of algebraic subgroups of GLpHsq lying in Ci. By +construction ηpMq is GSpCq-conjugate to a Mumford-Tate group, hence corresponds to a +point in the fibre above ξσ of one such family, but it might be in the wrong conjugacy class. +What one then wants is for each of the families yi to be naturally (using the K-algebraic +structure induced from H) defined over K, which occurs after passing to a finite extension. +Finally, to check that ξσ is atypical for M σ +ξ , we note that the quantity δpM σ +ξ , hξσq may +also be computed from the Hodge filtration on the Lie algebra of M σ +ξ (instead of the Hodge +direct sum decomposition). In particular, it can be computed using the de Rham realizations +of M σ +ξ and F ‚ +ξσ, and this data is σ-conjugate to the de Rham realizations of Mξ and F ‚ +ξ . +Corollary 7.14. Suppose that f : X Ñ S is an abelian family with HS equal to the derived +subgroup of GS. Let S Ă SpCq be a set of special points not lying in SHg,` which are defined +by, and atypical for, some subspace of their Hodge tensors of endomorphism type, and that +for each ξ there is a basis ϕ1, . . . , ϕℓ for a subspace of the Hodge endomorphisms of Vξ +defining ξ atypically such that +maxitdetrpϕiq, θisopϕiqu ď κ rKpξq : Ksa +for constants κ, a P Rą0 independent of ξ. Then S is finite. +Proof. It suffices to verify that the hypotheses of Corollary 7.7 hold, possibly passing to +a finite extension of K. Hypothesis (i) holds by assumption. Using Lemma 7.13 we may +replace S with its orbit under GalpQ{Kq after passing to a finite extension, showing (ii). For +hypothesis (iii) we use Lemma 7.12 to bound rθpTrξq in terms of the restricted determinant +and isogeny degree, and the result follows. +Corollary 7.15. Suppose that f : X Ñ S is an abelian family with HS equal to the derived +subgroup of GS. Let S Ă SpCq be a set of special points not lying in SHg,` which are defined +by, and atypical for, their Hodge idempotents. Suppose that +θpξq ď κ rKpξq : Ksa +49 + +for some κ, a P Rą0 independent of ξ, with θ some logarithmic Weil height. Then S is finite. +Proof. Using Corollary 7.14 it suffices to show that the isogeny heights of idempotents are +polynomially bounded by the Faltings height (which differs from any logarithmic Weil height +by a multiplicative constant) of the point ξ and the degree of the field of definition of ξ. +This is just the result of Masser-Wustholtz [MW95, Theorem 1.1]. Note that the statement +of [MW95, Theorem 1.1] is given (in our language) for ξ defined over a field of bounded +degree, but the constants depend polynomially on this degree as is explained at the end of +[MW95, pg.23]. +We note that Corollary 7.15 gives an interpretation of Theorem 1.18 in the introduction. +8 +Applications +In what follows we will use the following fact, which is part of the proof of Theorem 2 in +[And89, IX, §4.4]. +Theorem 8.1. Suppose that f : X Ñ S is a projective family of relative dimension n with +geometrically connected fibres and whose singular fibres all have simple normal crossings, +and let f 1 : X1 Ñ S1 be the base-change to the smooth locus S1 Ă S, which we assume is +non-empty. Then if s0 P SpCqzS1pCq, the vanishing cycles associated to the order-n normal +crossing singularities of Y “ Xs0 (in the sense of Definition 1.2) span a space of dimension +hnpΣY q, where hnpΣY q is the dimension of the n’th cohomology group of the dual graph ΣY +associated to Y (see [Kol14, §1]). +8.1 +Families of Curves +We now prove Theorem 1.1. We first observe that the case where g “ 2 was already proven +in [DO21, Theorem 1.1], albeit under different language. To reduce from our setup to the +setup of Daw and Orr in [DO21], note that the locus B Ă MgzMg in the case of g “ 2 +necessarily consists of singular curves C “ P1 Y ¨ ¨ ¨ Y P1. The stability condition requires +(see [DM69, Def. 1.1]) that each P1 component intersects each other component in at least 3 +nodes. The boundary strata of Mg are stratified by the number of nodes [Mum83], and the +stratum of curves with k nodes has dimension 3g ´ 3 ´ k; in particular, the only possibility +here is C “ P1 Y P1 with the components meeting at 3 nodes. But the extended Torelli +map [BPVdV84] sends this curve into the 0-dimensional Bailey-Borel stratum (c.f. +the +introduction of [Cap09]). +Remark. As the 0-dimensional Bailey Borel stratum of the boundary A2zA2 consists of a +single point, one could also reduce from the case of abelian surfaces to the case of curves +by considering a family of curves which induces, away from a finite set, a given family of +surfaces passing through the 0-dimensional stratum. +We now consider the general statement when g ě 3. +We first observe that we may +reduce to the case where S is defined over a number field K: indeed, the points of S are all +defined over Q, and if the set SX SpCq is infinite then S coincides with its Zariski closure +and hence is also defined over Q, and hence over some number field K. After replacing S +with a smooth finite covering we may reduce to the analogous problem for a K-algebraic +family f : X Ñ S of genus g curves and take S Ă SpQq to be the set of points where the +50 + +fibre of the associated variation V1 “ R1f 1an +˚ Q of Hodge structures admits a simple factor +with complex multiplication; here f 1 is the restriction of f to the fibre above S Ă S. +To get a polynomial bound on the heights of points of S in terms of the degree we +now wish to apply Theorem 1.17 to our situation. This means checking that integrating +around the vanishing cycles corresponding to two nodal singularities of Xs0 produces tuples +of non-constant G-functions satisfying the linear independence condition in Theorem 1.17. +This follows from Andr´e’s result Theorem 8.1 above, as well as the remark at [And89, +pg.192], which says that h1pΣXs0 q “ g ´ ř +i pgpCiq ě 2, where Xs0 “ C1 Y ¨ ¨ ¨ Y Cℓ is the +decomposition of the singular fibre into its components. +We are now in the situation where we have a logarithmic Weil height θ : SpQq Ñ Rą0 +for which there exists constants κ, a P Rą0 such that +θpξq ď κ rKpξq : Ksa +(25) +for all ξ P S. By the resolution of the Andr´e-Oort conjecture [PST`21], it suffices to replace +S with just those points for which the CM summand is a proper summand. After replacing +the problem in question for the equivalent problem for the associated Jacobian family, the +result then follows from Corollary 7.15 as soon as g ě 3. In particular Hodge-genericity +implies the monodromy assumption of Corollary 7.15, and the atypicality with respect to +the idempotents of the points in S holds under the assumption g ě 3 as a special locus of +Ag with a global isogeny factor has codimension at least two. +8.2 +Families of Abelian Varieties +We now prove Theorem 1.5. The proof is the same as the case for curves above, except we +do not even have to check that the vanishing cycles are independent because this has been +assumed. +8.3 +Degenerations to Hyperplanes +Lastly, we prove Corollary 1.10. We once again apply Theorem 1.17 using Andr´e’s result +Theorem 8.1 above. That Hodge-genericity implies the simplicity of the primitive varia- +tion and the atypicality of the CM points is a consequence of Beauville’s computation of +the monodromy groups of hypersurface variations in [Bea86]; here we note that a Hodge +generic curve on which the associated variation is non-constant will have maximum-possible +algebraic monodromy because its algebraic monodromy group must be non-trivial and Q- +normal in the Q-simple group of automorphisms preserving the polarization form. To apply +Theorem 1.17 it therefore suffices to compute the cohomology of the dual graph associated +to a generic hyperplane arrangement. 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Compositio Mathematica, 82(1):1–24, 1992. +55 + diff --git a/oNAzT4oBgHgl3EQf5P6V/content/tmp_files/load_file.txt b/oNAzT4oBgHgl3EQf5P6V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e23f3033445c612655e9eca90cd125881b072682 --- /dev/null +++ b/oNAzT4oBgHgl3EQf5P6V/content/tmp_files/load_file.txt @@ -0,0 +1,2998 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf,len=2997 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='01857v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='AG] 5 Jan 2023 Geometric G-functions and Atypicality David Urbanik January 6, 2023 Abstract In a seminal research manuscript, Andr´e showed how the theory of G-functions could be used to give height bounds on the moduli of smooth projective algebraic varieties acquiring non-generic algebraic cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The method was limited by the lack of a suitable cohomological interpretation for these G-functions at finite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In this paper we use recent developments in p-adic Hodge theory to remove this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With respect to Andr´e’s strategy for producing height bounds from algebraic rela- tions on G-function values at special points, we give a general method for producing relations that hold at all finite places, and show that producing relations at the infinite places is the essential difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This leads to new cases of the Zilber-Pink conjec- ture, as well as new height bounds on special moduli, including the first unconditional finiteness results for CM moduli in non-Shimura settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As a more technical contribution, we introduce a refinement of the Pila-Zannier strategy capable of handling Zilber-Pink-type atypical intersection problems in arbi- trary dimension and for arbitrary smooth projective families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 The G-function Method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': 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Extension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 51 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 Degenerations to Hyperplanes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 51 1 Introduction To motivate the more technical introduction that follows, we begin with some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fix some g ě 2, and suppose that Mg is the moduli stack of genus g curves with universal family C Ñ Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let Mg be its compactification which parameterizes stable curves, as constructed by Deligne-Mumford in [DM69], and let B Ă MgzMg be the locus of stable curves Cx “ C1 Y ¨ ¨ ¨ Y Cℓ with smooth components for which pgpC1q ` ¨ ¨ ¨ ` pgpCℓq ď g ´ 2, (1) where pgpCiq denotes the genus of the curve Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As explained in [BF22, §1], this condition is equivalent to δ ´ ℓ ě 1, where δ is the number of nodal singularities of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our first result concerns the Jacobians of points of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We recall that every abelian variety A over a field k admits a unique isogeny decomposition A « A1 ˆk ¨ ¨ ¨ ˆk Aℓ into simple factors, where the relation « is given by the existence of an isogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write S Ă MgpCq for the set of x P MgpCq for which the Jacobian JpCxq admits an isogeny factor with complex multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let S Ă Mg,C with g ě 2 be an irreducible Hodge-generic curve whose closure S Ă Mg intersects B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then SpCq X S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let us begin by explaining why results of the type described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 are difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The set S is infinite, and in particular contains the complex points of infinitely many subva- rieties of Mg which have dimension linear in g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' this can be seen, for instance, by intersecting 2 the image of the Torelli map with Hecke translates of loci like Ag´1 ˆ tyu Ă Ag, with y a point corresponding to a CM elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Understanding the geometry of the subvarieties that give rise to the points of S is a deep problem with numerous arithmetic implications, and it is difficult to rule out the possibility that infinitely many of these subvarieties might intersect some curve in Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because curves in Mg whose Jacobian has a global CM factor exist, some kind of Hodge- genericity hypothesis on S like the one given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 is clearly necessary, but the role of the hypothesis concerning the intersection of S with the boundary locus B is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This will turn out to be an artifact of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In fact, we will see that the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 makes little reference to curves at all, but is instead the consequence of a general theory of height bounds, in principle effective, for special moduli of one-dimensional smooth projective families of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For instance, the same argument will also give the following: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let V0 be a variety and x P V0 a closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We say that pV0, xq is an order-r normal crossing singularity if it is ´etale locally isomorphic to the locus z1 ¨ ¨ ¨ zr “ 0 in some neighbourhood of 0 P An for some r ď n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose V0 Ă V is an inclusion of complex analytic varieties whose germ at some x P V0 is isomorphic to the germ of tx1 ¨ ¨ ¨ xr “ 0u Ă An at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then a degree k vanishing cycle at pV, xq is a class rγs in HkpV, Zq obtained as the product of k simple loops around components xi1 “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xik “ 0 for some subset ti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , iku Ă t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the setting of an analytic family f : V Ñ Y with special fibre V0 above 0 P Y , we say that two vanishing cycles in V induced by V0 are distinct if for any analytic neighbourhood B of 0 they are the images of linearly independent cycles inside À i HipV1, Qq, where V1 is some fibre of f above B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let f : X Ñ S be a Hodge-generic family of abelian varieties of dimension g ě 3 over an irreducible complex algebraic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that at some point s0 P SpCq the fibre Xs0 has simple normal crossing singularities which induce at least two distinct degree one vanishing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then the set S Ă SpCq of points s P SpCq for which Xs admits an isogeny factor with complex multiplication is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A moduli interpretation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 in the style of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 is also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However compactified moduli spaces of abelian varieties are typically constructed using semiabelian schemes instead of singular normal crossing varieties as degeneration points, and although one can always construct a normal crossing compactification of a semiabelian variety, even in families [FC90], there is no canonical way to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Thus to understand which curves in Ag are covered by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 one has to undertake an analysis of the different normal crossing compactifications of the fibres of boundary strata in Ag, and as this will lead us too far astray we do not attempt this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The common theme connecting Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 will turn out to be the presence of two independent vanishing loops near a degeneration point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These are used in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using general moduli-theoretic arguments one can reduce both Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 to the case where we have a family f : X Ñ S defined over a number field K Ă C, S is a smooth curve, and the degeneration point s0 lies in SpKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We denote by S1 Ă S the locus over which the family f is smooth, and write f 1 for the map X1 Ñ S1, where X1 “ f ´1pSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Replacing S with a finite cover, we may moreover assume that the monodromy action on the local system V1 “ R1f 1an ˚ Z is unipotent near s0, and extend the 3 algebraic de Rham vector bundle H1 “ R1f 1 ˚Ω‚ X1{S1 canonically to a vector bundle H over a neighbourhood of s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given any section ω of H and vanishing cycle γ as above we may then consider a period function hpsq “ ż γs ωs for s an algebraic local parameter at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One checks that this function is given by a K- algebraic power series in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As one ranges over a basis of sections of H and a basis of vanishing cycles associated to s0, one obtains some finite set G“ thi : 1 ď i ď nu of period functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In his book [And89] on G-functions in geometry, Andr´e observed that one can potentially use the transcendence theory of such functions to control the heights of points in S at which the fibres acquire extra algebraic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The point is that algebraic cycles induce Q-algebraic relations between the values of the functions in G, and classical ideas from transcendence theory can be used to constrain the number of points in S at which such relations can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular Andr´e proves a result he calls a “Hasse principle for G-functions”, and shows that if one can construct certain “global” relations on the values of K-algebraic power series which are solutions to a differential system and which have appropriately bounded coefficients, then effective height bounds on the points at which those relations occur can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The problem is that the “globality” condition requires one to understand the functions hi, when regarded as power series over K, at each place of K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, one has to be able to give cohomological interpretations of the functions hi over p-adic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With a few exceptions (which we discuss in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6), this has proved elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One of our main contributions will be to give p-adic interpretations of these period functions in full generality, for which we use recent developments in p-adic Hodge theory due to Scholze [Sch13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With a p-adic understanding of period functions in place, we will then turn to techniques for constructing relations on periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In addition to the new insights needed to construct relations at the finite places, we also improve on existing techniques at the infinite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In fact, for constructing relations at the finite places we only need one vanishing cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the requirement that we have two will be necessary only at the infinite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then prove the following general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We say that a Q-Hodge structure V of algebro-geometric origin has complex multiplication if the algebra of endomorphisms of V generated by algebraic cycles has Q- vector space dimension equal to dimQ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One can also define what it means for a general Hodge structure to have complex multiplication in terms of abstract endomorphisms of Hodge structures, with the two defini- tions coinciding in the geometric setting under the Hodge conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As we are concerned explicitly with algebraic cycles in this paper we adopt the geometric definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that f : X Ñ S is a projective family of geometrically connected varieties over the number field K Ă C whose generic fibre is smooth and such that the primitive local subsystem V1 Ă Rwf 1an ˚ Q is simple, where f 1 is base-change of f to the locus S1 Ă S above which the fibres are smooth, and S is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that at some point s0 P SpKq the fibre Xs0 has simple normal crossing singularities which induce at least two distinct primitive degree w vanishing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write S Ă SpCq for the set of points x P SpCq at which: the Hodge conjecture for endomorphisms of the fibre V1 x holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and 4 the fibre V1 x has a Q-Hodge summand with complex multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then S Ă SpQq, and for any logarithmic Weil height θ : SpQq Ñ R there exists constants κ, a P Rą0 such that θpξq ď κ rKpξq : Ksa for all ξ P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The way that one goes from a result like Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 to a finiteness result for S is by apply- ing a Pila-Zannier strategy for problems of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that the usual Pila-Zannier strategy, used for instance to prove the Andr´e-Oort conjecture [PST`21], is insufficient here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The reason is that the approach used in Andr´e-Oort-style problems is to produce from S a large number of Q-algebraic points in a definable period image, but in our setting the analogous points only have lower transcendence degree than normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Instead we introduce a different approach which produces Q-algebraic points in a moduli space for varieties which intersect the definable period image, and eventually use this to obtain an algebraic inter- section to which the Ax-Schanuel Theorem applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This approach is capable of handling arbitrary smooth projective families and bases S of arbitrary dimension, and contains sev- eral new ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We give more details on our approach in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 below, and a brief comparison with previous approaches, including the one in [DR18], in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that, even without the Pila-Zannier strategy, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 already implies: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7, for any constant d ą 0, the number of points of S lying inside a number field of degree at most d is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 gives an absolute height bound in this case, so this is just the Northcott property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let us comment on the fact that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 produces unconditional results even in higher weight settings beyond the Shimura case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, the following is a formal conse- quence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let f, K, V1, f 1, s0, Xs0 and w be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7, and write S Ă SpCq for the set of points x P SpCq for which the Hodge structure V1 x has complex multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then for any logarithmic Weil height θ : SpQq Ñ R there exists constants κ, a P Rą0 such that θpξq ď κ rKpξq : Ksa for all ξ P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, for each d ą 0, the number of points of S lying a number field of degree at most d is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that the condition that we have two distinct vanishing cycles can be verified in explicit cases by computing the limiting mixed Hodge structure associated to the degeneration point s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for instance, Andr´e computes in [And89, IX, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4] the number of such vanishing cycles for cohomology in middle degree in terms of a cohomological invariant of a dual graph associated to the singularities in the special fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will use this ourselves when we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='10 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For instance, one can easily verify the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='9 to show the following: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The conclusion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='9 holds when f 1 is a family of smooth pro- jective hypersurfaces of degree d in Pn, with d ą n ` 1, and where the fibre Xs0 is a union of d hyperplanes in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 5 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The general position assumption is much too strong, one really only needs the hyperplane arrangement to not be overly degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' see the proof in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As far as we are aware, results like this beyond the setting of Shimura-type families have not appeared in the literature before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (Recent work in [Pap22] gives results under additional arithmetic hypotheses and conjectures near the degeneration point s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we refer to §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6 for more discussion of related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') We now give, in more detail, a description of the main technical achievements of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 The G-function Method Suppose that f : X Ñ S is a projective family of relative dimension n “ ν ´ 1 defined over a number field K Ă Q Ă C, with X and S both smooth, S a geometrically irreducible curve, and which has geometrically-connected fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Denote by f 1 : X1 Ñ S1 its base-change to the open locus S1 Ă S above which the fibres are smooth, and fix a degeneration point s0 P SpKqzS1pKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For each point ξ P S1pCq we have a smooth projective complex algebraic variety Xξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our goal, loosely phrased, will be as follows: Goal: Describe the set S Ă S1pCq where the fibre Xξ carries an algebraic sub- variety Yξ Ă Xξ which does not spread out to a family Y 1 Ă X1 lying over the generic point of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' More generally, one can also formulate this goal with Xξ replaced by all its self-products Xn ξ “ Xξ ˆ ¨ ¨ ¨ ˆ Xξ looooooomooooooon n times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the theory of relative Hilbert schemes one can show that S Ă SpQq, and so after choosing a (logarithmic) Weil height θ : SpQq Ñ R one obtains a function S Ñ R which we also denote by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our goal leads to the following natural question Question: How can one bound the heights θpξq for ξ P S?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In his research monograph [And89], Andr´e gave a method for producing such bounds, at least under certain assumptions on the degeneration point s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' He considers the case where monodromy around s0 acts via a unipotent linear transformation, and for which the fibre X0 Ă X at s0 degenerates via a reduced normal crossing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' this latter condition means that: there is an affine open subset U Ă X and functions z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν on U whose differentials trivialize Ω1 U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and the equation ze1 1 ¨ ¨ ¨ zeν ν “ s defines the graph of f ˇˇ U near s0, where s is a uniformizing function on S at s0, and ej P t0, 1u for all 1 ď j ď ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After reordering we obtain an integer µ such that that ej “ 0 for j ą µ and ej “ 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' On U one can then fix a point q in the locus z1 “ ¨ ¨ ¨ “ zµ “ 0 and consider, in an analytic neighbourhood of s0, complex analytic functions of s given by Ppsq “ 1 p2πiqµ´1 ż γs ι˚ s ˆ hq dz2 ¨ ¨ ¨ dzµ z2 ¨ ¨ ¨ zµ ˙ , (2) where 6 ιs : Xs X U ãÑ U is the inclusion of the fibre above s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' γs a “vanishing cycle” in the fibre Us obtained as a product of µ ´ 1 simple loops near q around the divisors zj “ 0 for j “ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and hq is a holomorphic function chosen so that hq dz2¨¨¨dzµ z2¨¨¨zµ represents the restriction of a relative class in the algebraic de Rham cohomology of X over S and whose power series representation in the coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zµ at q has coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As explained by Andr´e in [And89, IX,§4], the functions Ppsq are described by power series in s with coefficients in K when expanded around s0, and give, in a punctured neighbourhood around s0, a relative period of the Betti-algebraic de Rham comparison Rµ´1f 1 ˚Zpµ´1qbOS1an » pRµ´1f 1 ˚Ω‚ X1{S1qan C when restricted to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, this power series representation of P is a G-function in the sense of [And89, I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Andr´e actually assumes that µ “ ν, and assumes that X is covered by neighbour- hoods U of the above type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As it is not substantially more difficult, we will work in greater generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To explicate the relationship between these G-functions and the projective family f Andr´e classifies, at least in degree n, the period functions P of this form in terms of the monodromy around the degeneration point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To explain what we mean, let us fix an in- teger w and denote by V1 the variation of Hodge structure with underlying local system Rwf 1 ˚Zpwq{tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' modulo torsion, and let H1 “ Rwf 1 ˚Ω‚ X1{S1 be the associated algebraic de Rham cohomology vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The vector bundle H1 has a so-called canonical extension H to a vector bundle over S which we recall in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The sections of V1 that extend to sec- tions of Han C under the comparison isomorphism define a local subsystem M Ă V1ˇˇ B, where B Ă SpCq is a small analytic ball centered at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Poincar´e duality defines a natural pairing H1anˇˇ B b M Ñ OB, (3) and sections in the image of this pairing we refer to as non-degenerating (relative) periods at s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the functions Ppsq described above were of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Andr´e gives a description of the image in the case when w “ ν ´ 1, and for this w shows that when Xs0 has simple normal crossings the image of (3) is spanned by G-functions of the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In what follows we write MmˆnpAq for the space of m ˆ n matrices with values in the ring A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we will also write MmpAq for MmˆmpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Andr´e then uses the non-degenerating periods at s0 to give a method for bounding the heights of the points in the set S, based on the so-called Hasse principle for G-functions [And89, VII, §5], which may be stated as follows: Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a number field L, we write ΣL for the set of places of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='11 (Hasse Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that G “ pG1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Gmqt P Mmˆ1pKrrxssq sat- isfies the differential system d dxG “ ΓG for some Γ P MmpKpxqq, write Gi “ ř8 j“0 Gijxj, and suppose that lim sup nÑ8 ˜ 1 n ÿ vPΣK max iďm,jďn log` |Gij|v ¸ ă 8 where log`ptq “ log maxt1, tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let ΞpG, δq denote the set of ξ P Q satisfying the following property: there exists a homogeneous polynomial P P Kry1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , yms of degree at most δ such that: 7 (1) the relation PpG1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Gmq does not hold on the level of formal power series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and (2) for all v P ΣKpξq for which |ξ|v ă 1, either (i) at least one of the series Gi does not have v-adic convergence radius greater than |ξ|v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' or (ii) the relation PpG1pξq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Gmpξqq “ 0 holds v-adically at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then there exists an exponent e P Zě0 such that sup ξPΞpG,δq θpξq “ Opδeq, where e and the implied constant depend only on G, and θ is the standard logarithmic Weil height function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the context of an application of the Hasse principle, a point ξ P Q and a place v of Kpξq, we will say that v is relevant for ξ if |ξ|v ă 1 and the series G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Gm all have v-adic convergence radius greater than |ξ|v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, a place being relevant means that one has to demonstrate condition (2)(ii) holds if one wants to apply the Hasse principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The constants implicit in the term Opδeq can be made explicit, and even (in principle) effective, see the footnote in [And89, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='129] and the corresponding discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By taking G to be a vector consisting of G-functions at s0 arising from a basis of H1 and sections of M Andr´e uses this principle to bound the height of elements in certain subsets of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here the polynomials P are relations on periods coming from algebraic (or absolute Hodge) cycles associated to the cohomology groups of the fibre Xξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However in the absence of a p-adic interpretation of these power series, he is only able to bound those points ξ for which he can show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='i) holds at all finite places, which greatly restricts the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The possibility that p-adic cohomological input might remedy the problem is discussed in [And89, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8-10] and [And89, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='194, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1], but at the time of writing the availability of such techniques to Andr´e was limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The substantial growth in p-adic Hodge theory, and in particular the recent developments in p-adic Hodge theory due to Scholze [Sch13], provides an opportunity to revisit these ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our first main technical contribution will be to give a p-adic interpretation of Andr´e’s G-functions, and show how this greatly expands their applicability to arithmo-geometric problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 p-adic Interpretations of G-functions Our analysis will start by giving a purely algebraic-de-Rham description of Andr´e’s G- functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' a similar description already appears in the proof of [And89, IX, §4, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To do this, we once again fix an affine open subset U Ă X, with coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν inducing an ´etale map U Ñ Aν, and such that the map to S is given by s ÞÑ z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' zµ for some 1 ď µ ď ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we also set w “ µ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The vector bundle H1 extends canonically to a K-algebraic vector bundle H over S, as we review in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From our description of H we will see that any section ω of H then admits a restriction ωU to a relative de Rham sheaf on 8 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fixing a point q P UpKq in the locus z1 “ ¨ ¨ ¨ “ zµ “ 0 mapping to s0, we will further restrict ωU to a formal neighbourhood of q to obtain a unique representation i˚ q ωU “ hq dz2 ¨ ¨ ¨ dzµ z2 ¨ ¨ ¨ zµ , in a formal de Rham complex at q, with hq P Krrsss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This K-algebraic power series hq is obtained without leaving the algebro-geometric category, and its analytification agrees with the function P in (2) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our goal is to provide a p-adic interpretation of the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The basic difficulty is that there is no robust analogue of homology in the rigid-analytic setting, making it difficult to find a proper analogue of the integration pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However, because the cycles in question are of a particularly simple form, one can make do with less, as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' First, for a connected adic space Y defined over SpapCp, OCpq, we recall in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 the definition of the ´etale fundamental group π1 ´etpY, yq at a closed point y P Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider the case where Y “ ∆˝ “ SpapCpxT, T ´1y, OCpxT, T ´1yq is the rigid-analytic torus, and try to describe an element in π1 ´etp∆˝, yq giving a pro-p analogue of a rigid-analytic “loop” around the puncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Unfortunately, the space ∆˝ admits more rigid-analytic coverings than just those of Kummer type, making a na¨ıve approach difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our idea is basically to define this “loop” on just those coverings of Kummer type, and then to choose an extension to π1p∆˝, yq which will be compatible with the formalism of p-adic Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is not so easy to do in the (possibly non-abelian) setting of fundamental groups, so we instead work dually, viewing each element of π1p∆˝, yq through its induced functional on first-degree cohomology via the isomorphism H1p∆˝ p´et, ˆZpp1qq » Homcontpπ1p∆˝, yq, Zpp1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (The notation p´qp´et denotes the pro-´etale site introduced by Scholze in [Sch13], which we review in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') After fixing a compatible system of p’th roots of unity we obtain a functional α˚ : H1pGm,p´et, ˆZpp1qq Ñ Zpp1q, where Gm is the multiplicative group, which we then try to extend to a functional ˆα˚ : H1p∆˝ p´et, ˆZpp1qq Ñ Zpp1q compatible with pullback by the map H1pGm,p´et, ˆZpp1qq Ñ H1p∆˝ p´et, ˆZpp1qq induced by the embedding ∆˝ ãÑ Gm of adic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Of the possible extensions we could choose, we arrange for one satisfying the property that ˆα˚ is zero on the kernel of the natural map H1p∆˝ p´et, ˆZpp1qq Ñ H1p∆˝ p´et, BdRq, where BdR is the period sheaf induced by Scholze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the case of a more general space ∆a,b “ p∆˝qa ˆ ∆b embedding into Ga m ˆ Ab, where ∆ “ SpapCpxT y, OCpxT yq, one extends both α˚ and ˆα˚ to functionals α˚ a,b and ˆα˚ a,b on the cohomology in degree a using the Kunneth formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To apply this to the study of G-functions, we then consider the localized situation that arises from our choice of U Ă X and coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fixing again a point q in the lo- cus z1 “ ¨ ¨ ¨ “ zµ “ 0 mapping to s0 and a small rigid-analytic disk D “ SpapCpxsy, OCpxsyq around s0, we may consider the neighbourhood U “ f ´1pDq X U of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the coordi- nates on U induced by z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν one obtains a neighbourhood C Ă U near q of the form p∆˝qµ ˆ ∆ν´µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A similar local description applies to the fibres Us above each point s P D, giving neighbourhoods ∆w,ν´µ s Ă Us isomorphic to p∆˝qw ˆ ∆ν´µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For each closed point s P D one can then consider an evaluation functional ˆγ˚ s : Hwp∆w,ν´µ s , ˆZppwqq Ñ Zppwq obtained by pulling back ˆα˚ w,ν´µ along an isomorphism ∆w,ν´µ s » ∆w,ν´µ induced by the coordinates z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As the point s varies, the functionals ˆγ˚ s give a p-adic analogue of the family of vanishing cycles in the complex analytic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note, however, that these cycles are defined one fibre at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 9 Using our fixed compatible system of p’th roots of unity, one obtains a fundamental p-adic period t P BdR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then extend ˆγ˚ s to a map ˆγ˚ s,BdR : Hwp∆w,ν´µ s,p´et , ˆZppwqq b BdR Ñ BdR and obtain the following p-adic interpretation of Andr´e’s G-functions: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For each closed point s P D one has that hqpsq “ 1 tw ˆγ˚ s,BdR ´ ρ´1 ´ ω ˇˇ ∆w,ν´µ s ¯¯ , (4) where ρ´1 : Hw dRp∆w,ν´µ s q b BdR Ñ Hwp∆w,ν´µ s,p´et , ˆZppwqq b BdR is a p-adic period map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The notation “ρ´1” is formal, used for compatibility with other notation we will use later, and the map is constructed directly instead of as the inverse of a map “ρ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To make sense of the equality (4) we have fixed an embedding K ãÑ Cp, and the result holds for all such choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that period maps like ρ´1 are typically constructed to compare the cohomology of complete — or at least “nicely” completable — algebraic or rigid-analytic varieties, which is far from the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But the product of disks ∆w,ν´µ is simple enough that one can describe the map ρ´1 by an explicit ˇCech calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Showing that ρ´1 is sufficiently compatible with other p-adic period isomorphisms — in particular, enough to consistently pull back ˆγ˚ s to the cohomology of the ambient variety Xs — then amounts to us having chosen our extension of α˚ to not interact with the kernel of the map H1p∆˝ p´et, ˆZpp1qq Ñ H1p∆˝ p´et, BdRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For applications one actually needs something more precise than Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13, which we show in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The point is that one doesn’t just want an algebro-geometric interpretation of the function hq that holds in some unspecified neighbourhood D of the point s0, but instead an interpretation for hq that holds inside its entire radius of convergence after choosing an embedding of K into C or Cp (or at least when |s|v ă 1, in light of the condition in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This requires additional arguments in both the complex analytic and rigid analytic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, to make this work in the rigid analytic setting we will choose the coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν carefully so that neighbourhoods like the neighbourhood C mentioned above are sufficiently large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' see the discussion in §4 and §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 Relations on Periods The above considerations, of course, are purely local, and to produce algebraic relations between the values taken by Andr´e’s G-functions at a point s for the purpose of applying the Hasse principle one ultimately needs to relate these values to the cohomology of the projective fibre Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that we have an induced natural map HwpXs,p´et, ˆZppwqq Ñ Hwp∆w,ν´µ s,p´et , ˆZppwqq ˆγ˚ s ÝÝÑ Zppwq which by abuse of notation we also denote by ˆγ˚ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one additionally considers the compar- ison isomorphism of p-adic Hodge theory, one obtains a diagram 10 Hw dRpXsq b BdR Hw dRp∆w,ν´µ s q b BdR HwpXs,p´et, ˆZppwqq b BdR Hwp∆w,ν´µ s,p´et , ˆZppwqq b BdR „ ρ´1 , using which one may extend ˆγ˚ s,BdR consistently along both paths in the diagram to take values on Hw dRpXsq, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After fixing a frame ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm of H1ˇˇ D with corresponding representations i˚ q ωj “ hj dz2 ¨ ¨ ¨ dzµ z2 ¨ ¨ ¨ zµ , our result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13 can be interpreted as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The values h1psq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmpsq give the vector 1 tw ˆγ˚ s,BdR P HompHµ´1 dR pXsq b BdR, BdRq in the dual coordinates induced by ω1,s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To apply this fact we now consider the situation where Xs admits a non-trivial algebra E of algebraic self-correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If our point s P D comes from a point ξ of S defined over a finite extension L of the fixed number field K, our goal is to produce a K-algebraic relation on the coordinates h1psq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmpsq so that we can apply the Hasse principle for G-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But the strategy employed by Andr´e in [And89, X, §3] at the infinite places doesn’t work, as it involves the underlying Q-structure of the Betti cohomology of Xξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However as the p-adic case is enriched by the presence of a Galois action compatible with the period isomorphism, we may obtain relations using an entirely new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' First, we prove the following, which is almost immediate from the construction of ˆγ˚ s : Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let v be the place of K at which the functional ˆγ˚ s is defined, and GKv “ GalpKv{Kvq the associated local Galois group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then GKv acts on ˆγ˚ s through the character χ´w cycl, where χcycl : GKv Ñ Z˚ p is the usual cyclotomic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The point is that the only choice not invariant under the Galois action made in the con- struction of ˆγ˚ s is our choice of a non-trivial compatible system of p-power roots of unity corresponding to a p-adic “loop” inside the torus ∆˝, and if we “integrate” around w such loops then GK acts on the “integrals” through the w’th power of χcycl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (There is also a choice of splitting of the ´etale cohomology of ∆˝, but all such choices lead to the same functional on the cohomology of Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') We observe that this simple fact is already enough to produce non-trivial algebraic relations on the de Rham coordinates of ˆγ˚ s in the presence of an L-algebraic correspondence τ : Xs ��� Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, the cohomology class rτs in both de Rham and ´etale cohomology is fixed by a finite index subgroup of GKv, hence the functionals ˆγ˚ s , ˆγ˚ s ˝ rτs, ˆγ˚ s ˝ rτs2, ˆγ˚ s ˝ rτs3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (5) all lie in a subspace of the dual of HwpXs,p´et, ˆZppwqq on which a finite index subgroup of GK acts by χ´w cycl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one expresses the functionals in (5) in a fixed L-algebraic de Rham basis, then the coordinates of each element of the sequence (5) are L-linear combinations of the coordinates of ˆγ˚ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The Hodge-Tate comparison provides a simple way of bounding the 11 dimension of the χ´w cycl-character space in terms of the de Rham Hodge numbers, and hence one obtains an L-algebraic relation on the coordinates of ˆγ˚ s simply by taking determinants of minors of matrices constructed from the vectors in the sequence (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To make the relation K-algebraic one takes the product of this relation with all its Galois conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let us remark that, in the cases we consider, we often don’t expect relations like the ones we construct on the coordinates of ˆγs to actually exist for points s which are p-adically close to s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The reason is that one generally expects special moduli to be in some sense bounded away from singularities in p-adic metrics: this is the case for curves and abelian varieties with complex multiplication, all of which have potentially good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' So instead our construction of non-trivial relations on the coordinates of ˆγ˚ s under the (often counterfactual) assumption that s is a special modulus p-adically close to s0 is to be interpreted as a sort of integrality constraint on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4 Applications to Height Bounds The generality in which one can now construct algebraic relations on Andr´e’s G-functions at finite places eliminates a broad class of obstructions to applying the G-function method to problems in arithmetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed one can now show, in quite general settings, that polynomial height bounds on special moduli follow as soon as one can establish K-algebraic constraints at the infinite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now give a sample result of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Recall the variation of Hodge structure V1 “ Rwf an ˚ Z{tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' whose fibres have dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In what follows we write S Ă SpQq for the set of ξ P SpCq such that there exists a simple Hodge summand W Ă V1 Q,ξ of CM type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that V1 is simple and that the Hodge conjecture holds for en- domorphisms appearing in the fibres of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let s0 be a point in SzS1 at which the fibre Xs0 acquires a normal crossing singularity of order w at the point q P Xs0, and for which the associated tuple ph1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmq of G-functions is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then after replacing K with a finite extension, there exists (i) a finite covering c : C Ñ S, and a parameter s on C with simple zeros and vanishing exactly on the set c´1ps0q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and (ii) for all but finitely many ξ P c´1pSq a Kpξq-algebraic relation on the values h1pspξqq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmpspξqq, not induced by a functional relation on h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hm, and which holds at all finite places relevant for spξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Moreover, the degree of the relation in (ii) may be bounded independently of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16(ii) we actually mean to replace the original G-functions with the ones computed in terms of the parameter s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we give a more precise description in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This leads to the following theorem, which reinterprets Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 above: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that in the setting of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16) there exists an additional order w normal crossing point q1 P Xs0 with an associated non-constant tuple ph1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , h1 mq linearly independent from ph1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then for any logarithmic Weil height θ : SpQq Ñ Rą0 there exists constants κ, a P Rą0 such that θpξq ď κ rKpξq : Ksa for all ξ P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 Pila-Zannier for General Atypicality To get from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17 to the finiteness results in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5, one applies the Pila-Zannier strategy for obtaining finiteness results from lower bounds on Galois orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However the usual Pila-Zannier strategy, for instance used to prove the Andr´e-Oort conjecture, is insufficient here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In that setting one uses the fact that points ξ P S Ă SpCq above which the fibre Xξ has complex multiplication produce Q-algebraic points rt inside a definable period image I Ă D, where D is a period domain for the polarized Hodge structures appearing in the fibres of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By producing lots of Q-algebraic points in I of bounded height and over number fields of bounded degree one can apply a theorem of Pila- Wilkie to obtain an algebraic curve inside I, and from this functional transcendence results can be used to relate this to Hodge loci in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The notion of algebraicity here comes from the open embedding D Ă qD into a natural ambient flag variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If the points in S one is studying are not CM points, the situation becomes more compli- cated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' What happens in this case is that the points rt are no longer Q-algebraic, but merely have lower-than-expected transcendence degree on account of an intersection between Iand a Q-algebraic flag subvariety qDM Ă qD determined by the Mumford-Tate M of the Hodge structure rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our observation, which is related to ideas appearing in [DR18], is that one can obtain results in this more general setting by applying Pila-Zannier-type reasoning to the moduli of the varieties qDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' More specifically, one can reduce to the case where one considers only varieties qDM for which the associated Mumford-Tate groups M lie inside a single GSpCq-orbit for the generic Mumford-Tate group GS of the variation V, where the action on Mumford-Tate groups is by conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The situation one is then tasked with deal- ing with is the situation where there are many Q-algebraic translates g ¨ qDM of the variety qDM which intersect I atypically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One can understand the elements g that arise in terms of heights of Hodge tensors defining the associated Mumford-Tate groups gMg´1, and use this to bound both the heights of such g and the degree of their field of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The Pila-Wilkie theorem then produces, under appropriate bounds on the heights of some Hodge tensors associated to points of I, an algebraic family of subvarieties of qD which intersect I atypically, and from this one can run the usual functional transcendence arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We do not need any constraints on S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, we do not use that S is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As an application of this, we prove the following general result, which we state here somewhat informally (see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 for the relevant definitions and precise statements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that f : X Ñ S is a family of abelian varieties whose algebraic monodromy agrees with its derived Mumford-Tate group, and S Ă SpCq is the subset of points in the zero-dimensional Hodge locus which are defined by, and atypical for, the property of acquiring a non-generic isogeny summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then if there exists constants κ, a P Rą0 such that θpξq ď κ rKpξq : Ksa for all ξ P S, with θ some logarithmic Weil height, then S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6 Related Work As we have discussed in great detail, the G-function method for bounding heights on special moduli was introduced in Andr´e’s book [And89], but was limited by the lack of p-adic interpretations of these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Since then, two works of which we are aware of have managed to apply this method in a way which produces cohomological relations on values of G-functions at finite primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The first is follow-up work of Andr´e [And95], and in particular 13 [And95, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Here Andr´e works in the special setting of families of abelian varieties and gives a p-adic interpretation of values of G-functions using crystalline cohomological period matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However to obtain bounds on heights of special moduli, he is forced to restrict to moduli satisfying an integrality condition at all but finitely many primes, as the relations he obtains at the finite places do not hold for all finite places at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A second more recent approach is given by Daw and Orr in [DO22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Here they are able to use a concrete p-adic interpretation of certain G-functions coming from Tate’s p-adic uniformization of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' They are able to control all finite places at once, and prove results similar to us in the special case of curves in Y p1qn (the n-fold product of the moduli space of elliptic curves) intersecting a certain “very degenerate” point lying on the boundary of the compactified moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It seems to us difficult to extend this method to more general types of special moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With respect to obtaining height bounds on special moduli in more general settings, specifically beyond the case of abelian-type families, recent work of Papas [Pap22] gives general results under algebraic-cycle and arithmeticity conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have found his work useful for giving an outline of the general strategy of obtaining height bounds in settings beyond those considered by Andr´e, Daw and Orr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, he gives in [Pap22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] a general theorem for producing height bounds of the kind we are interested in under the Hodge conjecture and a conjecture on the existence of certain “good” arithmetic models for smooth projective families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The arithmetic models assumption has the effect of showing that the special points he studies do not lie in sufficiently small v-adic neighbourhoods near the degeneration point, and so is in effect a kind of integrality assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our approach gives a way of removing this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Finally, with respect to the Pila-Zannier strategy for Zilber-Pink-type atypical inter- sections, Daw and Ren in [DR18] give an approach for the special case of subvarieties of Shimura varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The basic idea is in some sense similar in that one tries to argue that having “many” special points in S will allow one to produce some low-dimension algebraic variety which interacts exceptionally with an analytic period image in order to contradict an Ax-Schanuel principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To do this they introduce and assume several conjectures relating to various notions of “complexity” of special subvarieties, and establish a Zilber-Pink-type theorem for point-like atypical intersections inside Shimura varieties under these assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our results are similar, except that we are able to work in a general algebro-geometric setting beyond the case of Shimura varieties, and various aspects of our approach seem sim- pler to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that aside from Galois-orbit lower bounds and that special varieties in the general-algebraic-family setting behave well under Galois-actions (a basic requirement fundamental to the study of atypical intersections), we do not need other conjectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' see for instance Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 Acknowledgements We thank Jacob Tsimerman, Chris Daw, Martin Orr, and Georgios Papas for comments on a draft of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We also thank Donu Arapura for a MathOverflow comment which suggested to the author the idea of considering hypersurface degenerations to hyperplanes as an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 2 Cohomological Preliminaries We continue with the notation and setup established in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 A Model for the Canonical Extension We begin by describing an explicit model for the canonical extension H of H1 referenced in the introduction, following Steenbrink [Ste76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As before we assume that E “ XzX1 “ f ´1ps0q, where ts0u “ SzS1 and E is a divisor with normal crossings, and define the de Rham complex Ω‚ Xplog Eq of algebraic differentials with logarithmic poles along E as follows: for an open set U Ă X the sections of Ωp Xplog Eq over U are the algebraic forms ω on UzE such that ω and dω have at most a simple pole along E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one chooses local coordinates pz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνq around a point q P E so that E is defined by z1 ¨ ¨ ¨ zµ “ 0 for some 1 ď µ ď ν, then the stalk Ω1 Xplog Eqpqq is a free module over OX,pqq with generators dz1{z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , dzµ{zµ, zµ`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν and Ωp Xplog Eq “ Źp Ω1 Xplog Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We further define Ωp X{Splog Eq as the pth exterior power of the quotient Ω1 Xplog Eq{f ˚Ω1 Splogts0uq, with Ω1 Splogts0uq defined analogously via differ- entials with at most a logarithmic pole at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The following is proven in [Ste76, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='18] (note that it makes no difference whether one uses the algebraic or analytic site, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4 below): Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For all w ě 0, the sheaf Rwf˚pΩ‚ X{Splog Eqq is locally free on S and for all s P S the canonical map Rwf˚pΩ‚ X{Splog Eqq bOS pOS,s{mS,sq Ñ HwpXs, Ω‚ X{Splog Eq bOX OXsq is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may therefore take H “ Rwf˚Ω‚ X{Splog Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let us now consider the setup in the introduction, where U Ă X was a fixed affine Zariski open subset with coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν trivializing Ω1 U and such that the map U Ñ S takes the form s ÞÑ z1 ¨ ¨ ¨ zµ for some 1 ď µ ď ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As before, we set w “ µ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have a natural map Rwf˚Ω‚ X{Splog Eq Ñ Rwf˚Ω‚ U{SplogpE X Uqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For S affine, the cohomology module Rwf˚Ω‚ U{SplogpE X Uqq may be identified with RwΓ Ω‚ U{SplogpE X Uqq, as follows from the Leray spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Moreover, because U is affine, this can in turn be identified with the cohomology in degree w of the complex 0 Ñ OU Ñ Ω1 U{SplogpE X Uqq Ñ ¨ ¨ ¨ Ñ Ωn U{SplogpU X Eqq, viewed as a module over OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Restricting to the completed stalk at a point q in the locus z1 “ ¨ ¨ ¨ “ zµ “ 0 mapping to s0 one obtains a complex of pOS,ps0q-modules 0 Ñ pOU,pqq Ñ pΩ1 U{SplogpE X Uqqpqq Ñ ¨ ¨ ¨ Ñ pΩn U{SplogpE X Uqqpqq, and by composition a restriction map η : ΓpRwf˚Ω‚ X{Splog Eq, Sq Ñ Cohomw ” pΩ‚ U{SplogpE X Uqqpqq ı , where we denote by Cohomw the na¨ıve cohomology in degree w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This map will be used in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 to construct G-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The following Lemma describes the form of the elements in Cohomw ” pΩ‚ U{SplogpE X Uqqpqq ı , and also the analogous modules obtained by considering convergent power series in the com- plex and rigid-analytic topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 15 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that A is any of the rings tKrrz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνss, Ctz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνu, kxz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνy, kxz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνyR1{µu which are (in order) formal power series over the characteristic zero field K, germs of complex analytic power series, germs of k-analytic power series over the non-archimedian valued field k, and power series convergent in the (closed or open) non-archimedian ball of radius R1{µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let B be the corresponding ring in tKrrsss, Ctsu, kxsy, kxsyRu and consider the map B Ñ A given by s ÞÑ z1 ¨ ¨ ¨ zµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Consider the complex 0 Ñ A Ñ Ω1 A{Bplog Eq Ñ ¨ ¨ ¨ Ñ ν ľ Ω1 A{Bplog Eq Ñ 0, (6) where Ω1 A{Bplog Eq is the quotient of Ω1 Aplog Eq and A bB Ω1 Bplogt0uq, with Ω1 Aplog Eq “ Adz1 z1 ‘ ¨ ¨ ¨ ‘ Adzµ zµ ‘ Adzµ`1 ‘ ¨ ¨ ¨ ‘ Adzν, and Ω1 Bplogt0uq “ B ds s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then every element α of Cohomw ” Ω‚ A{Bplog Eq ı admits a unique representation of the form hdz2 ¨ ¨ ¨ dzµ z2 ¨ ¨ ¨ zµ where h is a uniquely determined element of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the complex analytic setting the entire cohomology of the complex (6) is described in [Ste76, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13], and the same proof works in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For the convenience of the reader we give some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The relation s “ z1 ¨ ¨ ¨ zµ induces the relation dz1 z1 ` ¨ ¨ ¨ ` dzµ zµ “ 0 in the complex (6), which gives a natural presentation of the complex in terms of the forms dz2 z2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , dzµ zµ , dzµ`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , dzν only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The complex then reduces to a Kozul-type complex L‚ generated by Adz2 z2 ‘ ¨ ¨ ¨ ‘ Adzµ zµ ‘ A dzµ`1 ‘ ¨ ¨ ¨ ‘ A dzν, where the differential operators for L‚ are given by Di “ ziBi ´ z1B1 for 2 ď i ď µ and Di “ Bi for µ ` 1 ď i ď ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that an element β “ g dzi1 z ei1 i1 ^ ¨ ¨ ¨ ^ dzir zeir i1 in the complex L‚ lies in the kernel of the differential, with g P A a monomial, and where the exponents eij are chosen so that each factor in the wedge product is one of the specified generators for the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then from the construction of the Kozul complex we must have that Djpgq “ 0 for each j not appearing in the set ti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , iru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If we have ik ą µ for some k (and hence eik “ 0), and the variable zik occurs with exponent a ě 0 in the monomial g, we compute that d ˜ p´1qik zik a ` 1g dzi1 z ei1 i1 ^ ¨ ¨ ¨ ^ y dzik ^ ¨ ¨ ¨ ^ dzir zeir i1 ¸ “ β ` p´1qik zik a ` 1 ÿ j Djpgq dzj ^ ˜ dzi1 z ei1 i1 ^ ¨ ¨ ¨ ^ dzir zeir i1 ¸ loooooooooooooooooooooooomoooooooooooooooooooooooon “0 16 which shows that β can be integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Thus, non-trivial contributions to cohomology appear only when ti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , iru Ă t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , µu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In degree r “ w this means that ti1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , iwu “ t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , µu and that Djpgq “ 0 for j ą µ, meaning we can assume our class α is of the form h dz2¨¨¨dzµ z2¨¨¨zµ with h depending on z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zµ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To ensure that a monomial m in h cannot be integrated, one checks that we must have additionally that Djpmq “ 0 for 2 ď j ď µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But this means that h consists only of terms like c ¨ pz1 ¨ ¨ ¨ zµqe, hence g lies in the image of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The uniqueness claim is checked directly from the construction of the complex, as no non-zero elements of the specified form lie in the image of the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 ˇCech cohomological recollections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 ˇCech cohomology of complexes We now develop the general formalism of the ˇCech double complex associated to a complex pF‚, dFq of sheaves of abelian groups on a site C, generalizing the case of sheaves on a topological space which appears in [Sta20, Section 01ED] and [Sta20, Section 01FP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We assume that C has a final object X, and we let U “ tci : Ui Ñ XuiPI be a covering of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We first consider the case where F‚ “ F consists of a single sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We define CppU, Fq “ ź pi0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=',ipqPIp`1 FpUi0 ˆX ¨ ¨ ¨ ˆX Uipq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given s P CppU, Fq we will write si0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ip its value in the factor FpUi0 ˆX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ˆX Uipq, and we define the differential δ : CppU, Fq Ñ Cp`1pU, Fq by the formula δpsqi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ip`1 “ p`1 ÿ j“0 p´1qjsi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' pij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ip`1 ˇˇˇ Ui0 ˆX¨¨¨ˆXUip`1 where restriction comes from the natural fibre product projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One checks that pC‚pU, Fq, δq is a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The formation of the complex C‚pU, Fq is functorial in F, so given a complex pF‚, dFq one naturally obtains a double complex C‚pU, F‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write pL‚pU, F‚q, dq for the associ- ated total complex, with terms LnpU, F‚q “ à p`q“n ź i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ip FqpUi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ipq and with differential of an element α of degree n given by d “ δ ` p´1qp`1dF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Finally, we write qHppU, F‚q for the cohomology groups of this complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now compare the ˇCech cohomology to sheaf cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We denote the cohomology of a complex of sheaves F‚ computed on an object V of C using covers of C by H‚pVC, F‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then we have the following generalization of [Gro60, III, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6]: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let F‚ be a bounded below complex of abelian groups on C, and let U “ tci : Ui Ñ XuiPI be a covering of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then there exists a spectral sequence abutting to H‚pXC, F‚q whose second page is given by Epq 2 “ CohomppL‚pU, JqpF‚qqq, where JqpF‚q denotes the complex of presheaves whose j’th term is given by rV ÞÑ HqpVC, Fjqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One just has to check that all the steps in the argument in [Gro60, III, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6] generalize to this situation (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Sta20, Lemma 08BN]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider a Cartan-Eilenberg resolution L‚‚ of F‚ by injective sheaves, constructed as in [Sta20, Lemma 015I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the functorality of ˇCech cohomology we obtain a tricomplex C‚pU, L‚‚q “ rCipU, Ljkqs which we may regard as a bicomplex in degrees i and j ` k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because the sheaves Ljk, and hence the terms in the total complex of L‚‚, are all injective sheaves of abelian groups, the ˇCech complex C‚pU, L‚‚q computes the cohomology of L‚‚ regarded as a single complex: this follows by combining [Sta20, Lemma 03AW], which shows that the positive degree ˇCech cohomology on U of each injective sheaf is zero, with [Sta20, Lemma 0133] applied to the map L‚‚ Ñ C‚pU, L‚‚q, where we regard L‚‚ as a single complex, as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because the total complex of L‚‚ computes the cohomology of F‚, it follows that the map F‚ Ñ C‚pU, L‚‚q induces an isomorphism on cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now consider the tricomplex C‚pU, L‚‚q as a bicomplex in degrees i`j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then because Lj,‚ is an injective resolution of Fj for all j, the degree q cohomology of the complex CipU, Lj,‚q is then given by the ˇCech complex CipU, JqpFjqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Computing the second page then gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that for each U 1 obtained as a fibre product of objects in the cover U and for each k one has that HqpU 1 C, Fkq “ 0 for all q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then L‚pU, F‚q computes the cohomology of the complex F‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The assumption ensures that the spectral sequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 degenerates at the second page, hence the cohomology is computed by the first page, which is naturally identified with the cohomology of the total ˇCech complex L‚pU, F‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 Cup product in ˇCech cohomology We also recall how to define the cup product on the ˇCech complex, following [Sta20, Section 01FP] in the setting of complexes of sheaves on topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given two complexes of sheaves F‚ and G‚ of abelian groups on the site C, we write TotpF‚ b G‚q for the complex with terms À p`q“n Fp b Gq and where the differential is given by dpα b βq “ dpαq b β ` p´1qdegpαqα b dpβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a covering U “ tci : Ui Ñ XuiPI, our cup product is then a map Y : Tot pTotpC‚pU, F‚qq b TotpC‚pU, G‚qqq Ñ TotpC‚pU, TotpF‚ b G‚qqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It is given by the rule pα Y βqi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ip “ pÿ r“0 εpdeg α, deg β, p, rqαi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ir b βir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='ip, where εpn, m, p, rq “ p´1qpp`rqn`rp`r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The associativity of the cup product as well as the identity dpα Y βq “ dpαq Y β ` p´1qdegpαqα Y dpβq may be proved by explicit calculation, exactly as is done in [Sta20, Section 01FP] in the setting of complexes on topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Moreover, the cup product is compatible with a graded commutative structure on the complex F‚, as we now explain, following [Sta20, Section 01FP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 18 Suppose that we have a graded commutative multiplication map ^‚ : TotpF‚ b F‚q Ñ F‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is defined to mean that given sections s of Fa and t of Fb we obtain a section s ^ t of Fa`b in such a way that s^t “ p´1qabt^s, and that we have dps^tq “ dpsq^t`p´1qas^dptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may then consider the composition Tot pTotpC‚pU, F‚qq b TotpC‚pU, F‚qqq Y ÝÑ TotpC‚pU, TotpF‚ b F‚qqq ^ ÝÑ TotpC‚pU, F‚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It may be checked as in [Sta20, Section 01FP] that this induces a map on ˇCech cohomology HnpTotpC‚pU, F‚qqq ˆ HmpTotpC‚pU, F‚qqq Ñ Hn`mpTotpC‚pU, F‚qqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In our situation of interest, this will reproduce the cup product on both ´etale cohomology and (algebraic) de Rham cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 The pro-´etale site Let us fix an adic space X over Spapk, Okq, with k a perfectoid field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will assume that X is locally noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (This assumption will also continue to be in force in subsequent sections without further comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') We will begin by defining some categories (and sites) associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' First, one has the ´etale site X´et, whose objects consist of ´etale maps U Ñ X of adic spaces a morphisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Next we consider the category propX´etq: its objects consist of projective limits lim ÐÝiPI Ui of objects of X´et and its morphisms are the natural morphisms of limit diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A map of objects U Ñ V in the category propX´etq is called ´etale if it is induced by an ´etale morphism of objects U0 Ñ V0 in X´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A map of objects U Ñ V is called pro-´etale if we have U “ limi Ui in such a way so that U Ñ V is given by an inverse limit Ui Ñ V of objects ´etale over V , and such that Ui Ñ Uj is finite ´etale and surjective for large i ą j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The category Xp´et is then defined to be the full subcategory of propX´etq consisting of objects which are pro-´etale over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A covering U in propX´etq of an object U is given by a family of pro-´etale morphisms U “ tfi : Ui Ñ Uu such that |U| “ Ť i fip|Ui|q, where we give pro-objects the limit topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By [Sch13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='10], this defines a site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one instead starts with the category Xf´et of objects finite ´etale over X, one may carry out the analogous procedure to define a category Xpf´et, which we call the “pro-finite finite ´etale site”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It is naturally a subcategory of Xp´et, and we have a natural map of sites Xp´et Ñ Xpf´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The pro-finite ´etale site can be used to compute ´etale cohomology with coefficients in Zp as the cohomology of the sheaf ˆZppUq “ Homcontp|U|, Zpq, where we consider continuous morphisms of the underlying topological spaces, and Zp has the usual p-adic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now introduce some important sheaves on Xp´et, following [Sch13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The first is OX, the “uncompleted structure sheaf”, which is the pullback γ˚OX´et under the natural map γ : Xp´et Ñ X´et of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Likewise we have the subring of integral elements O` X “ γ˚O` X´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These sheaves can then be completed to obtain ˆO` X “ lim ÐÝ O` X{pn and ˆOX “ ˆO` X ” 1 p ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Next we have the tilted integral structure sheaf, defined as ˆO` X5 “ lim ÐÝΦ O` X{p, with the inverse limit over Frobenius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here we use the notion of the tilt X5 of X comes from Scholze’s theory 19 of perfectoid spaces [Sch11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We set ˆOX5 “ ˆO` X5 bk5` k5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then define Ainf “ WpˆO` X5q and Binf “ Ainf ” 1 p ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have a natural map θ : Ainf Ñ pO` X which extends to a map Binf Ñ ˆOX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To define it, we work locally, where the sheaf Ainf is represented by a ring WpA5q, and we wish to construct a map WpA5q Ñ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may represent an element x P WpA5q via its Witt vector components as a sum ř i pirxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then define θ ˜ÿ i pirxis ¸ “ ÿ i pix7 i, where the operation p´q7 is defined on y P A5, represented by the sequence py1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='q, by choosing lifts ˆyj for all j and setting y7 “ lim jÑ8 ˆyj pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then define B` dR “ lim ÐÝ Binf{pker θqn and BdR “ B` dRrt´1s, where t is any element gener- ating the kernel of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Finally we define OBinf “ OX bWpκq Binf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The map θ on Binf extends to a map θ : OBinf Ñ ˆOX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One then defines OB` dR “ lim ÐÝ OBinf{pker θqn and OBdR “ OB` dRrt´1s, where t is a generator of ker θ (this makes sense locally, as is checked in [Sch13, §6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lastly we define Ωi X “ OBdR bOX Ωi X as sheaves on Xp´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4 Coherent Cohomology on Various Sites An important sort of fact that we will use (often implicitly) throughout the paper is that it “doesn’t matter” on which site one computes the cohomology of coherent objects associated to a space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' What is meant by this is is that one has two sites associated to X, say X1 and X2, with a natural map of ringed sites τ : X2 Ñ X1, and given a complex of coherent sheaves F‚ on X1 the natural map HipX1, F‚q Ñ HipX2, τ ˚F‚q is an isomorphism for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that one typically only needs to check this when F‚ “ F is a single sheaf rather than a complex of such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The reason is that, in the situations of interest, the sites X1 and X2 will contain certain types of objects on which the coherent cohomology of any individual coherent sheaf vanishes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', affine, affinoid, Stein, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ), and using this fact for each Fi in the complex F‚ and an appropriate cover one learns that the “same” ˇCech complex computes cohomology on both X1 and X2, and the resulting fact is formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These facts we will only need for complexes of differentials (possibly relative, possibly logarithmic) and for sufficiently nice spaces X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (Our spaces or maps of spaces will also often be proper, which makes things even easier, see the remark below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') Nevertheless, we give some of the required facts in greater generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For F‚ a complex of coherent sheaves on X1 and τ : X2 Ñ X1 a map of sites, the natural map HipX1, F‚q Ñ HipX2, τ ˚F‚q is an isomorphism when (i) X is a scheme, X1 “ XZar, X2 “ X´et;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (ii) X is a proper C-scheme, X1 “ XZar, X2 “ Xan, and F‚ “ Ω‚ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (iii) X is a Cp-scheme, X1 “ XZar, X2 “ Xad, and F‚ “ Ω‚ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (iv) X is a rigid space, X1 “ Xad, X2 “ X´et;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 20 (v) X is a adic space, X1 “ X´et, X2 “ Xpro´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For (i) see [Sta20, Proposition 03DW];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for (ii) see the introduction to [Gro66];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for (iii) see [GK04, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for (iv) see [CT09, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for (v) see [Sch11, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that one obtains (ii) and (iii) for any coherent sheaf F‚ when X is proper by the relevant GAGA result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The same is also true if one considers the derived pushforward in the relative setting of a proper morphism f : X Ñ S of schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' see for instance [Ray71, Expose XII] and the appendix to [Con06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 The ´etale fundamental group and cohomology We now describe the ´etale fundamental group of X and its relation to the cohomology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write Xf´et for the category of adic spaces Y which are finite ´etale over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fixing a geometric point x of X, we obtain a natural fibre functor FX,x : Xf´et Ñ Set, and as usual the group π´et 1 pX, xq is defined as the group of automorphisms of this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For any finite abelian group Λ, we now describe a natural isomorphism Hompπ´et 1 pX, xq, Λq „ ÝÑ H1pXp´et, Λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that to compute H1pXp´et, Λq for a finite abelian group Λ it suffices to use the usual ´etale site X´et, since the natural map H1pXp´et, Λq Ñ H1pX´et, Λq induced by the map of sites X´et Ñ Xp´et is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' this is due to [Sch11, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17], as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The description of this isomorphism is essentially identical to the case of schemes, for which [Mil13, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='§11] is a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will give some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In what follows we also denote by Λ the constant sheaf on X´et it defines, and we use multiplicative notation for group multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A sheaf L of sets on X´et on which Λ acts is called a torsor for Λ if: (i) there exists a covering U “ tci : Ui Ñ XuiPI in X´et such that LpUiq ‰ ∅ for all i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and (ii) for every object U Ñ X in X´et and s P LpUq the map Λ ˇˇ U Ñ L ˇˇ U given by g ÞÑ gs is an isomorphism of sheaves over U´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A covering U “ tci : Ui Ñ XuiPI for which (i) holds is said to split L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Supposing we have such a covering, we construct a ˇCech cocycle rLs P H1pU, Λq as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Choose some sections si P LpUiq for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By (ii), on each “intersection” Uij arising from the cover U there exists a unique element λij P ΛpUijq such that λij ¨ si ˇˇ Uij “ sj ˇˇ Uij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then pgijqIˆI is a cocycle, and defines a class rLs in H1pU, Λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Moreover we have: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The map L ÞÑ rLs defines a bijection from the set of isomorphism classes of torsors for Λ split by U to H1pU, Λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the case of the ´etale site of a scheme this is [Mil13, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Prop 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1], and the proof is identical in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now use the fact that there is a further bijection tisom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' classes of Λ-torsorsu ÐÑ Hompπ1pX, xq, Λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (7) 21 This is true in a great deal of generality by the work of [AM69] (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the discussion in [H¨ub18, §9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We describe this correspondence in the special case where the Λ-torsor L is representable by a Galois covering Y Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' More precisely, we assume that Λ “ AutXpY q and that LpUq “ HomXpU, Y q for every U P Xf´et, with the natural action of Λ on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the fact π1pX, xq “ AutpFX,xq, we may define the map π1pX, xq Ñ AutXpY q by sending η P AutpFX,xq to the automorphism α P AutXpY q for which ηpyq “ αpyq for all y P FX,xpY q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' that such an element exists follows from the assumption that Y Ñ X be a Galois cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the situation where Λ “ lim ÐÝ Λn is a pro-finite group, one can take the limit of both sides of (8) to obtain a bijection tisom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' classes of Λ-torsorsu ÐÑ Homcontpπ1pX, xq, Λq, (8) with a similar explicit description in the case of a torsor coming from a limit of Galois coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 3 Cohomological Computations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 Basic ˇCech Computations Let k be a complete algebraically closed non-archimedian local field with ring of integers Ok and residue field κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write ∆˝ for the adic space given by SpapkxT ˘1y, OkxT ˘1yq, which can be thought of as a rigid-analytic annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider the natural cover of ∆˝ inside ∆˝ pro´et with the covering space modelled by the infinite tower r∆˝ “ lim ÐÝ T ÞÑT p ∆˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The space r∆˝ is then perfectoid of the form SpapkxT ˘1{p8y, OkxT ˘1{p8yq “ lim ÐÝ SpapkxT ˘1{pjy, OkxT ˘1{pjyq, and the covering map c : r∆˝ Ñ ∆˝ with respect to these presentations is simply given by T ÞÑ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Define Zpp1q “ lim ÐÝj µpj, where the transition maps are given by x ÞÑ xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider the self-product r∆˝2 “ r∆˝ˆ∆˝ r∆˝, and observe that its connected components are naturally indexed by Zpp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, one has that r∆˝2 “ lim T ÞÑT pj ∆˝ ˆ∆˝ ∆˝ looooomooooon ∆˝ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Where the fibre product ∆˝ j may be modelled as ∆˝ j “ SpapkxT ˘1 1 , T ˘1 2 y{pT pj 1 ´ T pj 2 q, OkxT ˘1 1 , T ˘1 2 y{pT pj 1 ´ T pj 2 qq, and the transition maps are given by pT1, T2q ÞÑ pT p 1 , T p 2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It is clear that the components of ∆˝ j are naturally identified with the group µpj of pj’th roots of unity, with ζj P µpj identified with the component on which T1 ´ ζjT2 “ 0, and hence the idempotents of the coordinate ring of r∆˝2 are identified with a compatible system of such roots and hence with Zpp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 22 Let F‚ “ Ω‚ ∆˝ be the sheaf of BdR-differentials on the pro-´etale site, as defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We let U “ tcu be our cover, and form the ˇCech complex C‚pU, F‚q and the associated total complex L‚ “ L‚pF‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will consider the cocycle sc “ logpT {rT 5sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To be more precise, [Sch13, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13] shows that the sequence 0 Ñ BdR Ñ OBdR dÝÑ Ω1 ∆˝ Ñ 0 (9) is exact, and does so by showing that the map OB` dR ˇˇ r∆˝ „ ÝÑ B` dR ˇˇ r∆˝rXs defined by X ÞÑ T b 1 ´ 1 b rT 5s is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here T 5 is defined as in [Sch13, §6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One does this by showing that B` dR ˇˇ r∆˝rXs admits the structure of an O∆˝ ˇˇ r∆˝-algebra, satisfying T ÞÑ rT 5s`X and compatible with the one on the quotient B` dRrrXss{pkerθq “ ˆO∆˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This then gives a natural map ´ O∆˝ bWpκq WpˆO` ∆˝5q ¯ ˇˇˇ r∆˝ „ ÝÑ B` dR ˇˇ r∆˝rrXss inducing the inverse of the map X ÞÑ T b 1 ´ 1 b rT 5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using this description, one can define logpT {rT 5sq by applying the inverse of the isomor- phism, computing logp1 ` X{rT 5sq (using the power series expansion) and then using the isomorphism to translate the resulting expression back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The resulting function satisfies the property that dplogpT {rT 5sqq “ dT {T , and that logpapT {rT 5sqq “ logpaq ` logpT {rT 5sq for any non-zero a P BdR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that we have two natural maps pi : r∆˝2 Ñ r∆˝ with i P t1, 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one models r∆˝2 as the space r∆˝2 “ SpapkxT ˘p8 1 , T ˘p8 2 y{pT1 ´ T2q, OkxT ˘p8 1 , T ˘p8 2 y{pT1 ´ T2qq, then these maps are given by T ÞÑ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The component r∆˝2 ζ‚ Ă r∆˝2 corresponding to the sequence ζ‚ “ pζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='q is then given by imposing the infinitely many relations T 1{p 1 “ ζ1T 1{p 2 , T 1{p2 1 “ ζ2T 1{p2 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We we consider the restrictions (isomorphisms) pi,ζ‚ : r∆˝2 ζ‚ „ ÝÑ r∆˝ associated to this compo- nent, they are, on the level of the ring maps ri,ζ‚, related by the fact that r1,ζ‚pT 1{pkq “ ζkr2,ζ‚pT 1{pkq for all k ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now apply the differential d ` δ to the cocycle sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The result is the direct sum of dT {T , regarded as a differential form on r∆˝, and the difference logpT2{rT 5 2sq ´ logpT1{rT 5 1sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If we compute this latter difference on the component r∆˝2 ζ‚ we obtain ´ logprζ‚sq, coming from the fact that rT 5 1s “ rζ‚srT 5 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We thus have that pd`δqpscq “ dT {T ´t, where t is the section of BdR ˇˇ r∆˝2 whose value on each component r∆˝2 ζ‚ is logprζ‚sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Our conclusion is that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the ˇCech complex associated to the sheaf Ω‚ ∆˝ and the cover U “ tcu the cycles dT {T and t are cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The class of dT {T is non-zero in the cohomology of the complex Ω‚ ∆˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the exactness of (9), it suffices to check the class of t is non-zero regarded as an element of H1p∆˝ p´et, BdRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that because our covering c is perfectoid, the cohomology of BdR in positive degree vanishes on this cover (see [Sch13, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5]), which means we can compute these groups using the above ˇCech complex as a consequence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Thus we are asking whether there is an element s of BdR ˇˇ r∆˝ such that s ˇˇ p1,ζ‚ ´ s ˇˇ p1,ζ‚ is a constant in Fil1B` dRzFil2B` dR for each choice of ζ‚, where the filtration is defined as in [Sch13, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because the restriction maps preserve the filtration it suffices to assume s P Fil1B` dR, and then to check that this is not even possible after passing to the quotient Fil1B` dR{Fil2B` dR – ˆO∆˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The two restrictions are related by an automorphism of r∆˝2 ζ‚ inducing an automorphism of BdR ˇˇ r∆˝2 ζ‚ and hence of pOr∆˝2 ζ‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But one easily checks that this automorphism, which is induced by scaling p-powers of T by the corresponding elements of ζ‚, cannot shift any function by a non-zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 Evaluation Functionals We now consider the more general setting where we have a product of annuli ∆a,b “ p∆˝qaˆ ∆b´a, corresponding to the adic space SpapkxT ˘1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , T ˘1 a , Ta`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Tby, OkxT ˘1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , T ˘1 a , Ta`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Tbyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In this section we will view all spaces, including algebraic varieties, as adic spaces over Spapk, Okq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular we consider the multiplicative group Gm and the affine line A1 as adic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We wish to define certain “evaluation functionals” ˆα˚ a,b : Hap∆a,b p´et, ˆZppaqq Ñ Zppaq and study their relationship with the p-adic Hodge comparisons and our calculation in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will actually have some freedom in the definition, as we will only require that ˆα˚ a,b has the “right” behaviour on Kummer-type covers, as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By a Kummer-type cover of ∆a,b we mean a cover obtained by pullback from a finite ´etale cover of V a,b “ Ga m ˆ Ab under the natural map ∆a,b Ñ V a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have a natural induced map HapV a,b pf´et, ˆZppaqq Ñ Hap∆a,b p´et, ˆZppaqq induced by the map ∆a,b p´et Ñ V a,b pf´et of sites, and we will denote by Ia,b its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that the etale cohomology of V a,b may be computed with only finite ´etale covers (products of tori are Kpπ, 1q spaces), so this agrees in particular with the image of HapV a,b p´et , ˆZppaqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For all k, the map HkpV a,b pf´et, ˆZpq Ñ Hkp∆a,b p´et, ˆZpq is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Both the cohomology of V a,b and the cohomology of ∆a,b are generated by the cohomology in degree one by the rigid-analytic K¨unneth formula (see [Ber93, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3] for a reference in the equivalent Berkovich setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As the K¨unneth maps are natural in the underlying site, it suffices to show that the maps H1pV a,b pf´et, ˆZpq Ñ H1p∆a,b p´et, ˆZpq in degree one are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From our discussion in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 we may identify these maps with the natural maps Hompπ1 ´etpV a,bq, Zpq Ñ Hompπ1 ´etp∆a,bq, Zpq, so we are reduced to showing that π1 ´etp∆a,bq Ñ π1 ´etpV a,bq is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the natural factorization, this reduces to the same statement for π1 ´etp∆˝q Ñ π1 ´etpGmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By [Sta20, Lemma 0BN6], we reduce to showing that every connected finite ´etale cover of Gm pulls back to a connected finite ´etale cover of ∆˝, which is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (Note that the rigid analytic finite ´etale coverings of Gm are just those of Kummer type as a consequence of the rigid analytic Riemann existence theorem [L¨ut93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') 24 Using the Lemma, we will define ˆα˚ a,b as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will first define a functional α˚ a,b on HapV a,b pf´et, ˆZppaqq, which induces a functional on Ia,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will then choose a splitting Hap∆a,b p´et, ˆZppaqq “ Ia,b ‘ Ja,b, and define ˆα˚ a,b by extending by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We begin by fixing a distinguished system tζ‹ i uiě1 of p-power roots of unity of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This induces the following data: A p-adic period t “ logprζ‹ ‚sq P BdR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Via the automorphisms T ÞÑ ζ˚ i T , an element, denoted α, of the pro-p fundamental group π1 ´etpGmqppq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' note that by the rigid analytic Riemann existence theorem [L¨ut93] the covers T ÞÑ T pi described above exhaust the connected finite ´etale coverings of Gm with degree a power of p, even on the adic finite ´etale site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A map α˚ : H1pGm,p´et, Zpp1qq Ñ Zpp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This uses the identification H1pGm,p´et, Zpp1qq » Homcontpπ1 ´etpGmq, Zpp1qq and is defined by evaluation on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the identification H1pGm,p´et, Zpp1qq » H1pGm,fp´et, Zpp1qq, a map, also denoted α˚, on the latter cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A map ˆα˚ : I1,0 Ñ Zpp1q, obtained by pulling back α˚ along the map I1,0 Ñ H1pGm,fp´et, Zpp1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A linear functional ˆα˚ BdR : I1,0 b BdR Ñ BdR, which is defined by evaluating on α and extending scalars along the map Zpp1q ãÑ BdR given by logpr´sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the coverings T ÞÑ T pj of ∆˝ in the previous section we obtain torsors Lj on ∆˝ p´et and a class rL8s “ plimjrLjsq P I1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To complete our definition we will need the following facts, both of which are formal verifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fix a, b ě 0, and let ˆα˚ i : H1pV a,b pf´et, Zpp1qq Ñ Zpp1q be the pullback of ˆα˚ along the i’th projection H1pV a,b pf´et, Zp1qq Ñ H1pGm,pf´et, Zp1qq induced by inclusion of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then the map ˆα˚ a,b :“ ˆα˚ 1 b ¨ ¨ ¨ b ˆα˚ a : H1pV a,b pf´et, Zpp1qq b ¨ ¨ ¨ b H1pV a,b pf´et, Zpp1qq loooooooooooooooooooooooooomoooooooooooooooooooooooooon “HapV a,b pf´et ,Zppaqq Ñ Zppaq takes the value tζ‹ ‚ub¨ ¨ ¨btζ‹ ‚u on the element rL8,1sb¨ ¨ ¨brL8,as, where rL8,is is induced from rL8s by the inclusion H1pGm,pf´et, Zp1qq Ñ H1pV a,b pf´et, Zp1qq coming from the projection onto the i’th factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Immediate from the definitions and functorality of cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Denote by ε the natural comparison map ε : Hap∆a,b p´et, Zppaqq b BdR Ñ Hap∆a,b p´et, Ω‚ ∆a,bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then ε maps the element prL1,8s Y ¨ ¨ ¨ Y rLa,8sq to the element dT1{T1 ^¨ ¨ ¨^dTa{Ta, and this element is non-zero in Hap∆a,b p´et, Ω‚ ∆a,bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If we define cup product using ˇCech cohomology on both sides and also on the cohomology of the sheaf BdR, the first part of the result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', ignoring the non-zeroness of dT1{T1^¨ ¨ ¨^dTa{Ta) will follow from the compatibility of cup product with the differential graded structure on Ω‚ ∆a,b (as discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2), as well as our result Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4 above, as long as we can show that rL1,8s maps to t1, where t1 is the class in H1p∆a,b p´et, BdRq obtained as the image of t under the map H1p∆˝ p´et, BdRq Ñ H1p∆a,b p´et, BdRq coming from projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From functoriality it suffices to show that rL8s maps to t under the natural map H1p∆˝ p´et, ˆZpp1qq Ñ H1p∆˝ p´et, BdRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Working on the level of ˇCech complexes with respect to the perfectoid cover c as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1, the gluing data for the torsor rL8s assigns the system of compatible roots ζ‚ to the component r∆˝2 ζ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As the map Zpp1q Ñ BdR is induced by pζiqiě1 ÞÑ logprζ‚sq, one obtains the cycle t as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For the non-zeroness of dT1{T1 ^ ¨ ¨ ¨ ^ dTa{Ta we may argue as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may first reduce to the case of b “ 0 by using the natural map Hap∆a,b p´et, BdRq Ñ Hap∆a,0 p´et, BdRq coming from inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 and the cup-product compatibility implies that the class of dT1{T1 ^ ¨ ¨ ¨ ^ dTa{Ta is cohomologous to the class of t1 Y ¨ ¨ ¨ Y ta, we may reduce to the same statement for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the Kunneth formula to compute the ´etale cohomology of ∆a,0 this then reduces to showing that each ti is non-zero, which is what we showed in the argument of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the fact that ε maps a class generating Ia,b to a non-zero element, it follows that we may choose a splitting Hap∆a,b p´et, ˆZppaqq “ Ia,b ‘ Ja,b such that Ja,b contains the kernel of Hap∆a,b p´et, ˆZppaqq Ñ Hap∆a,b p´et, BdRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then define ˆα˚ a,b on all of Hap∆a,b p´et, ˆZppaqq by extending by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Finally, we define: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The map ρ´1 : Hap∆a,b, Ω‚ ∆a,bq Ñ Ia,b b BdR is the unique BdR-linear map which sends the class of dT1{T1 ^ ¨ ¨ ¨ ^ dTa{Ta to the class rL1,8s Y ¨ ¨ ¨ Y rLa,8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 Extending to an ambient variety We now suppose that Y is a smooth proper algebraic variety over Cp, and that we have an adic neighbourhood ∆a,b Ă Y ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' There are two ways in which one could imagine extending the functional ˆα˚ a,b to the cohomology of Y : by pulling back along HapY ad p´et, Zppaqq Ñ Hap∆a,b p´et, Zppaqq, and by pulling back along ρ´1 and HapY ad, Ω‚ Y adq Ñ Hap∆a,b, Ω‚ ∆a,bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now check that these two extensions are compatible with the p-adic period isomorphism, meaning that we can consistently identify the extensions without issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 we may compute coherent cohomology on the pro-´etale site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the natural morphisms ˆZppaq Ñ BdR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' BdR Ñ Ω‚ p´q and Ω‚ p´q Ñ Ω‚ p´q of sheaves on the pro-´etale site,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' one obtains the following diagram: HapY ad p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ˆZppaqq b BdR HapY ad p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' BdRq HapY ad p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Ω‚ Y adq HapY ad p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Ω‚ Y adq b BdR Hap∆a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='b p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ˆZppaqq b BdR Hap∆a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='b p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' BdRq Hap∆a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='b p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Ω‚ ∆a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='bq Hap∆a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='b p´et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Ω‚ ∆a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='bq b BdR „ „ „ σ „ ρ´1 26 All the squares in the diagram are commutative by general cohomological principles ex- cept (a priori) possibly the ones involving ρ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' That the middle horizontal rightward arrows are isomorphisms is [Sch13, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' That the upper left horizontal arrow is an isomorphism is [Sch13, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4], and that the upper right horizontal arrow is shown in the proof of [Sch13, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After inverting the isomorphisms in the above diagram, the functional ˆα˚ a,b admits a pullback ˆγ˚ to any object in the top row of the diagram, and the resulting functional is independent of the path along which one takes the pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We check the consistency of the pullback to HapY ad p´et, Ω‚ Y adqbBdR, as this contains all the essential difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One may easily check that if x P HapY ad p´et, Ω‚ Y adqbBdR is an element in the top right group, then it admits a unique image in every object in the diagram except possibly the group Hap∆a,b p´et, ZppaqqbBdR in the bottom left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This in particular means that the difference between any two images of x inside Hap∆a,b p´et, Zppaqq b BdR lies in the kernel of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It then suffices to check that the kernel of σ lies in the kernel of ˆα˚ a,b, but this is by construction since ker σ Ă Ja,b b BdR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The induced functional ˆγ˚ on the cohomology of Y is independent of the summand Ja,b containing the kernel of σ chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Since, as a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7, the functional may be defined by pulling back first along ρ´1 and then the right vertical arrow, it only depends on the the restriction of ˆα˚ a,b to Ia,b Ă impρ´1q, and hence is independent of the choice of summand Ja,b whose intersection with Ia,b is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 4 Realizing G-functions We now give our main technical result, which will give a cohomological interpretation of Andr´e’s G-functions at all places of our fixed number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We recall the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have a projective family f : X Ñ S over K of relative dimension ν ´ 1, with S an algebraic curve, and which has geometrically connected fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' There is a fixed K-point s0 P S such that the family f 1 : X1 Ñ S1 with S1 “ Szts0u is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The fibre E Ă X above s0 is assumed to have simple normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have an affine Zariski open subset U Ă X with coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zν trivializing Ω1 U, a uniformizing parameter s at s0, all such that the map fU “ f ˇˇ U is surjective and s ÞÑ z1 ¨ ¨ ¨ zµ for some µ ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We let H “ Rwf˚Ω‚ X{Splog Eq where w “ µ ´ 1 and set H1 “ H ˇˇ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We additionally assume for simplicity that s is defined on all of S and that ds trivializes Ω1 S, which becomes true after removing finitely many points from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This in particular implies that S is affine and H may be identified with its module of global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Finally, write HU “ Rwf˚Ω‚ U{SplogpE X Uqq, and note that there is a natural restriction map H Ñ HU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We have a commuting diagram U Spec Krx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xνs S Spec Krts px1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=',xνqÞÑpz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=',zνq f ˇˇˇ U tÞÑx1¨¨¨xµ tÞÑs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (10) 27 We will denote the top arrow by g, the bottom arrow by u, and write j for the ar- row on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because g is ´etale, its image is an open K-algebraic subvariety V Ă Spec Krx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xνs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write T Ă Spec Krts for the image of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By a scaling of the coordinates pz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνq we mean coordinates pλz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , λzνq for some λ P Kˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one replaces the coordinates pz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνq with a scaling pλz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , λzνq and the coordinate s with λµs the diagram (10) continues to commute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' when say “replace pz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνq with a scaling” we mean to make such a change of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Choose a K-point q P g´1p0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After replacing pz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνq with a scaling by N ´1, where N P Z, the following property holds: for any embedding ι : K ãÑ Kv with v a finite place of K, the map g is invertible in the open ball of v-adic radius 1 around 0 P Spec Krx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xνs onto a neighbourhood containing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (In particular, this ball is contained inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') This property continues to hold if N is replaced by some N 1|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The idea is that one can write down a formal inverse to the map of germs pU, qq Ñ pV, 0q and have this inverse converge at each finite place in the desired neighbourhood after scaling coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' More explicitly, let us begin by embedding the affine variety U as a closed subvariety of SpecKry1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , yσs defined by polynomials p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , pℓ P Kry1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , yσs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After translation we may identify q with the origin in Spec Kry1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , yσs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The map g is then given by component polynomials g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , gν P Kry1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , yσs with no constant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The formal inverse A we wish to compute is then given by power series Aipx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xνq “ ÿ J Ai,JxJ (11) for 1 ď i ď σ, where J ranges over the set C of all appropriate compositions of integers ě 0 and we use multi-index notation to exponentiate the vector x “ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xνq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The fact that g ˝ A “ id and p ˝ A “ 0 gives a system of linear equations for each coefficient appearing in each Ai in terms of coefficients of the polynomials gj and pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let us suppose we can solve this system for a formal function A, and that this formal function converges in the open ball of v-adic radius 1 around 0 for all finite places of K outside of a finite set Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then it will suffice to scale the coordinates px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xνq by multiplying each xi by a sufficiently large integer N whose prime factors all lie above places of Σ: indeed, doing so does not affect the radius of convergence for finite places outside of Σ, and the radius of convergence of the resulting power series at a place v P Σ will increase by a factor of 1{|N|v and hence be greater than 1 as soon as |N|v is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We are reduced to the following more formal fact: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that we have formal power series (11) with coefficients in a number field K that are defined by the property that Bi ˝ A “ Ci, where B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Bc, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Cc P Kry1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , yσs are finitely many polynomials with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then A converges in the open v-adic ball of radius 1 away from finitely many places v of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To understand the system of equations defined by these polynomials we recall the multivariate Fa`a di Bruno formula [CS96], which says that the derivatives of a composition C “ B ˝ A of functions given by power series centred at zero are given by pBJCqp0q “ ÿ 1ď|λ|ď|J| pBλBqp0q |J| ÿ s“1 ÿ CspJ,λq J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' s ź j“1 rAℓjp0qskj pkj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='qrℓj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='s|kj| , (12) where we have made use of the following notation: 28 the vectors λ and kj come from Zσ ě0 and the vectors J and ℓj come from Zν ě0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for any vector u “ pu1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , urq P pZě0qr we have |u| “ u1 ` ¨ ¨ ¨ ` ur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we have CspJ, λq “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' pk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ℓsq : |ki|ą0, 0ăℓ1㨨¨ăℓs řs i“1 ki“λ and řs i“1 |ki|ℓi“J ) , for vectors u “ pu1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , urq and u1 “ pu1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , u1 rq, the symbol u ă u1 means that one of the following conditions holds: (i) |u| ă |u1|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (ii) |u| “ |u1| and u1 ă u1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' or (iii) |u| “ |u1|, u1 “ u1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , uk “ u1 k and uk`1 ă u1 k`1 for some 1 ď k ă r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the notation Aℓ for ℓ “ pℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ℓνq means pBℓA1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , BℓAνq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and for a vector u “ pu1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , urq, we have u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' “ u1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ¨ ¨ ¨ ur!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='. The terms on the right-hand side of the equation (12) involving the components of AJp0q are then J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' σÿ i“1 pBiBqp0qAi,J J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (13) Now let us suppose that B is a polynomial over K, and that pBJCqp0q “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' this is the case for B P tB1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Bcu and for |J| sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then at a finite place v, for all but finitely many v, the norms |pBλBqp0q|v are ď 1 if |λ| ď deg B, and equal to 0 if |λ| ą deg B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, the equation (12) induces the following linear equation for AJ{J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', at least when |J| ą deg B: σÿ i“1 pBiBqp0qAi,J J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' “ ´ ÿ 2ď|λ|ďdeg B pBλBqp0q |J| ÿ s“1 ÿ CspJ,λq ˜ s ź j“1 1 pkj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='q ¸ ˜ s ź j“1 rAℓjp0qskj rℓj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='s|kj| ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (14) We note that there are only finitely many possibilities for the coefficients śs j“1 1 pkj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='q which appear in (14), and these possibilities are independent of J and depend only on deg B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, because only finitely many λ ever occur in all such terms, the equation řs i“1 ki “ λ together with the condition |ki| ą 0 for all i ensures that only finitely values for the tuple ps, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ksq ever appear, and hence there are only finitely many śs j“1 1 pkj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='q which appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After excluding a further finite set of norms | ¨ |v, we may assume that all these coefficients have norm 1, and the non-zero coefficients of B and its derivatives also have norm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Letting B range over the finitely many polynomials in the set B P tB1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Bcu, we have proven that: For |J| ą deg B, and for all but finitely many places v, the vector 1 J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='AJ is the solution to a system of linear equations M AJ J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' “ N, (15) 29 where M is a c ˆ σ matrix, independent of J, whose non-zero entries all have unit v-norm, and N is a vector with c entries whose norms are at most max s,CpJ,λq ››››› s ź j“1 rAℓjp0qskj rℓj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='s|kj| ››››› v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (16) We may assume c ě σ or else the solution is not uniquely determined, and then c “ σ by choosing a linearly-independent subset of the rows of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now use this to show that, after possibly throwing out a further finite set of places v, one has }Ai,J{J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' }v ď 1 for all i and all J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We prove this by induction, starting from the case where |J| “ maxtdeg B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , deg Bcu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we note that the base cases with smaller |J| can be assumed after removing a further finite set of places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Removing a further finite set of places to ensure that } detpMq}v “ 1, we may use Cramer’s rule and the equation (15) to compute the entries of Ai,J{J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' as quotients detpM 1q{ detpMq, where M 1 is a matrix obtained from M by replacing a column with the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By induction the bound (16) is at most 1, so it follows that detpM 1q{ detpMq, and hence the entries of Ai,J{J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', have v-adic norm at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Scale coordinates as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1, and fix a point q in the common vanish- ing locus z1 “ ¨ ¨ ¨ “ zµ “ 0 with image s0 P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then q induces a K-linear map, compatible with base change along a finite extension L{K, Γ : HpSq Ñ Krrtss, whose image consists of G-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These G-functions satisfy the following two “realiza- tion” properties: (i) Fix an embedding ι : K ãÑ C, suppose that tωiuiPI is a subset of HpSq, and that R ą 0 is a real number such that Γpωiqι has radius of convergence at least R for all i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Denote by DR Ă San ι the component containing s0 of the complex analytic neighbourhood defined by |s| ă R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then for each point s1 P DRzts0u, there exists a linear functional γ˚ 1 : HwpXan s1 , Zpwqq Ñ Zpwq, such that if ρ is the natural isomorphism HwpXan s1 , Zpwqq b C „ ÝÑ Hw dRpXs1q, then we have 1 p2πiqw pγ˚ 1,C ˝ ρ´1qpωi,s1q “ pΓpωiqιqpups1qq (17) for all i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (ii) Fix an embedding ι : K ãÑ Cp for some prime p, suppose that tωiuiPI is a subset of HpSq, and that 1 ě R ą 0 is a real number such that Γpωiqι has radius of convergence at least R for all i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Denote by DR Ă Sad ι the component containing s0 of the adic neighbourhood defined by |s| ă R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then for each closed point s1 P DRzts0u there exists a neighbourhood ∆w,µ´ν s1 Ă Xad s1 such that the linear functional ˆγ˚ 1 : HwpXad s1,p´et, ˆZppwqq Ñ Zppwq, 30 constructed from this neighbourhood as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 satisfies the property that 1 tw pˆγ˚ 1,BdR ˝ ρ´1qpωi,s1q “ pΓpωiqιqpups1qq (18) for all i P I, where ρ is the natural isomorphism HwpXad s1,p´et, ˆZppwqq b BdR „ ÝÑ Hw dRpXs1q b BdR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The p-adic period t in the statement of (ii) is not to be confused with the coordinate t in the diagram (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The remainder of this section is devoted to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We begin by constructing the map Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From functoriality, the sections tωiuiPI all have restrictions to the sheaf RwfU,˚Ω‚ U{S, which as we explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 is represented by the cohomology in degree w of the complex 0 Ñ OU Ñ Ω1 U{SplogpE X Uqq Ñ ¨ ¨ ¨ Ñ Ωn U{SplogpU X Eqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (19) We may then further restrict to a formal neighbourhood of q and consider the complex 0 Ñ pOU,pqq Ñ pΩ1 U{SplogpE X Uqqpqq Ñ ¨ ¨ ¨ Ñ pΩn U{SplogpE X Uqqpqq, (20) and obtain a K-linear map η : HpSq Ñ Cohomw ” pΩ‚ U{SplogpE X Uqqpqq ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As we saw in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2, the target of η is naturally a 1-dimensional free module over pOS,ps0q, and so we may define Γ as the composition HpSq ηÝÑ Cohomw ” pΩ‚ U{SplogpE X Uqqpqq ı „ ÝÑ pOS,s0 uÝÑ pOSpec Krts,0 “ Krrtss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Before turning to the proof of (i), we briefly explain why the image consists of G- functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The point is that, within the degree-w cohomology of the complex (19), the each relative form ω is represented by h dz2¨¨¨dzµ z2¨¨¨zµ , where h is a function algebraic over a rational function field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, the power series hq in the coordinates z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zµ representing h at q is algebraic over a rational function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The calculation of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2, which we will see again in the proof of (i), computes Γpωq as the µ-diagonal of hq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', the power series in one variable t “ z1 ¨ ¨ ¨ zµ obtained by keeping all terms a ze1 1 ¨ ¨ ¨ zeµ µ with e1 “ ¨ ¨ ¨ “ eµ and discarding the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It is known (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [And89, I, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2] and the discussion on pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 965 of [AB13]) that any function obtained in this way is a G-function, which Andr´e himself uses in his proof in [And89, IX, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof of (i): We work entirely in the complex analytic category, and view the diagram (10) in the complex analytic category using base-change along ι : K ãÑ C and analytifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Denoting by UR the fibre of f over DR, we obtain a complex: 0 Ñ OUR Ñ Ω1 UR{DRplogpE X URqq Ñ ¨ ¨ ¨ Ñ Ωn UR{DRplogpUR X Eqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (21) If one restricts (21) to the stalk at q, one obtains a complex pΩ‚ UR{DRqpqq whose formal completion is naturally identified with the complexification of (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 to 31 this complex, one obtains, for each i, analytic functions Pi such that Pi dz2¨¨¨dzµ z2¨¨¨zµ represents the image of ωi,C in the cohomology Cohomw ” pΩ‚ UR{DRqpqq ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As the Pi are obtained using the same calculations that produced Γpωiqι, these functions agree at the formal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now give a geometric interpretation of the Pi following [And89, IX, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4], as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fix a sufficiently small disk D Ă DR centred at s0 and a neighbourhood U Ă f ´1pDq above D containing q such that the map g maps U isomorphically to a product of complex analytic disks ∆ν, where xi is the coordinate on the i’th factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The map f is then locally identified with the map ∆ν Ñ ∆ given by t ÞÑ x1 ¨ ¨ ¨ xµ, and the special fibre with the locus x1 ¨ ¨ ¨ xµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We can choose a family ε2,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , εµ,t of small loops inside ∆ν, with εi,t a loop around the divisor xi “ 0 inside the fibre ∆ν t defined by x1 ¨ ¨ ¨ xµ “ t, and consider the family of integrals P 1 iptq “ 1 p2πiqw ż εt ωi ˇˇ ∆ν t where εt “ ε2,t ˆ ¨ ¨ ¨ ˆ εµ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Representing the restriction ωi ˇˇ ∆ν t as a function of the form hi dx2¨¨¨dxµ x2¨¨¨xµ with h a power series in x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , xµ, one computes using the residue formula that P 1 iptq is the µ-diagonal of h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', the function whose power series is obtained from that of h by substituting in a te1 for all terms a xe1 1 ¨ ¨ ¨ xeµ µ with e1 “ ¨ ¨ ¨ “ eµ, and ignoring all other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is compatible with the calculation in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2, and we have that P 1 i “ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By taking the image of the cycles εt inside the fibres Xt, this calculation realizes the functions Pi (and hence the functions Γpωiqι) as functions inside the image of the integration pairing Rwf˚Zp´wq b Rwf˚Ω‚ X1{S1 Ñ OS restricted to the neighbourhood D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By analytic continuation, the cycles εt extend to a (possibly multi-valued) section rε of Rwf˚Z ˇˇ DR, and hence produce a (a priori possibly multi-valued) function rPi inside OS ˇˇ DR after pairing with ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But since rPi agrees with the analytification of Γpωiqι near s0 and its power series representation converges on DR, the analytic function rPi is single-valued, and gives an analytic realization of Γpωiqι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (In the case where the ωi’s give a frame of the de Rham cohomology over DR, one may also conclude that rεs is single-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') To complete the proof of (i), it suffices to define, for each s1 P DR, the functional γ˚ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This we define as the evaluation functional γ˚ 1,C : HwpXan s1 , Cq Ñ C, «ż p´q ω ÞÑ ż rεs1 ω ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To make sense of this definition, we are using the canonical isomorphism HwpXan s1 , Cq » Hw top-dRpXan s1 q b C (with topological de Rham cohomology) to represent each element of HwpXan s1 , Cq as an integration functional, and then defining γ˚ 1,C by evaluating this functional on rεs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that because the section rε may in principle by multi-valued, this is also true of the function γ˚ 1,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However the equality (17), which amounts to the above observation that the relative period with ωi is given by Γpωiq, holds regardless of which choice we make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To complete the proof we observe that this functional descends to Zpwq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 32 Proof of (ii): As before, we will regard all spaces as analytic spaces over Cp using the fixed embedding ι : K ãÑ Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We wish to mirror the argument we made in the complex analytic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The argument is in many respects simplified by our choice of coordinates, which ensure that the property in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 holds over the neighbourhood DR since R ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular if one writes D Ă S for the component of the neighbourhood |s| ă 1 containing s0, and one writes U Ă f ´1pDq for the neighbourhood around q defined by |pz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , zνq|ι ă 1, then we are guaranteed that the family U Ñ D is isomorphic to the family V Ñ E obtained by restricting j to the open ball of radius 1 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now let us construct a subspace isomorphic to ∆w,µ´ν inside Xs1X U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Letting t1 “ ups1q, we may instead construct such a subspace inside Vt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write R1 “ |t1|v ă R ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The fibre Vt1 is defined by the equation x1 ¨ ¨ ¨ xµ “ t1 inside the open ball defined by maxi|xi| ă 1, and we may embed ∆w,µ´ν inside this neighbourhood via the map xi ÞÑ R1{µ 1 Ti´1 for 2 ď i ď ν, and x1 ÞÑ R´w{µ 1 t1{pT1 ¨ ¨ ¨ Twq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This embedding identifies ∆w,µ´ν with the closed neighbourhood Vw,ν´µ t1 inside Vt1 defined by |xi| “ R1{µ 1 for 1 ď i ď µ, and |xi| ď R1{µ 1 for i ą µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may then define ˆγ˚ 1 by pulling back the functional ˆα˚ w,ν´µ on the cohomology of ∆w,ν´µ constructed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2, and then extend this to the entire fibre Xs1 as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8 ensures that the extension is uniquely determined by the neighbourhood Uw,ν´µ s1 isomorphic to Vw,ν´µ t1 » ∆w,µ´ν, and furthermore Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 ensures that the resulting map is compatible with the p-adic comparison isomorphism associated to the fibre Xs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It now suffices to verify the desired equality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' let UR be the fibre of U Ñ D above DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As before, we may restrict each ωi to the cohomology in degree w of the complex Ω‚ UR{DR to obtain, via Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2, sections hi dz2¨¨¨dzµ z2¨¨¨zµ , with hi a function on DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We first observe that the hi agree with the functions Γpωiqι ˝ u on DR: this is true by construction at the formal level, and that implies that these functions agree in a small enough ball around s0, so this follows by uniqueness of analytic continuation of rigid-analytic functions on affinoid balls (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Meh19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now the coherent cohomology group HwpUs1,p´et, Ω‚ Us1 q is computed by the cohomology of the na¨ıve de Rham complex Ω‚ Us1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The natural restriction map HwpUp´et, Ω‚ U{DplogpE X Uqqq Ñ HwpUs1,p´et, Ω‚ Us1q is then represented by the natural map Cohomw ” Ω‚ U{DplogpE X Uqq ı Ñ Cohomw ” Ω‚ Us1 ı , and this map sends hi dz2¨¨¨dzµ z2¨¨¨zµ to hips1q dz2¨¨¨dzµ z2¨¨¨zµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If we then evaluate this element with γ˚ 1,BdR, one sees by combining the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6) of ρ´1 and the computation Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4 that the result is simply twhips1q, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 5 Algebraic Relations on Functionals We now introduce a new way to obtain relations on Andr´e’s G-functions at finite places, which uses the explicitly p-adic nature of our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The first key observation is the following: 33 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let Y be a proper algebraic variety over the finite extension Kv of Qp, and let ˆγ˚ : HapYCp,p´et, ˆZppaqq Ñ Zppaq be constructed as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then the Galois group GKv “ GalpKv{Kvq acts on ˆγ˚ through χ´a cycl, where χcycl is the cyclotomic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write η for the map HapYp´et, ˆZppaqq Ñ Hap∆a,b p´et, ˆZppaqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We recall that ˆγ˚ was constructed from a functional ˆα˚ a,b on the GKv-invariant subspace Ia,b Ă Hap∆a,b p´et, ˆZpq after choosing an appropriate splitting Hap∆a,b p´et, ˆZppaqq “ Ia,b ‘ Ja,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write ˆβ˚ J for the extension-by-zero of ˆα˚ a,b with respect to this splitting, so that ˆγ˚ “ ˆβ˚ J ˝ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8, this is the same as ˆβ˚ g¨J ˝η for any g P GK, where ˆβ˚ g¨J is the extension-by-zero of ˆα˚ a,b with respect to the splitting Ia,b ‘ pg ¨ Ja,bq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here we use the fact that the kernel of σ : Hap∆a,b p´et, ˆZppaqq Ñ Hap∆a,b p´et, BdRq is GKv-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now given g P GK, the functional g ¨ ˆγ˚ is equal to pg ¨ ˆβ˚ Jq˝η “ pg ¨ ˆβ˚ g¨Jq˝η as the map η is Galois-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The map g ¨ ˆβ˚ g¨J is the extension-by-zero of g ¨ ˆα˚ a,b with respect to the splitting Ia,b ‘Ja,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a vector v P HapYp´et, ˆZppaqq, with decomposition ηpvq “ wI `wJ with respect to the splitting Ia,b ‘ Ja,b, we therefore have that pg ¨ ˆγ˚qpvq “ pˆβ˚ g¨J ˝ ηqpvq “ pˆβ˚ g¨JqpwIq ` 0 “ pg ¨ ˆα˚ a,bqpwIq “ ˆα˚ a,bpwIqχ´a cylcpgq “ ˆγ˚pvqχ´a cylcpgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that to show that g acts on ˆα˚ a,b via χ´a cycl we have used the definition of ˆα˚ as a product of individual maps ˆα˚ i in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4, as well as the fact that as Galois representations we have HompIa,b, Zppaqq » Zpp´aq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' this is true since Ia,b » HaprGa m ˆ Abs´et, Zppaqq » Zpp2aq as Galois representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now adopt a slightly more abstract perspective to produce relations on de Rham coordinates of ˆγ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the proposition below, the most typical application will be with Y a variety defined over a number field K, and V´et “ HapYK,´et, Qppaqq, VdR “ HwpY, Ω‚ Y q its ´etale and de Rham cohomology groups, although the extra generality will be useful to handle summands appearing in such cohomology groups as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let V´et be a BdR-admissible GKv-representation of weight 2a such that VdR,Kv “ pV´et bQp BdRqGKv admits a K-structure VdR bK Kv » VdR,Kv, where K Ă Kv is a finite extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that we have a GKv-invariant endomorphism τ : V´et Ñ V´et, whose de Rham realization is defined over K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' a functional ˆγ˚ : V´et Ñ Qppaq on which GKv acts through the ´a’th power of χcycl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and the dimension k of the GKv-invariant subspace of HompV´et, Qpp2aqq » HompV´et, Qppaqq bQp Qpp´aq is less than the degree of the minimal polynomial of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 34 Then with respect to any choice of basis ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm for VdR, there is a K-algebraic relation on the dual coordinates of ˆγ˚ BdR of degree equal to k`1, which does not hold for the coordinates of a generic functional on V´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The relation depends only on the coordinates of τ in the basis ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It suffices to show that the set ˆγ˚ BdR, ˆγ˚ BdR ˝ τ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ˆγ˚ BdR ˝ τ k (22) is linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, if we evaluate the vectors in this set on the basis ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm we will obtain a sequence of vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , vk`1 P Bm dR such that vi is obtained by applying a non-zero linear transformation to v1 whose matrix entries in the basis ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm lie in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But we may then get an K-algebraic relation on the coordinates of v1 by taking the determinant of a square submatrix of rv1| ¨ ¨ ¨ |vk`1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This relation has degree k ` 1 and depends only on the coordinates of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' That this relation does not hold generically follows from the definition of minimal polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To show this linear dependence, we observe that τ is invariant under GKv, and therefore the vectors (22) are all obtained from scalar extension of vectors in HompV´et, Qppaqq, and these vectors are all invariant under the action of GKv on HompV´et, Qpp2aqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But the space of GKv-invariants on HompV´et, Qpp2aqq has dimension at most k by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If in the setting of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 the characteristic polynomial P of τ is equal to its minimal polynomial, then one may assume that the relation is a product of linear relations, with each linear factor defined over KF, with F the splitting field of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' That the characteristic polynomial of τ is equal to its minimal polynomial means that V ˚ dR admits only finitely many τ-invariant subspaces, each of which is defined over the splitting field of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 shows that ˆγ˚ BdR lies inside one of these subspaces, so one may take a product of linear relations associated to each subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the setting of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2, the dimension of the GKv-invariant subspace of HompV´et, Qpp2aqq is at most the size of the first Hodge number of V ˚ dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Letting Cv be the completion of Kv, the module HompV´et, Qpp2aqq becomes isomor- phic to HompV´et b Cv, Cvq b Cvp´2aq after scalar extension, which the Hodge-Tate com- parison shows is isomorphic to a sum À i`j“´2a griV ˚ dR bK Cvpj ´ 2aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The GK-invariant subspace necessarily maps to the summand with indices pi, jq “ p0, 2aq, and this summand has dimension at most dimK gr0V ˚ dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the situation where the endomorphism τ appearing in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 is defined over a finite extension L of K, we will also want to control the degree of the extension rL : Ks, for which the following fact, proven in [Pap22], will be useful: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that Y is an algebraic variety defined over a number field K, and that the absolute Hodge conjecture holds in degree a for cohomological endomorphisms associated to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then the algebra of absolute Hodge endomorphisms may be identified with a subalgebra of EndpHapY, Ω‚ Y qqL, where L{K is a finite Galois extension with degree bounded only in terms of m “ dimK HapY, Ω‚ Y q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is [Pap22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] and [Pap22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that the absolute Hodge conjecture is only assumed for endomorphisms associated to Y in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 35 6 Height Bounds for Families over Curves In this section we prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This will require some addi- tional setup due to an important subtly that occurs in applications of the G-function method to bound heights on curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To understand why, suppose that s is some uniformizing pa- rameter on our curve S at s0 and our G-functions are, as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3, obtained from expanding periods near s0 in terms of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then if ξ P S is a special point and v is a place of Kpξq which is relevant for spξq, it could be that ξ does not lie sufficiently close to s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' more precisely, if D Ă A1 is a v-adic disk on which the G-functions are defined, ξ and s0 could lie in different components of s´1pDq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' What we need instead is to be in the situation where every component of s´1pDq contains an appropriate degeneration point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is the reason for passing to the finite cover in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 Setup To give the covering we use the following lemma, proven in [DO22, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1]: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let C1 be an irreducible projective curve over a characteristic zero field K with s0 P C1pKq a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then there exists a finite extension L{K, a smooth projective curve C over L, a finite map c : C Ñ C1 L, and a rational function s on C such that every zero ξ0 P CpLq of s is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' every zero ξ0 P CpLq of s maps under c to s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' s : C Ñ P1 L is a finite Galois covering (not necessarily ´etale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In our setting we may complete S a smooth projective curve S, and apply the Lemma to obtain a covering c : C Ñ SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By pullback we obtain a family Xc´1pSq Ñ c´1pSq and simi- larly for S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After replacing K with L and S with c´1pSq we have reduced Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 to the setting where: (A) we have a parameter s : S Ñ A1 with only simple zeros;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (B) for each zero ξ0 of s the fibre Xξ0 is isomorphic to Xs0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and (C) for any extension L of K, any place v of L, and any R ą 0, each connected component of the analytic neighbourhood |s|v ă R in the associated analytic space SL,v (complex or rigid) contains a zero of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To see the last property note that the number of components of |s|v ă R is bounded by the size of a generic fibre of s, and the automorphism group of the covering s acts transitively on such components because it acts transitively on fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now construct the set of G-functions Gwe will be working with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With an eye toward future applications, and so that our setup here is easily reused, we will do it in the general setting where we start with ℓ different order-w normal crossing singularities in the fibre Xs0, where ℓ is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then construct = G“ tG1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , GMu, which we regard as elements of the formal power series ring Krrtss, as a union G “ Gξ1 \\ ¨ ¨ ¨ \\ Gξk of G-functions associated to the elements of the fibre s´1p0q “ tξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ξku;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we may assume these points are defined over K after passing to a finite extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fixing a point s0 P s´1p0q, we will 36 further sub-divide the set Gs0 as a union Gs0 “ Gs0,q1 \\ ¨ ¨ ¨ \\ Gs0,qℓ, where q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , qℓ are the points of the fibre Xs0 where the normal crossings occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' again we can assume these are defined over K after passing to a finite extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Choosing a sufficiently small affine neighbourhood U qi Ă X containing qi we may find functions zi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ziν on U qi such that: the equation zi1 ¨ ¨ ¨ ziµ “ 0 defines Xs0 X U qi where µ “ w ` 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the point qi lies in the locus zi1 “ ¨ ¨ ¨ “ ziν “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the function s maps to z1i ¨ ¨ ¨ ziµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and the map U qi Ñ Aν K defined by pzi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ziνq is ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, the image of s inside U qi necessarily vanishes on Xs0, hence lies in the ideal defining Xs0 X U qi, which is necessarily locally principal generated near qi by a function zi1 ¨ ¨ ¨ ziµ such that the locus defined by zi “ 0 is smooth near qi for 1 ď i ď µ and and the differentials dzi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , dziµ are independent in a neighbourhood of qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After shrinking U qi we may then extend to a local ´etale coordinate system zi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ziν at qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With this setup we are now, after removing finitely many points from S so that ds trivializes Ω1 S, in the situation of the setup in §4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we define Gs0,qi “ tΓpωi1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Γpωimqu, where ωi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωim is a frame of HpSq over S and Γ is as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In general we will actually want to choose the frames ωi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωim to be compatible with certain monodromy data, as this will make it easier to analyze functional relations on the associated G-functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we give more details in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In general an application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 results in functions in a scaled parameter λs0is, where λs0i “ N ´1 s0i for Ns0i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However because, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1, any N 1 with N 1|Ns0i will suffice, we may arrange for there to be some common N for all elements of G by taking a product (or greatest common multiple) of the individual Ns0i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then replace s with N ´1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 Monodromy-Compatible Frames for H1 Recall that H1 carries a K-algebraic Gauss-Manin connection ∇1 such that the (analytic) flat sections of ∇1 may be identified with the sections of V1 C under the Betti-de Rham comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now explain how to use ∇1 to choose frames for H1 which make it easy to analyze functional relations on period matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write H1a,b “ pH1baq b pH1˚qbb with a, b, ě 0 integers for the associated tensor vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Write T Ă À a,b H1a,b for the subbundle of ∇1-flat tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write AutpH1q for the bundle whose fibre above s P S1 is GLpH1 sq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' note that AutpH1q acts naturally on each vector bundle H1a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then consider the subbundle M Ă AutpH1q which is defined fibrewise as the stabilizer of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fix a K-point x P S1pKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then M is a Mx-torsor with a canonical section 1 : S1 Ñ M given by the identity, so we obtain a trivialization M „ ÝÑ Mx ˆ S1 over S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (Note that it is clear from the local system picture that M is an Mx-torsor complex analytically, and one can descend the complex analytic torsor structure and hence the trivialization, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [BT22, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') Write FrpH1q for the frame bundle of H1, which is an algebraic vector bundle whose fibres FrpH1qs for s P S1 may be identified with the invertible maps in the set HomκpsqpH1 s, κpsqmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The bundle AutpH1q acts on FrpH1q fibrewise by pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given any section F of FrpH1q and any section t of H1a,b one obtains an element Fptq of pOm S1qa,b “ pOS1qbm b pO˚ S1qbb by evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 37 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A monodromy-compatible frame of H1 is a frame ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm for which the associated section F of FrpH1q satisfies the following property: for every a, b ě 0 and every global flat tensor t P H1a,b the image Fptq in pOm S1qa,b “ pOS1qbm b pO˚ S1qbb is a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Monodromy-compatible frames exist locally in the ´etale topology on S1, and given such a frame with associated section F and a section M of M the product M ¨ F is a monodromy-compatible frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The second property is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For the first property we may start by fixing a point Fx of FrpH1qx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then consider the algebraic locus F Ă FrpH1q defined by the property that for each section t of T and each point Fy P F above some y P S1 we have Fyptyq “ Fxptxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then the fibrewise action of M on FrpH1q preserves F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now show that the natural map F Ñ S1 is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, we can replace K with C and work locally on an analytic ball B inside S1, in which case we know that a monodromy- compatible frame F which agrees with Fx on global flat tensors exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The restriction F ˇˇ B is then isomorphic to M ˇˇ B via the map pg, sq ÞÑ pg ¨ F, sq, and the map Mx ˆ B Ñ B is clearly flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Knowing that FÑ S1 is flat we may consider FÑ S1 to be an fppf covering of S1, and observe that FˆS1 F is isomorphic to the trivial Mx-torsor MˆS1 F via the map pg, Fq ÞÑ pF, g ¨Fq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, Fis an fppf Mx-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For smooth algebraic groups every fppf torsor is an ´etale torsor by [hmb], so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Nothing in this subsection relied on the fact that S1 is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 Proof of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 From the results of the previous section we will assume that the frames of H used to define the functions in G restrict to monodromy-compatible frames of H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' There is no harm in this, as we are allowed to remove finitely many points from S1, and we could have started the argument by replacing the family X Ñ S with an ´etale base-change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Choose ξ P S and a finite place v of Kpξq which is relevant for ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let R be the minimum v-adic convergence radius of the functions in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then in particular ξ lies in some component DR of the neighbourhood of Sad Kpξqv defined by |s|v ă R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By (C) above, necessarily such a component must contain a point s0 in the fibre s´1p0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may then identify our neighbour- hood DR with the one in the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3, and work with the G-functions in the set Gs0 “ Gs0,q corresponding to s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we note that any relation satisfied by G-functions in this set can be interpreted as a relation satisfied by the G-functions in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that, by assumption, the G-functions in the set Gs0 “ th1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmu are not all constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here hi “ Γpωiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (This was assumed before we passed to the finite covering, but can be reinterpreted as a claim about the functionals γ˚ 1 and ˆγ˚ 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 in a neighbourhood of s0 and so continues to hold after passing to the covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') Write π : V1 ξ Ñ W for the Hodge-theoretic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By assumption, this projection is induced by an algebraic cycle class, and so has cohomological realizations in both ´etale and algebraic de Rham cohomology, compatibly with the comparison isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In partic- ular, the image of π corresponds to a summand W´et Ă HwpXξ,´et, Qppwqq and a summand WdR Ă HwpXξ, Ω‚ Xq which correspond under the p-adic Hodge comparison maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Possible vanishing of ˆγ˚ on W´et: We first consider the situation where the restriction ˆγ˚ˇˇ W´et vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the compatibility with the p-adic Hodge comparison, this means that 38 ˆγ˚ BdR vanishes when restricted to WdR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The subspace WdR Ă HwpXξ, Ω‚ Xξq is defined over an extension L of Kpξq which, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5, has degree bounded independently of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let ω P WdR be a vector expressed as an L-linear combination ω “ ř i aiωi,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3(ii) we find that ř i aihipspξqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Thus, we will be in the situation where we can apply the Hasse principle for G-functions if we can show that the functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hm do not satisfy any non-trivial linear relations at the functional level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Ruling out Functional Relations: The statement that h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hm are linearly indepen- dent as functions can be interpreted as a statement about the associated formal power series at s0, and deduced from the same statement about the associated complex analytic power series where we view K as a subfield of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In this setting the functionals γ˚ 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3(i) glue into a section (denoted by the same notation) of the degree w homology local system over a punctured neighbourhood of s0 in SpCq, and it will suffice to show that the germs of h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hm at some point s1 sufficiently close to s0 do not satisfy a non-zero linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After some setup, we will see that this is a consequence of the Ax-Schanuel theorem for periods, recently proved in [BT22] as an application of a general theorem of [BSCFN21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To set things up, fix a small ball B around s1 and extend γ˚ 1 to a global frame γ˚ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γ˚ m of the dual local system rV1ˇˇ Bs˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γm for the basis of V1ˇˇ B dual to γ˚ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γ˚ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider the analytically-varying period matrix M “ Mij on B defined by the formula Mij “ γ˚ i pωjq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' note that pM11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , M1mq “ ph1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now give a description of the Zariski closure of MpBq Ă GLmpCq using [BT22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because the frame ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm is monodromy-compatible, flat sections of tensor powers of H1 are constant in this frame, and hence MpBq lies inside the Ms1-torsor in GLmpCq consisting of the set of matrices which send flat tensors of H1 (in the ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm coordinates) to flat tensors of V1 (in the γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γm coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let P Ă GLmpCq be the connected component of this torsor containing the image of M, which is a torsor for the identity compoent M˝ s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The group M˝ s1 is naturally identified with the algebraic monodromy-group of the variation of Hodge structures V1, and it is then a consequence of [BT22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] that MpBq Zar “ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now suppose the functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hm satisfy a linear relation ř i aihi “ 0 in a neighbour- hood of s1, where ai P C are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This relation can be interpreted as a relation on the first row of points in GLmpCq, and from the above one learns that it necessarily vanishes on all of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, one learns that if g P M˝ s1 is any element then ř i aiγ˚ 1 pg ¨ ωi,s1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Interpreting this dually via the action of the (the Betti realization of) the group M˝ s1 on the local system V1˚, one learns that the vector γ˚ 1,s1 lies inside a proper monodromy-invariant subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Since γ˚ 1,s1 is rational, this means that V1˚ admits a non-trivial rational summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Via a choice of polarizing form one knows that V1˚ is abstractly isomorphic to V1, hence V1 also admits a non-trivial rational summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that any summand of the underlying local system of V1 is automatically a summand on the level of variations of Hodge structures as a consequence of the Theorem of the Fixed Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This contradicts the assumed simplicity of V1, so we conclude that no functional linear relations on the period functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hm exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Relations in the non-vanishing case: As W is a CM Hodge structure, its endomor- phism algebra is a field, hence has a primitive element ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, the characteristic polynomial P of ϕ is equal to its minimal polynomial, and this is true regardless of the cohomological realization of ϕ chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Combining Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4, we obtain a linear algebraic relation R on the dual coordinates of the restriction ˆγ˚ˇˇ W´et with respect 39 to a basis of WdR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' note here that W being a CM Hodge structure implies that it has more than one non-zero Hodge number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one expresses this basis as a linear combination of the basis vectors ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm, one then obtains a linear relation on the dual coordinates of ˆγ˚ in the basis ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This relation is defined over KFL, where F is the splitting field of P and L is the field of definition of the de Rham realization of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Moreover if one includes all factors of the relation given to us by the statement of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3, the resulting relation is defined over KL and holds at all finite places as it depends only the de Rham coordinates of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Conclusion: We have produced, by the arguments in the vanishing and non-vanishing cases, linear relations Rvan,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Rvan,j and Rnv on the coordinates of ˆγ˚ 1 in the basis ω1,ξ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ω1,ξ determined by π and ϕ and defined over a field KLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To obtain the conclusion we just have to argue that the relation obtained as the product of all Galois conjugates of Rvan,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Rvan,j and Rnv over Kpξq has degree bounded independently of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (Note that this replacement does not affect whether the G-functions satisfy this relation on the functional level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' if the G-functions satisfied a polynomial relation with linear factors they would satisfy a linear relation, and our above argument rules out this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') This is a matter of bounding rL : Kpξqs and rF : Qs independently of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But the field F is a splitting field of a polynomial of bounded degree, and the degree of rL : Kpξqs is bounded by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4 Proof of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17 Repeat the setup in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 using monodromy-compatible frames, except now with our set Gof G-functions also containing those G-functions associated to the normal crossing intersection at q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Repeat the argument of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 to establish relations exist at relevant finite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider a point ξ P S and a relevant infinite place v for spξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let R be the minimum of the convergence radii of the elements in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As before, using property (C), we can find a component DR of the complex analytic neighbourhood defined by |s|v ă R which contains both ξ and a degeneration point s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write ph1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmq and ph1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , h1 mq for the G- functions associated to the two normal crossing singularities in the fibres Xs0, which we regard as complex analytic power series using the embedding v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now construct a Kpξq-algebraic relation on the values h1pspξqq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , hmpspξqq, h1 1pspξqq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , h1 mpspξqq (23) We denote by γ˚ 1 and γ1˚ 1 the functionals given to us by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We consider two cases: Dependent Case: Suppose that γ˚ 1 ˇˇ W and γ1˚ 1 ˇˇ W are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then we can construct a linear relation among the values in (23) which, in the coordinates induced by ω1,ξ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm,ξ, is defined over the field of definition L of the de Rham realization of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (It is defined over Q in the Betti coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') One shows that such a linear relation does not extend to the functional level by arguing as in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 above, noting that if γ˚ 1 ´ λγ1˚ 1 vanishes on W for some scalar λ P Q, then this can be interpreted as a linear relation on the coordinates of γ˚ 1 ´ λγ1˚ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Independent Case: We now suppose that γ˚ 1 ˇˇ W and γ1˚ 1 ˇˇ W are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write γ˚ 2 “ γ1˚ 1 , and extend this to a basis γ˚ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γ˚ m with dual basis γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let E be the endomorphism algebra of W, which is a field of dimension dimQ W, and let F be its Galois 40 closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then W bQ F decomposes as a direct sum À σ:EãÑC Wσ of one-dimensional weight spaces for the action of E, so necessarily there is some F-linear combination γ “ λ1γ1`λ2γ2 and some τ : E ãÑ C such that γ lies inside À σ‰τ Wσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' On the de Rham side, the subspace W corresponds to a summand WdR of HwpXξ, Ω‚ Xξq defined over some number field L in the de Rham coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The weight space decompo- sition of W also admits a de Rham counterpart WdR “ À σ WdR,σ defined over LF in the de Rham coordinates, and choosing an LF-algebraic vector ωτ P WdR,τ one has γ˚pωτq “ 0 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Expressing ωτ in terms of the monodromy-compatible frame ω1,ξ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ωm,ξ one thus obtains a linear relation on the values in (23) defined over LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Functional Relations in the Independent Case: That the linear relation R obtained in the independent case does not hold on the functional level can be argued as in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 by working with the vector γ˚, however there is one additional subtly to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After showing that the relation R, if it were to hold at the functional level, would define a monodromy-invariant subspace of the monodromy-representation associated to V1˚, one cannot immediately conclude that V1˚ has a non-trivial Q summand because the vector γ˚ is defined over a possibly non-trivial extension F of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' However, the argument shows that one does obtain a monodromy-invariant subsystem M Ă V1˚ C defined by R in which γ˚ lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Letting L Ă V1˚ C be the simple summand containing γ˚, one may consider some non-trivial conjugate Lc of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because γ˚ 1 and γ˚ 2 are Q-vectors one necessarily learns that L‘Lc contains spanCtγ˚ 1 , γ˚ 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then because this span is defined over Q and V1˚ is Q-simple one necessarily finds that V1˚ “ L ‘ Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As the sub-local system M contained L one then finds L “ M, and finally because M was defined by a single linear relation one finds that rank V1 “ rankV1˚ “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But a rank two variation of Hodge structure with non-trivial monodromy is necessarily isomorphic to a representation coming from an elliptic family, and such variations have absolutely irreducible monodromy representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Conclusion: The above reasoning shows that at each relevant place v for spξq we may find a linear KLF-algebraic relation on the values of G at spξq which does not hold at the functional level, and such that the degrees rL : Kpξqs and rF : Qs are bounded independently of ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' note that to obtain the bound on rL : Kpξqs we are using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Taking the product of conjugates of these relations we obtain Kpξq-algebraic relations Rv of degree bounded by a constant κ independent of ξ and the place v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' as we saw in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='16 the relations Rv may be assumed to be the same for all finite places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The product R “ ś v Rv taken over the relevant infinite places and a relevant finite place (if it exists) is then a Kpξq-algebraic relation of degree at most κ prKpξq : Qs ` 1q which holds at all relevant places for spξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using the Weil height θ induced by the parameter s one concludes from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='11 that for all ξ P S θpξq ď κ1 rKpξq : Qsa, for some constants κ1, a P Rą0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 7 Pila-Zannier for General Intersections In this sections we drop all “primed” superscripts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', write S instead of S1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', as com- pactifications and degeneration points will not be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For a general overview of the ideas in this section, we refer the reader to §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We make an additional comment about our presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The Zilber-Pink conjecture has a “geometric” part and an “arithmetic” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' With reference to the formulation appearing 41 in [BKU21], which works in the setting of a general polarizable integral variation of Hodge structure V on a complex algebraic variety S, the difference between the two is whether the “atypical” Hodge loci one considers in the base of the variation of Hodge structures are positive or zero dimensional (in the sense of period dimension, see [BKU21, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It is known as a consequence of [BKU21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] that geometric Zilber-Pink statements — statements about non-Zarski density of positive-dimensional atypical Hodge loci — can be be proven without any arithmetic input, using only general Hodge-theoretic facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (There is an exception in that one cannot resolve Zilber-Pink-type statements for positive-dimensional Hodge loci coming from splittings of the generic Hodge datum associated to S, but this can be reinterpreted as a failure to resolve the conjecture for zero-dimensional Hodge loci for an auxiliary variation constructed from this datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') We therefore focus exclusively on the zero-dimensional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' arithmetic, part of Zilber-Pink here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 Reduction to Height Bounds on Tensors To begin this section we assume we have a smooth projective C-algebraic family f : X Ñ S over a smooth base S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' we note in particular we do not assume S is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We let π : rS Ñ S be the universal covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We fix a frame γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γm of rV “ π˚ pRwf an ˚ Zq, using which we make the identification of local systems rV » Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This give us a natural height function rθ on all tensor spaces associated to any fibre of rV induced by the standard Weil height on the affine space associated to Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fixing a polarization Q : Zm b Zm Ñ Z, we then consider the space D of all polarized Hodge structures on Zm with the same Hodge numbers as the variation V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We assume that map rϕ : rS Ñ D given by sending rs to its associated Hodge flag F ‚ rs has discrete fibres;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' this is the same as saying that, if we set Γ “ AutpZm, QqpZq, the associated period map ϕ : S Ñ ΓzD is quasi-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We fix a fundamental domain F Ă D for the Γ-action which intersects the image of rϕ, and we write I Ă F for the intersection rϕprSqXF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By [BKT20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3], this set is definable in the o-minimal structure Ran,exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In what follows we will freely use the notion of special subvariety of S for the variation V, defined for instance as in [BKU21, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A special point is a zero dimensional special subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We will also refer to points in I which are images of lifts of special points as special points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For an irreducible subvariety Y Ă S, the Mumford-Tate group of GY is the Mumford-Tate group of Vs above a generic point s P Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', a point outside the Hodge locus of V ˇˇ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We write HS Ă GS for the algebraic monodromy group of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the identity com- ponent of the Zariski closure of the image of the monodromy representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a special subvariety Y Ă S, we say that a Mumford-Tate group GY Ă M Ă GS defines Y if there does not exist Y Ĺ Y 1 Ă S such that GY 1 Ă M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' There are finitely many Mumford-Tate subgroups M Ă GS up to GSpCq- conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For GLmpCq-conjugacy this is [Voi10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the same proof works in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let M Ă GS be a Mumford-Tate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then the number of special points in I defined by M is bounded by a constant κ independent of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 42 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We emphasize here that we are using the basis γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , γm to identify the fibres of the local system on rS, and so we understand Mumford-Tate groups as subgroups of GLm,Q with respect to this choice of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because Mumford-Tate subgroups of GS lie in finitely many GSpCq-conjugacy classes, we may reduce to the case where we just consider special points x P I defined by Mumford- Tate groups M in a fixed GSpCq-conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These points satisfy the property that they are isolated in the subset IM Ă I of points whose Mumford-Tate group is contained in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Each Mumford-Tate group M defines a “Noether-Lefschetz” locus | NLM Ă qD consisting of all Hodge flags with Mumford-Tate group contained in M in the sense of [GGK12, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These loci can be extended to an algebraic family of subvarieties of qD over a Noetherian base, all of whose fibres are GSpCq-translates of | NLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If we consider the collection of points x P I which are isolated in IM1 for some M 1 “ gMg´1, where g P GSpCq, then all such points are isolated points in definable intersections of the form I X pg ¨ | NLMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But the number of isolated points in any fibre of the definable family tI X pg ¨ | NLMqugPGSpCq is bounded by a uniform constant κ as a consequence of definability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a Mumford-Tate group M Ă AutpZm, Qq defined as the stabilizer of a set T Ă À aě1pQmqba of tensors, we say that T is a set of minimal type defining M if: (i) T is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (ii) Suppose T is partitioned as T “ T1 \\ T2 \\ ¨ ¨ ¨ by tensor degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then subject to (i), the sequence p|T1|, |T2|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='q is maximized with respect to the natural lexicographic total order on elements of NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (We have as many low-order tensors as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') We additionally say that T is a minimal set defining M if in addition: (iii) Subject to (i) and (ii), the set T is chosen to have minimal total height, where the height rθpT q is the maximum of the heights of all its (necessarily finitely many) ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fix the data of: a Mumford-Tate subgroup M Ă GS and its GSpCq-conjugacy class C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' a set S Ă SpCq of special points defined by Mumford-Tate groups in C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' for each ξ P S, a lift rξ of ξ mapping to I and a Mumford-Tate group Mrξ Ă GLprVrξq » GLpZmq which defines ξ and belongs to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let | NLM Ă qD be the locus consisting of all Hodge flags with Mumford-Tate group contained in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then if P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Pr P | NLMpCq are representatives of the components of | NLM, there exists a map gp´q : SÑ GSpQq, denoted ξ ÞÑ gξ, with the following properties: (1) there exists a constant d such that each point in the image of gp´q lies inside a number field of degree at most d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (2) there exists a constant c such that each fibre of gp´q has cardinality at most c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 43 (3) the height rθpgξq is bounded by a polynomial in rθpTrξq, where Trξ is the minimal set of definition for Mrξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and (4) for each ξ P S we have gξ ¨ Mrξ ¨ g´1 ξ “ M and that rϕprξq P pg´1 ξ ¨ MpCq ¨ Pjq for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that, because all Mumford-Tate are defined by the tensor invariants they stabilize and the spaces of these tensor invariants are defined over Q, any two minimal sets defining Mumford-Tate groups in C have the same number of tensors in each degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Denoting by d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , dk the dimensions of the associated subspaces of tensors, where k is the largest k for which dk “ dimQpT XpQmqbkq is positive for such a minimal set T , we may the consider an affine parameter space T“ Aff rpQmqsd1 ˆ Aff “ pQmqb2‰d2 ˆ ¨ ¨ ¨ ˆ Aff “ pQmqbk‰dk , where for a vector space V we write AffrV s for the associated affine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To any point T P T we may associate a group MT defined as the stabilizer of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We then define T0 Ă T as the locus of T for which MT is conjugate to M, and consider the Q-algebraic family y : G Ñ T0 whose fibre above T consists of those g P GSpCq satisfying the property that g ¨ MT ¨ g´1 “ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It will suffice to show that, for any T P T0pQq, we can construct gpT q P y´1pT qpQq defined over a number field of uniformly bounded degree and whose height is bounded by a polynomial in rθpT q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, suppose we can do this, and define gξ “ gpTrξq where Trξ is a minimal set of definition for Mrξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We may then see that (1) is true by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For (2) we note that if gpTrξ1q “ gpTrξ2q then Mrξ1 “ Mrξ2, and the number of lifts rξ mapping to I of points in S for which this can occur is bounded by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then (3) is true by assumption, and as | NLM rξ “ g´1 ξ ¨ | NLM statement (4) reduces to the claim that rϕprξq has Mumford-Tate group contained in Mrξ, which is true because Mrξ defines ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Continuing now with our claim about the family y, we may consider a stratification T0 “ T1 \\¨ ¨ ¨\\ Tℓ such that for each i the base-change yi : Gi Ñ Ti along Ti ãÑ Tof y is flat and each Ti is smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' base-changing to one of these strata, we may assume that y : GÑ T0 is both smooth and flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because y is flat with smooth fibres, it is then necessarily a smooth morphism, and then this necessarily implies Gis a smooth variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Each fibre of y is naturally a torsor for the normalizer N of M in GS, and one has a natural map N ˆ G Ñ GˆT G bijective on complex points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' since this is a map of two smooth varieties, this means it is necessarily an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We conclude that y is an fppf N-torsor, and then because N is a smooth algebraic group, an ´etale N-torsor by [hmb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Choose an ´etale covering e : U Ñ T0 and a trivialization σ : N ˆ U „ ÝÑ G ˆT0 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For T P T0, construct gpT q by choosing any element ζ of the fibre e´1pT q and then defining gpT q “ σp1, ζq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' That the degree of the resulting field of definition is bounded is a consequence of the fact that the degree of e is bounded, and that the resulting height is polynomial in the height of T is an easy consequence of how heights behave under polynomial maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To give a finiteness criterion for special points, we now need to introduce some language to talk about atypical intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Recall that a Hodge structure h on a Q-vector space V can be thought of as a map h : S Ñ GLpV qR, where S “ ResC{RGm,R is the Deligne torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a Q-subgroup M Ă GLm whose real points contain the image of S, one obtains an induced Hodge structure on the Lie algebra m of M through the adjoint action;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular this is true for the Mumford-Tate group of the Hodge structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 44 Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a Hodge structure h : S Ñ GLpV qR factoring through a Mumford-Tate group M with Lie algebra m, we write δpM, hq for the sum of the positive Hodge numbers associated to the induced Hodge structure on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a point s P S, write hs (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Gs) for the Hodge structure (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Mumford- Tate group) at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given a complex subvariety Y Ă S and a Mumford-Tate group GY Ă M Ă GS, write δpM, Y q “ δpM, hsq for s P Y (the quantity is independent of the point chosen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here we understand M up to conjugacy by the monodromy on Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given some Mumford-Tate group GY Ă M Ă GS and a special subvariety Y Ă S, we say that Y is atypical for M if M defines Y and dim S ´ dim Y ă δpS, hsq ´ δpM, hsq for s P Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We say that Y is atypical if Y is atypical for GY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now restrict to the case where f : X Ñ S is defined over a number field K Ă C, and let SHg,` Ă SpCq be the positive dimensional Hodge locus: the collection of points which lie inside a special subvariety of positive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Fix a Mumford-Tate group M Ă GS, and let SĂ SpCqzSHg,` be a collection of special points which are defined by, and atypical for, some GSpCq-conjugate of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that (i) the group HS is equal to the derived subgroup of GS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (ii) each point of S is defined over Q, and the GalpQ{Kq-action preserves S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and (iii) that there exists constants a, b P Rą0 independent of ξ P S, and for each ξ P S a lift rξ mapping to I for which rθpTrξq ď a rKpξq : Ksb, where Trξ is a set of definition of minimal type for some Mumford-Tate group Mrξ which defines ξ atypically and is GSpCq-conjugate to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, we consider the definable locus G:“ rď j“1 tg P GSpCq : pg´1 ¨ MpCq ¨ Pjq X I ‰ ∅u looooooooooooooooooooooooooomooooooooooooooooooooooooooon Gj with P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Pj as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 and observe that the construction of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5 produces a point gξ of Gfor each ξ P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' These points are all defined over Q, and in particular over a number field of degree at most some fixed constant d by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5(3) and our assumption (iii), there exists constants a1, b1 ą 0 such that rθpgξq ď a1 rKpξq : Ksb1 for all ξ P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Assume for contradiction that S is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5(2) and (ii), one has for infinitely many positive integers N that ˇˇˇ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' gξ : rθpgξq ď a1N b1)ˇˇˇ ě 1 cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 45 Applying the Pila-Wilkie theorem [Pil09, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6] in the case of S infinite one finds that there exists an index j, an irreducible complex algebraic subvariety V Ă GSpCq, and an analytic open neighbourhood U1 intersecting V such that V X U1 Ă Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (A na¨ıve application of [Pil09, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='6] gives only a real-algebraic such V , but one can use [PT13, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] to ensure this is a complex algebraic subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') Morever V contains infinitely many elements gξ for ξ P S, hence V ´1 ¨ MpCq ¨ Pj intersects I in a locus of positive dimension (the intersection contains infinitely many points rϕprξq and is definable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now choose an irreducible algebraic curve C Ă V containing a point c “ gξ associated to some ξ P S subject to the conditions: there exists an analytic locus F Ă pC´1 ¨ MpCq ¨ Pjq X I such that dim rϕprξq F ą 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' and F does not lie inside any complex orbit in qD of a proper Q-normal factor of HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Both of these conditions may be understood in terms of tangency: because for any g P V X U1 near c the intersections pg´1 ¨ MpCq ¨ Pjq X I are non-empty, we may always choose C along a direction in V for which the intersections pg´1 ¨ MpCq ¨ Pjq X I are both non-empty and varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Similarly, because no germ of I lies inside any orbit of a proper normal factor of HS, and the orbits of such a normal factor foliate qD, one can even choose C so that the intersections C´1 ¨ MpCq ¨ Pj X I vary in a direction transverse to each such foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now in particular by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5(4), one has gξMrξg´1 ξ “ M and rϕprξq is a point of O “ c´1¨M ¨Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The fact that ξ is atypically defined by Mrξ means that dim O “ δpMrξ, rϕprξqq satisfies dim S ` dim O ă δpSq “ dim HSpCq ¨ ϕprξq, (24) where for the equality we use the fact that the derived subgroup of GS is HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By construc- tion, the (constructible) variety E “ C´1MpCq ¨ Pj has dimension dim O ` 1 and intersects I in a locus of dimension at least 1 at F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Letting qT “ HSpCq ¨ ϕprξq one has that codim qT F ă codim qT I` codim qT E as a consequence of (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the Ax-Schanuel Theorem for period mappings [BT17], one the learns that ξ, which lies inside ϕ´1pπDpFqq with πD : D Ñ ΓzD the natural projection, lies inside a (necessarily positive dimensional) weakly special subvariety W Ă S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (We note that in [BT17], the Ax-Schanuel theorem is formulated for intersections with graphs of period mappings, but one can recover our formulation by pulling back an algebraic intersection with the image to an algebraic intersection with the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') By assumption the point ξ does not lie inside SHg,`, so the weakly special subvariety W containing ξ does not either;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, the Mumford-Tate group of W is GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' On the other hand, because W contains the image of F, it is not defined by an orbit of a proper normal factor of HS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, its algebraic monodromy group is not a normal factor of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This contradicts [Yve92, §5], which ensures that algebraic monodromy groups of varieties are always Q-algebraic normal subgroups of their Mumford-Tate groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 Constraining Heights of Tensors We will now try to put ourselves in a situation where we can apply Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7, and in particular verify hypotheses (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We begin by giving a way to estimate the heights rθpTrξq that appear in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 using the theory of Siegel sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We set G “ AutpZ, Qq 46 and recall that D is naturally a homogeneous space for GpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one fixes a reference Hodge structure x0 P FĂ D we obtain an identification q : GpRq{H „ ÝÑ D, where H Ă GpRq is the centralizer of the map x0 : S Ñ GR which defines the Hodge structure x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' From the theory in [Bor69], there exists a Siegel set S Ă GpRq whose image is the fundamental domain Ffor the Γ “ GpZq-action fixed above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, every fibre of the map q above Fis contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (We refer to [Orr18, §2] for the definition of Siegel set, as we will only need their abstract properties here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, we will just use the fact that they are fundamental sets for the Γ-action compatibly with the map q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=') Consider now a rational element ϕ P GpQq which induces an endomorphism of a Hodge structure x P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then if gh P G is chosen such that x “ gx0, then g´1ϕg is an endomorphism of x0, and hence lies in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Writing h “ g´1ϕg, one then has ϕ “ ghg´1, which is a rational point contained in GG´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Orr [Orr18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] has shown the following quantitative result about Siegel sets: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8 (Orr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' There exists a constant C1 (depending on a choice of reductive Q- algebraic subgroup of G Ă GLm and Siegel set in G) such that for all ϕ P GG´1 X GpQq one has rθpϕq ď maxtC1ndm, du, where n “ |det ϕ|, and d is the maximum of the denominators in the entries of ϕ (regarded as a point in GLmpQq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As shown in [Orr18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2], one can also construct a Siegel set Gm for GLm,Q con- taining the centralizer Hm of x0 such that finitely many rational translates of Gm contain G, so one can even use this theorem to deal with endomorphisms ϕ that don’t preserve the polarization at the cost of changing the constant C1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' with respect to the above analysis all that changes is that H is replaced by Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using this, we now bound heights of endomorphisms of Hodge structures in F in terms of more geometric notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given any rational endomorphism ϕ of a Hodge structure x, the map ϕ is diagonalizable, hence is invertible when restricted to its image impϕq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We define its restricted determinant detrpϕq to be the determinant of this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We call a map y P Hompx, x1q between Hodge structures x and x1 on Zm an isogeny if the scalar extension yQ is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Its dual y_ is the map in Hompx1, xq induced from the natural map x1_ Ñ x_ by using the isomorphisms x1 » x1_ and x » x_ induced by the fixed polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given an integral Hodge structure x on the lattice Zm, a Hodge en- domorphism ϕ P EndpxqQ, and the Hodge idempotent πϕ defining the Hodge summand impϕq, we define the isogeny-height θisopϕq to be the minimum degree of an integral isogeny y P Hompx, x1q to some integral Hodge structure x1 on Zm such that both yQ ˝ ϕ ˝ y´1 Q and yQ ˝ πϕ ˝ y´1 Q are Q-scalar extensions of elements of Endpx1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The height of any rational endomorphism ϕ associated to some Hodge struc- ture x P F is bounded by a polynomial in its isogeny-height and the absolute value of its restricted determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let us begin by reducing to the case where ϕ and its associated idempotent πϕ are integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let y P Hompx, x1q be an integral isogeny to some x1 P Fsuch that ϕ1 Q “ yQ˝ϕ˝y´1 Q 47 and πϕ1,Q “ yQ ˝ πϕ ˝ y´1 Q are the Q-scalar extensions of integral endomorphisms ϕ1 and πϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As y is a homomorphism, it conjugates the actions of the tori associated to x and x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' If one chooses gx and gx1 such that x “ gxx0 and x1 “ gx1x0, where x0 is the reference Hodge structure fixed above, then g´1 x1 ygx centralizes x0 and hence lies in Hm, the centralizer of x0 in GLmpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It follows that y lies in Gm ¨ G´1 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The element y is assumed integral, so its height is bounded as a consequence of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8 by a linear multiple of its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The heights of ϕ is then bounded by a polynomial in its isogeny height (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', a polynomial in the minimum determinant of such a y) and the height of an integral endomorphism ϕ1 with the same restricted determinant and whose associated idempotent πϕ1 is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now assuming ϕ and πϕ are integral, we observe that, again by applying Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='8 and the reasoning at the beginning of this section, we are done in the case where ϕ has trivial kernel (and hence πϕ “ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In the cases where ϕ has non-trivial kernel we have ϕ “ pϕ ` p1 ´ πϕqq ´ p1 ´ πϕq, where we observe that ϕ ` p1 ´ πϕq has trivial kernel and an isogeny height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' A polynomial height bound for the sum follows from a polynomial height bound for its summands, so we are reduced to the case of integral idempotents, and in particular the case where ϕ “ πϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It now suffices to show that there are only finitely many ϕ which are integral polarization- preserving idempotents for Hodge structures x P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Indeed, because projections preserving a non-degenerate bilinear form are uniquely determined by their image, it suffices to show that only finitely many integral summands of Zm occur for Hodge structures in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Because Γ “ AutpZm, QqpZq is the discrete group defining the equivalence for which F is a fundamental set, the set F has only one Hodge structure from each integral isomorphism class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Given two distinct direct summand decompositions Zm “ V ‘ W and Zm “ V 1 ‘ W 1 where a “ dim V “ dim V 1 and b “ dim W “ dim W 1 respectively, any Hodge structure compatible with the first decomposition is isomorphic to one compatible with the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' One may then see that in fact only one such decomposition is associated to Hodge structures in F from the fact that, after fixing such a decomposition Zm “ V ‘ W, one has a natural map FV ˆ FW Ñ Fof fundamental sets sending a pair of polarized Hodge structures to its direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 Application in the Abelian Case The previous proposition allows one to bound the heights rθpTrξq appearing in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7, at least in the case where Trξ consists of endomorphisms, in terms of quantities of a geometric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now consider the more concrete case of Hodge structures arising from abelian varieties where this is easier to analyze;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, we start by verifying hypothesis (ii) of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose in the situation of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 that f is an abelian family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then after replacing K with a finite extension, hypothesis (ii) holds for the set S consisting of all special points not lying in SHg,` which are defined by a Mumford-Tate group which is GSpCq-conjugate to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is an easy consequence of Deligne’s verification [Del82] of the absolute Hodge conjecture for abelian families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' First, we note that the fact that the action of GalpQ{Kq preserves SHg,` is known, at least in the case where HS is Q-simple, in much greater gener- ality, see [KOU20, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To see that GalpQ{Kq will also preserve zero-dimensional Hodge loci outside of SHg,` it then suffices to check that it preserves Hodge loci, and that this is a consequence of the absolute Hodge conjecture is explained in [Voi10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 48 We must also check that if Mξ Ă GLpVξq is a Mumford-Tate group in the GSpCq- conjugacy class of M defining ξ atypically, and if σ P GalpQ{Kq is an automorphism, then ξσ admits a Mumford-Tate group M σ ξ Ă GLpVξσq which defines ξσ atypically and is GSpCq- conjugate to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For this consider the composition Vξ,C „ ÝÑ Hξ,C σÝÑ Hξσ,C „ ÝÑ Vξσ,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The absolute Hodge conjecture is the statement that the induced map on tensor spaces sends Hodge tensors to Hodge tensors, so in particular the induced map on spaces of linear transformations sends the Mumford-Tate group Mξ to some Mumford-Tate group M σ ξ Ă GLpVξσq which contains the Mumford-Tate group of Mξσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Moreover, this induced map also sends the realization of GS in GLpVξq to the realization of GS in GLpVξσq, hence there is an induced outer automorphism η of the abstract group GSpCq which sends Mξ (regarded as a subgroup through the realization of GS in the fibre at ξ) to M σ ξ (regarded as a subgroup through the realization of GS in the fibre at ξσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Now if gMξg´1 “ M for some g P GSpCq, one finds that ηpgqM σ ξ ηpgq´1 “ ηpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We claim that, after possibly replacing K with a finite extension, the group ηpMq is GSpCq-conjugate to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The point is that one can work entirely with the algebraic-de Rham realizations of these groups, in particular, we may denote by C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , Cℓ the GS-conjugacy classes of Mumford-Tate subgroup of GS and consider the algebraic vector bundles yi : Gi Ñ S whose fibres are the moduli of algebraic subgroups of GLpHsq lying in Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By construction ηpMq is GSpCq-conjugate to a Mumford-Tate group, hence corresponds to a point in the fibre above ξσ of one such family, but it might be in the wrong conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' What one then wants is for each of the families yi to be naturally (using the K-algebraic structure induced from H) defined over K, which occurs after passing to a finite extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Finally, to check that ξσ is atypical for M σ ξ , we note that the quantity δpM σ ξ , hξσq may also be computed from the Hodge filtration on the Lie algebra of M σ ξ (instead of the Hodge direct sum decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular, it can be computed using the de Rham realizations of M σ ξ and F ‚ ξσ, and this data is σ-conjugate to the de Rham realizations of Mξ and F ‚ ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that f : X Ñ S is an abelian family with HS equal to the derived subgroup of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let S Ă SpCq be a set of special points not lying in SHg,` which are defined by, and atypical for, some subspace of their Hodge tensors of endomorphism type, and that for each ξ there is a basis ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' , ϕℓ for a subspace of the Hodge endomorphisms of Vξ defining ξ atypically such that maxitdetrpϕiq, θisopϕiqu ď κ rKpξq : Ksa for constants κ, a P Rą0 independent of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' It suffices to verify that the hypotheses of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='7 hold, possibly passing to a finite extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Hypothesis (i) holds by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='13 we may replace S with its orbit under GalpQ{Kq after passing to a finite extension, showing (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' For hypothesis (iii) we use Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='12 to bound rθpTrξq in terms of the restricted determinant and isogeny degree, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that f : X Ñ S is an abelian family with HS equal to the derived subgroup of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Let S Ă SpCq be a set of special points not lying in SHg,` which are defined by, and atypical for, their Hodge idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that θpξq ď κ rKpξq : Ksa 49 for some κ, a P Rą0 independent of ξ, with θ some logarithmic Weil height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Using Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='14 it suffices to show that the isogeny heights of idempotents are polynomially bounded by the Faltings height (which differs from any logarithmic Weil height by a multiplicative constant) of the point ξ and the degree of the field of definition of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is just the result of Masser-Wustholtz [MW95, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Note that the statement of [MW95, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1] is given (in our language) for ξ defined over a field of bounded degree, but the constants depend polynomially on this degree as is explained at the end of [MW95, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We note that Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='15 gives an interpretation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='18 in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 8 Applications In what follows we will use the following fact, which is part of the proof of Theorem 2 in [And89, IX, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Suppose that f : X Ñ S is a projective family of relative dimension n with geometrically connected fibres and whose singular fibres all have simple normal crossings, and let f 1 : X1 Ñ S1 be the base-change to the smooth locus S1 Ă S, which we assume is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Then if s0 P SpCqzS1pCq, the vanishing cycles associated to the order-n normal crossing singularities of Y “ Xs0 (in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2) span a space of dimension hnpΣY q, where hnpΣY q is the dimension of the n’th cohomology group of the dual graph ΣY associated to Y (see [Kol14, §1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 Families of Curves We now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We first observe that the case where g “ 2 was already proven in [DO21, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1], albeit under different language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To reduce from our setup to the setup of Daw and Orr in [DO21], note that the locus B Ă MgzMg in the case of g “ 2 necessarily consists of singular curves C “ P1 Y ¨ ¨ ¨ Y P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The stability condition requires (see [DM69, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1]) that each P1 component intersects each other component in at least 3 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The boundary strata of Mg are stratified by the number of nodes [Mum83], and the stratum of curves with k nodes has dimension 3g ´ 3 ´ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' in particular, the only possibility here is C “ P1 Y P1 with the components meeting at 3 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' But the extended Torelli map [BPVdV84] sends this curve into the 0-dimensional Bailey-Borel stratum (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' the introduction of [Cap09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As the 0-dimensional Bailey Borel stratum of the boundary A2zA2 consists of a single point, one could also reduce from the case of abelian surfaces to the case of curves by considering a family of curves which induces, away from a finite set, a given family of surfaces passing through the 0-dimensional stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We now consider the general statement when g ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We first observe that we may reduce to the case where S is defined over a number field K: indeed, the points of S are all defined over Q, and if the set SX SpCq is infinite then S coincides with its Zariski closure and hence is also defined over Q, and hence over some number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After replacing S with a smooth finite covering we may reduce to the analogous problem for a K-algebraic family f : X Ñ S of genus g curves and take S Ă SpQq to be the set of points where the 50 fibre of the associated variation V1 “ R1f 1an ˚ Q of Hodge structures admits a simple factor with complex multiplication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here f 1 is the restriction of f to the fibre above S Ă S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To get a polynomial bound on the heights of points of S in terms of the degree we now wish to apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17 to our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This means checking that integrating around the vanishing cycles corresponding to two nodal singularities of Xs0 produces tuples of non-constant G-functions satisfying the linear independence condition in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This follows from Andr´e’s result Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 above, as well as the remark at [And89, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='192], which says that h1pΣXs0 q “ g ´ ř i pgpCiq ě 2, where Xs0 “ C1 Y ¨ ¨ ¨ Y Cℓ is the decomposition of the singular fibre into its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We are now in the situation where we have a logarithmic Weil height θ : SpQq Ñ Rą0 for which there exists constants κ, a P Rą0 such that θpξq ď κ rKpξq : Ksa (25) for all ξ P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' By the resolution of the Andr´e-Oort conjecture [PST`21], it suffices to replace S with just those points for which the CM summand is a proper summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' After replacing the problem in question for the equivalent problem for the associated Jacobian family, the result then follows from Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='15 as soon as g ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In particular Hodge-genericity implies the monodromy assumption of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='15, and the atypicality with respect to the idempotents of the points in S holds under the assumption g ě 3 as a special locus of Ag with a global isogeny factor has codimension at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='2 Families of Abelian Varieties We now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The proof is the same as the case for curves above, except we do not even have to check that the vanishing cycles are independent because this has been assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='3 Degenerations to Hyperplanes Lastly, we prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' We once again apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17 using Andr´e’s result Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' That Hodge-genericity implies the simplicity of the primitive varia- tion and the atypicality of the CM points is a consequence of Beauville’s computation of the monodromy groups of hypersurface variations in [Bea86];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' here we note that a Hodge generic curve on which the associated variation is non-constant will have maximum-possible algebraic monodromy because its algebraic monodromy group must be non-trivial and Q- normal in the Q-simple group of automorphisms preserving the polarization form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' To apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='17 it therefore suffices to compute the cohomology of the dual graph associated to a generic hyperplane arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' This is the same as the cohomology of the complement of this hyperplane arrangement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' one may deduce from [Bri73, Lemma 3] that the dimension of this cohomology group is `d´1 n ˘ , which under the assumption d ą n ` 1 is always at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' As is clear from the proof, a more detailed analysis of the cohomology of the dual graph associated to the singular fibre gives more results for other degeneration types as well, but we do not undertake this analysis here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 51 References [AB13] Boris Adamczewski and Jason P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Bell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Diagonalization and rationalization of algebraic Laurent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (4), 46(6):963–1004, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [AM69] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Artin and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Mazur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Etale homotopy, volume 100 of Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Notes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Springer, Cham, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [And89] Yves Andr´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' G-functions and geometry, volume E13 of Aspects Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Wies- baden etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' : Friedr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Vieweg &— Sohn, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [And95] Yves Andr´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The theory of motives and geometric interpretation of p-adic values of G-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' An introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In Number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' S´eminaire de th´eorie des nombres de Paris 1992-93, pages 37–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Cambridge: Cambridge University Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Bea86] Arnaud Beauville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections compl`etes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' In Complex analysis and algebraic geometry, pages 8–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Springer, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Ber93] Vladimir G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Berkovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ´Etale cohomology for non-Archimedean analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Publ.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='08838, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Bor69] Armand Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Introduction aux groupes arithm´etiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Actualit´es Scien- tifiques et Industrielles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 1341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Paris: Hermann & Cie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 125 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [BPVdV84] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Barth, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Peters, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Van de Ven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Compact complex surfaces, volume 4 of Ergeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Grenzgeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Folge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Springer, Cham, 1984.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Perfectoid spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Bonn: Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Bonn, Mathematisch- Naturwissenschaftliche Fakult¨at (Diss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' ), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Sch13] Peter Scholze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' p-adic Hodge theory for rigid-analytic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Pi, 1:77, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Id/No e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Sta20] The Stacks project authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' The stacks project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' https://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content='edu, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Ste76] Joseph Steenbrink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Limits of Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=', 31:229–257, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 54 [Voi10] Claire Voisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Hodge loci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Handbook of moduli, 3:507–546, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' [Yve92] Andr´e Yves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' Compositio Mathematica, 82(1):1–24, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} +page_content=' 55' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQf5P6V/content/2301.01857v1.pdf'} diff --git a/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf b/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8fd4c0bc02808f82abf80ed509df9970a47aa607 --- /dev/null +++ b/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0d219323cd7c382f49bd0443a71a6e2b3ce1ceec0136f3e405d9dfe3b96df026 +size 1617102 diff --git a/pdFST4oBgHgl3EQfNjhG/content/tmp_files/2301.13748v1.pdf.txt b/pdFST4oBgHgl3EQfNjhG/content/tmp_files/2301.13748v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3561bd1f2d62cdb56ad00a202666d01ab9e41fce --- /dev/null +++ b/pdFST4oBgHgl3EQfNjhG/content/tmp_files/2301.13748v1.pdf.txt @@ -0,0 +1,2010 @@ +� For correspondence: +sebastian.mair@it.uu.se +Funding: This work was partially +supported by the Wallenberg AI, +Autonomous Systems and Software +Program (WASP) funded by the +Knut and Alice Wallenberg Foun- +dation. +Code: The code is available +on request and will be published +on github once the paper is peer- +reviewed. +Archetypal Analysis++: Rethinking the +Initialization Strategy +Sebastian Mair +1 � and Jens Sjölund +1 +1Uppsala University, Sweden +Abstract +Archetypal analysis is a matrix factorization method with convexity constraints. Due to local +minima, a good initialization is essential. Frequently used initialization methods yield either sub- +optimal starting points or are prone to get stuck in poor local minima. In this paper, we propose +archetypal analysis++ (AA++), a probabilistic initialization strategy for archetypal analysis that +sequentially samples points based on their infuence on the objective, similar to 푘-means++. In fact, +we argue that 푘-means++ already approximates the proposed initialization method. Furthermore, +we suggest to adapt an efcient Monte Carlo approximation of 푘-means++ to AA++. In an extensive +empirical evaluation of 13 real-world data sets of varying sizes and dimensionalities and considering +two pre-processing strategies, we show that AA++ almost consistently outperforms all baselines, +including the most frequently used ones. +1 Introduction +Archetypal analysis (AA) (Cutler and Breiman, 1994) is a matrix factorization method with convexity +constraints. The idea is to represent every data point as a convex combination of points, called +archetypes, located on the boundary of the data set. Thus, archetypes can be seen as well-separated +observations that summarize the most relevant extremes of the data. The convexity constraints also +give archetypal analysis a natural interpretation. +Archetypal analysis has been applied, e.g., for global gene expression (Thøgersen et al., 2013), +bioinformatics (Hart et al., 2015), apparel design (Vinué et al., 2015), chemical spaces of small organic +molecules (Keller et al., 2021), geophysical data (Black et al., 2022), large-scale climate drivers (Hannachi +and Trendaflov, 2017; Chapman et al., 2022), and population genetics (Gimbernat-Mayol et al., 2022). +To improve the computation of archetypal analysis, various optimization approaches (Bauckhage +and Thurau, 2009; Mørup and Hansen, 2012; Chen et al., 2014; Bauckhage et al., 2015; Abrol and +Sharma, 2020) and approximations (Mair et al., 2017; Damle and Sun, 2017; Mei et al., 2018; Mair and +Brefeld, 2019; Han et al., 2022) have been proposed. However, the earliest point of attack for obtaining +a good solution is the initialization of the archetypes. Surprisingly, this has barely been investigated. +In the original paper, Cutler and Breiman (1994) use a random initialization, i.e., choosing random +points from the data set, which was adopted by many others, e.g., (Eugster and Leisch, 2011; Seth +and Eugster, 2015; Hinrich et al., 2016; Hannachi and Trendaflov, 2017; Mair et al., 2017; Mei et al., +2018; Krohne et al., 2019; Olsen et al., 2022) to name a few. Furthermore, Cutler and Breiman (1994) +state that a careful initialization improves the convergence speed and that archetypes should not be +initialized too close to each other. +The same idea serves as an argument for using the FurthestFirst approach (Gonzalez, 1985; +Hochbaum and Shmoys, 1985), yielding a well-separated initialization used, e.g., in 푘-means clustering +(Lloyd, 1982). Inspired by FurthestFirst, Mørup and Hansen (2010, 2012) propose a modifcation for +archetypal analysis called FurthestSum, which focuses on boundary points rather than well-separated +points. Here, boundary points refer to points on the boundary of the convex hull of the data. Since +then, FurthestSum has established itself as one of the default initialization strategies for archetypal +analysis (Thøgersen et al., 2013; Hinrich et al., 2016; Mair and Brefeld, 2019; Abrol and Sharma, 2020; +Beck et al., 2022; Black et al., 2022; Chapman et al., 2022; Gimbernat-Mayol et al., 2022). +Sebastian Mair et al. +| +arXiv +| +February 1, 2023 +| +1–20 +arXiv:2301.13748v1 [cs.LG] 31 Jan 2023 + +Despite its popularity, FurthestSum has also been criticized. For example, Suleman (2017) states +that FurthestSum is prone to selecting redundant archetypes, primarily when many archetypes are +used. Redundant archetypes lie in the convex hull of the already selected archetypes and thus do not +lower the overall error. In addition, Krohne et al. (2019) and Olsen et al. (2022) report better results +with a random initialization than with FurthestSum. A possible explanation is that FurthestSum’s +early focus on boundary points risks trapping it in poor local minima. +Contributions +In this paper, we motivate and propose archetypal analysis++ (AA++), an initialization strategy +inspired by 푘-means++ (Arthur and Vassilvitskii, 2007; Ostrovsky et al., 2013). Furthermore, we argue +that the 푘-means++ initialization can be seen as an approximation to the proposed AA++ strategy +and that a Monte Carlo approximation to the 푘-means++ initialization can be adapted for AA++ for a +more efcient initialization. Most importantly, we empirically demonstrate that our proposed AA++ +initialization for archetypal analysis outperforms almost consistently all baselines on 13 real-world +data sets. +2 Preliminaries +Before introducing archetypal analysis, we briefy revisit 푘-means clustering since we will build upon +similar ideas. +2.1 +푘-means Clustering +Let  ⊂ ℝ푑 be a data set of 푛 points in 푑 dimensions and let  = {퐳1, … , 퐳푘} be a set of 푘 cluster +centers. Consider the 푘-means clustering problem with the following objective +휙() = ∑ +퐱∈ +푑(퐱, )2 = ∑ +퐱∈ +min +퐪∈{퐳1,…,퐳푘} ‖퐱 − 퐪‖2 +2, +where 푑(퐱, )2 = min퐪∈ ‖퐱 − 퐪‖2 +2 is the minimal squared distance from a data point 퐱 to the closest +center in . +Often, the cluster centers  of 푘-means are initialized using the 푘-means++ initialization procedure +(Arthur and Vassilvitskii, 2007), which works as follows. The frst center is chosen uniformly at random. +Then, the remaining 푘 − 1 cluster centers are chosen according to a probability distribution where the +probability of choosing a point 퐱 is proportional to the closest distance to the already chosen cluster +centers, i.e., 푝(퐱) ∝ 푑(퐱, )2. The procedure is outlined in Algorithm 1. +2.2 +Archetypal Analysis +Let  = {퐱1, … , 퐱푛}푛 +푖=1 ⊂ ℝ푑 be a data set consisting of 푛 ∈ ℕ 푑-dimensional data points arranged +as rows in the design matrix 퐗 ∈ ℝ푛×푑. The idea in archetypal analysis (AA) (Cutler and Breiman, +1994) is to (approximately) represent every data point 퐱푖 as a convex combination of 푘 ∈ ℕ archetypes + = {퐳1, … , 퐳푘}, i.e., +퐱T +푖 ≈ 퐚T +푖 퐙, +퐚T +푖 ퟏ = 1, +퐚푖 ≥ 0, +Algorithm 1 푘-means++ Initialization +1: Input: Set of 푛 data points , number of clusters 푘 +2: Output: Initial set of clusters centers  +3: Sample index 푖 uniformly at random from [푛], i.e., +using 푝(푖) = 푛−1 +4: Append 퐱푖 to  +5: while || < 푘 do +6: +Sample 푖 using 푝(푖) ∝ min퐳∈ ‖퐱푖 − 퐳‖2 +2 +7: +Append 퐱푖 to  +8: end while +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +2 of 20 + +Convex Hull of X +Convex Hull of Z +Data Points xi +Initial Archetypes Zinit +Learning Path of Z +Learned Archetypes Z +Error +Figure 1. Archetypal analysis in two dimensions with 푘 = 4. +where the matrix 퐙 ∈ ℝ푘×푑 contains the archetypes as rows and the vector 퐚푖 ∈ ℝ푘 defnes the weights +for the 푖th data point. Besides, ퟏ denotes the vector of ones and 퐚푖 ≥ 0 is meant element-wise. The +archetypes 퐳푗 (푗 = 1, … , 푘) themselves are also represented (exactly) as convex combinations, but of +the data points, i.e., +퐳T +푗 = 퐛T +푗 퐗, +퐛T +푗 ퟏ = 1, +퐛푗 ≥ 0, +where 퐛푗 ∈ ℝ푛 is the weight vector of the 푗th archetype. Let 퐀 ∈ ℝ푛×푘 and 퐁 ∈ ℝ푘×푛 be the matrices +consisting of the weights 퐚푖 (푖 = 1, … , 푛) and 퐛푗 (푗 = 1, … , 푘). Then, archetypal analysis yields an +approximate factorization of the design matrix as follows +퐗 ≈ 퐀퐁퐗 = 퐀퐙, +(1) +where 퐙 = 퐁퐗 ∈ ℝ푘×푑 is the matrix of archetypes. Due to the convexity constraints, the weight +matrices 퐀 and 퐁 are row-stochastic. The weight matrices 퐀 and 퐁 are typically determined by +minimizing the approximation error in the Frobenius norm, resulting in the optimization problem +minimize +퐀,퐁 +‖퐗 − 퐀퐁퐗‖2 +퐹 +subject to +퐀ퟏ = 1, 퐀 ≥ 0, 퐁ퟏ = 1, 퐁 ≥ 0. +(2) +This can be equivalently expressed as minimizing the sum of projections of the data points on the +archetype-induced convex hull as follows +‖퐗 − 퐀퐙‖2 +퐹 = ∑ +퐱∈ +min +퐪∈conv({퐳1,…,퐳푘}) ‖퐱 − 퐪‖2 +2, +(3) +where conv(푆) refers to the convex hull of a set 푆. An example of archetypal analysis and projection +errors is depicted in Figure 1. The optimization problem is a generalized low-rank problem (Udell +et al., 2016), which is biconvex but not convex. Because it is biconvex, a local optimum can be found +via an alternating optimization scheme such as the standard one outlined in Algorithm 2. However, +the quality of this local optimum is directly dependent on the initialization. +2.3 +Archetype Initializations +Popular ways of initializing the archetypes 퐙 is by using uniformly at random chosen data points or +the FurthestSum procedure (Mørup and Hansen, 2010, 2012), but we also introduce FurthestFirst. +FurthestFirst +Originally proposed for the metric 푘-center problem, the FurthestFirst algorithm (Gonzalez, 1985; +Hochbaum and Shmoys, 1985) selects the frst center/archetype at random and selects every consecu- +tive point which is furthest away from the closest already selected center/archetype. Mathematically, +the index of the next point is +푗next = arg max푖∈[푛] (min +퐪∈ ‖퐱푖 − 퐪‖훼 +2 ) , +where [푛] = {1, 2, … , 푛} and 훼 = 1. Note that in 푘-means++, the distances are squared, i.e., 훼 = 2, and +that points are sampled. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +3 of 20 + +Algorithm 2 Archetypal Analysis (Cutler and Breiman, 1994) +1: Input: data matrix 퐗, number of archetypes 푘 +2: Output: weight matrices 퐀 and 퐁 +3: 퐙 ← initialization of the archetypes 퐙 +4: while not converged do +5: +퐚푖 = arg min +퐚T +푖 ퟏ=1, 퐚푖≥0 +‖퐙T퐚푖 − 퐱푖‖2 +2 +∀푖 = 1, … , 푛 +6: +퐙 = (퐀T퐀)−1퐀T퐗 +7: +퐛푗 = arg min +퐛T +푗 ퟏ=1, 퐛푗≥0 +‖퐗T퐛푗 − 퐳푗‖2 +2 +∀푗 = 1, … , 푘 +8: +퐙 = 퐁퐗 +9: end while +Algorithm 3 Archetypal Analysis++ Initialization +1: Input: Set of 푛 data points , number of archetypes 푘 +2: Output: Initial set of archetypes  +3: Sample index 푖 uniformly at random from [푛], i.e., +using 푝(푖) = 푛−1 +4: Append 퐱푖 to  +5: while || < 푘 do +6: +Sample 푖 using 푝(푖) ∝ min퐪∈conv() ‖퐱푖 − 퐪‖2 +2 +7: +Append 퐱푖 to  +8: end while +FurthestSum +Specifcally for archetypal analysis, Mørup and Hansen (2010, 2012) propose a modifcation of Fur- +thestFirst called FurthestSum, which sums over the distances of the already selected points, i.e., +푗next = arg max푖∈[푛] (∑ +퐪∈ +‖퐱푖 − 퐪‖2) . +To increase its performance, the frst point, which was chosen uniformly at random, is usually +discarded in the end and replaced by a new point chosen via the criteria outlined above. +3 Archetypal Analysis++ +The idea of the proposed Archetypal Analysis++ initialization procedure is very similar to the one +from 푘-means++. We begin with choosing the frst archetype uniformly at random. The second +archetype is chosen according to a distribution that assigns probabilities proportional to the distance +from the frst archetype, i.e., 푝(퐱) ∝ ‖퐱 − 퐳1‖2 +2. The remaining 푘 − 2 archetypes are chosen according to +a probability distribution where the probability of choosing a point 퐱 is proportional to the minimum +distance to the convex hull of the already chosen archetypes, i.e., 푝(퐱) ∝ min퐪∈conv({퐳1,…,퐳푘}) ‖퐱 − 퐪‖2 +2. +This procedure is depicted in Figure 2 and outlined in Algorithm 3. +With every additional point sampled, the convex hull of the initialized factors, i.e., conv(), +expands. This is because selecting a point outside the convex hull of , i.e., a point in {퐱 ∈  ∣ 퐱 ∉ +conv()}, is by defnition not contained in the convex hull of  and hence expands it. In contrast, a +point within the convex hull of  would not increase its volume but has zero probability of being +selected. However, selecting a new point can make a previously selected point redundant, i.e., it then +lies in the convex hull of all selected initial archetypes and does not help to increase it. However, this +is not a problem, as our empirical evaluation later shows. +Note that line 6 of Algorithm 3 solves the same optimization problem as line 5 in Algorithm 2. +Thus, parts of the archetypal analysis implementations can be re-used, simplifying the implementation +of AA++. Besides, AA++ has no hyperparameters, and line 6 can be trivially parallelized. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +4 of 20 + +MSE=39.40 +Uniform +k=1 +MSE=8.55 +k=2 +MSE=4.60 +k=3 +MSE=4.55 +k=4 +MSE=59.11 +FurthestSum +MSE=15.03 +MSE=0.09 +MSE=0.08 +MSE=39.40 +AA++ +MSE=12.54 +MSE=0.54 +MSE=0.31 +Figure 2. A comparison of Uniform, FurthestSum, and the proposed AA++ when consecutively initializing 푘 = 4 +archetypes. +3.1 +Theoretical Analysis +We frst show that by adding a new archetype 퐳 to the set of archetypes  in the while loop of AA++ +in Algorithm 3, the objective function decreases or remains unchanged. +Lemma 3.1. Adding a point 퐱 to the set of archetypes  either decreases the objective function or leaves +it unchanged. +Proof. Let  = conv() be the polytope corresponding to the convex hull of the archetypes. There are +only three scenarios. First, the added point 퐱 lies within . Then,  remains unchanged and so do the +projections of the points outside of . Hence, the value of the objective function remains unchanged. +Second, the added point 퐱 lies outside of  and 퐱 is the only point projected on that face of . Then, 퐱 +increases the volume of the convex hull but the projections and thus the value of the objective function +remain the same. Third, the added point 퐱 lies outside of  and there are other points that lie on the +same face of  as 퐱. Then, 퐱 increases the volume of the convex hull and thus decreases the projections +of the other points that lie on the same face. Hence, the value of the objective function decreases. +Note that using a random initialization samples archetypes that fall in all three categories men- +tioned in the proof. In comparison, the proposed approach only considers points from the latter two +categories when adding a new point. +We now show that sampling 푘 points according to our data-dependent sampling procedure results +in a larger (in terms of volume) convex hull of sampled points compared to a uniform sample in +expectation. Thus, the projection of points onto this convex hull (cost) is expected to be smaller. +Proposition 3.2. AA++ (Algorithm 3) grows the volume of the convex hull of  faster than a uniform +initialization in expectation. +The proof of Proposition 3.2 can be found in the appendix. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +5 of 20 + +Convex Hull of Z +True Distance +Approximated Distance +Initialized Archetypes Z +Figure 3. Approximation of the distance function. The true distance of the green point is depicted using a solid +line whereas the approximation is shown as a (larger) dashed line. The red point has no distance to the convex +hull, but the approximation yields a positive distance. +3.2 +Complexity Analysis +The proposed initialization strategy outlined in Algorithm 3 selects the frst archetype uniformly at +random. The remaining 푘 − 1 archetypes are chosen according to a probability proportional to the +squared distance between the candidate point and the convex hull of the already chosen archetypes. +To compute this projection, a quadratic program (QP) has to be solved. Thus, the complexity of +Algorithm 3 is (푛 ⋅ 푘 ⋅ QP). It depends not only on the size of the data set 푛 and the number of +archetypes 푘 to be initialized but also on the complexity of solving the quadratic program (QP), +which is often cubic in the number of variables 푘 (Goldfarb and Liu, 1990). +4 Approximating the Archetypal Analysis++ Initialization +The proposed initialization procedure AA++ has to solve 푛 ⋅ 푘 quadratic programs, which can be +time-consuming. Thus, we provide two strategies to approximate the initialization procedure. +4.1 +Approximating the Distance Computation +Mair and Brefeld (2019) show that the objective function of 푘-means upper bounds the objective of +archetypal analysis, which is due to the per-point projections, i.e., +min +퐪∈conv() ‖퐱 − 퐪‖2 +2 ≤ min +퐪∈ ‖퐱 − 퐪‖2 +2 +for a set of clusters/archetypes . Since the computationally most expensive operation in the proposed +Algorithm 3 is the projection onto the convex hull in line 6, it can be approximated by the distance +to the closest point within the already chosen archetypes. See Figure 3 for an example. Note that the +distance is then always over-estimated and even points within the convex hull of the already chosen +points might have a distance, although the projection should have a length of zero. This is depicted +in Figure 3 for the red point. Following this approach boils down to the 푘-means++ initialization +procedure. Hence, the new complexity is (푛 ⋅ 푘 ⋅ 푑), thus avoiding the cost of solving the QP. +4.2 +Approximating the Sampling Procedure +Another approach is to approximate the sampling procedure by avoiding to consider every of the 푛 +data points which is especially benefcial in large-scale scenarios. To initialize 푘-means++ in sublinear +time, Bachem et al. (2016) leverage a Markov Chain Monte Carlo (MCMC) sampling procedure. This +procedure is based on the Metropolis-Hastings algorithm (Hastings, 1970) with an independent and +uniform proposal distribution. We adapt this idea for AA++ as follows. The frst archetype is still +sampled uniformly at random. For every following archetype, a Markov chain of length 푚 ≪ 푛 is +constructed iteratively. We begin by sampling an initial point 퐱푖. Then, in every step of the chain, we +sample a candidate point 퐱푗 and compute the acceptance probability, which is given by +휋 = min (1, +min퐪∈conv() ‖퐱푗 − 퐪‖2 +2 +min퐪∈conv() ‖퐱푖 − 퐪‖2 +2 ) . +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +6 of 20 + +Algorithm 4 AA++ Monte Carlo Initialization +1: Input: Set of 푛 data points , number of archetypes 푘, chain length 푚 +2: Output: Initial set of archetypes  +3: Sample index 푖 uniformly at random from [푛], i.e., +using 푝(푖) = 푛−1 +4: Append 퐱푖 to  +5: while || < 푘 do +6: +Sample 푖 uniformly at random from [푛], i.e., +using 푝(푖) = 푛−1 +7: +Compute the distance to the convex hull, i.e., +푑2 +푖 = min퐪∈conv() ‖퐱푖 − 퐪‖2 +2 +8: +for 푙 = 2, 3, … , 푚 do +9: +Sample 푗 uniformly at random from [푛], i.e., +using 푝(푗) = 푛−1 +10: +Compute the distance to the convex hull, i.e., +푑2 +푗 = min퐪∈conv() ‖퐱푗 − 퐪‖2 +2 +11: +if 푑2 +푗 /푑2 +푖 > Unif(0, 1) then +12: +푖 = 푗 +13: +푑2 +푖 = 푑2 +푗 +14: +end if +15: +end for +16: +Append 퐱푖 to  +17: end while +With probability 휋, we update the current state from 퐱푖 to 퐱푗, otherwise we keep 퐱푖. After 푚 steps, we +add the current 퐱푖 as the next archetype to the set of initial archetypes . This approach is summarized +in Algorithm 4, and the complexity of this strategy is (푚 ⋅ 푘 ⋅ QP). +Bachem et al. (2016) also provide a theoretical result that bounds the error in terms of the total +variation distance of the approximate sampling distribution to the true sampling distribution. Here, +‖푝 − 푞‖TV denotes the total variation distance between two distributions 푝 and 푞 which is defned as +‖푝 − 푞‖TV = 1 +2 ∑ +퐱∈Ω +|푝(퐱) − 푞(퐱)|, +where Ω is a fnite sample space on which both distributions are defned on. The following bound +on the error shows that the longer the chain length 푚, the smaller the error 휖. +Theorem 4.1 (Bachem et al. (2016)). Let 푘 > 0 and 0 < 휖 < 1. Let 푝++ be the probability distribution +over  defned by using AA++ (Algorithm 3) and 푝MCMC be the probability distribution over  defned +by using AA++MC (Algorithm 4). Then, +‖푝MCMC − 푝++‖TV ≤ 휖 +for a chain length 푚 = (훾′ log 푘 +휖 ), where +훾′ = +max +⊂,||≤푘 max +퐱∈ 푛 +푑(퐱, )2 +∑퐱′∈ 푑(퐱′, )2 , +and 푑(퐱, )2 = min퐪∈conv() ‖퐱 − 퐪‖2 +2. +Note that 훾′ is a property of the data set. +5 Experiments +Data +We use the following six real-world data sets of varying sizes and dimensionalities. Additional eight +real-world data sets are considered in the appendix. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +7 of 20 + +A frst large data set is Covertype (Blackard and Dean, 1999), which consists of 푛 = 581, 012 +instances in 푑 = 54 dimensions. The Ijcnn1 data set has 푛 = 49, 990 points in 푑 = 22 dimensions +and was used in the IJCNN 2001 neural network competition.1 We employ the same pre-processing +as Chang and Lin (2001). KDD-Protein2 has 푛 = 145, 751 data points, each represented with 푑 = 74 +dimensions measuring the match between a protein and a native sequence. The data set Pose is a +subset of the Human3.6M data set (Catalin Ionescu, 2011; Ionescu et al., 2014). Pose was used in the +ECCV 2018 PoseTrack Challenge and deals with 3D human pose estimation.3 Each of the 푛 = 35, 832 +poses is represented as 3D coordinates of 16 joints. Thus, the problem is 48-dimensional. Another +larger data set we use is a subset of the Million Song Dataset (Bertin-Mahieux et al., 2011), which is +called Song. It has 푛 = 515, 345 data points in 푑 = 90 dimensions. Furthermore, we utilize the MNIST +(LeCun et al., 2010) data of size 푛 = 60, 000, which contains gray-scale images size 28 × 28 showing +handwritten digits, thus having ten classes. We use the subset of images showing the digit 4. This +subset has 푛 = 5, 842 data points in 푑 = 784 dimensions. +Data Pre-processing +We apply pre-processing to avoid numerical problems during learning. In total, we consider two +diferent approaches: (i) CenterAndMaxScale, in which the data set is frst centered and then the data +matrix is divided by the largest element, and (ii) Standardization which also centers the data set but +then divides every dimension by its standard deviation. Results on CenterAndMaxScale are presented +in the main paper, while results on Standardization are presented in the appendix. +Baselines +We compare our proposed initialization strategy AA++ against a Uniform subsample of all data points, +FurthestFirst (Gonzalez, 1985; Hochbaum and Shmoys, 1985), and FurthestSum (Mørup and Hansen, +2010, 2012). In addition, we evaluate two approximations, the frst of which is equivalent to 푘-means++ +and the second one, which is AA++MC. For the latter, we evaluate 1%, 5%, 10%, and 20% of the data set +size as chain lengths 푚. Note that by 푘-means++ we only refer to the initialization strategy. +Setup +For various numbers of archetypes 푘 depending on the data set, we initialize archetypal analysis +according to each of the baseline strategies and compute the Mean Squared Error (MSE), i.e., the +objective in Equation (2) normalized by 푛−1. In addition, we perform a fxed number of ten iterations +of archetypal analysis based on those initializations. Here, we use the vanilla version of archetypal +analysis according to Cutler and Breiman (1994). For the optimization problem within AA++, we +utilize the non-negative least squares (NNLS) method (Lawson and Hanson, 1995) and enforce the +summation constraint by adding another equation in the linear system. +We compute statistics over 50 seeds, except for the larger data sets (푛 > 500, 000) for which we only +use 15 seeds. We report on median performances and depict the 75% and 25% quantiles. Furthermore, +we track the time it takes to initialize the archetypes. The code is implemented in Python using numpy +(Harris et al., 2020).4 All experiments run on an Intel Xeon machine with 28 cores with 2.60 GHz and +256 GB of memory. +Results on MNIST +We frst analyze the behavior of the most commonly used baselines Uniform and FurthestSum, and +compare it to our proposed AA++ strategy. To do so, we use the MNIST data restricted to the digit 4 +and visualize the 푘 = 3 chosen initial archetypes 퐳1, 퐳2, 퐳3 per initialization method in Figure 4. We can +see that Uniform picks three very similar samples, whereas FurthestSum rather chooses outliers. In +comparison, AA++ yields a selection with more variation yet without outliers. This is also refected in +the MSE right after initialization. Here, AA++ yields the lowest MSE and thus explains the data best. +Figure 5 shows the MSE right after initialization as a small straight line on the left-hand-side and +then ten iterations of archetypal analysis for 푘 = 3 and 푘 = 9 archetypes. Note that the 푦-axis is in +log-scale and that the lines refect median values over 50 seeds. In both cases, FurthestSum yields the +1htps://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ +2htp://osmot.cs.cornell.edu/kddcup/datasets.html +3htp://vision.imar.ro/human3.6m/challenge_open.php +4We will release the code on github after the paper is accepted. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +8 of 20 + +Archetype 1 +Uniform +MSE=5.68e+01 +FurthestSum +MSE=8.10e+01 +AA++ +MSE=4.69e+01 +Archetype 2 +Archetype 3 +Figure 4. Three initial archetypes +on MNIST restricted to the digit 4 +chosen by two baselines and our +proposed AA++ method. +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Iterations of AA +4 × 101 +5 × 101 +6 × 101 +7 × 101 +MSE +MNIST Digit 4 k=3 +Uniform +FurthestFirst +FurthestSum +K-Means++ +AA++ +AA++MC 1% +AA++MC 5% +AA++MC 10% +AA++MC 20% +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Iterations of AA +3 × 101 +4 × 101 +5 × 101 +MSE +MNIST Digit 4 k=9 +Figure 5. Optimization trajectories for 푘 = 3, 9 on MNIST restricted to the +digit 4. The part from iterations 8 to 10 is enlarged. +10−3 +10−2 +MSE +Covertype k=15 +10−3 +10−2 +Covertype k=25 +10−4 +10−3 +Covertype k=50 +10−4 +10−3 +Covertype k=75 +10−4 +10−3 +Covertype k=100 +3 × 10−1 +4 × 10−1 +6 × 10−1 +MSE +Ijcnn1 k=15 +2 × 10−1 +3 × 10−1 +4 × 10−1 +6 × 10−1 +Ijcnn1 k=25 +10−1 +2 × 10−1 +3 × 10−1 +4 × 10−1 +Ijcnn1 k=50 +10−1 +2 × 10−1 +3 × 10−1 +4 × 10−1 +Ijcnn1 k=75 +10−1 +6 × 10−2 +2 × 10−1 +3 × 10−1 +Ijcnn1 k=100 +10−4 +10−3 +MSE +KDD-Protein k=15 +10−4 +10−3 +KDD-Protein k=25 +10−4 +10−3 +KDD-Protein k=50 +10−4 +10−3 +KDD-Protein k=75 +10−4 +10−3 +KDD-Protein k=100 +3 × 10−1 +4 × 10−1 +6 × 10−1 +MSE +Pose k=15 +3 × 10−1 +4 × 10−1 +6 × 10−1 +Pose k=25 +2 × 10−1 +3 × 10−1 +4 × 10−1 +Pose k=50 +2 × 10−1 +3 × 10−1 +4 × 10−1 +Pose k=75 +2 × 10−1 +3 × 10−1 +4 × 10−1 +Pose k=100 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−3 +10−2 +MSE +Song k=15 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−3 +3 × 10−4 +4 × 10−4 +6 × 10−4 +Song k=25 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−3 +2 × 10−4 +3 × 10−4 +4 × 10−4 +6 × 10−4 +Song k=50 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−3 +2 × 10−4 +3 × 10−4 +4 × 10−4 +6 × 10−4 +Song k=75 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−3 +Song k=100 +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +AA++MC 20% +AA++MC 10% +AA++MC 5% +AA++MC 1% +Figure 6. Results on Covertype, Ijcnn1, KDD-Protein, Pose, and Song. +worst initialization and also performs worse than most of its competitors during optimization, while +AA++ and Uniform perform much better. +Performance Results +We evaluate the initialization strategies on fve further data sets to get a better impression of the +performance. Figure 6 depicts the results on Covertype, Ijcnn1, KDD-Protein, Pose, and Song. Although +being the most commonly used initialization methods, Uniform (red line) and FurthestSum (blue line) +often yield the worst initializations which can be seen by the short straight line on the left-hand-side. +During the frst ten iterations of archetypal analysis, the error decreases. However, there is frequently +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +9 of 20 + +Uniform +FurthestFirst +FurthestSum +K-Means++ +AA++ +k=15 +k=25 +k=50 +k=75 +k=100 +0 +0 +0 +0 +0 +2 +3 +3 +1 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +10 +9 +10 +12 +13 +Best Initialization +Uniform +FurthestFirst +FurthestSum +K-Means++ +AA++ +0 +0 +0 +0 +0 +3 +2 +2 +1 +1 +1 +0 +1 +1 +1 +1 +1 +0 +0 +0 +8 +10 +10 +11 +11 +Best Overall +Uniform +FurthestFirst +FurthestSum +K-Means++ +AA++ +0 +0 +0 +0 +0 +3 +1 +0 +1 +1 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +9 +11 +12 +12 +12 +Median Initialization +Uniform +FurthestFirst +FurthestSum +K-Means++ +AA++ +0 +0 +0 +0 +0 +3 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +10 +11 +12 +13 +13 +Median Overall +1 +3 +5 +7 +9 +11 +13 +Figure 7. Aggregated statistics over 13 data sets (fve data sets from above excluding MNIST and eight data sets +from the appendix). Each table shows how often each initialization method yields the best result for various +settings of 푘 under diferent settings. Best refers to the lowest single seed and median refers to the median over +many seeds. We report on the performance after initialization and overall during the optimization. +15 +25 +50 +75 +100 +k +10−3 +10−1 +101 +103 +Initialization Time in Seconds +Covertype +Uniform +FurthestFirst +FurthestSum +K-Means++ +AA++ +AA++MC 20% +AA++MC 10% +AA++MC 5% +AA++MC 1% +15 +25 +50 +75 +100 +k +10−3 +10−1 +101 +Pose +Figure 8. The median time it takes to initialize archetypal analysis. +a signifcant performance gap between Uniform and FurthestSum when compared to AA++ (green +line), especially for Covertype, KDD-Protein, and Pose. The baseline 푘-means++ (orange line), which +can be seen as an approximation to AA++, often performs similar to AA++ and almost always better +than FurthestFirst (purple line), except on Covertype. As for the Monte Carlo approximations of AA++ +using 1%, 5%, 10%, and 20% of the data set size as chain lengths 푚, we can see that they approximate +AA++ sufciently well on these data sets. Overall, our proposed AA++ initialization performs best in +almost all cases. +This is confrmed in the aggregated statistics in Figure 7. Each table shows how often each +initialization method (approximations excluded) performs best on the evaluated data sets. In total, +we consider 13 data sets: all previously introduced data sets excluding MNIST and eight data sets +which are considered in the appendix. Thus, 13 is the highest and best number that can appear in +those tables. Within Figure 7, best refers to the lowest outcome among all seeds, and median refers to +the median over all seeds per method. Besides, initialization only considers the performance after +initialization, whereas overall refects the best performance across iterations of AA, which is typically +at the last iteration. Once again, AA++ clearly yields the best results in all scenarios, especially for +larger values of 푘. +Timing Results +Unfortunately, the increase in performance comes at a cost. Figure 8 shows the time needed for +initializing the archetypes on the Covertype and Pose data sets for various choices of 푘. As expected, +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +10 of 20 + +the fastest-performing method is Uniform since it uses no information about the data except the +number of data points. FurthestSum is slower than Uniform, and the proposed approach AA++ is +consistently the slowest. However, note that initializing 푘 points is still much cheaper than running +푘 AA iterations. Using the MCMC-based approximation reduces the initialization time of AA++ +drastically, e.g., on Covertype, AA++MC using 1% of the data points as the chain length 푚 takes +approximately as much time as FurthestSum. +Infuence of the Pre-processing Scheme +A surprising fnding is that FurthestSum is especially sensitive to the pre-processing scheme, as shown +in Figure 11 in the appendix. Standardization clearly degrades the performance of FurthestSum but +does not afect AA++ and its approximations. +6 Related Work +Other variants exist besides the classical archetypal analysis (Cutler and Breiman, 1994). Moving +archetypes (Cutler and Stone, 1997) defnes AA for moving targets. There are adaptations of archety- +pal analysis for missing data (Epifanio et al., 2020) and interval data (D’Esposito et al., 2012), and +probabilistic archetypal analysis (Seth and Eugster, 2016) rephrases the factorization problem in a +probabilistic way. More recently, approaches based on deep learning have been considered Keller +et al. (2019); van Dijk et al. (2019); Keller et al. (2021). For those, the initialization of archetypes is +irrelevant, as considered in this paper. Furthermore, archetypoid analysis (Vinué et al., 2015) restricts +the archetypes to be data points instead of convex combinations of data points. Hence, the idea is +similar to 푘-medoids (Kaufman and Rousseeuw, 1990) and intends to add more interpretability. Note +that AA++ can also be used as an initialization for archetypoid analysis. +Suleman (2017) stresses that an improper initialization of archetypal analysis is problematic and +that the FurthestSum method is prone to selecting redundant archetypes, especially when having +many archetypes. This is contradicted by our experiments, which show that FurthestSum’s problem is +rather the poor choice of boundary points. Nascimento and Madaleno (2019) consider an anomalous +pattern initialization algorithm for initializing archetypal analysis. However, their focus was on +inferring the number of archetypes 푘 and then using archetypal analysis for fuzzy clustering. +Less common initialization strategies for archetypal analysis include 푘-means, as used by Han +et al. (2022), and a coreset (Mair and Brefeld, 2019), as used by Black et al. (2022) and Chapman et al. +(2022). Note that the coreset was proposed as a way to condense the data set into a smaller set for a +more efcient training of archetypal analysis rather than as a way for initializing it. Hence, we don’t +compare to it. +For the related non-negative matrix factorization (NMF) (Lee and Seung, 1999) problem, several +initialization techniques based on randomization, other low-rank decompositions, clusterings, heuris- +tics, and even learned approaches (Sjölund and Bånkestad, 2022) are used. A summary of various NMF +initialization methods is provided by Esposito (2021). Among the considered baselines in this paper, all +are randomized. However, FurthestSum can be seen as a heuristic, and FurthestFirst and 푘-means++ +can be classifed as clusterings. The proposed AA++ approach is randomized just as Uniform, with +the diference that AA++ is data-dependent and Uniform is data-independent. +7 Conclusion +We introduced archetypal analysis++ (AA++), an initialization method for archetypal analysis inspired +by 푘-means++. The proposed method does not have any hyperparameters and is straightforward to +implement, as it re-uses already existing subroutines of any implementation of archetypal analysis. +Furthermore, we showed that the 푘-means++ initialization method can be seen as an approximation +to AA++, and we also proposed an MCMC approximation of AA++ to reduce the computational +efort. 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AA++ (Algorithm 3) grows the volume of the convex hull of  faster than a uniform +initialization in expectation. +Within the proof we use Chebyshev’s sum inequality which states that if 푝1 ≥ 푝2 ≥ ⋯ ≥ 푝푛 and +푟1 ≥ 푟2 ≥ ⋯ ≥ 푟푛, then 1 +푛 ∑푛 +푖=1 푝푖푟푖 ≥ ( 1 +푛 ∑푛 +푖=1 푝푖) ( 1 +푛 ∑푛 +푖=1 푟푖). +Proof of Proposition 3.2. Without loss of generality we focus on the two-dimensional setting since +similar arguments hold in higher dimensions. AA++ (Algorithm 3) as well as a uniform initialization +select the frst point uniformly at random. Assume both algorithms start with the same point 퐱 and +call it 퐳1. +After sampling a second point 퐳2, we obtain a line. We frst show that the expected length of the +line is larger when using AA++. Without loss of generality subtract 퐳1 from all 퐱푖 such that the line is +equivalent to the norm of the points. Reorder the points 퐱푖 according to their norms 푟푖 = ‖퐱푖‖2 and let +푝푖 = +‖퐱푖‖22 +∑푛 +푗=1 ‖퐱푗‖22 be the probability of choosing the point according to AA++. Furthermore, let 푢푖 = 푛−1 be +the probability of choosing the point according to the uniform initialization. Thus, 푝1 ≥ 푝2 ≥ ⋯ ≥ 푝푛 +and 푟1 ≥ 푟2 ≥ ⋯ ≥ 푟푛. Due to Chebyshev’s sum inequality, we obtain +1 +푛 +푛 +∑ +푖=1 +푝푖푟푖 ≥ ( +1 +푛 +푛 +∑ +푖=1 +푝푖) ( +1 +푛 +푛 +∑ +푖=1 +푟푖) ⟺ +푛 +∑ +푖=1 +푝푖푟푖 ≥ 1 +푛 +푛 +∑ +푖=1 +푟푖 ⟺ 피푝[‖퐱푖‖] ≥ 피푢[‖퐱푖‖], +by using ∑푛 +푖=1 푝푖 = 1 and by multiplying both sides by 푛. Hence, showing that the expected length +using AA++ is larger than by using a uniform sample. +Now consider the third chosen point which spans conv() to a triangle, given the point is not +chosen on the line between 퐳1 and 퐳2. The growth of volume of the convex hull is thus the area +of the triangle, which is 1 +2 base ⋅ height. We already know that the base length is larger for AA++ +than for a uniform initialization in expectation. However, we again assume they are the same. Let +푟푖 = min퐪∈conv() ‖퐱푖 − 퐪‖2 +2 be the projecting of each point 퐱푖 to the line, where  = {퐳1, 퐳2}, and 푝푖 +the normalized probability of choosing the point according to Algorithm 3, i.e., 푝푖 = +푟푖 +∑푛 +푗=1 푟푗 . Then, by +reordering the points as before and applying Chebyshev’s sum inequality again, we have that the +expected height is larger for AA++ than for uniform. Hence, the area of the triangle and thus the +volume of the convex hull grows faster for AA++. Every further point opens another triangle which +contributes in area to the overall volume of the convex hull. +B Pre-processing +In the main body of the paper, we pre-processed the data by frst centering the data set and then +dividing it by the maximum value. Another frequently used pre-processing scheme is standardization. +That is, per dimension, we subtract the mean and divide it by the standard deviation. Since both +are linear transformations, they do not change the membership of the points being on the border of +the convex hull (Ziegler, 2012). However, the former scheme maintains the shape of the data set in +terms of its convex hull, whereas the latter scheme changes it. This is depicted in Figure 9, where the +original data is shown in the middle, the pre-processing of the main body of the paper is shown on +the left-hand-side and a data standardization is shown on the right-hand-side. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +15 of 20 + +−2 +0 +2 +4 +−2 +−1 +0 +1 +2 +−2 +0 +2 +4 +−2 +−1 +0 +1 +2 +Convex Hull of X +Data Set X +−2 +0 +2 +4 +−2 +−1 +0 +1 +2 +Appendix B—Figure 9. Comparison between the two pre-processing approaches. Left: Center data and divide +by maximum value. Middle: Original data set. Right: Standardized data set. +Appendix C—Table 1. An overview of the data sets used in this paper. The upper part is considered in the +main body of the paper and the lower part is discussed in the appendix. +Data Set Name +Number of Data Points +Number of Dimensions +Main Paper +Covertype +581,012 +54 +Ijcnn1 +49,990 +22 +KDD-Protein +145,751 +74 +Pose +35,832 +48 +Song +515,345 +90 +MNIST 4 +5,842 +784 +Appendix +Airfoil +1,503 +5 +California Housing +20,640 +8 +Concrete +1,030 +8 +Banking1 +4,971 +7 +Banking2 +12,456 +8 +Banking3 +19,939 +11 +MiniBooNE +130,064 +50 +RNA +488,565 +8 +C Results on Additional Data Sets +We further evaluate the initialization methods on the following data sets. Airfoil (Brooks et al., 1989) +has 푛 = 1503 data points represented in 푑 = 5 dimensions. The California Housing (Pace and Barry, +1997) data set has 푛 = 20, 640 examples in 푑 = 8 dimensions. Concrete (Yeh, 1998) has 푛 = 1030 +instances in 푑 = 8 dimensions. The data sets Banking1, Banking2, and Banking3 (Dulá and López, +2012) have 4971, 12456, and 19939 points in 7, 8, and 11 dimensions, respectively. MiniBooNE (Dua and +Graf, 2017) consists of 푛 = 130, 064 data points in 푑 = 50 dimensions. The data set RNA (Uzilov et al., +2006) contains 푛 = 488, 565 RNA input sequence pairs with 푑 = 8 features. A summary of all used +data sets is provided in Table 1. Note that the appendix contains mainly data sets that are either small +in terms of dimensions or number of data points. For that reason, they are arguably less relevant for +archetypal analysis than the ones in the main body of the paper. +Figure 10 depicts the performance for the additional eight data sets. Note that we omit AA++MC 1% +for small data sets, i.e., if 푛 < 25, 000. Again, we can see that Uniform and the FurthestSum initialization +perform worse than the proposed approach and its approximation. AA++ is almost consistently best, +except on some rare occasions, such as for 푘 = 25 on Concrete. While the MCMC approximations of +AA++ performed very close to AA++ itself, we can see some more signifcant performance gaps on +these additional data sets. Especially on BankProblem1, most Monte Carlo versions fail to approximate +AA++ properly. For other data sets such as MiniBooNE and RNA, AA++MC using only 1% of the data +as a chain length is a sub-optimal choice. However, note that all approximations are still better than +the Uniform baseline. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +16 of 20 + +10−7 +10−5 +10−3 +MSE +Airfoil k=15 +10−8 +10−6 +10−4 +10−2 +Airfoil k=25 +10−10 +10−7 +10−4 +Airfoil k=50 +10−11 +10−8 +10−5 +Airfoil k=75 +10−13 +10−10 +10−7 +10−4 +Airfoil k=100 +10−6 +10−4 +MSE +BankProblem1 k=15 +10−9 +10−7 +10−5 +10−3 +BankProblem1 k=25 +10−14 +10−10 +10−6 +BankProblem1 k=50 +10−15 +10−11 +10−7 +10−3 +BankProblem1 k=75 +10−16 +10−12 +10−8 +10−4 +BankProblem1 k=100 +10−6 +10−5 +10−4 +10−3 +MSE +BankProblem2 k=15 +10−6 +10−5 +10−4 +10−3 +BankProblem2 k=25 +10−6 +10−4 +BankProblem2 k=50 +10−7 +10−5 +10−3 +BankProblem2 k=75 +10−8 +10−6 +10−4 +BankProblem2 k=100 +10−4 +10−3 +MSE +BankProblem3 k=15 +10−5 +10−4 +10−3 +BankProblem3 k=25 +10−5 +10−4 +10−3 +BankProblem3 k=50 +10−5 +10−4 +10−3 +BankProblem3 k=75 +10−6 +10−5 +10−4 +10−3 +BankProblem3 k=100 +10−8 +10−6 +10−4 +MSE +California k=15 +10−8 +10−6 +10−4 +California k=25 +10−9 +10−7 +10−5 +California k=50 +10−9 +10−7 +10−5 +California k=75 +10−9 +10−7 +10−5 +California k=100 +10−1 +MSE +Concrete k=15 +10−2 +Concrete k=25 +10−2 +Concrete k=50 +10−3 +10−2 +Concrete k=75 +10−3 +10−2 +Concrete k=100 +10−10 +10−8 +10−6 +10−4 +MSE +MiniBooNE k=15 +10−10 +10−8 +10−6 +10−4 +MiniBooNE k=25 +10−10 +10−8 +10−6 +10−4 +MiniBooNE k=50 +10−10 +10−7 +10−4 +MiniBooNE k=75 +10−10 +10−7 +10−4 +MiniBooNE k=100 +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Iterations of AA +10−5 +10−3 +MSE +RNA k=15 +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Iterations of AA +10−7 +10−5 +10−3 +RNA k=25 +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Iterations of AA +10−7 +10−5 +10−3 +RNA k=50 +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Iterations of AA +10−8 +10−6 +10−4 +RNA k=75 +init. 1 +2 +3 +4 +5 +6 +7 +8 +9 10 +Iterations of AA +10−8 +10−6 +10−4 +RNA k=100 +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +AA++MC 20% +AA++MC 10% +AA++MC 5% +AA++MC 1% +Appendix C—Figure 10. Results on Airfoil, California Housing, Concrete, Banking1, Banking2, Banking3, +MiniBooNE, and RNA using the CenterAndMaxScale pre-processing as in the main body of the paper. +D Results on Standardized Data Sets +We also conduct the same set of experiments on all data sets using standardization as pre-processing. +In Figure 11, we can see for the frst set of data sets that FurthestSum usually performs worst, often +by a large margin. In contrast, the most consistent behavior has the proposed AA++ method, which +is often the best. Besides, the proposed approximations of AA++ perform sufciently close to AA++ +itself. +The performance on the second set of data sets is depicted in Figure 12. On those, Uniform +is usually yielding the worst results. However, FurthestSum is also occasionally underperforming, +especially on the RNA data set. Once again, the most consistent behavior is achieved by AA++, which +is also often best. While the MCMC-based approximations are usually good, 푘-means++ still has a +gap compared to AA++. +The results in Figures 11 and 12 are summarized in Figure 13. As expected, AA++ wins on most +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +17 of 20 + +102 +MSE +Covertype k=15 +102 +Covertype k=25 +101 +102 +Covertype k=50 +101 +102 +Covertype k=75 +101 +102 +Covertype k=100 +101 +7 × 100 +8 × 100 +9 × 100 +MSE +Ijcnn1 k=15 +101 +4 × 100 +6 × 100 +Ijcnn1 k=25 +101 +3 × 100 +4 × 100 +6 × 100 +Ijcnn1 k=50 +101 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Ijcnn1 k=75 +101 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +Ijcnn1 k=100 +102 +4 × 101 +6 × 101 +MSE +KDD-Protein k=15 +102 +3 × 101 +4 × 101 +6 × 101 +KDD-Protein k=25 +3 × 101 +4 × 101 +6 × 101 +KDD-Protein k=50 +3 × 101 +4 × 101 +KDD-Protein k=75 +3 × 101 +4 × 101 +KDD-Protein k=100 +2 × 101 +3 × 101 +MSE +Pose k=15 +101 +2 × 101 +Pose k=25 +101 +6 × 100 +2 × 101 +Pose k=50 +101 +6 × 100 +2 × 101 +Pose k=75 +101 +6 × 100 +Pose k=100 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +102 +103 +MSE +Song k=15 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +102 +Song k=25 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +102 +Song k=50 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +102 +Song k=75 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +102 +Song k=100 +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +AA++MC 20% +AA++MC 10% +AA++MC 5% +AA++MC 1% +Appendix D—Figure 11. Results on Covertype, Ijcnn1, KDD-Protein, Pose, and Song using standardization as +pre-processing. +the data sets irrespective of the applied setting. However, the numbers are weaker than in Figure 7, +which summarizes results using the CenterAndMaxScale pre-processing. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +18 of 20 + +10−1 +100 +MSE +Airfoil k=15 +10−1 +Airfoil k=25 +10−2 +10−1 +Airfoil k=50 +10−3 +10−2 +10−1 +Airfoil k=75 +10−4 +10−3 +10−2 +10−1 +Airfoil k=100 +10−1 +100 +101 +102 +MSE +BankProblem1 k=15 +10−2 +10−1 +100 +101 +BankProblem1 k=25 +10−2 +10−1 +100 +BankProblem1 k=50 +10−3 +10−2 +10−1 +100 +BankProblem1 k=75 +10−3 +10−2 +10−1 +100 +BankProblem1 k=100 +10−1 +100 +MSE +BankProblem2 k=15 +10−1 +100 +BankProblem2 k=25 +10−2 +10−1 +100 +BankProblem2 k=50 +10−2 +10−1 +100 +BankProblem2 k=75 +10−3 +10−2 +10−1 +100 +BankProblem2 k=100 +100 +101 +MSE +BankProblem3 k=15 +100 +101 +BankProblem3 k=25 +10−1 +100 +BankProblem3 k=50 +10−1 +100 +BankProblem3 k=75 +10−1 +100 +BankProblem3 k=100 +10−1 +100 +MSE +California k=15 +10−1 +100 +California k=25 +10−1 +100 +California k=50 +10−1 +100 +California k=75 +10−2 +10−1 +100 +California k=100 +100 +6 × 10−1 +2 × 100 +3 × 100 +MSE +Concrete k=15 +100 +3 × 10−1 +4 × 10−1 +6 × 10−1 +2 × 100 +Concrete k=25 +10−1 +100 +Concrete k=50 +10−1 +Concrete k=75 +10−1 +Concrete k=100 +100 +101 +MSE +MiniBooNE k=15 +10−1 +100 +101 +MiniBooNE k=25 +10−1 +100 +101 +MiniBooNE k=50 +10−1 +100 +101 +MiniBooNE k=75 +10−1 +100 +101 +MiniBooNE k=100 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +100 +MSE +RNA k=15 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +100 +RNA k=25 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−1 +100 +RNA k=50 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−1 +100 +RNA k=75 +init. 1 2 3 4 5 6 7 8 9 10 +Iterations of AA +10−1 +100 +RNA k=100 +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +AA++MC 20% +AA++MC 10% +AA++MC 5% +AA++MC 1% +Appendix D—Figure 12. Results on Airfoil, California Housing, Concrete, Banking1, Banking2, Banking3, +MiniBooNE, and RNA using standardization as pre-processing. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +19 of 20 + +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +k=15 +k=25 +k=50 +k=75 +k=100 +0 +0 +0 +0 +0 +3 +3 +2 +1 +1 +0 +0 +0 +0 +0 +2 +0 +0 +0 +2 +8 +10 +11 +12 +10 +Best Initialization +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +0 +0 +0 +0 +0 +4 +3 +2 +1 +1 +1 +0 +1 +0 +0 +0 +0 +0 +0 +1 +8 +10 +10 +12 +11 +Best Overall +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +0 +0 +1 +0 +0 +0 +2 +1 +1 +1 +0 +0 +0 +0 +0 +4 +3 +1 +3 +1 +9 +8 +10 +9 +11 +Median Initialization +Uniform +FurthestFirst +FurthestSum +k-Means++ +AA++ +1 +0 +0 +0 +1 +3 +2 +1 +0 +1 +2 +1 +0 +0 +0 +1 +1 +1 +3 +2 +6 +9 +11 +10 +9 +Median Overall +1 +3 +5 +7 +9 +11 +13 +Appendix D—Figure 13. Aggregated statistics over 13 data sets (fve data sets from above excluding MNIST +and eight data sets from the appendix) using standardization as pre-processing. Each table shows how often each +initialization method yields the best result for various settings of 푘 under diferent settings. Best refers to the +lowest single seed and median refers to the median over many seeds. We report on the performance after +initialization and overall during the optimization. +Sebastian Mair et al. 2023 +| +Archetypal Analysis++: Rethinking the Initialization Strategy +arXiv +| +20 of 20 + diff --git a/pdFST4oBgHgl3EQfNjhG/content/tmp_files/load_file.txt b/pdFST4oBgHgl3EQfNjhG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..846ebccc2daba8320502da86dff0757c35389bf0 --- /dev/null +++ b/pdFST4oBgHgl3EQfNjhG/content/tmp_files/load_file.txt @@ -0,0 +1,1759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf,len=1758 +page_content='� For correspondence: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='mair@it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='se Funding: This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foun- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Code: The code is available on request and will be published on github once the paper is peer- reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Archetypal Analysis++: Rethinking the Initialization Strategy Sebastian Mair 1 � and Jens Sjölund 1 1Uppsala University, Sweden Abstract Archetypal analysis is a matrix factorization method with convexity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Due to local minima, a good initialization is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Frequently used initialization methods yield either sub- optimal starting points or are prone to get stuck in poor local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In this paper, we propose archetypal analysis++ (AA++), a probabilistic initialization strategy for archetypal analysis that sequentially samples points based on their infuence on the objective, similar to 푘-means++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In fact, we argue that 푘-means++ already approximates the proposed initialization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, we suggest to adapt an efcient Monte Carlo approximation of 푘-means++ to AA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In an extensive empirical evaluation of 13 real-world data sets of varying sizes and dimensionalities and considering two pre-processing strategies, we show that AA++ almost consistently outperforms all baselines, including the most frequently used ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 Introduction Archetypal analysis (AA) (Cutler and Breiman, 1994) is a matrix factorization method with convexity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The idea is to represent every data point as a convex combination of points, called archetypes, located on the boundary of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, archetypes can be seen as well-separated observations that summarize the most relevant extremes of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The convexity constraints also give archetypal analysis a natural interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Archetypal analysis has been applied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', for global gene expression (Thøgersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2013), bioinformatics (Hart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2015), apparel design (Vinué et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2015), chemical spaces of small organic molecules (Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2021), geophysical data (Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022), large-scale climate drivers (Hannachi and Trendaflov, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022), and population genetics (Gimbernat-Mayol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' To improve the computation of archetypal analysis, various optimization approaches (Bauckhage and Thurau, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mørup and Hansen, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Bauckhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Abrol and Sharma, 2020) and approximations (Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Damle and Sun, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mair and Brefeld, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022) have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, the earliest point of attack for obtaining a good solution is the initialization of the archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Surprisingly, this has barely been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In the original paper, Cutler and Breiman (1994) use a random initialization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', choosing random points from the data set, which was adopted by many others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', (Eugster and Leisch, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Seth and Eugster, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hinrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hannachi and Trendaflov, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Krohne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022) to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, Cutler and Breiman (1994) state that a careful initialization improves the convergence speed and that archetypes should not be initialized too close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The same idea serves as an argument for using the FurthestFirst approach (Gonzalez, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hochbaum and Shmoys, 1985), yielding a well-separated initialization used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', in 푘-means clustering (Lloyd, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Inspired by FurthestFirst, Mørup and Hansen (2010, 2012) propose a modifcation for archetypal analysis called FurthestSum, which focuses on boundary points rather than well-separated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Here, boundary points refer to points on the boundary of the convex hull of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Since then, FurthestSum has established itself as one of the default initialization strategies for archetypal analysis (Thøgersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hinrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mair and Brefeld, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Abrol and Sharma, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Gimbernat-Mayol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' | arXiv | February 1, 2023 | 1–20 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='13748v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='LG] 31 Jan 2023 Despite its popularity, FurthestSum has also been criticized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For example, Suleman (2017) states that FurthestSum is prone to selecting redundant archetypes, primarily when many archetypes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Redundant archetypes lie in the convex hull of the already selected archetypes and thus do not lower the overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In addition, Krohne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2019) and Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2022) report better results with a random initialization than with FurthestSum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' A possible explanation is that FurthestSum’s early focus on boundary points risks trapping it in poor local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Contributions In this paper, we motivate and propose archetypal analysis++ (AA++), an initialization strategy inspired by 푘-means++ (Arthur and Vassilvitskii, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Ostrovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, we argue that the 푘-means++ initialization can be seen as an approximation to the proposed AA++ strategy and that a Monte Carlo approximation to the 푘-means++ initialization can be adapted for AA++ for a more efcient initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Most importantly, we empirically demonstrate that our proposed AA++ initialization for archetypal analysis outperforms almost consistently all baselines on 13 real-world data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2 Preliminaries Before introducing archetypal analysis, we briefy revisit 푘-means clustering since we will build upon similar ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='1 푘-means Clustering Let \ue244 ⊂ ℝ푑 be a data set of 푛 points in 푑 dimensions and let \ue246 = {퐳1, … , 퐳푘} be a set of 푘 cluster centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Consider the 푘-means clustering problem with the following objective 휙\ue244(\ue246) = ∑ 퐱∈\ue244 푑(퐱, \ue246)2 = ∑ 퐱∈\ue244 min 퐪∈{퐳1,…,퐳푘} ‖퐱 − 퐪‖2 2, where 푑(퐱, \ue246)2 = min퐪∈\ue246 ‖퐱 − 퐪‖2 2 is the minimal squared distance from a data point 퐱 to the closest center in \ue246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Often, the cluster centers \ue246 of 푘-means are initialized using the 푘-means++ initialization procedure (Arthur and Vassilvitskii, 2007), which works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The frst center is chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, the remaining 푘 − 1 cluster centers are chosen according to a probability distribution where the probability of choosing a point 퐱 is proportional to the closest distance to the already chosen cluster centers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푝(퐱) ∝ 푑(퐱, \ue246)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The procedure is outlined in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 Archetypal Analysis Let \ue244 = {퐱1, … , 퐱푛}푛 푖=1 ⊂ ℝ푑 be a data set consisting of 푛 ∈ ℕ 푑-dimensional data points arranged as rows in the design matrix 퐗 ∈ ℝ푛×푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The idea in archetypal analysis (AA) (Cutler and Breiman, 1994) is to (approximately) represent every data point 퐱푖 as a convex combination of 푘 ∈ ℕ archetypes \ue246 = {퐳1, … , 퐳푘}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 퐱T 푖 ≈ 퐚T 푖 퐙, 퐚T 푖 ퟏ = 1, 퐚푖 ≥ 0, Algorithm 1 푘-means++ Initialization 1: Input: Set of 푛 data points \ue244, number of clusters 푘 2: Output: Initial set of clusters centers \ue246 3: Sample index 푖 uniformly at random from [푛], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', using 푝(푖) = 푛−1 4: Append 퐱푖 to \ue246 5: while |\ue22f| < 푘 do 6: Sample 푖 using 푝(푖) ∝ min퐳∈\ue246 ‖퐱푖 − 퐳‖2 2 7: Append 퐱푖 to \ue246 8: end while Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 2 of 20 Convex Hull of X Convex Hull of Z Data Points xi Initial Archetypes Zinit Learning Path of Z Learned Archetypes Z Error Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Archetypal analysis in two dimensions with 푘 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' where the matrix 퐙 ∈ ℝ푘×푑 contains the archetypes as rows and the vector 퐚푖 ∈ ℝ푘 defnes the weights for the 푖th data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Besides, ퟏ denotes the vector of ones and 퐚푖 ≥ 0 is meant element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The archetypes 퐳푗 (푗 = 1, … , 푘) themselves are also represented (exactly) as convex combinations, but of the data points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 퐳T 푗 = 퐛T 푗 퐗, 퐛T 푗 ퟏ = 1, 퐛푗 ≥ 0, where 퐛푗 ∈ ℝ푛 is the weight vector of the 푗th archetype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Let 퐀 ∈ ℝ푛×푘 and 퐁 ∈ ℝ푘×푛 be the matrices consisting of the weights 퐚푖 (푖 = 1, … , 푛) and 퐛푗 (푗 = 1, … , 푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, archetypal analysis yields an approximate factorization of the design matrix as follows 퐗 ≈ 퐀퐁퐗 = 퐀퐙, (1) where 퐙 = 퐁퐗 ∈ ℝ푘×푑 is the matrix of archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Due to the convexity constraints, the weight matrices 퐀 and 퐁 are row-stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The weight matrices 퐀 and 퐁 are typically determined by minimizing the approximation error in the Frobenius norm, resulting in the optimization problem minimize 퐀,퐁 ‖퐗 − 퐀퐁퐗‖2 퐹 subject to 퐀ퟏ = 1, 퐀 ≥ 0, 퐁ퟏ = 1, 퐁 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2) This can be equivalently expressed as minimizing the sum of projections of the data points on the archetype-induced convex hull as follows ‖퐗 − 퐀퐙‖2 퐹 = ∑ 퐱∈\ue244 min 퐪∈conv({퐳1,…,퐳푘}) ‖퐱 − 퐪‖2 2, (3) where conv(푆) refers to the convex hull of a set 푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' An example of archetypal analysis and projection errors is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The optimization problem is a generalized low-rank problem (Udell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2016), which is biconvex but not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Because it is biconvex, a local optimum can be found via an alternating optimization scheme such as the standard one outlined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, the quality of this local optimum is directly dependent on the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 Archetype Initializations Popular ways of initializing the archetypes 퐙 is by using uniformly at random chosen data points or the FurthestSum procedure (Mørup and Hansen, 2010, 2012), but we also introduce FurthestFirst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' FurthestFirst Originally proposed for the metric 푘-center problem, the FurthestFirst algorithm (Gonzalez, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hochbaum and Shmoys, 1985) selects the frst center/archetype at random and selects every consecu- tive point which is furthest away from the closest already selected center/archetype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Mathematically, the index of the next point is 푗next = arg max푖∈[푛] (min 퐪∈\ue246 ‖퐱푖 − 퐪‖훼 2 ) , where [푛] = {1, 2, … , 푛} and 훼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that in 푘-means++, the distances are squared, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 훼 = 2, and that points are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 3 of 20 Algorithm 2 Archetypal Analysis (Cutler and Breiman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1994) 1: Input: data matrix 퐗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' number of archetypes 푘 2: Output: weight matrices 퐀 and 퐁 3: 퐙 ← initialization of the archetypes 퐙 4: while not converged do 5: 퐚푖 = arg min 퐚T 푖 ퟏ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 퐚푖≥0 ‖퐙T퐚푖 − 퐱푖‖2 2 ∀푖 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' … ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 푛 6: 퐙 = (퐀T퐀)−1퐀T퐗 7: 퐛푗 = arg min 퐛T 푗 ퟏ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 퐛푗≥0 ‖퐗T퐛푗 − 퐳푗‖2 2 ∀푗 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' … ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 푘 8: 퐙 = 퐁퐗 9: end while Algorithm 3 Archetypal Analysis++ Initialization 1: Input: Set of 푛 data points \ue244,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' number of archetypes 푘 2: Output: Initial set of archetypes \ue246 3: Sample index 푖 uniformly at random from [푛],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', using 푝(푖) = 푛−1 4: Append 퐱푖 to \ue246 5: while |\ue246| < 푘 do 6: Sample 푖 using 푝(푖) ∝ min퐪∈conv(\ue246) ‖퐱푖 − 퐪‖2 2 7: Append 퐱푖 to \ue246 8: end while FurthestSum Specifcally for archetypal analysis, Mørup and Hansen (2010, 2012) propose a modifcation of Fur- thestFirst called FurthestSum, which sums over the distances of the already selected points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푗next = arg max푖∈[푛] (∑ 퐪∈\ue246 ‖퐱푖 − 퐪‖2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' To increase its performance, the frst point, which was chosen uniformly at random, is usually discarded in the end and replaced by a new point chosen via the criteria outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 3 Archetypal Analysis++ The idea of the proposed Archetypal Analysis++ initialization procedure is very similar to the one from 푘-means++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We begin with choosing the frst archetype uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The second archetype is chosen according to a distribution that assigns probabilities proportional to the distance from the frst archetype, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푝(퐱) ∝ ‖퐱 − 퐳1‖2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The remaining 푘 − 2 archetypes are chosen according to a probability distribution where the probability of choosing a point 퐱 is proportional to the minimum distance to the convex hull of the already chosen archetypes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푝(퐱) ∝ min퐪∈conv({퐳1,…,퐳푘}) ‖퐱 − 퐪‖2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This procedure is depicted in Figure 2 and outlined in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' With every additional point sampled, the convex hull of the initialized factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', conv(\ue246), expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This is because selecting a point outside the convex hull of \ue246, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', a point in {퐱 ∈ \ue244 ∣ 퐱 ∉ conv(\ue246)}, is by defnition not contained in the convex hull of \ue246 and hence expands it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In contrast, a point within the convex hull of \ue246 would not increase its volume but has zero probability of being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, selecting a new point can make a previously selected point redundant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', it then lies in the convex hull of all selected initial archetypes and does not help to increase it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, this is not a problem, as our empirical evaluation later shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that line 6 of Algorithm 3 solves the same optimization problem as line 5 in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, parts of the archetypal analysis implementations can be re-used, simplifying the implementation of AA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Besides, AA++ has no hyperparameters, and line 6 can be trivially parallelized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 4 of 20 MSE=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='40 Uniform k=1 MSE=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='55 k=2 MSE=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='60 k=3 MSE=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='55 k=4 MSE=59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='11 FurthestSum MSE=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='03 MSE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='09 MSE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='08 MSE=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='40 AA++ MSE=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='54 MSE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='54 MSE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='31 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' A comparison of Uniform, FurthestSum, and the proposed AA++ when consecutively initializing 푘 = 4 archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='1 Theoretical Analysis We frst show that by adding a new archetype 퐳 to the set of archetypes \ue246 in the while loop of AA++ in Algorithm 3, the objective function decreases or remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Adding a point 퐱 to the set of archetypes \ue246 either decreases the objective function or leaves it unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Let \ue23c = conv(\ue246) be the polytope corresponding to the convex hull of the archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' There are only three scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' First, the added point 퐱 lies within \ue23c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, \ue23c remains unchanged and so do the projections of the points outside of \ue23c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, the value of the objective function remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Second, the added point 퐱 lies outside of \ue23c and 퐱 is the only point projected on that face of \ue23c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, 퐱 increases the volume of the convex hull but the projections and thus the value of the objective function remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Third, the added point 퐱 lies outside of \ue23c and there are other points that lie on the same face of \ue23c as 퐱.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, 퐱 increases the volume of the convex hull and thus decreases the projections of the other points that lie on the same face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, the value of the objective function decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that using a random initialization samples archetypes that fall in all three categories men- tioned in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In comparison, the proposed approach only considers points from the latter two categories when adding a new point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We now show that sampling 푘 points according to our data-dependent sampling procedure results in a larger (in terms of volume) convex hull of sampled points compared to a uniform sample in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, the projection of points onto this convex hull (cost) is expected to be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' AA++ (Algorithm 3) grows the volume of the convex hull of \ue246 faster than a uniform initialization in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 5 of 20 Convex Hull of Z True Distance Approximated Distance Initialized Archetypes Z Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Approximation of the distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The true distance of the green point is depicted using a solid line whereas the approximation is shown as a (larger) dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The red point has no distance to the convex hull, but the approximation yields a positive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 Complexity Analysis The proposed initialization strategy outlined in Algorithm 3 selects the frst archetype uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The remaining 푘 − 1 archetypes are chosen according to a probability proportional to the squared distance between the candidate point and the convex hull of the already chosen archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' To compute this projection, a quadratic program (QP) has to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, the complexity of Algorithm 3 is \ue23b(푛 ⋅ 푘 ⋅ QP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' It depends not only on the size of the data set 푛 and the number of archetypes 푘 to be initialized but also on the complexity of solving the quadratic program \ue23b(QP), which is often cubic in the number of variables 푘 (Goldfarb and Liu, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 4 Approximating the Archetypal Analysis++ Initialization The proposed initialization procedure AA++ has to solve 푛 ⋅ 푘 quadratic programs, which can be time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, we provide two strategies to approximate the initialization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='1 Approximating the Distance Computation Mair and Brefeld (2019) show that the objective function of 푘-means upper bounds the objective of archetypal analysis, which is due to the per-point projections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', min 퐪∈conv(\ue246) ‖퐱 − 퐪‖2 2 ≤ min 퐪∈\ue246 ‖퐱 − 퐪‖2 2 for a set of clusters/archetypes \ue246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Since the computationally most expensive operation in the proposed Algorithm 3 is the projection onto the convex hull in line 6, it can be approximated by the distance to the closest point within the already chosen archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' See Figure 3 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that the distance is then always over-estimated and even points within the convex hull of the already chosen points might have a distance, although the projection should have a length of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This is depicted in Figure 3 for the red point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Following this approach boils down to the 푘-means++ initialization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, the new complexity is \ue23b(푛 ⋅ 푘 ⋅ 푑), thus avoiding the cost of solving the QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 Approximating the Sampling Procedure Another approach is to approximate the sampling procedure by avoiding to consider every of the 푛 data points which is especially benefcial in large-scale scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' To initialize 푘-means++ in sublinear time, Bachem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2016) leverage a Markov Chain Monte Carlo (MCMC) sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This procedure is based on the Metropolis-Hastings algorithm (Hastings, 1970) with an independent and uniform proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We adapt this idea for AA++ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The frst archetype is still sampled uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For every following archetype, a Markov chain of length 푚 ≪ 푛 is constructed iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We begin by sampling an initial point 퐱푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, in every step of the chain, we sample a candidate point 퐱푗 and compute the acceptance probability, which is given by 휋 = min (1, min퐪∈conv(\ue246) ‖퐱푗 − 퐪‖2 2 min퐪∈conv(\ue246) ‖퐱푖 − 퐪‖2 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 6 of 20 Algorithm 4 AA++ Monte Carlo Initialization 1: Input: Set of 푛 data points \ue244, number of archetypes 푘, chain length 푚 2: Output: Initial set of archetypes \ue246 3: Sample index 푖 uniformly at random from [푛], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', using 푝(푖) = 푛−1 4: Append 퐱푖 to \ue246 5: while |\ue246| < 푘 do 6: Sample 푖 uniformly at random from [푛], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', using 푝(푖) = 푛−1 7: Compute the distance to the convex hull, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푑2 푖 = min퐪∈conv(\ue246) ‖퐱푖 − 퐪‖2 2 8: for 푙 = 2, 3, … , 푚 do 9: Sample 푗 uniformly at random from [푛], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', using 푝(푗) = 푛−1 10: Compute the distance to the convex hull, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푑2 푗 = min퐪∈conv(\ue246) ‖퐱푗 − 퐪‖2 2 11: if 푑2 푗 /푑2 푖 > Unif(0, 1) then 12: 푖 = 푗 13: 푑2 푖 = 푑2 푗 14: end if 15: end for 16: Append 퐱푖 to \ue246 17: end while With probability 휋, we update the current state from 퐱푖 to 퐱푗, otherwise we keep 퐱푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' After 푚 steps, we add the current 퐱푖 as the next archetype to the set of initial archetypes \ue246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This approach is summarized in Algorithm 4, and the complexity of this strategy is \ue23b(푚 ⋅ 푘 ⋅ QP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Bachem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2016) also provide a theoretical result that bounds the error in terms of the total variation distance of the approximate sampling distribution to the true sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Here, ‖푝 − 푞‖TV denotes the total variation distance between two distributions 푝 and 푞 which is defned as ‖푝 − 푞‖TV = 1 2 ∑ 퐱∈Ω |푝(퐱) − 푞(퐱)|, where Ω is a fnite sample space on which both distributions are defned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The following bound on the error shows that the longer the chain length 푚, the smaller the error 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='1 (Bachem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Let 푘 > 0 and 0 < 휖 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Let 푝++ be the probability distribution over \ue246 defned by using AA++ (Algorithm 3) and 푝MCMC be the probability distribution over \ue246 defned by using AA++MC (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, ‖푝MCMC − 푝++‖TV ≤ 휖 for a chain length 푚 = \ue23b(훾′ log 푘 휖 ), where 훾′ = max \ue246⊂\ue244,|\ue246|≤푘 max 퐱∈\ue244 푛 푑(퐱, \ue246)2 ∑퐱′∈\ue244 푑(퐱′, \ue246)2 , and 푑(퐱, \ue246)2 = min퐪∈conv(\ue246) ‖퐱 − 퐪‖2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that 훾′ is a property of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 5 Experiments Data We use the following six real-world data sets of varying sizes and dimensionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Additional eight real-world data sets are considered in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 7 of 20 A frst large data set is Covertype (Blackard and Dean, 1999), which consists of 푛 = 581, 012 instances in 푑 = 54 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The Ijcnn1 data set has 푛 = 49, 990 points in 푑 = 22 dimensions and was used in the IJCNN 2001 neural network competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='1 We employ the same pre-processing as Chang and Lin (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' KDD-Protein2 has 푛 = 145, 751 data points, each represented with 푑 = 74 dimensions measuring the match between a protein and a native sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The data set Pose is a subset of the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6M data set (Catalin Ionescu, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Ionescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Pose was used in the ECCV 2018 PoseTrack Challenge and deals with 3D human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 Each of the 푛 = 35, 832 poses is represented as 3D coordinates of 16 joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, the problem is 48-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Another larger data set we use is a subset of the Million Song Dataset (Bertin-Mahieux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2011), which is called Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' It has 푛 = 515, 345 data points in 푑 = 90 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, we utilize the MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2010) data of size 푛 = 60, 000, which contains gray-scale images size 28 × 28 showing handwritten digits, thus having ten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We use the subset of images showing the digit 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This subset has 푛 = 5, 842 data points in 푑 = 784 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Data Pre-processing We apply pre-processing to avoid numerical problems during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In total, we consider two diferent approaches: (i) CenterAndMaxScale, in which the data set is frst centered and then the data matrix is divided by the largest element, and (ii) Standardization which also centers the data set but then divides every dimension by its standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Results on CenterAndMaxScale are presented in the main paper, while results on Standardization are presented in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Baselines We compare our proposed initialization strategy AA++ against a Uniform subsample of all data points, FurthestFirst (Gonzalez, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hochbaum and Shmoys, 1985), and FurthestSum (Mørup and Hansen, 2010, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In addition, we evaluate two approximations, the frst of which is equivalent to 푘-means++ and the second one, which is AA++MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For the latter, we evaluate 1%, 5%, 10%, and 20% of the data set size as chain lengths 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that by 푘-means++ we only refer to the initialization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Setup For various numbers of archetypes 푘 depending on the data set, we initialize archetypal analysis according to each of the baseline strategies and compute the Mean Squared Error (MSE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', the objective in Equation (2) normalized by 푛−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In addition, we perform a fxed number of ten iterations of archetypal analysis based on those initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Here, we use the vanilla version of archetypal analysis according to Cutler and Breiman (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For the optimization problem within AA++, we utilize the non-negative least squares (NNLS) method (Lawson and Hanson, 1995) and enforce the summation constraint by adding another equation in the linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We compute statistics over 50 seeds, except for the larger data sets (푛 > 500, 000) for which we only use 15 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We report on median performances and depict the 75% and 25% quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, we track the time it takes to initialize the archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The code is implemented in Python using numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 All experiments run on an Intel Xeon machine with 28 cores with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='60 GHz and 256 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Results on MNIST We frst analyze the behavior of the most commonly used baselines Uniform and FurthestSum, and compare it to our proposed AA++ strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' To do so, we use the MNIST data restricted to the digit 4 and visualize the 푘 = 3 chosen initial archetypes 퐳1, 퐳2, 퐳3 per initialization method in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We can see that Uniform picks three very similar samples, whereas FurthestSum rather chooses outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In comparison, AA++ yields a selection with more variation yet without outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This is also refected in the MSE right after initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Here, AA++ yields the lowest MSE and thus explains the data best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Figure 5 shows the MSE right after initialization as a small straight line on the left-hand-side and then ten iterations of archetypal analysis for 푘 = 3 and 푘 = 9 archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that the 푦-axis is in log-scale and that the lines refect median values over 50 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In both cases, FurthestSum yields the 1htps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='csie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='tw/~cjlin/libsvmtools/datasets/ 2htp://osmot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='edu/kddcup/datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='html 3htp://vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='imar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='ro/human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6m/challenge_open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='php 4We will release the code on github after the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 8 of 20 Archetype 1 Uniform MSE=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='68e+01 FurthestSum MSE=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10e+01 AA++ MSE=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='69e+01 Archetype 2 Archetype 3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Three initial archetypes on MNIST restricted to the digit 4 chosen by two baselines and our proposed AA++ method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 4 × 101 5 × 101 6 × 101 7 × 101 MSE MNIST Digit 4 k=3 Uniform FurthestFirst FurthestSum K-Means++ AA++ AA++MC 1% AA++MC 5% AA++MC 10% AA++MC 20% init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 3 × 101 4 × 101 5 × 101 MSE MNIST Digit 4 k=9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Optimization trajectories for 푘 = 3, 9 on MNIST restricted to the digit 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The part from iterations 8 to 10 is enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Covertype k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Covertype k=25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Covertype k=50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Covertype k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Covertype k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='KDD-Protein k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='KDD-Protein k=25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='KDD-Protein k=50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='KDD-Protein k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='KDD-Protein k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−3 10−2 MSE Song k=15 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−3 3 × 10−4 4 × 10−4 6 × 10−4 Song k=25 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−3 2 × 10−4 3 × 10−4 4 × 10−4 6 × 10−4 Song k=50 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−3 2 × 10−4 3 × 10−4 4 × 10−4 6 × 10−4 Song k=75 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−3 Song k=100 Uniform FurthestFirst FurthestSum k-Means++ AA++ AA++MC 20% AA++MC 10% AA++MC 5% AA++MC 1% Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Results on Covertype, Ijcnn1, KDD-Protein, Pose, and Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' worst initialization and also performs worse than most of its competitors during optimization, while AA++ and Uniform perform much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Performance Results We evaluate the initialization strategies on fve further data sets to get a better impression of the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Figure 6 depicts the results on Covertype, Ijcnn1, KDD-Protein, Pose, and Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Although being the most commonly used initialization methods, Uniform (red line) and FurthestSum (blue line) often yield the worst initializations which can be seen by the short straight line on the left-hand-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' During the frst ten iterations of archetypal analysis, the error decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, there is frequently Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Archetypal Analysis++: Rethinking the Initialization Strategy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='arXiv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='9 of 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Uniform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='FurthestFirst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='FurthestSum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='K-Means++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='AA++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='k=25 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Aggregated statistics over 13 data sets (fve data sets from above excluding MNIST and eight data sets from the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Each table shows how often each initialization method yields the best result for various settings of 푘 under diferent settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Best refers to the lowest single seed and median refers to the median over many seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We report on the performance after initialization and overall during the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 15 25 50 75 100 k 10−3 10−1 101 103 Initialization Time in Seconds Covertype Uniform FurthestFirst FurthestSum K-Means++ AA++ AA++MC 20% AA++MC 10% AA++MC 5% AA++MC 1% 15 25 50 75 100 k 10−3 10−1 101 Pose Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The median time it takes to initialize archetypal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' a signifcant performance gap between Uniform and FurthestSum when compared to AA++ (green line), especially for Covertype, KDD-Protein, and Pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The baseline 푘-means++ (orange line), which can be seen as an approximation to AA++, often performs similar to AA++ and almost always better than FurthestFirst (purple line), except on Covertype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' As for the Monte Carlo approximations of AA++ using 1%, 5%, 10%, and 20% of the data set size as chain lengths 푚, we can see that they approximate AA++ sufciently well on these data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Overall, our proposed AA++ initialization performs best in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This is confrmed in the aggregated statistics in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Each table shows how often each initialization method (approximations excluded) performs best on the evaluated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In total, we consider 13 data sets: all previously introduced data sets excluding MNIST and eight data sets which are considered in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, 13 is the highest and best number that can appear in those tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Within Figure 7, best refers to the lowest outcome among all seeds, and median refers to the median over all seeds per method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Besides, initialization only considers the performance after initialization, whereas overall refects the best performance across iterations of AA, which is typically at the last iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Once again, AA++ clearly yields the best results in all scenarios, especially for larger values of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Timing Results Unfortunately, the increase in performance comes at a cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Figure 8 shows the time needed for initializing the archetypes on the Covertype and Pose data sets for various choices of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' As expected, Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 10 of 20 the fastest-performing method is Uniform since it uses no information about the data except the number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' FurthestSum is slower than Uniform, and the proposed approach AA++ is consistently the slowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, note that initializing 푘 points is still much cheaper than running 푘 AA iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Using the MCMC-based approximation reduces the initialization time of AA++ drastically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', on Covertype, AA++MC using 1% of the data points as the chain length 푚 takes approximately as much time as FurthestSum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Infuence of the Pre-processing Scheme A surprising fnding is that FurthestSum is especially sensitive to the pre-processing scheme, as shown in Figure 11 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Standardization clearly degrades the performance of FurthestSum but does not afect AA++ and its approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 6 Related Work Other variants exist besides the classical archetypal analysis (Cutler and Breiman, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Moving archetypes (Cutler and Stone, 1997) defnes AA for moving targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' There are adaptations of archety- pal analysis for missing data (Epifanio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2020) and interval data (D’Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2012), and probabilistic archetypal analysis (Seth and Eugster, 2016) rephrases the factorization problem in a probabilistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' More recently, approaches based on deep learning have been considered Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' van Dijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For those, the initialization of archetypes is irrelevant, as considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, archetypoid analysis (Vinué et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2015) restricts the archetypes to be data points instead of convex combinations of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, the idea is similar to 푘-medoids (Kaufman and Rousseeuw, 1990) and intends to add more interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that AA++ can also be used as an initialization for archetypoid analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Suleman (2017) stresses that an improper initialization of archetypal analysis is problematic and that the FurthestSum method is prone to selecting redundant archetypes, especially when having many archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This is contradicted by our experiments, which show that FurthestSum’s problem is rather the poor choice of boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Nascimento and Madaleno (2019) consider an anomalous pattern initialization algorithm for initializing archetypal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, their focus was on inferring the number of archetypes 푘 and then using archetypal analysis for fuzzy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Less common initialization strategies for archetypal analysis include 푘-means, as used by Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2022), and a coreset (Mair and Brefeld, 2019), as used by Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2022) and Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that the coreset was proposed as a way to condense the data set into a smaller set for a more efcient training of archetypal analysis rather than as a way for initializing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, we don’t compare to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For the related non-negative matrix factorization (NMF) (Lee and Seung, 1999) problem, several initialization techniques based on randomization, other low-rank decompositions, clusterings, heuris- tics, and even learned approaches (Sjölund and Bånkestad, 2022) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' A summary of various NMF initialization methods is provided by Esposito (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Among the considered baselines in this paper, all are randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, FurthestSum can be seen as a heuristic, and FurthestFirst and 푘-means++ can be classifed as clusterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The proposed AA++ approach is randomized just as Uniform, with the diference that AA++ is data-dependent and Uniform is data-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 7 Conclusion We introduced archetypal analysis++ (AA++), an initialization method for archetypal analysis inspired by 푘-means++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The proposed method does not have any hyperparameters and is straightforward to implement, as it re-uses already existing subroutines of any implementation of archetypal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, we showed that the 푘-means++ initialization method can be seen as an approximation to AA++, and we also proposed an MCMC approximation of AA++ to reduce the computational efort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We empirically verifed, for two pre-processing schemes, that AA++ essentially consistently outperforms all baselines, including the most frequently used ones, namely Uniform and FurthestSum, on 13 real-world data sets of varying sizes and dimensionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' A Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' AA++ (Algorithm 3) grows the volume of the convex hull of \ue246 faster than a uniform initialization in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Within the proof we use Chebyshev’s sum inequality which states that if 푝1 ≥ 푝2 ≥ ⋯ ≥ 푝푛 and 푟1 ≥ 푟2 ≥ ⋯ ≥ 푟푛, then 1 푛 ∑푛 푖=1 푝푖푟푖 ≥ ( 1 푛 ∑푛 푖=1 푝푖) ( 1 푛 ∑푛 푖=1 푟푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Without loss of generality we focus on the two-dimensional setting since similar arguments hold in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' AA++ (Algorithm 3) as well as a uniform initialization select the frst point uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Assume both algorithms start with the same point 퐱 and call it 퐳1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' After sampling a second point 퐳2, we obtain a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We frst show that the expected length of the line is larger when using AA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Without loss of generality subtract 퐳1 from all 퐱푖 such that the line is equivalent to the norm of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Reorder the points 퐱푖 according to their norms 푟푖 = ‖퐱푖‖2 and let 푝푖 = ‖퐱푖‖22 ∑푛 푗=1 ‖퐱푗‖22 be the probability of choosing the point according to AA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Furthermore, let 푢푖 = 푛−1 be the probability of choosing the point according to the uniform initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Thus, 푝1 ≥ 푝2 ≥ ⋯ ≥ 푝푛 and 푟1 ≥ 푟2 ≥ ⋯ ≥ 푟푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Due to Chebyshev’s sum inequality, we obtain 1 푛 푛 ∑ 푖=1 푝푖푟푖 ≥ ( 1 푛 푛 ∑ 푖=1 푝푖) ( 1 푛 푛 ∑ 푖=1 푟푖) ⟺ 푛 ∑ 푖=1 푝푖푟푖 ≥ 1 푛 푛 ∑ 푖=1 푟푖 ⟺ 피푝[‖퐱푖‖] ≥ 피푢[‖퐱푖‖], by using ∑푛 푖=1 푝푖 = 1 and by multiplying both sides by 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, showing that the expected length using AA++ is larger than by using a uniform sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Now consider the third chosen point which spans conv(\ue246) to a triangle, given the point is not chosen on the line between 퐳1 and 퐳2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The growth of volume of the convex hull is thus the area of the triangle, which is 1 2 base ⋅ height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We already know that the base length is larger for AA++ than for a uniform initialization in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, we again assume they are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Let 푟푖 = min퐪∈conv(\ue246) ‖퐱푖 − 퐪‖2 2 be the projecting of each point 퐱푖 to the line, where \ue246 = {퐳1, 퐳2}, and 푝푖 the normalized probability of choosing the point according to Algorithm 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 푝푖 = 푟푖 ∑푛 푗=1 푟푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Then, by reordering the points as before and applying Chebyshev’s sum inequality again, we have that the expected height is larger for AA++ than for uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Hence, the area of the triangle and thus the volume of the convex hull grows faster for AA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Every further point opens another triangle which contributes in area to the overall volume of the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' B Pre-processing In the main body of the paper, we pre-processed the data by frst centering the data set and then dividing it by the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Another frequently used pre-processing scheme is standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' That is, per dimension, we subtract the mean and divide it by the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Since both are linear transformations, they do not change the membership of the points being on the border of the convex hull (Ziegler, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, the former scheme maintains the shape of the data set in terms of its convex hull, whereas the latter scheme changes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' This is depicted in Figure 9, where the original data is shown in the middle, the pre-processing of the main body of the paper is shown on the left-hand-side and a data standardization is shown on the right-hand-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 15 of 20 −2 0 2 4 −2 −1 0 1 2 −2 0 2 4 −2 −1 0 1 2 Convex Hull of X Data Set X −2 0 2 4 −2 −1 0 1 2 Appendix B—Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Comparison between the two pre-processing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Left: Center data and divide by maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Middle: Original data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Right: Standardized data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Appendix C—Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' An overview of the data sets used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The upper part is considered in the main body of the paper and the lower part is discussed in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Data Set Name Number of Data Points Number of Dimensions Main Paper Covertype 581,012 54 Ijcnn1 49,990 22 KDD-Protein 145,751 74 Pose 35,832 48 Song 515,345 90 MNIST 4 5,842 784 Appendix Airfoil 1,503 5 California Housing 20,640 8 Concrete 1,030 8 Banking1 4,971 7 Banking2 12,456 8 Banking3 19,939 11 MiniBooNE 130,064 50 RNA 488,565 8 C Results on Additional Data Sets We further evaluate the initialization methods on the following data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Airfoil (Brooks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 1989) has 푛 = 1503 data points represented in 푑 = 5 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The California Housing (Pace and Barry, 1997) data set has 푛 = 20, 640 examples in 푑 = 8 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Concrete (Yeh, 1998) has 푛 = 1030 instances in 푑 = 8 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The data sets Banking1, Banking2, and Banking3 (Dulá and López, 2012) have 4971, 12456, and 19939 points in 7, 8, and 11 dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' MiniBooNE (Dua and Graf, 2017) consists of 푛 = 130, 064 data points in 푑 = 50 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The data set RNA (Uzilov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', 2006) contains 푛 = 488, 565 RNA input sequence pairs with 푑 = 8 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' A summary of all used data sets is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that the appendix contains mainly data sets that are either small in terms of dimensions or number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For that reason, they are arguably less relevant for archetypal analysis than the ones in the main body of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Figure 10 depicts the performance for the additional eight data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Note that we omit AA++MC 1% for small data sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=', if 푛 < 25, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Again, we can see that Uniform and the FurthestSum initialization perform worse than the proposed approach and its approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' AA++ is almost consistently best, except on some rare occasions, such as for 푘 = 25 on Concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' While the MCMC approximations of AA++ performed very close to AA++ itself, we can see some more signifcant performance gaps on these additional data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Especially on BankProblem1, most Monte Carlo versions fail to approximate AA++ properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' For other data sets such as MiniBooNE and RNA, AA++MC using only 1% of the data as a chain length is a sub-optimal choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, note that all approximations are still better than the Uniform baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Archetypal Analysis++: Rethinking the Initialization Strategy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='arXiv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='16 of 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Airfoil k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Airfoil k=25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Airfoil k=50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Airfoil k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−13 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MiniBooNE k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−5 10−3 MSE RNA k=15 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−7 10−5 10−3 RNA k=25 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−7 10−5 10−3 RNA k=50 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−8 10−6 10−4 RNA k=75 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 10−8 10−6 10−4 RNA k=100 Uniform FurthestFirst FurthestSum k-Means++ AA++ AA++MC 20% AA++MC 10% AA++MC 5% AA++MC 1% Appendix C—Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Results on Airfoil, California Housing, Concrete, Banking1, Banking2, Banking3, MiniBooNE, and RNA using the CenterAndMaxScale pre-processing as in the main body of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' D Results on Standardized Data Sets We also conduct the same set of experiments on all data sets using standardization as pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In Figure 11, we can see for the frst set of data sets that FurthestSum usually performs worst, often by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' In contrast, the most consistent behavior has the proposed AA++ method, which is often the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Besides, the proposed approximations of AA++ perform sufciently close to AA++ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The performance on the second set of data sets is depicted in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' On those, Uniform is usually yielding the worst results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, FurthestSum is also occasionally underperforming, especially on the RNA data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Once again, the most consistent behavior is achieved by AA++, which is also often best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' While the MCMC-based approximations are usually good, 푘-means++ still has a gap compared to AA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' The results in Figures 11 and 12 are summarized in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' As expected, AA++ wins on most Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Archetypal Analysis++: Rethinking the Initialization Strategy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='arXiv ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='2 × 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='3 × 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Ijcnn1 k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='4 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='KDD-Protein k=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='102 ' metadata={'source': 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+page_content='2 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='6 × 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Pose k=100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 102 103 MSE Song k=15 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 102 Song k=25 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 102 Song k=50 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 102 Song k=75 init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Iterations of AA 102 Song k=100 Uniform FurthestFirst FurthestSum k-Means++ AA++ AA++MC 20% AA++MC 10% AA++MC 5% AA++MC 1% Appendix D—Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Results on Covertype, Ijcnn1, KDD-Protein, Pose, and Song using standardization as pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' the data sets irrespective of the applied setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' However, the numbers are weaker than in Figure 7, which summarizes results using the CenterAndMaxScale pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='| ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content='Appendix D—Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Aggregated statistics over 13 data sets (fve data sets from above excluding MNIST and eight data sets from the appendix) using standardization as pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Each table shows how often each initialization method yields the best result for various settings of 푘 under diferent settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Best refers to the lowest single seed and median refers to the median over many seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' We report on the performance after initialization and overall during the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' Sebastian Mair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} +page_content=' 2023 | Archetypal Analysis++: Rethinking the Initialization Strategy arXiv | 20 of 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFST4oBgHgl3EQfNjhG/content/2301.13748v1.pdf'} diff --git a/ptFRT4oBgHgl3EQfdjek/content/tmp_files/2301.13568v1.pdf.txt b/ptFRT4oBgHgl3EQfdjek/content/tmp_files/2301.13568v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d55d9277695525b7a867800a3e3b363673371f5 --- /dev/null +++ b/ptFRT4oBgHgl3EQfdjek/content/tmp_files/2301.13568v1.pdf.txt @@ -0,0 +1,5595 @@ +Astronomy & Astrophysics manuscript no. 45342corr +©ESO 2023 +February 1, 2023 +Stable accretion and episodic outflows in the young transition disk +system GM Aurigae. +A semester-long optical and near-infrared spectrophotometric monitoring +campaign⋆,⋆⋆ +J. Bouvier1, A. Sousa1, K. Pouilly2, J.M. Almenara1, J.-F. Donati3, S. Alencar4, A. Frasca5, K. Grankin6, A. Carmona1, +G. Pantolmos1, B. Zaire4, X. Bonfils1, A. Bayo7, 8, L.M. Rebull9, J. Alonso-Santiago5, J. F. Gameiro10, 11, N. J. Cook12, +E. Artigau12, and the Spirou Legacy Survey (SLS) consortium +1 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +2 Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Sweden +3 Univ. de Toulouse, CNRS, IRAP, 14 avenue Belin, 31400 Toulouse, France +4 Departamento de Fisica – ICEx – UFMG, Av. Antonio Carlos 6627, 30270-901 Belo Horizonte, MG, Brazil +5 INAF – Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy +6 Crimean Astrophysical Observatory, Nauchny, 298409, Republic of Crimea +7 Instituto de Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Chile +8 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany +9 Infrared Science Archive (IRSA), IPAC, 1200 E. California Blvd., California Institute of Technology, Pasadena, CA 91125, USA +10 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal +11 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, rua do Campo Alegre 687, 4169-007 Porto. +Portugal +12 Research on Exoplanets, Université de Montréal, Département de Physique, Montréal, QC H3C 3J7, Canada +Received 2 November 2022; accepted 5 January 2023 +ABSTRACT +Context. Young stellar systems actively accrete from their circumstellar disk and simultaneously launch outflows. The physical link +between accretion and ejection processes remains to be fully understood. +Aims. We investigate the structure and dynamics of magnetospheric accretion and associated outflows on a scale smaller than 0.1 au +around the young transitional disk system GM Aur. +Methods. We devised a coordinated observing campaign to monitor the variability of the system on timescales ranging from days to +months, including partly simultaneous high-resolution optical and near-infrared spectroscopy, multiwavelength photometry, and low- +resolution near-infrared spectroscopy, over a total duration of six months, covering 30 rotational cycles. We analyzed the photometric +and line profile variability to characterize the accretion and ejection processes. +Results. The optical and near-infrared light curves indicate that the luminosity of the system is modulated by surface spots at the +stellar rotation period of 6.04 ± 0.15 days. Part of the Balmer, Paschen, and Brackett hydrogen line profiles as well as the HeI 5876 +Å and HeI 10830 Å line profiles are modulated on the same period. The Paβ line flux correlates with the photometric excess in the +u’ band, which suggests that most of the line emission originates from the accretion process. High-velocity redshifted absorptions +reaching below the continuum periodically appear in the near-infrared line profiles at the rotational phase in which the veiling and +line fluxes are the largest. These are signatures of a stable accretion funnel flow and associated accretion shock at the stellar surface. +This large-scale magnetospheric accretion structure appears fairly stable over at least 15 and possibly up to 30 rotational periods. In +contrast, outflow signatures randomly appear as blueshifted absorption components in the Balmer and HeI 10830 Å line profiles. They +are not rotationally modulated and disappear on a timescale of a few days. The coexistence of a stable, large-scale accretion pattern +and episodic outflows supports magnetospheric ejections as the main process occurring at the star-disk interface. +Conclusions. Long-term monitoring of the variability of the GM Aur transitional disk system provides clues to the accretion and +ejection structure and dynamics close to the star. Stable magnetospheric accretion and episodic outflows appear to be physically +linked on a scale of a few stellar radii in this system. +Key words. Stars: pre-main sequence – Stars: variables: T Tauri – Stars: magnetic field – Protoplanetary disks – Stars: individual: +GM Aurigae +⋆ Based on observations obtained at the Canada-France-Hawaii Tele- +scope (CFHT), at the Observatoire de Haute-Provence (OHP), at the Eu- +ropean Organisation for Astronomical Research in the Southern Hemi- +sphere (ESO), and at the Las Cumbres Observatory global telescope +network (LCOGT). +⋆⋆ Tables +containing +the +u’g’r’i’ +LCOGT +photometric +mea- +surements +are +only +available +in +electronic +form +at +the +CDS +via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via +https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/ +Article number, page 1 of 30 +arXiv:2301.13568v1 [astro-ph.SR] 31 Jan 2023 + +A&A proofs: manuscript no. 45342corr +1. Introduction +Accretion and ejection processes are at the origin of most of the +peculiar properties of young stellar systems. The structure and +dynamics of the accretion flows within the disk and from the +inner disk to the star, as well as the properties of the multiple +outflows arising from the disk, from the star-disk interface, and +from the stellar surface, remain to be fully deciphered, however. +Low-mass pre-main-sequence stars, the so-called T Tauri stars +(TTS), accrete from their circumstellar disks for a few million +years, while contemporaneous planet formation impacts the disk +structure and evolution. In the inner regions of the system, the +disk is disrupted by the strong stellar magnetosphere that chan- +nels the accretion flow toward the star along magnetic field lines +(see, e.g., the review by Hartmann et al. 2016). Thus, accretion +funnel flows develop that connect the inner disk to the stellar +surface, where the material is accreted at nearly free-fall veloc- +ity and is eventually halted in a strong accretion shock. Simulta- +neously, outflows are produced at the star-disk interface close to +the magnetospheric truncation radius through the inflation and +reconnection of magnetic field lines that are twisted by differen- +tial rotation (e.g., Zanni & Ferreira 2013). Ultimately, the release +of gravitational energy delivered by the accretion process may +trigger accretion-powered stellar winds (Matt & Pudritz 2005). +The torque balance between accretion and ejection processes is a +central issue for understanding the spin evolution of young stars +(e.g., Pantolmos et al. 2020; Ireland et al. 2021) +The star-disk interaction takes place on a distance of a few +stellar radii (e.g., Bessolaz et al. 2008), that is, on a scale of about +0.1 au or smaller. MHD models developed by several groups +predict the structure and dynamics of the magnetospheric ac- +cretion region and associated outflows (see, e.g., the review by +Romanova & Owocki 2015). Observationally, two main direc- +tions have been explored so far to investigate the properties of +this region. On one hand, monitoring the spectroscopic and pho- +tometric variability of the system over a few rotational periods, +that is, typically over a few weeks, allows identifying the signa- +ture of funnel flows, hot spots, and outflows, and relating them +to the strength and topology of the surface magnetic field that +is measured from spectropolarimetry (e.g., Pouilly et al. 2020, +2021; Bouvier et al. 2020a; Donati et al. 2019, 2020a; Alencar +et al. 2018). On the other hand, a direct approach attempts to +spatially resolve the star-disk interaction region on a scale of a +few milliarcsecond on the sky, using long-baseline near-infrared +interferometry (e.g., Eisner et al. 2014; Gravity Collaboration +et al. 2020; Bouvier et al. 2020b). Both approaches have been +successful in mapping the inner region of accreting systems and +have provided strong support to the magnetospheric accretion +scenario and its MHD modeling. Following previous studies, we +report here the results of a new observing campaign devoted to +the young stellar system GM Aur. +GM Aur (RA = 04h55, Dec = +30◦21, V = 12.1 mag) is a +solar-type pre-main-sequence star located in the Taurus-Auriga +molecular cloud at a distance of 157.9 ± 1.2 pc (Gaia Collabo- +ration et al. 2021). This classical T Tauri star (cTTS) has a spec- +tral type K6 (Herczeg & Hillenbrand 2014) and is surrounded +by a circumstellar disk from which it actively accretes material +at a rate of 0.6-2.0·10−8 M⊙yr−1 (Robinson & Espaillat 2019). +Based on its spectral energy distribution, which exhibits a small +near-infrared excess compared to a significant mid-infrared one, +the system has long been suspected to be in a transitional stage, +that is, that it is surrounded by a disk whose inner regions are +relatively devoid of matter (Strom et al. 1989). High-resolution +ALMA images of the circumstellar disk indeed reveal that it is +highly structured. The large-scale disk, inclined at ∼53◦ on the +line of sight, features a large inner dust cavity extending over +∼35-40 au and a succession of annular gaps and dusty rings on a +wider scale up to 200 au (Macías et al. 2018; Huang et al. 2020). +Much closer to the central star, long-baseline VLTI/GRAVITY +interferometric observations unveil a compact dusty disk, whose +inner edge was recently reported to be located at rin = 0.013+0.015 +−0.008 +au from the central star (Bohn et al. 2022) and that extends over +at least a few 0.1 au (Akeson et al. 2005) and possibly up to 6.6 +au (Varga et al. 2018; Woitke et al. 2019). The gaseous com- +ponent of the inner disk has been detected from CO 4.7 micron +emission down to 0.5 ± 0.2 au (Salyk et al. 2009). The inclina- +tion and position angle of the major axis of the inner dusty disk +(i=68◦+16 +−28, PA=37◦+31 +−22) are found to be consistent with those of +the outer disk, which suggests that the inner and outer disks are +aligned (Bohn et al. 2022). +In an attempt to decipher the physical processes at work +at the heart of the system, GM Aur has been the subject of +several multiwavelength monitoring campaigns. The long-term +light curve presented by Grankin et al. (2007) over the period +1986-1995 exhibits relatively low-level variability, with a V- +band magnitude ranging from 11.74 to 12.35 mag. Photometric +variations are modulated by surface spots at the stellar rotation +period of 6.0-6.1 days (Percy et al. 2010; Artemenko et al. 2012). +Ingleby et al. (2015) reported variability over the full wavelength +range from the far-UV to the near-infrared, which they attributed +in part to an accretion rate that varies by about a factor of 2 to 3 +on a timescale of months, and for another part to dust inhomo- +geneities that are located in the inner disk close to the truncation +radius. Variations in the mass accretion rate of similar ampli- +tude have also been reported on a shorter timescale of about a +week by Robinson & Espaillat (2019), and a connection between +mass loss and mass accretion has been further suggested by Es- +paillat et al. (2019). McGinnis et al. (2020) presented the results +of a high-resolution optical spectroscopic monitoring campaign +performed on a timescale of a week that illustrated the variabil- +ity of the Hα, Hβ, and HeI emission line profiles of the system. +From the measured radial velocity variations of the HeI 5876 Å +line profile, whose narrow component traces the accretion shock, +they deduced that GM Aur accretes material from its circumstel- +lar disk through an inclined magnetosphere, whose axis is tilted +by about 13◦ relative to the stellar rotational axis. GM Aur in- +deed harbors a strong surface magnetic field, with a mean value +of 2.2 kG (Johns-Krull 2007; Symington et al. 2005). Finally, +from a recent multiwavelength X-UV-optical campaign, Espail- +lat et al. (2021) reported evidence for a transverse density strat- +ification within the accretion shock at the base of the magnetic +funnel flow. +We report here the results of a new coordinated monitor- +ing campaign on GM Aur that combines high-resolution optical +spectroscopy and near-infrared spectropolarimetry, multiwave- +length optical and near-infrared photometry, and long-term low- +resolution near-infrared spectroscopy. Part of the observations +have been obtained simultaneously over a timescale of a few +weeks, while the total duration of the campaign amounted to six +months. The goal of the campaign was to investigate the phys- +ical processes that cause variability in GM Aur on a scale of a +few stellar radii, and in particular, to constrain the structure and +dynamics of the magnetospheric accretion flow from the inner +disk to the star. We devised a long-term campaign in order to +be able to probe various timescales, from days to months, and +obtain a sufficiently long temporal baseline to investigate the re- +lation between accretion and ejection processes on small spatial +scales from the stellar surface to the inner disk regions. +Article number, page 2 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +The campaign whose results are reported here took place in +the framework of a larger project led by the ODYSSEUS team1 +(see Espaillat et al. 2022), which uses the Hubble UV Legacy +Library of Young Stars as Essential Standards program (ULL- +YSES2, Roman-Duval et al. 2020), on HST Director’s Discre- +tionary time, to monitor a sample of T Tauri stars in the UV +domain, which includes GM Aur. Additional follow-up observa- +tions were acquired for this project at ESO in the framework of +the PENELLOPE Large Program3 (Manara et al. 2021). +In Section 2 we describe the observational techniques we im- +plemented to perform the campaign. In Section 3 we derive the +properties of the system and analyze its photometric and spectro- +scopic variability over timescales from days to months, includ- +ing veiling measurements and emission line profiles. We infer +the global structure of the magnetospheric accretion flow from +the observed variability and characterize associated outflows. In +Section 4 we discuss the dynamics of the accretion and ejec- +tion structure and show that short-lived episodic outflows coex- +ist with a stable magnetospheric accretion pattern. In Section 5 +we conclude on the ability of multiwavelength, multi-technique +coordinated observational campaigns to unveil the physical pro- +cesses at work in young stellar systems at the sub-au scale. +2. Observations +In this section, we describe the acquisition and data-reduction +processes of photometric, spectroscopic, and spectropolarimet- +ric datasets obtained during the large-scale campaign we per- +formed on the cTTS GM Aur from September 6, 2021, to March +8, 2022, using CFHT/SPIRou, OHP/SOPHIE, ESO/ExTrA, +LCOGT, and ESO/REM. A summary plot of the GM Aur ob- +serving campaign reported here is provided in Figure 1. +2.1. LCOGT: Multiwavelength optical photometry +GM Aur was observed at Las Cumbres Observatory Global Net- +work (LCOGT, Brown et al. 2013) from September 6 to De- +cember 30, 2021. We acquired 850 images in the Sloan u’g’r’i’ +filters over two runs with a sub-day cadence (LCO2021B-001, +PI L. Rebull; CLN2021B-003, PI A. Bayo). The u’ images were +obtained with the Sinistro 1m telescopes of the network using +an exposure time of 180 seconds and reading the 2Kx2K cen- +tral window of the detector with a 2x2 binning, resulting in a +13x13 arcmin field of view on the sky. The g’r’i’ images were +obtained with the 0.4m SBIG telescopes, offering a field of view +of 29.2x19.5 arcmin, with exposure times of 60, 20, and 20 sec- +onds, respectively. We retrieved the BANZAI-reduced images +from the LCOGT archive service and the noncalibrated photo- +metric catalogs provided in the image headers for all detected +stars in the field. +In order to compute differential photometry, we considered +two stars, HD 282625 and HD 282626, both located within 3 ar- +cmin of GM Aur. The first star was used as a reference star to +calibrate the differential light curve, and the second was used as +a check star to assess that these are nonvariable sources. These +field stars have spectral types F2 and F5, respectively, and are +only slightly brighter than GM Aur. We confirmed that the two +stars are nonvariable from their differential light curves, and we +deduced a mean rms photometric error of 0.025 mag in the +u’g’r’ filters and 0.033 mag in the i’ filter. We proceeded to +1 https://sites.bu.edu/odysseus/ +2 https://ullyses.stsci.edu/ +3 https://sites.google.com/view/cfmanara/penellope +compute the differential light curve between GM Aur and HD +282625 in all four filters. We adopted the mean magnitude of HD +282625 listed in the APASS and Pan-STARRS surveys, namely +g’=11.331 mag, r’=10.916 mag, and i’=10.756 mag to calibrate +the GM Aur light curve in the g’r’i’ filters to within an accuracy +of 0.02 mag. +We were not able to find an estimate of the u’-band magni- +tude for the comparison stars in the literature. Instead, we as- +sumed the intrinsic (u’-g’) colors of an F2 star (Covey et al. +2007; Kraus & Hillenbrand 2007) for HD 282625, to which we +applied interstellar reddening. From the observed versus intrinsic +color indices of the HD 282625 (g’-r’) and (r’-i’) bands, we de- +rived AV = 0.8±0.1 mag, using the R = 3.1 reddening law from +Fiorucci & Munari (2003). This procedure yielded an estimate +of the reddened (u’-g’) color of the comparison star from which +we derived its u’-band magnitude. The photometric calibration +of the GM Aur u’-band light curve is thus relatively indirect and +probably not accurate to better than 0.1 mag4. +2.2. REM: Optical and near infrared photometry +Observations were performed with the 60 cm robotic REM tele- +scope located at the ESO La Silla Observatory (Chile), on 15 +nights from JD 2,459,497 to JD 2,459,520 (October 9 to Novem- +ber 1, 2021). By means of a dichroic, REM simultaneously +feeds two cameras at the two Nasmyth focal stations, one cam- +era for the near-infrared (REMIR), and the other for the opti- +cal (ROSS2). The cameras have nearly the same field of view +of about 10′ × 10′ and use wide-band filters (J, H, and K′ for +REMIR and Sloan/SDSS g′, r′, i′, and z′ for ROSS2). +Exposure times were 60 s for ROSS2, which simultaneously +acquires images in the four Sloan bands, and five ditherings of +3 s each were adopted for each filter of REMIR. For the ROSS2 +camera, we generated master flats using the twilight flat-fields +taken during the observing run, which are available in the REM +archive. The latter were used to correct for pixel-to-pixel sensi- +tivity variations, as well as for the vignetting and illumination of +the field of view. After subtracting the dark-frame, each scientific +image was divided by the proper master-flat, depending on the +filter. The prereduction of the REMIR images is automatically +done by the AQuA pipeline (Testa et al. 2004), and the coadded +and sky-subtracted frames, resulting from five individual dither- +ings, are made available to the observer. +The adopted comparison stars are reported in Table A.1 +along with their griz (Tonry et al. 2018) and JHK′ (Cutri et al. +2003) magnitudes. Aperture photometry for all the stars listed +in Table A.1 was performed with DAOPHOT by using the IDL5 +routine Aper. For each frame and filter, we used the instrumental +magnitudes of the stars listed in Table A.1 to generate an artifi- +cial comparison, weighting them with the flux corresponding to +their standard magnitude in a way similar to the ensemble pho- +tometry. This procedure also allowed us to evaluate a standard +error based on the differences between the magnitudes calculated +with different comparison stars. +The optical photometry gathered at REM is listed in Ta- +ble A.2 of Appendix A. In the common g’r’i’ bands, we found it +to agree well with that obtained at LCOGT with a tighter sam- +pling rate, and therefore, we did not use it further in the analysis +below. The individual JHK’ measurements are listed in Table 1. +4 The table of photometric measurements is available electronically at +CDS, Strasbourg. +5 Interactive Data Language (IDL) is a registered trademark of Harris +Corporation. +Article number, page 3 of 30 + +A&A proofs: manuscript no. 45342corr +REM +LCOGT +OHP/SOPHIE +CFHT/SPIRou +ESO/ExTrA +Fig. 1. Temporal sampling of the GM Aur campaign from September 6, 2021, to March 8, 2022. Bottom: g’-band light curve from LCOGT (black +dots) and from REM (magenta crosses). The REM g’-band magnitudes are offset in this figure by +0.1 mag to match the LCOGT measurements. +The mean photometric error on both the LCOGT and REM g’-band measurements is 0.025 mag. Vertical arrows: The vertical arrows show the +dates of OHP/SOPHIE (red), CFHT/SPIRou (blue), and ESO/ExTrA (black) observations. The core of the campaign took place during October +2021 with contemporaneous measurements from the five instruments. +The median errors on JHK’ measurements are 0.05, 0.05, and +0.06 mag, respectively. However, some measurements were af- +fected by the nearby bright moon around JD 2,459,512, and had +an error larger than 0.1 mag. We chose to discard these measure- +ments. We eventually derived the following values for the me- +dian near-infrared magnitudes of the GM Aur system and their +rms variations at the time of the observations: J=9.41 ± 0.10 +mag, H= 8.71 ± 0.03 mag, and K’=8.40 ± 0.07 mag. +2.3. OHP SOPHIE: High-resolution optical spectroscopy +Observations were carried out from October 12 to 29, 2021, +at Observatoire de Haute-Provence using the fiber-fed SOPHIE +spectrograph (Perruchot et al. 2008) in high-efficiency mode, +which delivers a spectral resolution of R ∼ 40,000 over the wave- +length range 387-694 nm. We obtained 15 spectra over 18 nights, +with an exposure time of 3600 s, yielding a signal-to-noise ra- +tio ranging from 42 to 67 at 600 nm. The raw spectra were +fully reduced at the telescope by the SOPHIE real-time pipeline +(Bouchy et al. 2009). The data products include a resampled +1D spectrum with a constant wavelength step of 0.01 Å, cor- +rected for barycentric radial velocity, an order-by-order estimate +of the signal-to-noise ratio, and a measurement of the source ra- +dial velocity, Vr, and projected rotational velocity, v sin i. The +latter two quantities are derived from a cross-correlation anal- +ysis of nearly 7,000 spectral lines between the observed spec- +trum and a K5 spectral mask template (e.g., Boisse et al. 2010). +We list the values of these parameters in Table 2. The mean for- +mal error provided by the SOPHIE pipe-line on the Vr measure- +ment is 0.013 km s−1. This accuracy is well suited to investigat- +ing the significantly larger amplitude of photospheric line pro- +file variability induced by surface spots and/or accretion flows +in young stars (e.g., Petrov et al. 2001). No error is provided +by the pipeline for the v sin i measurements, and we assumed +that an upper limit is given by the rms deviation of the individ- +ual measurements, excluding JD 2,459,512 (see below), namely +0.38 km s−1. +The SOPHIE spectrograph includes a second fiber that si- +multaneously records the spectrum of the nearby sky. Inspection +of the cross-correlation function (CCF) of the sky fiber with a +synthetic mask of spectral type G2 revealed that the signature +of the moon becomes apparent at the expected barycentric Earth +radial velocity from JD 2,459,505 onward because the growing +moon approaches the target. The lunar contamination culminates +Article number, page 4 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Table 1. REM JHK photometry. +Julian date +J +Julian date +H +Julian date +K +(2,450,000+) +(mag) +(2,450,000+) +(mag) +(2,450,000+) +(mag) +9497.73017 +9.4 +9497.73222 +8.73 +9497.73431 +8.43 +9497.73084 +9.41 +9497.73289 +8.69 +9497.73497 +8.39 +9500.80130 +9.39 +9500.80335 +8.62 +9500.80543 +8.37 +9500.80197 +9.36 +9500.80402 +8.67 +9500.80609 +8.36 +9501.80968 +9.41 +9501.81171 +8.67 +9501.81379 +8.39 +9501.81034 +9.38 +9501.81238 +8.69 +9501.81445 +8.4 +9502.83416 +9.57 +9502.83554 +8.69 +9502.83764 +8.56 +9503.83843 +9.66 +9502.83621 +8.71 +9502.83830 +8.49 +9504.84267 +9.72 +9503.83980 +8.72 +9507.79811 +8.34 +9507.79393 +9.43 +9503.84047 +8.72 +9507.79877 +8.35 +9507.79459 +9.4 +9504.84406 +8.75 +9509.81133 +8.23 +9509.80714 +9.28 +9504.84473 +8.74 +9509.81198 +8.27 +9513.71118 +9.36 +9506.79183 +8.75 +9513.71545 +8.26 +9515.79210 +9.35 +9506.79250 +8.73 +9513.71611 +8.3 +9515.79276 +9.41 +9517.72696 +8.72 +9515.79695 +8.33 +9517.72494 +9.42 +9517.72762 +8.7 +9517.72906 +8.42 +9517.72559 +9.42 +9518.73394 +8.7 +9517.72972 +8.4 +9518.73191 +9.43 +9518.73461 +8.69 +9518.73604 +8.4 +9518.73256 +9.44 +9519.74207 +8.71 +9518.73670 +8.43 +9519.74005 +9.41 +9519.74274 +8.71 +9519.74416 +8.43 +9519.74071 +9.41 +9520.74628 +8.68 +9519.74482 +8.41 +9520.74424 +9.38 +9520.74695 +8.68 +9520.74837 +8.42 +9520.74490 +9.37 +— +— +9520.74903 +8.39 +Table 2. Journal of OHP/SOPHIE observations. +Julian date +S/N +Vr +v sin i +CCF span +(2,450,000+) +km s−1 +km s−1 +km s−1 +9499.5168 +52 +14.03 +12.12 +0.39 +9500.5044 +44 +15.17 +12.04 +-0.25 +9501.6560 +44 +14.94 +11.99 +0.36 +9502.5564 +65 +15.14 +12.3 +-0.77 +9503.6296 +59 +14.91 +12.24 +-0.44 +9504.6045 +67 +14.5 +12.78 +0.21 +9505.5450 +63 +14.89 +12.49 +-0.22 +9506.4838 +59 +15.59 +12.46 +-0.67 +9510.5691 +61 +15.07 +12.96 +-0.33 +9511.5284 +67 +15.23 +12.08 +-0.75 +9512.5162 +50 +16.85† +9.74† +-2.19† +9513.6282 +52 +15.13 +11.76 +0.09 +9514.5918 +59 +14.98 +12.06 +-0.85 +9515.6107 +42 +15.19 +11.53 +-1.02 +9516.5321 +49 +14.46 +12.7 +0.24 +† Uncertain value due to lunar contamination (see text). +on JD 2,459,512, as the bright moon is located about 10 degrees +away from GM Aur, which explains the discrepant values mea- +sured for Vr and v sin i on this date. Table 2 suggests that except +for JD 2,459,512, the contamination of the CCF by the moon +only marginally impacts the Vr and v sin i measurements. How- +ever, to be conservative, we only considered the Vr and v sin i +measurements obtained from the first six spectra of the observ- +ing run for the subsequent analysis, from JD 2,459,499 to JD +2,459,504, where no lunar contamination is present. +The journal of observations is given in Table 2. It lists the +Julian date, the signal-to-noise ratio of individual spectra at 600 +nm, the radial and rotational velocities derived from each spec- +trum, and the bisector span computed from the cross-correlation +function (Queloz et al. 2001). +2.4. CFHT SPIRou: Near-infrared spectropolarimetry +Near-infrared spectropolarimetric observations of GM Aur were +performed at CFHT using the SPIRou near-infrared spectropo- +larimeter. It has a spectral range covering from 0.95 to 2.50 µm +in a single exposure at a spectral resolution of 70,000 (Donati +et al. 2020b). The observations were completed in the frame- +work of the CFHT SPIRou Legacy Survey over four observing +runs extending from September 15 to December 18, 2021. Each +monthly run was scheduled around the full moon, and we aimed +at obtaining one spectrum per night during the run. An additional +single spectrum was obtained on January 6, 2022. We thus gath- +ered 34 spectra, whose temporal sampling is shown in Fig. 1. +Each spectrum consists of four polarimetric sub-exposures6 +for a total integration time of 2,200 s. Individual exposures were +combined to yield a single spectrum with a signal-to-noise ratio +ranging from ∼60 to 100. In one instance, on JD 2,459,503.08, +the polarimetric sequence was aborted, and the spectrum con- +sists of a single sub-exposure. The raw data were reduced within +the SPIRou consortium, using version V6.132 of the APERO +pipeline (Cook et al. 2022). Spectra were cross-correlated with +a K2 spectral mask template over about 6,700 spectral lines, and +the radial velocity of the object was derived with sub-km s−1 +accuracy (0.08 km s−1 mean rms uncertainty) by fitting a Gaus- +sian to the resulting CCF. The median Vr amounts to 14.65 ± +0.27 km s−1. Using TWA 9A, a WTTS of spectral type K5, as a +template, the veiling was derived in the JHK bands following the +procedure described in Sousa et al. (2023) The median rms er- +rors on veiling measurements in the JHK bands are 0.011, 0.025, +and 0.027, respectively. +The journal of observations is presented in Table 3. It lists the +Julian date, the signal-to-noise ratio at 2.16 µm, the photospheric +radial velocity and its uncertainty, and the veiling in the JHK +bands. +2.5. ESO ExTrA: Low-resolution near-infrared spectroscopy +The ExTrA facility (Bonfils et al. 2015), located at La Silla Ob- +servatory in Chile, consists of three 60 cm telescopes and a sin- +gle near-infrared (0.88 to 1.55 µm) fibre-fed spectrograph. We +observed GM Aur on 89 nights between October 13, 2021, and +March 8, 2022, using either one telescope (21 nights) or two tele- +scopes simultaneously (68 nights). Five fiber units are located +at the focal plane of each telescope, each consisting of two 8′′ +aperture fibers. One fiber is used to observe a star and the other +is used to observe the nearby sky background. We observed GM +Aur with one fiber unit and used another fiber unit to simultane- +ously observe 2MASS J04535474+3021441 (J = 8.450 ± 0.027 +mag) as a comparison star to compute differential photome- +try. We used the higher-resolution mode of the spectrograph +(R∼200) and 300-second exposures. We obtained between 1 and +30 exposures per night for a total of 1898 spectra with a median +signal-to-noise ratio of 105 at 1.05 µm for GM Aur. The ExTrA +data were corrected for dark current, extracted using the flat- +field, corrected for sky background emission, and were wave- +length calibrated using custom data reduction software. Median +spectra of GM Aur were computed for each night and telescope, +yielding a total of 157 spectra with a median signal-to-noise ratio +of 179 and a standard deviation of 62. +We computed differential photometry of GM Aur relative to +the comparison star by integrating the individual ExTrA spectra +6 The polarimetric analysis of the dataset will be published in a com- +panion paper (Zaire et al., in prep.). +Article number, page 5 of 30 + +A&A proofs: manuscript no. 45342corr +Table 3. Journal of CFHT/SPIRou observations. +Julian date +S/N +Vr +σVr +rJ +rH +rK +(2,450,000+) +km s−1 +September 2021 +9473.066 +101 +14.77 +0.06 +0.05 +0.05 +0.27 +9475.043 +109 +14.09 +0.17 +0.05 +0.06 +0.22 +9476.969 +122 +14.88 +0.16 +0.12 +0.09 +0.27 +9478.027 +119 +14.85 +0.04 +0.17 +0.14 +0.38 +9480.082 +121 +14.62 +0.09 +0.05 +0.09 +0.29 +9481.086 +113 +14.42 +0.08 +0.06 +0.10 +0.23 +9482.086 +120 +14.70 +0.06 +0.05 +0.07 +0.23 +October 2021 +9502.074 +112 +14.97 +0.06 +0.15 +0.14 +0.32 +9503.074† +64 +14.48 +— +0.26 +0.15 +0.07 +9504.098 +94 +14.61 +0.05 +0.10 +0.11 +0.30 +9506.082 +112 +14.65 +0.01 +0.04 +0.12 +0.29 +9508.082 +113 +14.68 +0.08 +0.18 +0.24 +0.43 +9509.086 +110 +14.85 +0.06 +0.32 +0.26 +0.51 +9510.086 +92 +14.40 +0.12 +0.41 +0.29 +0.58 +9511.074 +116 +14.33 +0.05 +0.02 +-0.04 +0.28 +9513.090 +98 +14.70 +0.25 +0.09 +0.13 +0.29 +9514.051 +105 +14.66 +0.16 +0.18 +0.12 +0.36 +9515.078 +108 +14.66 +0.03 +0.10 +0.11 +0.28 +9516.102 +104 +14.73 +0.16 +0.10 +0.09 +0.26 +November 2021 +9535.102 +102 +14.27 +0.09 +0.13 +0.07 +0.25 +9537.098 +101 +14.95 +0.15 +0.11 +0.08 +0.26 +9538.039 +104 +14.22 +0.10 +0.05 +0.11 +0.22 +9538.984 +95 +14.40 +0.10 +0.12 +0.08 +0.28 +9539.988 +107 +14.51 +0.03 +0.14 +0.13 +0.27 +9541.012 +110 +13.96 +0.10 +0.14 +0.14 +0.38 +December 2021 +9557.980 +110 +14.42 +0.11 +0.22 +0.19 +0.50 +9558.949 +110 +14.12 +0.03 +0.21 +0.19 +0.45 +9559.961 +95 +14.71 +0.11 +0.09 +0.11 +0.35 +9560.965 +94 +15.04 +0.06 +0.07 +0.12 +0.35 +9563.078 +106 +14.73 +0.06 +0.21 +0.06 +0.43 +9564.059 +102 +14.34 +0.10 +0.32 +0.11 +0.43 +9566.051 +96 +14.37 +0.13 +0.07 +0.03 +0.34 +9566.953 +95 +14.93 +0.03 +0.06 +0.02 +0.25 +January 2022 +9585.941 +112 +14.30 +0.06 +0.11 +0.12 +0.25 +† Single sub-exposure. +over the J-filter passband. The UKIRT-WFCAM filter transmis- +sion curves were retrieved from the SVO Filter Profile Service7 +(Rodrigo et al. 2012; Rodrigo & Solano 2020). We multiplied +each corrected individual spectrum of GM Aur and the compar- +ison star by the filter transmission curve, integrated the flux, and +obtained a magnitude difference from the flux ratio. We com- +puted a differential magnitude measurement for each night as +the mean and standard deviation of the individual measurements +taken on that night. We then derived the J-band magnitude of +GM Aur from the 2MASS magnitude of the comparison star. +We obtained a median value and 68.3% confidence interval of +J = 9.417 ± 0.061 mag for the ExTrA observations. The results +are listed in Table C.1 of Appendix C. +7 http://svo2.cab.inta-csic.es/theory/fps/ +3. Results +3.1. Multicolor photometry +The LCOGT u’g’r’i’ light curves of GM Aur span nearly four +months and are shown in Fig. 2. A TESS light curve extending +over 50 days, from JD 2,459,474 to JD 2,459,524, is also shown. +We derive a mean magnitude and rms variability of u’=14.32 +± 0.35 mag, g’=12.77 ± 0.17 mag, r’=11.64 ± 0.11 mag, and +i’=11.21 ± 0.09 mag, respectively. The variability amplitude is +much larger than the photometric errors in all bands; it amounts +to 0.025 mag in the g’r’i’ bands and 0.033 mag in the u’ band, +and it decreases toward longer wavelengths, from u’ to i’. We +ran two period-search algorithms on the light curves: a CLEAN +periodogram analysis (Roberts et al. 1987), which is conceptu- +ally similar to a Fourier transform, and the String-Length method +(Dworetsky 1983), which finds the period that minimizes the av- +erage distance between consecutive points in the phased light +curve. Both yielded the same period at all wavelengths, namely +P=6.04 ± 0.15 days, with the uncertainty being estimated from +the standard deviation of a Gaussian fitted to the periodogram +peak. The results of the period search are shown in Fig. 3. The +light curves folded in phase at the 6.04 d period are shown in +Fig. 2. They clearly show the modulation of the brightness level +at this period, particularly in the u’ band, where the amplitude +is largest. We ascribe this low-level modulation to surface spots +and the P=6.04 d period to stellar rotation. This estimate agrees +within the error bars with the previously reported periods for +GM Aur, namely P=6.1 d by Percy et al. (2010) and P=6.02 d +by Artemenko et al. (2012). We therefore adopt the following +ephemeris: +HJD(d) = 2, 459, 460.80 + 6.04 × E, +(1) +which +defines +the +rotational +phase +Φrot=(HJD- +2,459,460.80)/Prot (modulo Prot), where phase zero (Φrot= +0) is chosen as the epoch of maximum optical brightness in the +rotational cycle. +Superimposed onto spot modulation, additional signs of in- +trinsic variability are visible. This is the case, for instance, of +an apparent brightness event that is centered on JD 2,459,509 +and lasted for a few days, as well as more pronounced wide +dips toward the end of the observing period, from JD 2,459,544 +on. Interestingly, the first half of the light curve exhibits several +day-long brightening events, while the last third is dominated +by wide dimming events. We note that most of the brighten- +ing events, with g’≤12.5 mag, tend to occur at rotational phases +shorter than 0.15 or longer than 0.85, that is, close to the maxi- +mum brightness of the spot modulation. If the photometric vari- +ations of the system are modulated by the visibility of a bright +accretion spot at the stellar surface, these brightening events seen +around the time of maximum accretion shock visibility most +likely reflect varying accretion on the stellar surface. The wide +dips in the last part of the light curve exhibit the same period- +icity and phase as the spot modulation. They reach a minimum +brightness close to phase 0.5, with an amplitude that steadily de- +creases from 0.6 mag to 0.3 mag in the g’ band over a timescale +of a few weeks, from JD 2,459,550 to JD 2,459,570. +Fig. 4 shows the color behavior of the system. As the sys- +tem fades, it becomes redder, with a color slope larger than ex- +pected from ISM-like extinction at least in the (u’-g’) and (g’-r’) +color indices. According to Venuti et al. (2015), a large color +slope at short wavelengths is characteristics of accretion-driven +photometric variability. However, both the large scatter seen in +the (u’-g’) color index at low brightness levels and the changing +Article number, page 6 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. 2. GM Aur light curves. Top: GM AUr’s u’g’r’i’ light curves from LCOGT observations that extended over nearly four months. The mean +photometric error is 0.025 mag in the g’r’i’ bands and 0.033 mag in the u’ band. A scaled TESS light curve (black) obtained contemporaneously +is overplotted onto the LCOGT r’-band light curve. Middle: LCOGT light curves folded in phase with a period of 6.04 days with the ephemeris +of Eq.1. Bottom: Same as above, with a color scale for data points that reflects the Julian date. The amplitude of the light variations appears to +increase slightly toward the end of the observing run. In all panels, the error bars on the measurements are smaller than the symbol size. +Article number, page 7 of 30 + +11 +12 +Magnitude +TESS +13 +14 +15 +2459480 +2459500 +2459520 +2459540 +2459560 +2459580 +JulianDate11 +12 +Magnitude +13 +14 +15 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Phase2459560 +12 +Magnitude +2459540 +13 +2459520 +14 +2459500 +2459480 +15 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +PhaseA&A proofs: manuscript no. 45342corr +Fig. 3. Period search results. Top: CLEAN periodogram analysis of the +u’g’r’i’ light curves. A peak occurs at the frequency of 0.166 day−1, +which corresponds to a period of 6.04 ± 0.15 days. Bottom: String- +length analysis of the u’g’r’i’ light curves. A clear minimum appears +for a period of 6.03 ± 0.15 days. +Fig. 4. Color-magnitude relation. The (u’-g’), (g’-r’), and (r’-i’) colors +are plotted as a function of the system brightness in the g’-band. The +dashed lines indicate the expected ISM reddening slope computed from +Fiorucci & Munari (2003) with R=3.1. +shape of the light curve during the semester suggest that several +sources of variability might be present, such as a combination of +accretion and obscuration events. +3.2. High-resolution optical spectroscopy +We took advantage of the wide wavelength range covered by +the SOPHIE spectrograph at high spectral resolution to derive +the stellar parameters of GM Aur, to measure optical veiling, +and to investigate the emission line profiles and their variability +across the optical range. These results are described in the next +subsections. +Table 4. Properties of the GM Aur system. +Parameter +Value +SpT +K4-K5 +AV +0.3 ± 0.3 mag +Teff +4287 ± 35 K +L⋆/L⊙ +0.9 ± 0.2 +R⋆/R⊙ +1.7 ± 0.2 +M⋆/M⊙ +0.95 ± 0.13 +˙Macc +0.7 ± 0.3 10−8 M⊙yr−1 +EW(LiI) +420 ± 23 mÅ +v sin i +14.9 ± 0.3 km s−1 +Prot +6.04 ± 0.15 d +rcor +8.3 ± 0.5 R⋆(0.064 au) +3.2.1. Stellar parameters +To derive the stellar parameters (Teff, v sin i, Vr, and Vmic), we fit +synthetic ZEEMAN spectra (Landstreet 1988; Wade et al. 2001; +Folsom et al. 2012) to the average of the first six SOPHIE spec- +tra, which are not contaminated by the moon. Synthetic spec- +tra were computed from MARCS atmosphere models (Gustafs- +son et al. 2008) and the VALD line list database (Ryabchikova +et al. 2015). We applied a χ2 minimization procedure based on a +Levenberg-Marquart algorithm over seven wavelength windows +ranging from 455 to 649 nm, excluding the regions affected by +tellurics, emission, or molecular lines. We set the macroturbu- +lent velocity to 2.0 km s−1 and the surface gravity log g to 4.0, +and we assumed solar metallicity. These are typical parameters +for low-mass TTSs. During the fitting procedure, we applied the +mean veiling value derived below for the GM Aur mean opti- +cal spectrum (r0.55 = 0.3, see Sect. 3.2.2) to the synthetic spec- +tra. Finally, we averaged the results obtained from the various +wavelength windows, except for one window that yielded values +higher than 2σ from the mean. We thus derived Teff = 4287 ± 35 +K, Vr = 14.94 ± 0.14 km s−1, Vmic= 3.3 ± 0.4 km s−1, and v sin i += 14.9 ± 0.3 km s−1. We also used the ROTFIT package (Frasca +et al. 2003, 2006) applied to the mean GM Aur’s SOPHIE spec- +trum, from which we derive Teff = 4505 ± 53 K, Vr = 15.37 ± +0.26 km s−1, and v sin i = 13.0 ± 0.7 km s−1. +While all these values are consistent within 3σ, the large +Teff difference derived from ZEEMAN and ROTFIT illustrates +model-dependent uncertainties that are likely related to the use +of different model templates (MARCs for ZEEMAN vs. BTSetll +for ROTFIT) and possibly to wavelength-dependent systemat- +ics induced by starspots (Gangi et al. 2022; Flores et al. 2022). +For consistency with similar observing campaigns that we pre- +viously performed on young stars (e.g. Pouilly et al. 2020, 2021; +Bouvier et al. 2020a; Alencar et al. 2018), we adopted the results +of the ZEEMAN fitting. This estimate also agrees better with the +K6 spectral type derived by Herczeg & Hillenbrand (2014), from +which they deduced Teff= 4115 K. When the Teff-SpT conversion +tables from Herczeg & Hillenbrand (2014) and Pecaut & Mama- +jek (2013) are used, the Teff value derived above corresponds to a +K4-K5 spectral type, which agrees fairly well with previous esti- +mates obtained from optical and near-infrared spectroscopy (K5- +K6; e.g., Espaillat et al. 2010; Herczeg & Hillenbrand 2014). +The Vr and v sin i values can be compared to those derived +from the uncontaminated CCF of the first six SOPHIE spectra, +namely = 14.79 ± 0.40 km s−1 and = 12.25 ± +0.26 km s−1. The quoted uncertainties are the rms of the six mea- +surements and therefore include intrinsic variability of the CCF +profiles due to spot modulation, for example. The two estimates +Article number, page 8 of 30 + +u +g +0.10 +Power +0.05 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Frequency (1/d)20 +length +StringI +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Frequency (1/d)"r +0.5 +b-,r +1.5 +2.0 +12.2 +12.4 +12.6 +12.8 +13.0 +13.2 +g'(mag)J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +of Vr agree within the errors as well as with the median Vr value +derived from the SPIRou spectra (see Section 2.4), while the +v sin i value derived from the CCF is significantly lower than the +value deduced from spectral fitting. We suspect that the discrep- +ancy may arise from the color-dependent FWHM-v sin i relation +used for SOPHIE CCFs, which is calibrated on main-sequence +stars (Boisse et al. 2010) and may not be fully adequate for pre- +main-sequence objects. +From the average g’r’i’ colors reported for GM Aur in Sec- +tion 2.1, the intrinsic colors of a K4-K5 dwarf listed by Kraus +& Hillenbrand (2007) and Covey et al. (2007), and using ISM +extinction coefficients from Fiorucci & Munari (2003), we de- +rive a visual extinction on the line of sight AV = 0.3 ± 0.3 mag. +This agrees with previous determinations (e.g., Herczeg & Hil- +lenbrand 2014). The REM photometry reported above yields a +median J-band magnitude of 9.42 ± 0.11 mag, which is close +to that of 2MASS (J=9.34 mag). We dereddened the median +J-band magnitude with A j = 0.28 × AV = 0.084 ± 0.084 mag +and used the J-band bolometric correction listed in Pecaut & +Mamajek (2013) for a K4-K5 dwarf, BCJ=1.55 ± 0.03 mag, +to derive the stellar luminosity, L⋆= 0.9 ± 0.2 L⊙, assuming +the Gaia distance of 157.9 ± 1.2 pc (Gaia Collaboration et al. +2021). We thus derive a stellar radius R⋆= 1.7 ± 0.2 R⊙ from +Stefan’s law, and a stellar mass M⋆= 1.05 ± 0.05 M⊙ from the +Siess et al. (2000) pre-main-sequence evolution models, while +CESAM models (Marques et al. 2013) yield M⋆= 0.88 ± 0.12 +M⊙ (E. Alécian, priv. comm.). We therefore adopt M⋆= 0.95 ± +0.13 M⊙, in agreement with Baraffe et al. (2015) models. This +estimate is also consistent within the errors with the dynamical +mass estimate, Mdyn = 1.00 ± 0.02 M⊙, reported by Guilloteau +et al. (2014), which was later revised to Mdyn = 1.14 ± 0.02 M⊙ +by Simon et al. (2019). Table 4 summarizes the derived stellar +parameters. +Finally, we combined the v sin i and rotational period mea- +surements with the stellar radius estimate to derive the stellar +inclination sin i = Prot × v sin i / (2 π R⋆) = 1.05 with the val- +ues listed in Table 4. Accounting for 1σ uncertainties on the +stellar parameters, we derive a lower limit of i⋆ ≥ 63◦ for the +stellar rotational axis onto the line of sight. This value is signif- +icantly higher than the inclination inferred from high-resolution +ALMA images of the outer disk of GM Aur observed at mil- +limeter wavelength, which yield idisk = 53.2 deg (Huang et al. +2020). It agrees better, however, with the inclination value de- +rived for the disk seen in scattered light with adaptive optics +on a scale of a few dozen au, for which Oh et al. (2016) ob- +tained idisk = 64 ± 2 deg. On the much smaller scale of 0.013 +au, Bohn et al. (2022) measured an inner-disk inclination idisk += 68+16 +−28 deg from long-baseline K-band interferometry using +VLTI/GRAVITY. They did not find evidence for an inner and +outer disk misalignment in this system. As outlined by Appen- +zeller & Bertout (2013), the uncertainty on the determination of +stellar inclinations from rotation measurements rapidly increases +at large angles and is prone to systematic errors. For GM Aur, +inferring the stellar inclination from the disk inclination might +therefore be more reliable than estimating it from rotation mea- +surements, owing in particular to the significant uncertainty on +the stellar radius. Nevertheless, all independent measurements +indicate a moderate to high inclination for the system. +3.2.2. Veiling +At optical wavelengths, accreting T Tauri stars exhibit an addi- +tional source of continuum flux, which presumably arises from +the accretion shock at the stellar surface. This optical excess +Fig. 5. Optical veiling measurements from the OHP/SOPHIE spectra. +Top: Part of a spectral window showing the template spectrum (gray), +the observed spectrum (black), and the velocity broadened and veiled +template spectrum (red) that fits the observed spectrum. Bottom: Veiling +measured in several spectral windows (see text) as a function of Julian +date. The central wavelength of the spectral windows is indicated in the +top left corner of the panel. +Fig. 6. Mean optical veiling plotted as a function of rotational phase. +The color code indicates the Julian date. +Fig. 7. LiI 6707 Å line profile. Left: The 34 line profile measurements +from SOPHIE spectra are shown superimposed. The color code cor- +responds to successive rotational cycles. Right: 2D periodogram across +the line profile. The dotted horizontal line drawn at a frequency of 0.166 +day−1 indicates the stellar rotational period. The white curve displays +the mean line profile. The color code reflects the periodogram power. +continuum partly veils the stellar photospheric spectrum and is +therefore referred to as "veiling" (Hartigan et al. 1995). We mea- +sured the optical veiling from the high-resolution OHP/SOPHIE +spectra by comparing the photospheric spectrum of GM Aur +to that of the unveiled nonaccreting template V819 Tau, a +WTTS with Teff= 4250 ± 50 K, Vr = 16.6 km s−1, and v sin i +Article number, page 9 of 30 + +0.8 +2459515 +0.6 +2459510 +Optical +2459505 +0.2 +2459500 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Phase1.0 +0.9 +0.8 +Flux +0.7 +0.6 +0.5 +0.4 +0.3 +-40 +-20 +0 +20 +40 +v (km/s)0.50 +0.45 +0.5 +0.40 +0.4 +0.35 +requency +Power +0.3 +0.30 +0.25 +0.2 +0.20 +0.1 +0.15 +0.10 +-40 +-20 +0 +20 +40 +v (km/s)1.4 +No veiling nor broadening +Observed +1.2 +Fit +xnl +Normalized +0.8 +0.6 +0.4 +0.2 +459.0 +459.5 +460.0 +460.5 +461.0 +461.5 +462.0 +Wavelength (nm)1.0 +460nm +547nm +573nm +644nm +495nm +561nm +605nm +0.8 +ul +0.4 +0.2 +0.0 +9500.0 +9502.5 +9505.0 +9507.5 +9510.0 +9512.5 +9515.0 +Julian Date -2,450,000 (d)A&A proofs: manuscript no. 45342corr += 9.5 km s−1 (Donati et al. 2015). We retrieved an archival +CFHT/ESPaDOnS spectrum of V819 Tau, which we resampled +at the spectral resolution of OHP/SOPHIE spectra, translated +into the radial velocity of GM Aur, and rotationally broadened +using the rotational function from Gray (1973) to match the ro- +tational velocity of GM Aur. The template spectrum was then +fit to the GM Aur spectra over the same wavelength windows +as discussed in Section 3.2.1 by adjusting the veiling using the +following formula: +I = I0 + r +1 + r , +(2) +where I is the veiled spectrum, I0 is the spectrum without veil- +ing, and r is the continuum veiling. Rei et al. (2018) showed that +an additional line-dependent veiling component may be present +in the strongest photospheric lines. We therefore retained only +weak to moderate lines with an EW between 0.01 and 0.1 Å to +perform the fit. The veiling measured for each spectrum in each +spectral window is shown in Figure 5. Systematic offsets are +clearly seen between the veiling values measured in the different +spectral windows. These offsets may partly reflect a wavelength- +dependent veiling, as veiling seems to decrease toward longer +wavelengths for all but the bluest spectral window, but it may +also result from systematic errors depending on the specific sam- +ple of lines included in each window. We therefore computed an +average veiling value, r0.55, over all spectral windows for individ- +ual GM Aur spectra. This value is listed in Table 5 with its asso- +ciated uncertainty. The optical veiling is moderate, ranging from +0.17 to 0.55 at 5500 Å. Similar mean values were derived from +ROTFIT, namely r = 0.5, 0.4, and 0.3 at 4500, 6000, and 6500 +Å, respectively, using a library of 400 spectral templates with +spectral types FGKM from the OHP/ELODIE database, which +yielded a best χ2 fit for spectral types ranging from K3.5 to K5. +Figure 6 shows the mean optical veiling plotted along the rota- +tional phase. A hint of higher veiling values towards phases 0 +and 1, and minimum values around phase 0.5-0.6 appears. A pe- +riodogram analysis of the mean veiling variation did not yield +significant results, however. +The lack of significant veiling variability is confirmed by the +examination of the LiI 6707 Å photospheric line profile shown +in Fig. 7. The shape and depth of the line appear to remain quite +stable over the duration of the observing run, and a periodogram +analysis across the line profile (Giampapa et al. 1993) revealed +no signs of periodic modulation. This suggests that the source +of optical veiling remains at least partly in view throughout the +rotational cycle. This might arise from a high-latitude accretion +shock. +3.2.3. Emission lines +The main emission lines seen in the optical spectrum of GM +Aur, namely Hα, Hβ, and HeI 5876 Å, are displayed in Fig- +ure 88. The Balmer lines show a broad emission peak and a +slightly blueshifted absorption component, whose depth varies +from being hardly discernable to reaching below the continuum +8 The [OI] 6300 Å line is also seen in emission in the spectrum. How- +ever, it is significantly contaminated by sky, which cannot be easily cor- +rected for due to the different response of the object and sky fibers of the +SOPHIE spectrograph. The other emission lines seen in the high signal- +to-noise ratio mean SOPHIE spectrum are Ca II H&K, Hγ, Hδ, Hϵ, and +He I 6678 Å, the latter being weak and affected by a deep photospheric +FeI line. +Table 5. Optical line EW and veiling measurements from the +OHP/SOPHIE spectra. +Julian date +EW +Veiling +Hα +Hβ +HeI +r0.55 +rms +(2,450,000+) +Å +Å +Å +9499.51683266 +92.3 +9.2 +0.31 +0.35 +0.19 +9500.50445956 +81.1 +6.8 +0.37 +0.17 +0.15 +9501.65604543 +86 +10.3 +0.5 +0.26 +0.16 +9502.55640645 +73.3 +7.4 +0.45 +0.46 +0.17 +9503.62969684 +79.6 +8.5 +0.4 +0.4 +0.15 +9504.60450624 +88.1 +10.5 +0.44 +0.3 +0.14 +9505.54502964 +85.8 +10.2 +0.23 +0.27 +0.13 +9506.48382743 +86.4 +9 +0.33 +0.37 +0.15 +9510.56919423 +102 +15.1 +0.74 +0.51 +0.18 +9511.52849738 +91.1 +12.6 +0.34 +0.38 +0.16 +9512.51625045 +68.3† +3.7† +0.18† +0.29† +0.13† +9513.62825832 +82.6 +10.1 +0.36 +0.37 +0.15 +9514.59185217 +81.5 +9 +0.5 +0.55 +0.23 +9515.61072618 +76.8 +7.7 +0.42 +0.3 +0.19 +9516.53216217 +74.7 +9.6 +0.36 +0.27 +0.15 +† Uncertain value due to lunar contamination (see text). +in the Hβ profile. The wide, slightly blueshifted absorption com- +ponent peaks at -30 to -20 km s−1and covers a velocity range +from about -90 to +40 km s−1 in the Hβ line profile. Additional +absorption components appear at higher blueshifted velocities, +peaking from -110 to -90 km s−1, and extending down to about +-160 km s−1. These blueshifted absorption components cause +most of the variability in the Balmer line profile. The Hβ pro- +file also exhibits significant variability over the red wing, up to +velocities of +200 km s−1. However, none of the profiles exhibits +redshifted absorption components reaching below the continuum +level. +Owing to the complex line shapes, equivalent widths (EW) +of the Hα, Hβ, and HeI 5876 Å lines were computed by directly +integrating below the line profile. The results are listed in Ta- +ble 5. The measurement accuracy is 10% or better for EW(Hα) +and EW(Hβ), and about 20% for EW(HeI) due to the more un- +certain continuum location. We note that on JD 2,459,512, the +EWs measurements are systematically lower than during the rest +of the observations, which might be due to lunar contamina- +tion, as discussed in Section 2.3. The average and rms values +we obtain are EW(Hα) = 83 ± 8 Å, EW(Hβ) = 9.3 ± 2.5 Å, and +EW(HeI) = 0.40 ± 0.13 Å. +The HeI 5876 Å line profile is roughly symmetric and con- +sists of a narrow component (FWHM ∼ 30-40 km s−1) superim- +posed on a broad pedestal, as previously reported by McGinnis +et al. (2020). The peak intensity of the narrow component varies +significantly, while the broad component appears relatively sta- +ble (see Fig. 8). We fit the HeI line profile with a two-component +Gaussian model to extract the properties of the narrow (NC) and +broad (BC) components. The EWs were derived from the Gaus- +sian fit of the components. The radial velocity, FWHM, and EW +of the NC and BC, as well as their uncertainty derived from the +covariance matrix of the Levenberg-Marquart fitting algorithm, +are listed in Table 6. +We derived the radial velocity of the narrow component and +found it to be variable and redshifted by ∼5-10 km s−1 relative +to the stellar velocity. Figure 9 shows the HeI NC radial velocity +curve plotted as a function of Julian date and rotational phase. +As previously reported by McGinnis et al. (2020), HeI NC Vr +Article number, page 10 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. 8. Optical line profile variability. Top: Series of optical line profiles Hα, Hβ, and HeI plotted as a function of Julian date (left subpanels) and +rotational phase (right subpanels). The color code corresponds to successive rotational cycles. Middle: Same profiles, superimposed to illustrate +their variability. Bottom: 2D periodograms across the line profiles. The color code reflects the periodogram power, from zero (blue) to 0.6 (red). +The dotted horizontal red line drawn at a frequency of 0.166 day−1 indicates the stellar rotational period. The white curve is the mean line profile. +appears to be rotationally modulated with a full amplitude of +∼6 km s−1, as expected if the NC component of the HeI 5876 Å +line profile were produced in a high-latitude accretion shock at +the stellar surface. We fit the observed NC Vr curve with the +geometrical accretion shock model described in Pouilly et al. +(2021). The model computes the variation of the HeI NC radial +velocity as the combination of the rotational modulation of the +accretion shock and the intrinsic inflow velocity. The free pa- +rameters of the model are the inflow velocity, the latitude of the +accretion shock, the phase at which it faces the observer, and the +Article number, page 11 of 30 + +Hα +day +phase +9499.5 +3.07 +9500.5 +1.09 +9501. +3.23 +A +9502.6 +2.24 +9503.6 +1.25 +9504.6 +2.4 +9505.5 +1.41 +9506.5 +0.41 +9510.6 +2.56 +9511.5 +1.56 +9512.5 +0.57 +9513.6 +2.75 +9514.6 +0.76 +9515.6 +2.91 +9516.5 +0.91 +-500 +0 +500-500 +0 +500 +v (km/s) +v (km/s)Hβ +phase +day +3.07 +9499.5 +1.09 +9500. +3.23 +9501. +2.24 +9502. +9503.6 +1.25 +2.4 +9504.6 +1.41 +9505.5 +9506.5 +0.41 +2.56 +9510.6 +1.56 +9511.5 +0.57 +9512.5 +9513. +2.75 +9514.6 +0.76 +9515.6 +2.91 +9516.5 +0.91 +-500 +0 +500-500 +0 +500 +v (km/s) +v (km/s)Hel +day +hase +9499.5 +3.07 +9500.5 +9501.7 +9502.6 +9503.6. +9504.6 +9505.5 +9506.5 +9510.6 +9511.5 +9512.5 +9513.6 +9514.6 +9515.6 +2.91 +9516.5 +0.91 +-200 +0 +200-200 +0 +200 +v (km/s) +v (km/s)15 +10 +5 +0 +-400 +-200 +0 +200 +400 +v (km/s)3 +2 +Flux +1 +0 +-400 +-200 +0 +200 +400 +v (km/s)1.6 +1.4 +1.2 +F +1.0 +0.8 +-100 +-50 +0 +50 +100 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +Frequency (1/d) +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-150 +-100 +-50 +0 +50 +100 +150 +v (km/s)A&A proofs: manuscript no. 45342corr +Table 6. Properties of the narrow (NC) and broad (BC) components of the HeI 5876 Å line profile and their 1σ error. +Julian date +Vr +FWHM +EW +(2,450,000+) +km s−1 +km s−1 +Å +NC +err +BC +err +NC +err +BC +err +NC +err +BC +err +9499.51683 +11.8 +1.0 +14.8 +1.8 +27.7 +3.6 +100.7 +6.7 +0.07 +0.02 +0.30 +0.05 +9500.50445 +3.3 +1.3 +14.5 +3.1 +38.7 +4.5 +106.6 +9.6 +0.13 +0.03 +0.30 +0.07 +9501.65604 +4.6 +0.8 +20.0 +3.7 +40.7 +2.7 +91.0 +7.4 +0.28 +0.04 +0.29 +0.08 +9502.55640 +7.7 +0.7 +20.0 +2.5 +36.2 +2.4 +81.2 +5.5 +0.22 +0.03 +0.27 +0.06 +9503.62969 +8.6 +0.7 +12.9 +2.3 +37.6 +2.8 +113.2 +9.4 +0.18 +0.03 +0.30 +0.06 +9504.60450 +9.2 +0.8 +10.1 +1.8 +34.0 +2.9 +111.4 +7.3 +0.13 +0.02 +0.36 +0.06 +9505.54503 +11.9 +1.4 +8.7 +2.5 +28.2 +4.9 +101.5 +9.3 +0.06 +0.02 +0.22 +0.05 +9506.48382 +9.1 +1.3 +11.9 +1.9 +31.3 +4.6 +104.3 +7.7 +0.08 +0.02 +0.31 +0.06 +9510.56919 +9.0 +0.4 +11.0 +1.3 +36.3 +1.4 +120.7 +5.4 +0.29 +0.02 +0.51 +0.06 +9511.52849 +10.0 +0.8 +14.5 +1.8 +30.9 +3.0 +97.3 +7.0 +0.11 +0.02 +0.27 +0.05 +9512.51624 +6.7 +1.8 +7.6 +4.5 +38.6 +7.7 +99.3 +21.6 +0.08 +0.04 +0.14 +0.09 +9513.62825 +4.6 +0.8 +20.0 +4.6 +38.3 +2.6 +89.3 +9.2 +0.21 +0.03 +0.18 +0.06 +9514.59185 +9.5 +0.6 +20.0 +1.8 +33.6 +2.3 +90.1 +5.0 +0.19 +0.03 +0.36 +0.06 +9515.61072 +7.7 +0.7 +20.0 +2.0 +31.4 +2.5 +84.4 +4.9 +0.15 +0.03 +0.32 +0.05 +9516.53216 +8.9 +1.0 +15.5 +2.0 +29.4 +3.3 +102.5 +7.1 +0.10 +0.02 +0.32 +0.05 +9500.0 +9502.5 +9505.0 +9507.5 +9510.0 +9512.5 +9515.0 +HJD (-2,450,000 d) +2.5 +5.0 +7.5 +10.0 +12.5 +Vr HeI (km.s +1) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +2.5 +5.0 +7.5 +10.0 +12.5 +Vr HeI (km.s +1) +Fig. 9. Radial velocity of the narrow component of the HeI 5876 Å line +profile in the stellar rest frame plotted as a function of Julian date (top) +and rotational phase computed from the ephemeris of Eq.1 (bottom). +The color code corresponds to successive rotational cycles. The fit by +a geometrical accretion shock model (see text) is shown (dash-dotted +curve) together with its 1σ uncertainty (gray area). +stellar inclination. The HeI NC Vr curve is best reproduced with +an accretion shock located at a latitude of 83 ± 1.5◦ that faces +the observer at phase 0.2 ± 0.08 and has a radial post-shock ve- +locity of 18.3 ± 1.0 km s−1 in the stellar rest frame. Because the +model now includes the inflow velocity, the HeI NC radial ve- +locity curve does not reach the mean velocity at the time when +the spot faces the observer, as it would for the case of static stel- +lar spots. The stellar inclination we derive from the model is i += 64 ± 2.2◦, which is consistent with the inner disk inclination +derived from K-band VLTI/GRAVITY data (see Section 3.2.1). +According to the model, the accretion shock faces the observer +close to the origin of the phase, which is consistent with the +photometric behavior described above (see Sect. 3.1). The HeI +line profile is also strongest over rotational phases ranging from +0.75 to 1.09 (excluding the probable accretion burst occuring at +JD 2,459,510, see Sect. 3.1), and this is also when the highest +veiling values are measured in the JHK bands (see Fig. D.1 in +Appendix D). These two results further support a maximum vis- +ibility of the accretion shock close to Φrot= 0. +A bidimensional periodogram analysis (Giampapa et al. +1993) of the Balmer and HeI 5876 Å line profiles reveals a peri- +odic modulation of part of the profiles (see Fig. 8). The intensity +of the narrow component of the HeI line profile is modulated at +a frequency of ∼0.15 d−1, corresponding to a period of 6.7 d, +with a large uncertainty due to the limited temporal sampling of +the spectral series, however, that translates into a poor frequency +resolution. As the HeI line profile NC component arises in the +accretion shock (Beristain et al. 2001), it is expected to be mod- +ulated at the stellar rotational period or close to it in case of lat- +itudinal differential rotation at the stellar surface. In the Balmer +line profiles, a modulation close to the stellar period also ap- +pears over three distinct locations: in the highly redshifted wing +over the velocity range ∼200-400 km s−1, at slightly redshifted +velocities between 0 and ∼50 km s−1, and at highly blueshifted +velocities from -400 to -200 km s−1. While the maximum power +of the periodogram in the blue wing appears at the stellar rota- +tion period, it seems to drift to longer periods, similar to that seen +in the HeI NC component, toward the red wing. If the red wing +modulation of Balmer line profiles is caused by the absorption +of shock emission by a funnel flow crossing the line of sight, +this might be an indication of differential rotation along the fun- +nel flow. Noticeably, no sign of periodic variability is seen in the +Balmer line profiles in the velocity channels in which the vari- +able blueshifted absorption components arise. This suggests that +these components either result from sporadic ejection processes +or that they vary on periods longer than ten days. +Correlation matrices (Johns & Basri 1995) between line pro- +files were computed and are presented in Appendix B. They dis- +play the degree to which temporal flux variations in a pair of +spectral lines are correlated. Matrices can be computed for a sin- +gle profile (autocorrelation), for instance, Hα⋆Hα, to investigate +how the different parts of the profile vary with respect to each +other, or between two profiles (cross-correlation), for example, +Hα⋆Hβ, to compare the intensity variations of different lines. +The Hα⋆Hα and Hβ⋆Hβ matrices shown in Fig. B.1 are quite +similar. The blue and red wings of the profiles vary in a corre- +lated way. The high-velocity red wings are anticorrelated with +the emission peak region. This may be a sign that high-velocity +redshifted absorption components appear when the peak inten- +sity is higher, although the absorptions do not reach below the +continuum. Many absorption features are superimposed on the +emission line component. These features do not present a peri- +odicity and are not correlated with each other, nor with the emis- +sion part of the profile. The Hβ⋆Hα matrix mimics the matrices +of Hα⋆Hα and the Hβ⋆Hβ, showing that both lines are formed +in the same region, as expected from magnetospheric accretion +models (e.g., Muzerolle et al. 2001). +Article number, page 12 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +3.3. High-resolution near-infrared spectroscopy +The main emission lines seen in the high-resolution near-infrared +spectrum of GM Aur, namely HeI 10830 Å, Paβ, and Brγ, are +depicted in Figures 10 and 119. The near-infrared hydrogen +lines exhibit broad emission profiles, with FWHM ∼200 km s−1, +whose peaks are slightly blueshifted compared to the stellar ve- +locity, and are located at -30 and -10 km s−1 for Paβ and Brγ, +respectively. The profiles are roughly symmetric, but have pro- +nounced high-velocity redshifted absorptions that extend up to ++400 km s−1 and reach below the continuum level. Paβ and Brγ +exhibit inverse P Cygni (IPC) profiles of varying depth in most +SPIRou spectra, which is in stark contrast to the optical hydro- +gen line profiles, Hα and Hβ, which do not exhibit such features +(see Sect. 3.2.3). +Correlation matrices for the near-infrared hydrogen line +profiles are shown in Fig. B.2. The Paβ⋆Paβ, Brγ⋆Brγ, and +Paβ⋆Brγ matrices are quite similar. The emission part of the +profiles is overall correlated, and anti-correlated with the high- +velocity redshifted absorption component. This indicates that the +redshifted absorption is deeper when the emission line is more +intense. Cross-correlation matrices between optical and near- +infrared line profiles observed in the same night during the Oc- +tober runs are shown in Fig. B.3. The Brγ and the Paβ emis- +sion components correlate well with the main Hα and Hβ emis- +sion components (-100 km s−1< v < 200 km s−1), which sug- +gests that at least part of the line emission forms in the same +extended region. The Paβ redshifted absorption, and to a lesser +extent, the Brγ redshifted absorption, correlates well with the +Hα and Hβ blue and red high-velocity wings. This indicates +that they all form in the same region close to the star. The near- +infrared profiles are strongly anticorrelated with the highly vari- +able blueshifted absorption components that appear at velocities +of about -100 km s−1 in the Balmer line profiles and presumably +trace outflows. Except for this feature, the overall correlated vari- +ations of optical and near-infrared line profiles suggest that they +are all good diagnostics of the accretion flow, as previously sug- +gested by Alcalá et al. (2014), for example. Detailed modeling +of the line profile shapes and variability is needed, however, to +ascertain the exact location from which they arise (Tessore et al. +2023). +The HeI 10830 Å line profile is more complex. It is usu- +ally dominated by an emission peak at low velocities that be- +comes quite weak at specific times, however. The profile also +often displays highly redshifted absorption features, similar to +the IPC components seen in the Paβ and Brγ lines. Unlike the +near-infrared hydrogen profiles however, the HeI line addition- +ally exhibits various absorption components in the blue wing of +the profile. At least two systems of blueshifted absorptions can +be identified: a low-velocity system extending between -20 to +-80 km s−1, and a high-velocity system ranging from -100 to - +250 km s−1. Figures 10 and 11 show that the two systems are +quite variable. These two absorption systems are reminiscent of +those observed in the Hα and Hβ lines, although they occur at +slightly bluer velocities in the HeI profile, with an offset of about +-40 km s−1 compared to the optical profiles. +Correlation matrices involving the HeI 10830 Å line are dis- +played in Fig. B.4. The HeI autocorrelation matrix shows sev- +9 Paγ and Paδ also appear in SPIRou spectra. Their shape and variabil- +ity behavior is similar to that of Paβ and Brγ. Brδ is also included in the +SPIRou wavelength range, but lies at the intersection of spectral orders. +In all SPIRou spectra, we also detect a weak H2 2.12 µm line, with EW += 0.22 ± 0.04 Å, and FWHM = 1.40 ± 0.05 Å (≃ 20 km s−1). This line +is located at the stellar rest velocity (Vr= 0.88 ± 0.75 km s−1). +eral correlated components. Over the velocity channels extend- +ing from -100 km s−1 up to 200 km s−1, the line flux varies in a +correlated fashion, but this region of the line does not correlate +with the rest of the profile. Similarly, the redshifted absorption +region, which extends from 200 km s−1 to 400 km s−1, varies as +a whole and does not correlate with the rest of the profile. The +blue wing of the profile presents many short-duration absorp- +tion components superimposed on the emission component, and, +probably due to this, each velocity bin varies independently of +the other. Comparison of the HeI 10830 Å line profile to optical +and near-infrared hydrogen profiles indicates that the emission +core and peak intensity are correlated, as are the high-velocity +redshifted absorptions seen in the HeI, Paβ, and Brγ line pro- +files. However, the HeI blue wing shows a strong anticorrelation +with the core of the Hβ profile. +The periodogram analysis of the line profiles is shown in Fig- +ure 11. In the three lines, the IPC components appear to be peri- +odically modulated at the stellar rotation period. As the SPIRou +dataset extends over four months, this suggests that a quite sta- +ble structure, presumably the accretion funnel flow, gives rise +to this spectral feature10. The redshifted absorption component +is deeper when the emission line is more intense, which agrees +with the assumption that the two components trace the densest +part of the accretion column close to the star. The low-velocity +red wing of the HeI line profile also shows a periodic modula- +tion at the same period, which might trace the visibility of the +accretion shock, as seen in the optical HeI line profile. The pe- +riodogram power in the blue wing of the lines is weaker, ex- +cept perhaps over the high-velocity channels of the Paβ line. In +particular, the variable blueshifted absorption systems seen in +the HeI line profile are not periodically modulated on this long +timescale. We performed a similar analysis of each SPIRou run +individually, and the results are shown in Appendix E. Fig. E.1 +to E.4 reveal that the most conspicuous high-velocity redshifted +absorptions in the HeI, Paβ, and Brγ line profiles occur prefer- +entially around Φrot=0. During the SPIRou October run, which +was simultaneous with the OHP/SOPHIE observations, we re- +alized that the highly blueshifted velocity channels of the near- +infrared lines appear to be modulated at the stellar rotation pe- +riod (see Fig. E.2). Hence, while the modulation of high-velocity +blueshifted absorptions seen in the HeI profile disappear on a +timescale of several months, the modulation may survive over +a few rotational periods. Finally, similar to what is seen in the +Balmer line profiles over a much shorter time span (see Sec- +tion 3.2.3), there is a marginal indication from the periodograms +of the near-infrared lines that the modulation period drifts from +the blue to the red wing of the profiles. This might be a sign of +differential rotation in the source of the variability. +The EW of the HeI 10830 Å, Paβ, and Brγ lines was com- +puted on the residual profiles using two methods. The first +method consists of adjusting a Gaussian fit to the line profile, +and the second method is integrating below the line profile. The +two methods were applied to the Paβ and Brγ lines that most +often exhibit a strong Gaussian-like emission and, at times, a +pronounced redshifted absorption. The difference between the +EW derived from the Gaussian fit and from the profile integra- +tion then provides an estimate of the strength of the redshifted +absorption component. The HeI line exhibits a complex profile, +with pronounced absorptions appearing in both the blue and red +wings, and it cannot be fit by a Gaussian. We report here HeI +10 The long time coverage of the SPIRou observations enabled us to +explore periods up to 100 days. However, we did not find significant +periods longer than those reported here. +Article number, page 13 of 30 + +A&A proofs: manuscript no. 45342corr +Fig. 10. Near-infrared line profiles: HeI (left), Paβ (center), and Brγ (right), plotted with arbitrary offsets as a function of Julian date. Each color +represents a SPIRou run, namely September (blue), October (orange), November (green), December 2021 (red), and January 2022 (purple). The +October SPIRou run is contemporaneous to the OHP/SOPHIE observations. +Article number, page 14 of 30 + +Hel +Paβ +Bry +day +day +day +9473.1 +9473.1 +9473.1 +9475.1 +9475.1 +9475.1 +9477.0 +0'LL v6 +9477.0 +9478.0. +9478.0 +9478.0 +9480.1 +9480.1 +9480.1 +9481.1 +9481.1 +9481.1 +9482.1 +9482.1 +9482.1 +9502.1 +9502.1 +9502.1 +T'E0S6 +9503.1 +9504.1 +9504.1 +9504.1 +9506.1 +9506.1 +9506.1 +9508.1 +9508.1 +9508.1 +9509.1 +9509.1 +9509.1 +9510.1 +9510.1 +9510.1 +9511.1 +9511.1 +9511.1 +9513.1 +9513.1 +9513.1 +9514.1 +9514.1 +9514.1 +9515.1 +9515.1 +9515.1 +9516.1 +9516.1 +9516.1 +9535.1 +9535.1 +9535.1 +9537.1 +9537.1 +9537.1 +9538.1 +T:8Es6 +9538.1 +9539.0 +0:6ES6 +9539.0 +9540.0 +9540.0 +9540.0 +9541.0 +9541.0 +9541.0 +VA +9558.0 +9558.0 +9558.0 +9559.0 +9559.0 +9559.0 +9560.0 +9560.0 +9560.0 +9561.0 +9561.0 +9561.0 +9563.1 +9563.1 +9563.1 +9564.1 +9564.1 +9564.1 +9566.1 +9566.1 +9566.1 +9567.0 +9567.0 +9567.0 +9586.0 +9586.0 +9586.0 +一 +一 +-500 +0 +500 +-500 +0 +500 +-500 +0 +500 +v (km/s) +v (km/s) +v (km/s)J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. 11. Near-infrared line profile variability. Top: Series of residual near-infrared line profiles HeI (left), Paβ (center), and Brγ (right), plotted +superimposed to illustrate their variability. Each color represents a SPIRou run as in Fig. 10. Bottom: 2D periodograms across the line profiles. +The color code reflects the periodogram power from zero (blue) to 0.5 (red). The dotted red horizontal line drawn at a frequency of 0.166 day−1 +indicates the stellar rotational period. The white curve displays the mean line profile. +EWs measured through profile integration only. EW measure- +ments are usually accurate to within 5% because of the well- +defined adjacent stellar continuum level. +The results are listed in Table 7, and the evolution of EW +during the observing campaign is illustrated on Fig. 12. While +the EW variations of the three lines are usually correlated, there +are notable exceptions. For instance, during the first SPIRou run +around JD 2,459,478, the HeI line goes into absorption, driven +by a broad redshifted absorption component that reaches half the +continuum value, while the Paβ and Brγ lines reach a local inten- +sity maximum, even though they also exhibit a pronounced red- +shifted absorption (see Fig. E.1). In contrast, during the second +SPIRou run, the variations of the three lines are well correlated. +Line EWs and near-infrared veiling measurements (listed in Ta- +ble 3) are both shown as a function of Julian date and rotational +phase in Fig. D.1 (Appendix D). Both quantities appear to be ro- +tationally modulated, with maximum values occurring close to +Φrot=0. +Finally, the photospheric radial velocity curve derived from +SPIRou spectra is shown in Figure 13. Vr is found to vary +between 13.96 and 15.04 km s−1, with a median value of +14.65 km s−1. A CLEAN periodogram analysis reveals a period +P=5.94 ± 0.11 d, and the String-Length method yields P = 5.98 +± 0.12 d, both consistent with the 6.04 ± 0.15 d photometric +period derived in Section 3.1. The photospheric radial velocity +curve folded in phase at the stellar rotation period is shown in +Fig. 13. Vr exhibits a roughly sinusoidal variations in rotational +phase, which suggests that it is modulated by a surface spot. It +reaches the median velocity going blueward around Φrot= 0, as +expected from a stellar spot facing the observer at this phase. Be- +cause this is also the phase of maximum brightness of the system +(see Section 3.1), this suggests that a hot spot modulates the pho- +tospheric Vr curve. The Vr curve derived from the near-infrared +photospheric lines (see Fig. 13) is inverted compared to the Vr +curve derived for the HeI 5876 Å NC line profile (see Fig. 9). +Similar antiphase radial velocity variations between absorption +Fig. 12. EW of near-infrared lines. Top: EW of the HeI, Paβ, and Brγ +lines measured on SPIRou spectra plotted as a function of Julian date. +The measurement uncertainties are about the symbol size. Bottom: Dif- +ference between the EWs measured by direct line profile integration +(i.e., including the redshifted absorption component) and the EW de- +rived from the Gaussian fitting of the emission component only plot- +ted as a function of Julian date. This differential quantity measures the +strength of the redshifted absorption component in the Paβ and Brγ line +profiles. The more negative the differential quantity, the deeper the red- +shifted absorption. +and emission lines have been reported for other accreting T Tauri +stars and were interpreted as caused by the modulation of the line +profiles by a hot spot at the stellar surface (Petrov et al. 2001, +2011; Gahm et al. 2013). Alternatively, the photospheric radial +velocity variations may also be produced by a cool spot that co- +exists with the accretion shock at the same location on the stellar +Article number, page 15 of 30 + +Hel +2.5 +2.0 +E 1.5 +1.0 +0.5 +-400 +-200 +0 +200 +400 +v (km/s)Paβ +3.0 +2.5 +2.0 +1.5 +1.0 +-400 +-200 +0 +200 +400 +v (km/s)Bry +1.6 +1.4 +xn +1.2 +1.0 +0.8 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +(1/d) +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +(1/d) +0.35 +Frequency ( +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)20 +Hel +PaB +BrG +3 10 +EW +C +9480 +9500 +9520 +9540 +9560 +9580 +JulianDate-2,450,000PaB +BrG +dEW +9480 +9500 +9520 +9540 +9560 +9580 +JulianDate-2,450,000A&A proofs: manuscript no. 45342corr +Fig. 13. Radial velocity variations measured in the near-infrared photo- +spheric lines of the SPIRou spectra. Top: Radial velocity as a function of +Julian date. Bottom: Radial velocity curve folded in phase at the stellar +rotational period P=6.04 days. The color code reflects the Julian date. +Table 7. Near-infrared line EW measurements from the CFHT/SPIRou +spectra. +Julian date +EW (Å) +(2,450,000+) +HeIint +Paβg +Paβint +Paβdif f +Brγg +Brγint +Brγdif f +9473.066 +2.88 +8.29 +7.41 +-0.87 +3.26 +2.94 +-0.32 +9475.043 +0.98 +7.58 +7.40 +-0.18 +3.07 +2.73 +-0.34 +9476.969 +-0.76 +6.05 +5.82 +-0.23 +2.60 +1.95 +-0.65 +9478.027 +-2.45 +11.58 +9.64 +-1.93 +6.97 +5.70 +-1.26 +9480.082 +2.57 +10.74 +9.76 +-0.97 +4.56 +4.05 +-0.51 +9481.086 +1.86 +9.90 +9.65 +-0.25 +4.15 +3.80 +-0.35 +9482.086 +0.98 +5.18 +5.53 +0.35 +1.72 +1.47 +-0.25 +9502.074 +3.33 +10.82 +10.02 +-0.80 +5.37 +4.73 +-0.63 +9503.074 +2.43 +9.02 +8.12 +-0.89 +3.02 +2.17 +-0.85 +9504.098 +2.58 +12.18 +11.48 +-0.70 +5.09 +4.63 +-0.45 +9506.082 +2.19 +11.36 +11.80 +0.44 +6.54 +6.29 +-0.24 +9508.082 +7.33 +16.93 +16.12 +-0.81 +9.00 +8.48 +-0.51 +9509.086 +9.45 +19.84 +19.11 +-0.73 +9.68 +9.16 +-0.51 +9510.086 +10.53 +17.07 +16.80 +-0.27 +8.27 +7.57 +-0.70 +9511.074 +4.73 +14.56 +14.20 +-0.36 +7.48 +7.48 +0.00 +9513.090 +0.73 +8.00 +8.32 +0.31 +4.68 +4.82 +0.13 +9514.051 +4.82 +13.19 +12.92 +-0.27 +6.88 +6.37 +-0.51 +9515.078 +1.03 +12.18 +11.88 +-0.30 +5.81 +5.71 +-0.09 +9516.102 +0.51 +11.26 +10.78 +-0.48 +5.02 +4.62 +-0.39 +9535.102 +5.37 +11.00 +10.67 +-0.33 +6.03 +5.83 +-0.20 +9537.098 +0.96 +9.65 +9.96 +0.30 +4.93 +5.04 +0.11 +9538.039 +2.84 +6.93 +6.13 +-0.80 +2.74 +2.31 +-0.42 +9538.984 +4.10 +7.99 +6.49 +-1.50 +3.12 +2.23 +-0.88 +9539.988 +3.45 +9.12 +8.52 +-0.59 +3.08 +2.63 +-0.45 +9541.012 +5.32 +12.48 +12.53 +0.05 +5.46 +5.46 +0.00 +9557.980 +3.13 +9.12 +8.55 +-0.57 +3.96 +3.42 +-0.53 +9558.949 +4.88 +13.61 +13.41 +-0.20 +6.25 +6.14 +-0.10 +9559.961 +1.81 +10.18 +10.16 +-0.02 +5.77 +5.63 +-0.14 +9560.965 +1.29 +6.34 +6.67 +0.33 +3.85 +3.82 +-0.02 +9563.078 +7.11 +12.10 +11.03 +-1.07 +5.48 +4.56 +-0.92 +9564.059 +7.27 +10.85 +10.41 +-0.44 +5.09 +4.72 +-0.37 +9566.051 +2.98 +6.62 +6.78 +0.15 +2.62 +2.44 +-0.17 +9566.953 +1.05 +7.17 +7.57 +0.39 +4.39 +4.47 +0.08 +9585.941 +6.31 +13.67 +13.56 +-0.11 +6.47 +6.50 +0.02 +Note: EWg is obtained from a Gaussian fit to the line profile, while EWint is measured from +line profile integration. EWdif f is the difference between EWint and EWg. +surface, as Doppler images of accreting T Tauri stars suggest +(e.g., Donati et al. 2010, 2011, 2019, 2020a). +3.4. Low-resolution near-infrared spectroscopy +The median nightly spectra obtained from the ExTrA-T2 and +ExTrA-T3 telescopes are shown in Figure 14. We computed the +0.9 +1.0 +1.1 +1.2 +1.3 +Wavelength [µm] +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Relative intensity T2 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Relative intensity T3 +HeI +Paβ +Paγ +Paδ +Paϵ +Paζ +Paη +Paθ +9500 +9525 +9550 +9575 +9600 +9625 +9650 +BJD - 2,450,000 +0 +10 +20 +EW(Paβ) [˚A] +Fig. 14. Median spectrum for each night from the ExTrA-T2 telescope +(lower part, 88 spectra) and ExTrA-T3 telescope (upper part, 69 spec- +tra). The main emission lines are indicated. The color is proportional to +the EW of the Paβ line shown in the top panel as a function of Julian +date. +EW of the HeI 10830 Å, Paβ, Paγ, and Paδ lines from the ExTrA +spectra using specutils11. We analyzed each telescope inde- +pendently because the point spread function (and therefore the +resolution) of the spectrograph depends on the position on the +detector. First, we fit a Gaussian to each line on the median spec- +tra to measure the line center and full width, the latter we defined +as amounting to six times the standard deviation of the Gaussian +fit in order to isolate the line profile from the nearby continuum. +The local continuum around each line was modeled with a third- +degree polynomial, adjusted on three line widths centered on the +line, but excluding the line. Then, for each line in each of the +1898 individual spectra, we computed the EW by integrating the +flux over the full line width using the parameters and the nor- +malization region defined from the median spectra. Finally, we +computed the median of the individual measurements for each +night, regardless of the telescope. The median is less affected +by outliers than the mean, and the differences between the mean +and the median values are within the error bars. Table C.1 in +Appendix C lists the results. +In order to estimate the reliability of the procedure, we com- +pared EWs derived from ExTrA and from SPIRou spectra for +the Paβ line using 20 measurements obtained on each instru- +ment less than one day apart. The results show an excellent cor- +relation, with a slight tendency for the EW measured from Ex- +TrA to exceed those measured from SPIRou, with a mean dif- +ference of 0.7±1.1 Å. A similar result is obtained for the HeI +line, with a mean difference of 0.31 Å and an rms of 1.1 Å be- +tween ExTrA and SPIRou estimates. The comparison with high- +resolution measurements thus validates the EWs obtained from +low-resolution spectra. +The line variability is illustrated on Figure 14, where the me- +dian nightly spectra are superimposed. The median and extreme +11 https://github.com/astropy/specutils +Article number, page 16 of 30 + +9580 +15.0 +9560 +(km/s) +9540 +14.5 +D +Vrad +9520 +9500 +14.0 +9480 +9480 +9500 +9520 +9540 +9560 +9580 +JulianDate-2.450.0009580 +15.0 +9560 +(km/s) +9540 +14.5 +D +Vrad +9520 +9500 +14.0 +9480 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +PhaseJ. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. 15. EW variability of the near-infrared line profiles. Left: EW measurements plotted as a function of Julian date for the HeI, Paβ, Paγ, and +Paδ lines from the ExTrA spectra. Right: GLS periodogram of the EW measurements. The period on the x-axis is displayed on a log scale. +Fig. 16. EWs of near-infrared line profiles and J-band magnitude vari- +ability. Top: EW of the near-infrared HeI and Paβ lines folded in phase +at the stellar rotational period. EW(HeI) is offset by -10 Å for clar- +ity. Bottom: J-band light curve, deduced from ExTrA spectra, folded in +phase at the stellar period. The brightness modulation is similar to that +observed at optical wavelengths (see Fig. 2). +Fig. 17. Mid-term variation of EW(Paβ) (red) compared to the system +u’-band light curve (blue) for measurements taken less than one day +apart. To facilitate the comparison, the EW measurements are plotted +on a magnitude scale and are offset, namely -2.5 log EW(Paβ) + 5. +Table 8. Minimum, median, and maximum near-infrared line EW mea- +surements from the ExTrA spectra. +EW (Å) +HeI +Paβ +Paγ +Paδ +Min. +-1.9 +2.8 +3.4 +0.2 +Med. +2.8 +10.5 +7.0 +2.1 +Max. +11.8 +18.6 +11.4 +4.8 +values of EWs measured for the HeI, Paβ, Paγ, and Paδ lines are +listed in Table 8, and the night-by-night measurements are listed +in Table C.1. We note that the HeI line at times appears to be in +absorption at this low spectral resolution. +Figure 15 shows the EW measurements plotted as a function +of time and its generalized Lomb-Scargle periodogram (GLS; +Zechmeister & Kürster 2009). The EW of the four lines is +found to be modulated with a period of 6.028 ± 0.087 days, +consistent with the stellar rotation period, where the error es- +timate is the standard deviation of a Gaussian fit to the peri- +odogram peak. Fig. 16 shows the HeI and Paβ line EWs folded +in phase at the stellar rotation period. The modulation of the line +strength clearly appears, with maximum flux around phase zero. +As shown in the same figure, these variations follow the modu- +lated brightness level of the system in the J band. +Longer-term EW variations of higher amplitudes are also +clearly seen in Fig. 15. These variations seem to be correlated +with the multicolor photometry presented in Section 3.1, in par- +ticular, during the brightness event centered on JD 2,459,509 and +the wide dip around JD 2,459,549. In Figure 17, a clear correla- +tion appears between the photometric variations in the u’ band +and the Paβ line variations. If the brightening of the system in +the u’ band is linked to accretion, this correlation suggests that +most of the emission line flux is connected to the same process. +3.5. Mass-accretion rate +The combination of optical and near-infrared spectroscopy of- +fers a number of emission lines from which we can estimate +the mass accretion rate onto the star, using the empirical re- +lations between line luminosity and accretion luminosity pro- +posed by Alcalá et al. (2017). Combining the range of Hα and +Hβ EWs reported above with the nearby continuum fluxes com- +puted from the r’ and g’ magnitudes corrected for extinction, +we obtain the line fluxes and luminosities as follows: Fline = +Article number, page 17 of 30 + +20 +Hel +Paβ +Pa +Pad +15 +A +10 +EW +5 +0 +9500 +9520 +9540 +9560 +9580 +9600 +9620 +9640 +BJD - 2.450.0000.25 +0.20 +Normalized power +0.15 +0.10 +0.05 +0.00 +2 +3 4567 10 +20 +30 4060 +Period [d]20 +EW(PaB) +EW(Hel)-10 +10 +3 +2459600 +2459550 +-10 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase9.2 +2459600 +9.4 +(mag +2459550 +9.6 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase11 +PaB +12 +M +14 +15 +9500 +9520 +9540 +9560 +9580 +Juliandate-2.450.000A&A proofs: manuscript no. 45342corr +Fo +λ × EW(line) × 10−0.4(mλ−Aλ) and Lline = 4πd2Fline, where Fo +λ is +the flux of a zero-magnitude star12, mλ and Aλ are the magnitude +and extinction in the photometric band of interest, and d is the +distance to the star. The accretion luminosity was derived from +the line luminosity using the relation reported by Alcalá et al. +(2017), and ˙Macc was deduced from Lacc assuming a magneto- +spheric radius of 5 R⋆ (see Alcalá et al. 2017, Eq.(1)). Taking +the uncertainties on all involved quantities into account, we thus +derive ˙Macc = 0.7 ± 0.3 and 0.5 ±0.4 × 10−8 M⊙yr−1 from the +Hα and Hβ line fluxes, respectively. +We performed a similar analysis using the extensive mea- +surements of Paβ EWs obtained from the ExTrA spectra. From +the extreme values, namely EW(Paβ) = 2.8 –18.6 Å, and the +mean REM J-band magnitude of the system during the observ- +ing period, J = 9.41 ± 0.10, we derive +˙Macc = 0.3 – 2.0 × +10−8 M⊙yr−1, with a median value of ˙Macc = 1.0 × 10−8 M⊙yr−1. +Finally, we derived additional estimates of ˙Macc from the Brγ +line flux, using the median and extreme values of EW(Brγ) mea- +sured on SPIRou spectra, namely 5.1, 1.7, and 9.7 Å, which +we calibrated with the mean REM K’-band magnitude of the +system, K’= 8.40. Using the relations reported by Alcalá et al. +(2017) between line and accretion luminosity, we thus derived +˙Macc = 0.2-1.8 × 10−8 M⊙yr−1, with a median value of ˙Macc = +0.8 ×10−8 M⊙yr−1. +The dispersion in the ˙Macc estimates partly results from in- +trinsic ˙Macc variability. For instance, the SOPHIE spectra were +obtained during a brightening of the system that occurred around +JD 2,459, 512, at a time at which all optical and infrared lines +were relatively strong. We thus derive a relatively high value of +˙Macc from these spectra compared, for example, to the smaller +˙Macc estimate derived from the Brγ line in the SPIRou spec- +tra that were obtained during more quiescent phases of the sys- +tem. We cannot exclude, however, that some of the dispersion +may also arise from systematic uncertainties in the empirical +line-to-accretion luminosity relations over the optical and near- +infrared wavelength ranges. In any case, the various estimates +agree globally, with ˙Macc typically varying between 0.2 and 2.0 +×10−8 M⊙yr−1. +4. Discussion +The monitoring campaign we performed on GM Aur reveals +significant but relatively low-level temporal variability over a +timescale of six months. The photometric variations are mild, as +might be expected for this moderately accreting young system +( ˙Macc ∼ 0.8 × 10−8 M⊙yr−1), with amplitudes ranging from 1.5 +mag in the u’ band to 0.3 mag in the i’ band, and 0.1 mag in the +J band. The brightness of the system varies smoothly because it +is modulated by the visibility of surface spots at the stellar rota- +tional period of 6.04 d. Except for the HeI 10830 Å line profile, +the spectral appearance of the system does not change drasti- +cally on a timescale of months, and the veiling is low and stable, +amounting to about 0.3 in the optical range. The strength of the +main emission lines (Hα, Hβ, HeI 5876Å, Paβ, and Brγ) varies +by a factor of 2 to 3 over the course of the semester. In contrast, +the HeI 10830 Å line profile exhibits extreme variability and is +sometimes barely noticeable in emission. The line profile shapes +are strongly variable on a timescale of days, and the develop- +ment of both blueshifted and redshifted absorption components +is superimposed onto a broad emission component. Blueshifted +12 Fo +λ = 2.43 10−9 and 5.27 10−9 erg s−1 cm−2 Å−1 in the r’ and g’ bands, +respectively. +absorption components indicate outflows (e.g., Edwards et al. +2003; Kwan et al. 2007), and redshifted components probe fun- +nel flows (e.g., Edwards et al. 2006; Fischer et al. 2008). Re- +markably, the redshifted absorption features that reach below the +continuum level, that is, inverse P Cygni profiles (IPC) (see Cal- +vet & Hartmann 1992), which are seen in the high-resolution +near infrared line profiles (HeI 10830 Å, Paβ, Brγ), are steadily +modulated by stellar rotation over an extended observing time +span of 3 months. Rotational modulation is also clearly detected +in the strength of the near-infrared lines and is continuously ob- +served at low resolution for more than five months. Assuming +that most of the line flux arises from the magnetospheric accre- +tion region, as suggested by the periodic appearance of the IPCs +and the correlation between line flux and u’-band excess, this in- +dicates a stable large-scale accretion structure on this timescale. +In contrast, high-velocity blueshifted absorption components +are neither periodic nor stable on this timescale. While they are +ubiquitous in the optical line profiles, most notably Hα and Hβ, +and are also quite conspicuous in the HeI 10830 Å line profile, +their signature evolves on a timescale of a few days, sometimes +drifting in velocity before disappearing altogether. As an exam- +ple, Fig. E.3 shows the evolution of the blueshifted absorption +components in the HeI line profile over the course of the SPIRou +November run. On JD 9537, a high-velocity blueshifted feature +appears in the profile and drifts toward lower velocities over the +next several days, from -200 km s−1 to -120 km s−1. Soon after, +on JD 9540, a new high-velocity component appears and follows +the same trend. A similar behavior is seen in the high-velocity +blueshifted component appearing in the HeI 10830 Å line pro- +file during the SPIRou September run, which drifted from -240 +km s−1 to -150 km s−1 on a timescale of five days, from JD 9475 +to JD 9480 (see Fig. E.1). This suggests episodic outflows lasting +for a few days only. We see no sign of a steady, constant veloc- +ity wind in the HeI line profiles over the observing period, nor +do we find evidence for a rotational modulation of the outflow +signatures. The only stable outflow signature seen in the optical +and near-infrared line profiles consists of a narrow, low-velocity +blueshifted absorption in the Hα profile that peaks at -20 km s−1 +and remains visible over more than two weeks. +Observations thus suggest that we witness a globally stable +accretion structure and a succession of short-lived episodic out- +flows. The contrasting behavior of accretion and outflow diag- +nostics observed on a timescale of months thus raises the ques- +tion whether the two processes are physically connected on the +scale of a few stellar radii that we probe here. To examine this is- +sue, we investigated the aftermath of the brightening event GM +Aur underwent around JD 2,459,509. On this date, the system +exhibited a significant brightening at optical and near-infrared +wavelengths, most notably in the u’ band (∼1 mag), as well as +some of the strongest line fluxes and highest optical and near- +infrared veiling values measured during the campaign. A simul- +taneous TESS light curve of the system recorded the brightening +event (see Fig. 2), which started on JD 2,459,508 and ended on +JD 2,459,511, and exhibited a flat peak lasting for two days with +a 20% flux increase. We therefore interpret this episode as an +accretion burst that occurred around the rotational phase Φrot=0, +corresponding to the maximum visibility of the accretion shock, +and lasted for several days. From the measured continuum level +and Balmer line EW during the burst, we derive an increase of +a factor of 2 in the accretion rate, reaching ∼2×10−8 M⊙yr−1. +Inspection of the optical and near-infrared line profiles during +and after this event reveals high-velocity blueshifted absorption +components that appear in the Balmer and HeI 10830 Å line pro- +Article number, page 18 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. 18. Magnetospheric ejection model. Development of a magnetospheric ejection computed from star-disk interaction MHD simulations. The +snapshots shown here are extracted from model 3 of Pantolmos et al. (2020), where the magnetospheric truncation radius amounts to 54% of the +corotation radius. The three snapshots are shown at 0.2, 0.5, 0.8, and 1.1 Prot. The white curves indicate expanding magnetic field lines that give +rise to the ejection of plasmoids. The color scale indicates density. The green arrows show the velocity field of the stellar and disk winds and of +MEs at their interface. +Fig. 19. Magnetospheric ejection model. Velocity of the gas in the +ejected plasmoid at a distance of 10 (magenta), 20 (blue), and 30 (cyan) +stellar radii as a function of time in units of the stellar rotational pe- +riod. Successive ejections of plasmoids are featured. The vertical lines +correspond to the snapshots shown in Fig.18. +files. On JD 2,459,509 a new blueshifted absorption component +appears in the HeI infrared line at a velocity of -280 km s−1and +extends down to -350 km s−1 before it reaches the continuum +level again, and it drifts to lower velocities (∼ -150 km s−1) over +the next few days. The closest optical spectrum was recorded +only toward the end of the accretion burst, on JD 2,459,510.6. In +this spectrum, the Hα and Hβ profiles feature a new blueshifted +absorption component at a velocity of -110 km s−1 that lasts for +a few days. The contemporaneous occurrence of an accretion +burst rapidly followed by outflow signatures in the line profiles +therefore suggests that the accretion and ejection processes are +physically connected on small scales. +We propose that the most likely scenario that accounts for +these episodic events is magnetospheric ejections, possibly trig- +gered by magnetic reconnections in the accreting magneto- +sphere. Magnetospheric ejections (Zanni & Ferreira 2013; Sauty +et al. 2022), or nonstationary conical winds (Romanova et al. +2009), are caused by the expansion and reconnection of the field +lines that connect the star with the disk. The inflation process +is the result of the star-disk differential rotation and the conse- +quent build-up of toroidal magnetic field pressure (e.g., Good- +son et al. 1997). Quasi-periodic ejections of plasmoids are pre- +dicted to occur throughout the magnetospheric inflation cycle on +a timescale of about the rotational period (Hayashi et al. 1996). +The speed and variability of the outflows likely depend on var- +ious parameters, such as the magnetic field strength and topol- +ogy, the thermal disk pressure, nonideal MHD effects, and the +interaction of the magnetospheric-ejection region with the sur- +rounding outflows, that is, stellar and disk winds (Miller & Stone +1997; Romanova et al. 2009; Zanni & Ferreira 2013). Previous +monitoring campaigns on young stellar objects have reported ev- +idence for magnetospheric inflation cycles (Bouvier et al. 2003; +Alencar et al. 2018). We show a 2.5D MHD simulation of the +interaction between an inner accretion disk and a dipolar mag- +netosphere from Pantolmos et al. (2020) in Fig. 18. The figure +illustrates the ejection of plasmoids along expanding field lines +at the distance of a few stellar radii from the stellar surface. The +timescale for successive ejections is about that of the stellar ro- +tation period. The outflow speed, shown in Fig 19, reaches more +than 200 km s−1 in the early phases of the ejection, then deceler- +ates to about 150 km s−1 on a timescale of days (0.3×Prot), and +finally vanishes altogether. A direct comparison of the model to +observations is not straightforward. The terminal speed we de- +rive from observations is higher than predicted by the simula- +tion, and its temporal evolution on a timescale of a few days +might be dominated by projection effects as the system rotates +and does not trace the evolution of the plasmoid velocity field. +Moreover, the sporadic ejections we observe do not appear to +have the quasi-periodic character of the model. Both the wind +speed and ejection timescale may, however, depend on numerical +effects. In any case, the behavior of the magnetospheric ejection +model qualitatively matches the dynamics of the high-velocity +blueshifted absorptions seen in the line profiles of GM Aur on a +timescale of days to weeks. Moreover, in the scenario of a mag- +netospheric inflation cycle, magnetic reconnection leads to an +accretion burst and simultaneously triggers an ejection episode +Article number, page 19 of 30 + +9- +-5 +-3 +-2 +-1 +-4 +0 +1 +Log10(p/p*) +t= 0.2P* +t= 0.5P* +t= 0.8P* +t= 1.1P* +326 km/s +30 +20 +R/R* +10 +十0 +0 +10 +20 +30 0 +10 +20 +30 0 +10 +20 +30 0 +10 +20 +30 +R/R* +R/R* +R/R* +R/R*250 +200 +Speed [km/s] +S +150 +10R* +20R* +30R* +100 +0.0 +0.5 +1.0 +1.5 +2.0 +Time [Prot]A&A proofs: manuscript no. 45342corr +(e.g., Goodson & Winglee 1999). This is quite reminiscent in- +deed of what is suggested by the variability of GM Aur. +We also noted a change in the GM Aur light curve whose first +part is dominated by successive low-level brightening events, up +to the major accretion burst described above, while it exhibits +luminosity dips toward the end of the observations. It is unclear +whether the contrasting behavior of the system luminosity ob- +served before and after the main accretion burst is a consequence +of the burst itself, perhaps inducing a structural change in the +star-disk interaction process. It is conceivable that the rearrange- +ment of the magnetic topology after the inflation or reconnection +event that led to the burst has impacted the large-scale geometry +of the star-disk interaction region. Either a modest increase in +the inner disk scale-height (Nagel et al. 2017) or a reduction of +the extent of the truncation radius is prone to trigger a periodic +obscuration of the central star by circumstellar dust, that is, a dip- +per phenomenon (Cody et al. 2014), especially in young systems +seen at high inclination (McGinnis et al. 2015; Bodman et al. +2017). The magnetic topology and the mass accretion rate may +thus have slightly evolved over the six-month time span of the +campaign, as suggested by the long-term variations of the emis- +sion line EWs shown in Fig. 15. However, the results we report +here clearly indicate that the large-scale geometry of the star- +disk interaction was not drastically modified over this timescale. +This is evidenced by the phase stability of the modulated light +curve, the smooth variability of the emission line profiles around +their mean shape, and the strictly periodic appearance of IPC +profiles over at least three months. All these accretion diagnos- +tics support a globally stable magnetospheric accretion structure +during the campaign. +Finally, it is interesting to compare the line profile shape and +variability reported here to those reported by McGinnis et al. +(2020), which were obtained in 2011, that is, ten years prior to +our observing campaign. The shapes of the Hα and Hβ profiles +are quite different in the two studies. In McGinnis et al. (2020), +these are pure emission profiles with a triangular shape, with- +out any significant absorption components, and they exhibit lit- +tle variability over the timescale of a week (see their Figures +S5, S6). Here, the same profiles appear to be much more struc- +tured, with highly variable absorption components on the same +timescale. The HeI 5876Å line profile variability also differs be- +tween the two studies, but in the opposite direction. In both stud- +ies, the line profile consists of a broad and a narrow component. +However, in McGinnis et al. (2020), the intensity of the broad +component clearly varies, especially on the blue wing, while +we found it to be quite stable here. It seems that the behavior +of the system was different between the two epochs. The Hα +and Hβ line EWs are indeed systematically higher in McGinnis +et al. (2020) and were comparable to the highest values we mea- +sured here during the JD 2,459,509 accretion burst. It is therefore +likely that GM Aur was in a state of more active accretion during +the 2011 observations. This is consistent with the more triangular +shape and lack of structure of the Balmer emission line profiles, +which are predicted to become more optical thick as the funnel +flow density increases (Muzerolle et al. 2001). It is also consis- +tent with the higher level of variability seen in the blue wing of +the broad component of the HeI 5876 Å line profile that may +betray the existence of a hot accretion-driven wind at times of +enhanced accretion (Beristain et al. 2001). Long-term changes +in the system behavior driven by a varying mass accretion rate +and/or a change in the magnetic topology are therefore likely to +occur. Over the six-month span of our observing campaign, the +significant variation observed in the EW of the Paβ line profile, +which by the end of the campaign reaches similar levels to those +measured during the JD 2,459,509 accretion burst (see Fig. 15), +suggests that such changes may occur on a timescale of a few +weeks to months. +5. Conclusion +By combining optical and near-infrared high-resolution spectro- +scopic time series, seconded by a long-term monitoring of the +photometric variability of the system and low-resolution near- +infrared spectrophotometry, we were able to characterize the ac- +cretion and ejection process occurring in the young system GM +Aur on a timescale ranging from days to months. We report a +stable accretion pattern according to which the large-scale mag- +netic field of the star controls the accretion of gas from the in- +ner disk onto the central star along funnel flows. The appear- +ance of inverse P Cygni profiles that signal the crossing of fun- +nel flows on the line of sight is remarkably periodic at the stel- +lar rotation period of 6.04 days. Similarly, the photometric and +line flux variations, both driven by the visibility of the accre- +tion shock located at the foot of the main accretion column, are +modulated at the same period. While the amplitude varies, the +phase of variability of all these accretion diagnostics remains +stable over the 30 rotational periods covered by the campaign. +This suggests that the underlying magnetic topology that con- +trols the non-axisymmetric accretion flow, presumably an in- +clined dipole on the large scale, did not undergo major changes +over a timescale of six months. In stark contrast, high-velocity +blueshifted absorption components that indicate outflows appear +at random times in the emission line profiles. They are not rota- +tionally modulated, and their signatures last for a few days only. +We argue that these transient outflows associated with a stable +accretion pattern are best accounted for by magnetospheric ejec- +tion models, as predicted by MHD simulations. Thus, by prob- +ing the dynamics of the star-disk interaction region, these results +show that the physical connection between accretion and ejec- +tion processes that has long been established on large scales also +appears to be valid on the much smaller sub-au scales. +Acknowledgements. We thank the referee for a prompt and detailed report. This +study is based on observations obtained at the Canada-France-Hawaii Telescope +(CFHT) which is operated by the National Research Council (NRC) of Canada, +the Institut National des Sciences de l’Univers of the Centre National de la +Recherche Scientifique (CNRS) of France, and the University of Hawaii. The +observations at the CFHT were performed with care and respect from the sum- +mit of Maunakea which is a significant cultural and historic site; based on ob- +servations made at Observatoire de Haute Provence (CNRS), France; based on +data collected under the ExTrA project at the ESO La Silla Paranal Observatory. +ExTrA is a project of Institut de Planétologie et d’Astrophysique de Grenoble +(IPAG/CNRS/UGA), funded by the European Research Council under the ERC +Grant Agreement n. 337591-ExTrA. We thank Ágnes Kóspál for providing a +reduced TESS light curve of GM Aur. Funding for the TESS mission is pro- +vided by NASA’s Science Mission directorate. This project has received fund- +ing from the European Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agreement no. 742095; +SPIDI: Star-Planets-Inner Disk-Interactions, http://www.spidi-eu.org). We ac- +knowledge funding from the French National Research Agency (ANR) under +contract number ANR-18-CE31-0019 (SPlaSH). SHPA acknowledges financial +support from CNPq, CAPES and Fapemig. JFD acknowledges funding from the +European Research Council (ERC) under the H2020 research & innovation pro- +gramme (grant agreement no. 740651 NewWorlds). AF acknowledges support +by the PRIN-INAF 2019 STRADE (Spectroscopically TRAcing the Disk dis- +persal Evolution) and by the Large Grant INAF YODA (YSOs Outflow, Disks +and Accretion). JFG was supported by fundação para a Ciência e Tecnologia +(FCT) through the research grants UIDB/04434/2020 and UIDP/04434/2020. +This work benefited from discussions with the ODYSSEUS (HST AR-16129) +and PENELLOPE teams. Some of the plots presented in this paper were built +using TOPCAT (Taylor 2005). +Article number, page 20 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +References +Akeson, R. L., Boden, A. F., Monnier, J. 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Fig. B.1 to B.4 present the correlation +matrices of the optical and near-infrared line profiles studied +here. +Appendix C: EWs and J-band measurements from +the ExTrA spectra +We provide the HeI 10830 Å, Paβ, Paγ, and Paδ line EWs and +the J-band photometry measured from the ExTrA spectra in Ta- +ble C.1. For each night, the table lists the mean observation time, +the median EW measurement and its standard deviation for each +spectral line, and the J-band magnitude and its error. +Appendix D: nIR veiling and line EWs +We present in Figure D.1 the evolution of near-infrared veiling +in the JHK bands and of the HeI, Paβ, and Brγ line EWs over the +course of the campaign. +Appendix E: Line profile variability on successive +SPIRou runs +Figures E.1 to E.4 show the line profile variability of the HeI +10830 Å, Paβ, and Brγ line profiles for the successive SPIRou +runs in September, October, November, and December 2021. +The figures also include 2D periodograms for each line. We note, +however, that only the October SPIRou run is long enough, ex- +tending over 14 days, to yield significant results when search- +ing for a modulation of the line profiles around the rotational +period of 6.04 days. The September, November, and December +runs lasted for only 9, 6, and 9 days, respectively, which is too +short to reliably investigate periods longer than 5 days. +Article number, page 22 of 30 + +J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Table A.1. Literature data for comparison stars in the field of GM Aur on the REM cameras. +Id +Name +2MASS +g +r +i +z +J +H +K′ +(mag) +(mag) +(mag) +(mag) +(mag) +(mag) +(mag) +*2 +HD 282625 +J04551650+3022369 +11.310 +10.914 +10.772 +10.707 +9.781 +9.554 +9.487 +*3 +TIC 96533048 +J04551015+3021333 +12.361 +11.399 +10.936 +10.668 +9.295 +8.716 +8.563 +*4 +HD 282626 +J04550536+3020382 +11.747 +11.375 +11.238 +11.174 +10.239 +10.052 +9.987 +*5 +. . . +J04551400+3020168 +13.279 +12.637 +12.312 +12.109 +10.986 +10.601 +10.501 +*6 +. . . +J04550078+3020090 +13.351 +12.288 +11.604 +11.135 +9.564 +8.828 +8.608 +*7 +. . . +J04550253+3019228 +14.529 +13.494 +12.892 +12.496 +10.994 +10.264 +10.086 +Notes: griz magnitudes from Pan-STARRS (Tonry et al. 2018); JHK′ magnitudes from 2MASS (Cutri et al. 2003). +Fig. B.1. Correlation matrices for hydrogen optical line profiles computed from the 15 OHP/SOPHIE spectra obtained during the campaign: +Hα⋆Hα (left), Hβ⋆Hβ (center), and Hα⋆Hβ (right). +Fig. B.2. Correlation matrices for the hydrogen near-infrared line profiles computed from the 34 CFHT/SPIRou spectra obtained during the +campaign : Paβ⋆Paβ (left), Brγ⋆Brγ (center), and Paβ⋆Brγ (right). +Article number, page 23 of 30 + +1.0 +400 +0.8 +0.6 +200 +v(km/s)- ha +0.4 +0.2 +0 +0.0 +-200 +0.2 +0.4 +-400 +0.6 +-400 +-200 +0 +200 +400 +v (km/s) - ha1.00 +400 +0.75 +0.50 +200 +v (km/s) - hb +0.25 +0. +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +v (km/s) - hb400 +0.75 +0.50 +200 +v (km/s) - ha +0.25 +0. +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +v (km/s) - hb1.0 +400 +0.8 + 0.6 +200 +v (km/s) -pab +0.4 +0.2 +0 +0.0 +-200 +0.2 +0.4 +-400 +0.6 +-400 +-200 +0 +200 +400 +v (km/s) - pab1.0 +400 +0.8 +0.6 +200 +v (km/s) - brg +0.4 +0.2 +0 +0.0 +-200 +0.2 +0.4 +-400 +0.6 +-400 +-200 +0 +200 +400 +v (km/s) - brg400 +0.8 +0.6 +200 +v (km/s) -pab + 0.4 +0.2 +0. +0.0 +-200 +-0.2 +0.4 +-400 +0.6 +-400 +-200 +0 +200 +400 +v (km/s) - brgA&A proofs: manuscript no. 45342corr +Fig. B.3. Correlation matrices between optical and near-infrared hydrogen line profiles computed from ten OHP/SOPHIE and ten CFHT/SPIRou +spectra obtained over the same nights during the October runs: Hα⋆Paβ (left), Hβ⋆Paβ (center left), Hα⋆Brγ (center right), and Hβ⋆Brγ (right). +Fig. B.4. Correlation matrices for the HeI 10830 Å and hydrogen lines: HeI⋆HeI (left), Paβ⋆HeI (center), and Hβ⋆HeI (right). The Brγ⋆HeI and +Hα⋆HeI matrices are not shown as they are similar to those of Paβ⋆HeI and Hβ⋆HeI, respectively. +Article number, page 24 of 30 + +400 +0.75 +0.50 +200 +v (km/s) - ha +0.25 +0 +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +v (km/s)- pab400 +0.75 +0.50 +200 +v (km/s) - hb +0.25 +0 +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +v (km/s) - pab400 +0.75 +0.50 +200 +v (km/s) - ha +0.25 +0 +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +y (km/s) - brg400 +0.75 +0.50 +200 +v (km/s) - hb +0.25 +0 +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +v (km/s) - brg1.00 +400 +0.75 +0.50 +200 +v (km/s) - heiir +0.25 +0 +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +v (km/s) - heiir400 +0.8 +0.6 +200 +v (km/s) -pab +0.4 +0.2 +0 +0.0 +0.2 +-200 +0.4 +0.6 +-400 +0.8 +-400 +-200 +0 +200 +400 +y (km/s) - heiir400 +0.75 +0.50 +200 +v (km/s) - hb +0.25 +0 +0.00 +0.25 +-200 +0.50 +-400 +0.75 +-400 +-200 +0 +200 +400 +y (km/s) - heiirJ. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Table A.2. REM g’r’i’z’ photometry +Julian date +g’ +err +r’ +err +i’ +err +z’ +err +(2,450,000+) +(mag) +(mag) +(mag) +(mag) +(mag) +(mag) +(mag) +9497.72986 +12.525 +0.045 +11.566 +0.008 +11.161 +0.006 +10.861 +0.012 +9497.73190 +12.559 +0.042 +11.587 +0.011 +11.171 +0.007 +10.872 +0.017 +9497.73396 +12.566 +0.051 +11.601 +0.019 +11.193 +0.004 +10.892 +0.007 +9500.80099 +12.67 +0.016 +11.589 +0.005 +11.174 +0.016 +10.827 +0.004 +9500.80302 +12.667 +0.02 +11.592 +0.001 +11.175 +0.013 +10.838 +0.006 +9500.80505 +12.682 +0.01 +11.608 +0.007 +11.182 +0.019 +10.853 +0.005 +9501.80935 +12.617 +0.02 +11.58 +0.01 +11.168 +0.01 +10.871 +0.007 +9501.81138 +12.626 +0.024 +11.582 +0.01 +11.169 +0.007 +10.871 +0.006 +9501.81344 +12.635 +0.019 +11.585 +0.013 +11.173 +0.008 +10.868 +0.004 +9502.83317 +12.5 +0.022 +11.544 +0.003 +11.144 +0.012 +10.825 +0.01 +9502.83523 +12.52 +0.024 +11.548 +0.003 +11.149 +0.01 +10.834 +0.011 +9502.83729 +12.519 +0.021 +11.55 +0.004 +11.149 +0.012 +10.843 +0.011 +9503.83744 +12.494 +0.035 +11.522 +0.007 +11.129 +0.014 +10.812 +0.009 +9503.83947 +12.494 +0.032 +11.515 +0.004 +11.127 +0.013 +10.802 +0.009 +9503.84153 +12.505 +0.033 +11.521 +0.004 +11.121 +0.013 +10.816 +0.01 +9504.84169 +12.75 +0.028 +11.656 +0.008 +11.223 +0.004 +10.902 +0.02 +9504.84375 +12.749 +0.029 +11.653 +0.008 +11.222 +0.007 +10.897 +0.019 +9504.84578 +12.75 +0.027 +11.654 +0.009 +11.223 +0.006 +10.89 +0.02 +9506.78944 +12.81 +0.041 +11.67 +0.01 +11.239 +0.009 +10.907 +0.006 +9506.79150 +12.812 +0.031 +11.671 +0.006 +11.243 +0.012 +10.906 +0.01 +9506.79353 +12.797 +0.034 +11.679 +0.008 +11.233 +0.011 +10.886 +0.009 +9507.79360 +12.503 +0.045 +11.53 +0.014 +11.134 +0.008 +10.814 +0.01 +9507.79566 +12.495 +0.043 +11.534 +0.011 +11.131 +0.009 +10.822 +0.015 +9507.79772 +12.477 +0.044 +11.519 +0.013 +11.127 +0.009 +10.82 +0.012 +9508.80257 +12.135 +0.048 +11.284 +0.006 +10.957 +0 +10.661 +0 +9508.80461 +12.141 +0.042 +11.277 +0.001 +10.953 +0 +10.65 +0.005 +9508.80664 +12.13 +0.037 +11.278 +0.004 +10.952 +0.002 +10.653 +0.01 +9509.80682 +12.039 +0.042 +11.218 +0 +10.929 +0.013 +10.627 +0.008 +9509.80888 +12.021 +0.035 +11.209 +0.001 +10.924 +0.017 +10.63 +0.013 +9509.81092 +12.027 +0.033 +11.217 +0.001 +10.929 +0.014 +10.618 +0.012 +9510.86553 +12.498 +0.025 +11.493 +0.001 +11.105 +0.016 +10.788 +0.008 +9510.86759 +12.511 +0.024 +11.494 +0.002 +11.123 +0.013 +10.787 +0.008 +9510.86963 +12.517 +0.022 +11.495 +0.001 +11.109 +0.008 +10.801 +0.019 +9513.71086 +12.66 +0.017 +11.618 +0.002 +11.202 +0.011 +10.866 +0.013 +9513.71300 +12.659 +0.017 +11.615 +0.001 +11.209 +0.013 +10.867 +0 +9513.71503 +12.653 +0.016 +11.601 +0.001 +11.216 +0.009 +10.868 +0.011 +9514.78750 +12.531 +0.034 +11.536 +0.001 +11.169 +0.013 +10.841 +0.004 +9514.78954 +12.543 +0.028 +11.534 +0.001 +11.158 +0.013 +10.832 +0.013 +9514.79157 +12.544 +0.026 +11.535 +0.003 +11.17 +0.01 +10.819 +0.008 +9515.79177 +12.677 +0.016 +11.617 +0.003 +11.184 +0.002 +10.861 +0.025 +9515.79383 +12.693 +0.017 +11.622 +0.006 +11.191 +0.007 +10.868 +0.017 +9515.79586 +12.689 +0.019 +11.621 +0.008 +11.184 +0 +10.854 +0.02 +9517.72459 +12.762 +0.019 +11.641 +0.007 +11.252 +0.008 +10.898 +0.009 +9517.72665 +12.774 +0.014 +11.642 +0.01 +11.249 +0.002 +10.904 +0.004 +9517.72869 +12.769 +0.016 +11.638 +0.01 +11.247 +0.006 +10.9 +0.009 +9518.73156 +12.774 +0.021 +11.663 +0.002 +11.228 +0.012 +10.898 +0.008 +9518.73362 +12.778 +0.019 +11.659 +0.002 +11.222 +0.012 +10.887 +0.008 +9518.73568 +12.771 +0.017 +11.659 +0.001 +11.227 +0.013 +10.892 +0.011 +9519.73968 +12.76 +0.051 +11.693 +0.012 +11.223 +0.01 +10.879 +0.006 +9519.74174 +12.758 +0.051 +11.683 +0.011 +11.218 +0.008 +10.877 +0.006 +9519.74378 +12.766 +0.051 +11.682 +0.011 +11.214 +0.01 +10.882 +0.011 +9520.74391 +12.641 +0.016 +11.597 +0.001 +11.176 +0.012 +10.846 +0.008 +9520.74595 +12.651 +0.016 +11.596 +0.001 +11.177 +0.015 +10.866 +0.007 +9520.74801 +12.662 +0.014 +11.603 +0.001 +11.177 +0.011 +10.867 +0.008 +Table C.1. EW measurements and J-band photometry from the ExTrA +spectra. +Julian date +EW(HeI) +EW(Paβ) +EW(Paγ) +EW(Paδ) +J +errJ +(2,450,000+) +(Å) +(Å) +(Å) +(Å) +(mag) +(mag) +9501.89138 +2.95 ± 0.80 +10.45 ± 0.28 +8.23 ± 0.88 +2.37 ± 0.80 +9.487 +0.028 +9504.85514 +2.43 ± 0.50 +9.33 ± 0.61 +4.93 ± 0.70 +1.48 ± 0.67 +9.499 +0.027 +9506.86219 +2.43 ± 0.58 +10.91 ± 0.72 +7.28 ± 0.60 +2.80 ± 0.64 +9.451 +0.027 +9507.81204 +7.87 ± 0.70 +14.65 ± 1.00 +9.57 ± 0.62 +3.92 ± 0.73 +9.411 +0.027 +9509.84834 +10.06 ± 0.61 +16.56 ± 0.82 +9.05 ± 0.60 +4.23 ± 0.69 +9.316 +0.027 +9510.88153† +5.91 ± 0.61 +12.95 ± 0.93 +7.62 ± 0.58 +2.42 ± 0.59 +— +— +9512.79907 +1.87 ± 0.30 +7.09 ± 1.20 +7.14 ± 0.27 +1.14 ± 0.02 +9.461 +0.027 +9513.79425 +5.50 ± 0.69 +10.50 ± 1.07 +8.73 ± 1.39 +2.90 ± 1.29 +9.407 +0.027 +9514.84366 +2.16 ± 0.67 +10.56 ± 0.83 +7.42 ± 0.71 +2.56 ± 0.68 +9.444 +0.027 +9517.84528 +4.26 ± 0.74 +11.42 ± 0.78 +7.85 ± 0.95 +2.74 ± 0.83 +9.495 +0.037 +9518.84178 +1.08 ± 0.55 +7.88 ± 0.73 +7.77 ± 0.50 +2.08 ± 0.82 +9.482 +0.027 +9519.86259 +4.16 ± 0.75 +9.48 ± 0.71 +7.76 ± 0.56 +3.05 ± 1.01 +9.464 +0.027 +9520.85926 +4.42 ± 0.66 +13.95 ± 0.57 +8.53 ± 0.78 +2.29 ± 0.35 +9.444 +0.027 +9521.87770 +5.36 ± 0.48 +12.38 ± 0.25 +9.63 ± 0.94 +1.63 ± 0.23 +9.424 +0.027 +9522.79321 +3.04 ± 0.31 +14.32 ± 0.41 +7.54 ± 0.56 +2.51 ± 0.48 +9.448 +0.027 +9524.87753 +1.01 ± 0.84 +12.81 ± 1.07 +8.01 ± 0.69 +1.40 ± 0.27 +9.467 +0.027 +9525.86383 +2.47 ± 0.42 +11.31 ± 0.86 +6.61 ± 0.49 +1.13 ± 0.76 +9.473 +0.027 +9526.86452 +2.24 ± 0.47 +8.44 ± 0.63 +5.00 ± 0.70 +1.17 ± 1.18 +9.476 +0.027 +9527.86505 +2.71 ± 0.64 +10.61 ± 0.76 +5.91 ± 0.46 +1.08 ± 0.74 +9.442 +0.027 +9528.82389 +1.59 ± 0.52 +8.35 ± 1.12 +4.64 ± 0.72 +0.02 ± 0.51 +9.475 +0.027 +9529.85920 +0.03 ± 0.65 +6.70 ± 0.70 +3.97 ± 0.85 +0.33 ± 0.59 +9.463 +0.027 +9530.85025 +0.30 ± 0.81 +9.43 ± 0.96 +5.83 ± 1.17 +1.13 ± 0.75 +9.477 +0.029 +9531.81805 +2.70 ± 0.46 +8.98 ± 0.62 +6.50 ± 0.49 +2.82 ± 0.46 +9.460 +0.027 +9532.82519 +3.27 ± 0.61 +10.56 ± 0.82 +5.81 ± 0.57 +2.67 ± 0.65 +9.429 +0.035 +9533.85615 +4.04 ± 0.65 +6.78 ± 0.60 +4.30 ± 0.61 +1.31 ± 0.63 +9.428 +0.027 +9534.84045 +4.57 ± 0.86 +8.43 ± 0.95 +5.70 ± 0.60 +2.12 ± 0.74 +9.425 +0.090 +9535.83697 +1.85 ± 0.52 +8.08 ± 0.71 +5.85 ± 0.68 +1.84 ± 0.62 +9.456 +0.053 +9536.86516 +0.15 ± 0.64 +6.93 ± 2.51 +6.94 ± 0.30 +2.34 ± 0.03 +9.358 +0.027 +9537.78315 +3.45 ± 1.60 +6.97 ± 0.98 +5.43 ± 1.00 +2.06 ± 0.69 +9.545 +0.027 +9538.80149 +2.83 ± 1.64 +5.43 ± 1.42 +4.09 ± 1.20 +1.59 ± 0.86 +9.531 +0.027 +9539.81823 +3.43 ± 0.68 +7.42 ± 1.27 +5.50 ± 0.60 +1.48 ± 0.65 +9.396 +0.028 +9540.84522 +3.61 ± 0.72 +10.38 ± 2.17 +6.80 ± 0.84 +2.58 ± 0.70 +9.313 +0.037 +9543.79258 +2.94 ± 0.46 +10.10 ± 0.79 +6.48 ± 0.40 +1.94 ± 0.61 +9.419 +0.027 +9544.73410 +2.34 ± 0.48 +9.12 ± 0.59 +5.92 ± 0.42 +1.68 ± 0.52 +9.376 +0.027 +9547.68435 +0.08 ± 0.37 +4.16 ± 0.69 +4.52 ± 0.58 +0.46 ± 0.36 +9.443 +0.027 +9548.72798 +-1.19 ± 0.62 +2.79 ± 0.90 +4.06 ± 0.62 +0.30 ± 0.68 +9.568 +0.027 +9549.73881 +1.79 ± 0.63 +6.06 ± 1.24 +5.67 ± 0.82 +1.21 ± 0.69 +9.537 +0.027 +9550.69491 +0.74 ± 0.41 +3.22 ± 0.74 +3.84 ± 0.59 +1.16 ± 0.44 +9.492 +0.027 +9551.70856 +2.48 ± 0.64 +5.54 ± 0.70 +4.66 ± 0.48 +1.22 ± 0.54 +9.474 +0.028 +9553.69774 +1.13 ± 0.64 +6.05 ± 0.62 +4.50 ± 0.63 +0.87 ± 0.61 +9.443 +0.028 +9555.72248 +4.05 ± 0.75 +12.83 ± 1.35 +7.78 ± 0.94 +2.04 ± 0.72 +9.266 +0.028 +9556.67400 +1.85 ± 0.57 +11.64 ± 1.85 +6.23 ± 1.62 +1.48 ± 0.61 +9.333 +0.027 +9560.71477 +0.99 ± 0.56 +6.39 ± 0.68 +5.64 ± 0.63 +1.31 ± 0.70 +9.507 +0.028 +9561.73574 +2.54 ± 0.44 +7.87 ± 0.52 +7.21 ± 0.44 +2.16 ± 0.57 +9.481 +0.028 +9564.76316 +6.31 ± 0.66 +11.98 ± 0.74 +6.73 ± 0.67 +1.83 ± 0.66 +9.394 +0.029 +9566.66048 +-1.91 ± 0.85 +3.89 ± 0.75 +3.44 ± 0.59 +0.30 ± 0.51 +9.505 +0.028 +9567.68335 +2.96 ± 0.69 +8.80 ± 0.92 +5.48 ± 0.39 +1.18 ± 0.64 +9.470 +0.027 +9569.70386 +2.77 ± 0.56 +7.92 ± 0.78 +5.88 ± 0.90 +1.35 ± 0.75 +9.397 +0.029 +9570.68277 +4.73 ± 0.48 +10.35 ± 1.11 +7.26 ± 0.55 +2.46 ± 0.79 +9.399 +0.027 +9573.66353 +4.33 ± 0.48 +10.45 ± 0.41 +9.01 ± 0.66 +2.64 ± 0.41 +9.432 +0.027 +9577.61918 +2.11 ± 0.36 +8.54 ± 0.65 +5.62 ± 0.51 +1.43 ± 0.57 +9.401 +0.027 +9578.66562 +7.11 ± 0.65 +12.34 ± 0.76 +7.62 ± 0.91 +3.88 ± 0.55 +9.368 +0.027 +9580.62157 +4.88 ± 0.50 +10.48 ± 0.72 +7.06 ± 0.53 +2.42 ± 0.56 +9.410 +0.027 +9581.66843 +6.99 ± 0.42 +12.16 ± 0.54 +7.38 ± 0.64 +2.40 ± 0.46 +9.331 +0.027 +9582.60702 +3.56 ± 0.56 +10.93 ± 0.72 +6.81 ± 0.86 +1.99 ± 0.52 +9.390 +0.027 +9586.69987 +6.29 ± 0.58 +13.34 ± 1.26 +9.30 ± 0.66 +3.94 ± 0.51 +9.406 +0.029 +9588.70981 +2.08 ± 0.26 +6.93 ± 0.69 +6.47 ± 0.55 +1.77 ± 0.67 +9.426 +0.027 +9589.60234 +2.20 ± 0.68 +9.23 ± 1.17 +7.62 ± 0.58 +2.38 ± 0.52 +9.429 +0.027 +9596.62015 +2.49 ± 0.58 +9.68 ± 1.33 +6.58 ± 0.47 +2.26 ± 0.62 +9.458 +0.028 +9597.67444 +4.58 ± 0.65 +11.33 ± 0.95 +9.61 ± 0.56 +2.91 ± 0.64 +9.450 +0.028 +9598.59529 +4.30 ± 0.54 +10.94 ± 0.86 +7.80 ± 0.48 +2.43 ± 0.63 +9.397 +0.028 +9600.55490 +7.14 ± 0.48 +16.46 ± 0.86 +10.65 ± 0.68 +4.19 ± 0.58 +9.424 +0.028 +9601.56718 +2.41 ± 0.72 +8.23 ± 1.12 +6.02 ± 0.59 +1.59 ± 0.60 +9.456 +0.028 +9602.53523 +1.35 ± 0.63 +9.00 ± 0.75 +6.44 ± 0.80 +2.51 ± 0.61 +9.486 +0.027 +9605.54455 +10.34 ± 0.74 +17.45 ± 1.23 +11.39 ± 0.82 +4.42 ± 0.90 +9.372 +0.027 +9606.55663 +4.71 ± 0.58 +12.46 ± 1.33 +7.92 ± 0.82 +2.47 ± 0.82 +9.409 +0.028 +9607.55053 +2.61 ± 0.51 +11.21 ± 1.02 +7.28 ± 0.60 +2.37 ± 0.57 +9.443 +0.027 +9608.52952 +-0.27 ± 0.75 +7.99 ± 1.16 +5.62 ± 1.00 +1.60 ± 0.60 +9.484 +0.027 +9615.52033 +5.17 ± 0.56 +13.40 ± 1.17 +8.94 ± 0.97 +3.34 ± 0.62 +9.409 +0.027 +9616.55954 +6.72 ± 0.64 +13.72 ± 1.09 +9.56 ± 0.88 +3.19 ± 0.62 +9.360 +0.027 +9617.55930 +6.54 ± 0.92 +15.62 ± 0.69 +9.79 ± 0.72 +3.73 ± 0.72 +9.354 +0.030 +9618.55535 +8.27 ± 1.11 +18.41 ± 1.05 +11.10 ± 0.65 +4.79 ± 0.72 +9.332 +0.028 +9619.56342 +7.84 ± 0.76 +14.54 ± 0.90 +9.04 ± 0.72 +3.59 ± 0.78 +9.417 +0.028 +9623.52755 +11.79 ± 0.81 +17.61 ± 1.58 +10.38 ± 0.73 +4.42 ± 0.75 +9.320 +0.027 +9624.55795 +9.62 ± 0.78 +14.33 ± 1.01 +8.17 ± 0.75 +3.10 ± 0.65 +9.365 +0.028 +9625.53847 +6.42 ± 0.92 +10.27 ± 1.41 +6.52 ± 0.85 +1.64 ± 0.70 +9.438 +0.028 +9626.54670 +-0.82 ± 0.70 +5.39 ± 1.01 +4.91 ± 0.92 +1.34 ± 0.86 +9.527 +0.030 +9629.53317 +7.30 ± 0.63 +12.52 ± 0.87 +7.25 ± 0.65 +2.74 ± 0.60 +9.336 +0.028 +9630.53061 +11.17 ± 0.86 +18.58 ± 0.89 +10.69 ± 0.81 +4.68 ± 0.87 +9.342 +0.027 +9631.50698 +8.20 ± 0.41 +15.59 ± 1.21 +9.15 ± 0.94 +3.27 ± 0.30 +9.409 +0.027 +9632.52380 +1.15 ± 0.82 +9.75 ± 1.99 +5.78 ± 0.74 +1.04 ± 0.73 +9.627 +0.028 +9633.54300 +1.99 ± 0.41 +10.77 ± 1.32 +6.97 ± 0.60 +1.70 ± 0.31 +9.445 +0.027 +9634.53461 +2.68 ± 0.59 +8.75 ± 1.34 +5.18 ± 0.56 +1.25 ± 0.63 +9.413 +0.027 +9643.52095 +1.44 ± 0.58 +14.07 ± 1.00 +10.88 ± 0.90 +3.03 ± 0.62 +9.393 +0.028 +9644.52110 +1.53 ± 0.68 +10.47 ± 1.09 +8.49 ± 0.89 +1.87 ± 0.83 +9.467 +0.029 +9645.51054 +2.58 ± 0.50 +10.74 ± 2.45 +9.04 ± 0.41 +2.61 ± 0.71 +9.416 +0.027 +9646.52604 +6.67 ± 1.02 +14.56 ± 1.53 +9.61 ± 0.92 +2.84 ± 0.83 +9.375 +0.028 +9647.50937 +2.58 ± 0.66 +10.79 ± 1.33 +7.04 ± 1.01 +1.48 ± 0.64 +9.392 +0.028 +† No J-band photometry could be obtained for that night due to poor seeing. +Article number, page 25 of 30 + +A&A proofs: manuscript no. 45342corr +Fig. D.1. Veiling measured on SPIRou spectra in the JHK bands (tri- +angles) and the EW of the HeI, Paβ, and Brγ lines (dots) plotted as a +function of Julian date (top) and rotational phase (bottom). +Article number, page 26 of 30 + +EW(Hel) +10 +EW(PaB) +EW(BrG) +EWs +and +△ rh +rk +IR veiling +near +0.1 +9480 +9500 +9520 +9540 +9560 +9580 +Juliandate-2.450.000EW(Hel) +10 +EW(PaB) +g and Ew's (A) +EW(BrG) +rij +△ rh +1 +rk +IRveiling +AA +near +0.1 +A +A +A +0 +0.2 +0.4 +0.6 +0.8 +1.0 +PhaseJ. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. E.1. Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over nine days during the September 2021 SPIRou run. +Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels). The colors represent successive +rotational cycles. Bottom: 2D periodograms across the line profiles. The dotted horizontal red line drawn at a frequency of 0.166 day−1 indicates +the stellar rotational period. The white curve displays the mean line profile. The color code reflects the periodogram power from zero (blue) to 1 +(red). +Article number, page 27 of 30 + +day +phase +0.03 +9475.1 +9480. +948 +482 +-500 +500-500 +0 +500 +v (km/s) +v (km/s)day +phase +9473.7 +0.03 +947 +9480 +9481. +9482. +500 +0 +500-500 +o +500 +v (km/s) +v (km/s)day +phase +9473.1 +0.03 +94/5 +9477.0 +9478. +9480.1 +9481. +0.68 +9482. +0.85 +500 +500-500 +0 +500 +v (km/s) +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +Frequency (1/d) +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)A&A proofs: manuscript no. 45342corr +Fig. E.2. Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over 14 days during the October 2021 SPIRou run. Top: Line +profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels). The colors represent successive rotational +cycles. Bottom: 2D periodograms across the line profiles. The dotted horizontal red line drawn at a frequency of 0.166 day−1 indicates the stellar +rotational period. The white curve displays the mean line profile. The color code reflects the periodogram power from zero (blue) to 0.9 (red). +Article number, page 28 of 30 + +day +phase +9502. +5.0 +9503.1 +7.16 +9504.1 +9506.1 +9508, +5.33 +9509.1 +9510.1 +5.66 +9511.1 +8 +9513 +.83 +9514 +84 +9515. +9516.1 +500 +0 +500-500 +0 +500 +v (km/s) +v (km/s)day +phase +9502.1 +5.0 +9503.1 +7.16 +9504.1, +5.16 +9506.1 +9508.1 +9509.2 +5.5 +9510.1 +6.66 +9511.1 +6.8 +9513.1 +5.83 +9514. +4.84 +9515.1 +6.99 +9516.1 +6.0 +500 +500-500 +0 +500 +v (km/s) +v (km/s)day +phase +9502. +5.0 +9503. +7.16 +9504. +6.16 +506 +5.1 +6.33 +6.6 +6.82 +9513 +5.83 +9514. +9515.1 +0.95 +9516.1 +6.0 +500 +0 +500-500 +0 +500 +v (km/s) +v (km/s)0.50 +0.45 +0.40 +Frequency (1/d) +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +Frequency (1/d) +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)J. Bouvier, A. Sousa, K. Pouilly, et al.: GMAur +Fig. E.3. Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over six days during the November 2021 SPIRou run. +Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels). The colors represent successive +rotational cycles. Bottom: 2D periodograms across the line profiles. The dotted horizontal red line drawn at a frequency of 0.166 day−1 indicates +the stellar rotational period. The white curve displays the mean line profile. The color code reflects the periodogram power from zero (blue) to 1 +(red). +Article number, page 29 of 30 + +day +phase +-500 +500-500 +0 +500 +v (km/s) +v (km/s)day +phase +9535.1 +11.1 +537 +9538 +9541. +-500 +500-500 +500 +v (km/s) +v (km/s)phase +day +11.11 +9535.1 +11.28 +9537. +10.3 +538.1 +9540.0 +9541.0 +500 +0 +500-500 +0 +500 +v (km/s) +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)A&A proofs: manuscript no. 45342corr +Fig. E.4. Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over nine days during the December 2021 SPIRou run. +Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels). The colors represent successive +rotational cycles. Bottom: 2D periodograms across the line profiles. The dotted horizontal red line drawn at a frequency of 0.166 day−1 indicates +the stellar rotational period. The white curve displays the mean line profile. The color code reflects the periodogram power from zero (blue) to 1 +(red). +Article number, page 30 of 30 + +day +phase +9558.0 +4.09 +9559. +9560.0 +9561.0 +9563. +9564.1 +567 +-500 +500-500 +0 +500 +v (km/s) +v (km/s)day +phase +9558.0 +14.09 +9559.0 +.5 +9560.0 +9561.0 +9563. +9564.2 +9566.1 +9567.0 +4.94 +-500 +0 +500-500 +0 +500 +v (km/s) +v (km/s)a +onas +9558.0 +559.0 +9560.0 +9561.0 +9563.1 +564. +5.58 +o6 +4.59 +14.94 +9567.0 +-500 +500-500 +0 +500 +v (km/s) +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s)0.50 +0.45 +0.40 +Frequency (1/d) +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +-400 +-200 +0 +200 +400 +v (km/s) \ No newline at end of file diff --git a/ptFRT4oBgHgl3EQfdjek/content/tmp_files/load_file.txt b/ptFRT4oBgHgl3EQfdjek/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7467d6f738be7a28c2c41e31d095980461239373 --- /dev/null +++ b/ptFRT4oBgHgl3EQfdjek/content/tmp_files/load_file.txt @@ -0,0 +1,5250 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf,len=5249 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr ©ESO 2023 February 1, 2023 Stable accretion and episodic outflows in the young transition disk system GM Aurigae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A semester-long optical and near-infrared spectrophotometric monitoring campaign⋆,⋆⋆ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Almenara1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Donati3, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alencar4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Frasca5, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Grankin6, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Carmona1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pantolmos1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Zaire4, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bonfils1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bayo7, 8, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Rebull9, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alonso-Santiago5, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Gameiro10, 11, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Cook12, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Artigau12, and the Spirou Legacy Survey (SLS) consortium 1 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 2 Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Sweden 3 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' de Toulouse, CNRS, IRAP, 14 avenue Belin, 31400 Toulouse, France 4 Departamento de Fisica – ICEx – UFMG, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Antonio Carlos 6627, 30270-901 Belo Horizonte, MG, Brazil 5 INAF – Osservatorio Astrofisico di Catania, via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sofia 78, 95123 Catania, Italy 6 Crimean Astrophysical Observatory, Nauchny, 298409, Republic of Crimea 7 Instituto de Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Chile 8 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany 9 Infrared Science Archive (IRSA), IPAC, 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' California Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', California Institute of Technology, Pasadena, CA 91125, USA 10 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal 11 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, rua do Campo Alegre 687, 4169-007 Porto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Portugal 12 Research on Exoplanets, Université de Montréal, Département de Physique, Montréal, QC H3C 3J7, Canada Received 2 November 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' accepted 5 January 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Young stellar systems actively accrete from their circumstellar disk and simultaneously launch outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The physical link between accretion and ejection processes remains to be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We investigate the structure and dynamics of magnetospheric accretion and associated outflows on a scale smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 au around the young transitional disk system GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We devised a coordinated observing campaign to monitor the variability of the system on timescales ranging from days to months, including partly simultaneous high-resolution optical and near-infrared spectroscopy, multiwavelength photometry, and low- resolution near-infrared spectroscopy, over a total duration of six months, covering 30 rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We analyzed the photometric and line profile variability to characterize the accretion and ejection processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The optical and near-infrared light curves indicate that the luminosity of the system is modulated by surface spots at the stellar rotation period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Part of the Balmer, Paschen, and Brackett hydrogen line profiles as well as the HeI 5876 Å and HeI 10830 Å line profiles are modulated on the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Paβ line flux correlates with the photometric excess in the u’ band, which suggests that most of the line emission originates from the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' High-velocity redshifted absorptions reaching below the continuum periodically appear in the near-infrared line profiles at the rotational phase in which the veiling and line fluxes are the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These are signatures of a stable accretion funnel flow and associated accretion shock at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This large-scale magnetospheric accretion structure appears fairly stable over at least 15 and possibly up to 30 rotational periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In contrast, outflow signatures randomly appear as blueshifted absorption components in the Balmer and HeI 10830 Å line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' They are not rotationally modulated and disappear on a timescale of a few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The coexistence of a stable, large-scale accretion pattern and episodic outflows supports magnetospheric ejections as the main process occurring at the star-disk interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Long-term monitoring of the variability of the GM Aur transitional disk system provides clues to the accretion and ejection structure and dynamics close to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Stable magnetospheric accretion and episodic outflows appear to be physically linked on a scale of a few stellar radii in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Stars: pre-main sequence – Stars: variables: T Tauri – Stars: magnetic field – Protoplanetary disks – Stars: individual: GM Aurigae ⋆ Based on observations obtained at the Canada-France-Hawaii Tele- scope (CFHT), at the Observatoire de Haute-Provence (OHP), at the Eu- ropean Organisation for Astronomical Research in the Southern Hemi- sphere (ESO), and at the Las Cumbres Observatory global telescope network (LCOGT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' ⋆⋆ Tables containing the u’g’r’i’ LCOGT photometric mea- surements are only available in electronic form at the CDS via anonymous ftp to cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='fr (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5) or via https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='fr/cgi-bin/qcat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='J/A+A/ Article number, page 1 of 30 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='13568v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='SR] 31 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Introduction Accretion and ejection processes are at the origin of most of the peculiar properties of young stellar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The structure and dynamics of the accretion flows within the disk and from the inner disk to the star, as well as the properties of the multiple outflows arising from the disk, from the star-disk interface, and from the stellar surface, remain to be fully deciphered, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Low-mass pre-main-sequence stars, the so-called T Tauri stars (TTS), accrete from their circumstellar disks for a few million years, while contemporaneous planet formation impacts the disk structure and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In the inner regions of the system, the disk is disrupted by the strong stellar magnetosphere that chan- nels the accretion flow toward the star along magnetic field lines (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', the review by Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Thus, accretion funnel flows develop that connect the inner disk to the stellar surface, where the material is accreted at nearly free-fall veloc- ity and is eventually halted in a strong accretion shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Simulta- neously, outflows are produced at the star-disk interface close to the magnetospheric truncation radius through the inflation and reconnection of magnetic field lines that are twisted by differen- tial rotation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Zanni & Ferreira 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Ultimately, the release of gravitational energy delivered by the accretion process may trigger accretion-powered stellar winds (Matt & Pudritz 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The torque balance between accretion and ejection processes is a central issue for understanding the spin evolution of young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Pantolmos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Ireland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2021) The star-disk interaction takes place on a distance of a few stellar radii (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Bessolaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2008), that is, on a scale of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 au or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' MHD models developed by several groups predict the structure and dynamics of the magnetospheric ac- cretion region and associated outflows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', the review by Romanova & Owocki 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Observationally, two main direc- tions have been explored so far to investigate the properties of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' On one hand, monitoring the spectroscopic and pho- tometric variability of the system over a few rotational periods, that is, typically over a few weeks, allows identifying the signa- ture of funnel flows, hot spots, and outflows, and relating them to the strength and topology of the surface magnetic field that is measured from spectropolarimetry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Pouilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2019, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' On the other hand, a direct approach attempts to spatially resolve the star-disk interaction region on a scale of a few milliarcsecond on the sky, using long-baseline near-infrared interferometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Eisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Both approaches have been successful in mapping the inner region of accreting systems and have provided strong support to the magnetospheric accretion scenario and its MHD modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Following previous studies, we report here the results of a new observing campaign devoted to the young stellar system GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' GM Aur (RA = 04h55, Dec = +30◦21, V = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 mag) is a solar-type pre-main-sequence star located in the Taurus-Auriga molecular cloud at a distance of 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 pc (Gaia Collabo- ration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This classical T Tauri star (cTTS) has a spec- tral type K6 (Herczeg & Hillenbrand 2014) and is surrounded by a circumstellar disk from which it actively accretes material at a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0·10−8 M⊙yr−1 (Robinson & Espaillat 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Based on its spectral energy distribution, which exhibits a small near-infrared excess compared to a significant mid-infrared one, the system has long been suspected to be in a transitional stage, that is, that it is surrounded by a disk whose inner regions are relatively devoid of matter (Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' High-resolution ALMA images of the circumstellar disk indeed reveal that it is highly structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The large-scale disk, inclined at ∼53◦ on the line of sight, features a large inner dust cavity extending over ∼35-40 au and a succession of annular gaps and dusty rings on a wider scale up to 200 au (Macías et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Much closer to the central star, long-baseline VLTI/GRAVITY interferometric observations unveil a compact dusty disk, whose inner edge was recently reported to be located at rin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='013+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='008 au from the central star (Bohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022) and that extends over at least a few 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 au (Akeson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2005) and possibly up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 au (Varga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Woitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The gaseous com- ponent of the inner disk has been detected from CO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 micron emission down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 au (Salyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The inclina- tion and position angle of the major axis of the inner dusty disk (i=68◦+16 −28, PA=37◦+31 −22) are found to be consistent with those of the outer disk, which suggests that the inner and outer disks are aligned (Bohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In an attempt to decipher the physical processes at work at the heart of the system, GM Aur has been the subject of several multiwavelength monitoring campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The long-term light curve presented by Grankin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2007) over the period 1986-1995 exhibits relatively low-level variability, with a V- band magnitude ranging from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='74 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Photometric variations are modulated by surface spots at the stellar rotation period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 days (Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Artemenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Ingleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2015) reported variability over the full wavelength range from the far-UV to the near-infrared, which they attributed in part to an accretion rate that varies by about a factor of 2 to 3 on a timescale of months, and for another part to dust inhomo- geneities that are located in the inner disk close to the truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Variations in the mass accretion rate of similar ampli- tude have also been reported on a shorter timescale of about a week by Robinson & Espaillat (2019), and a connection between mass loss and mass accretion has been further suggested by Es- paillat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020) presented the results of a high-resolution optical spectroscopic monitoring campaign performed on a timescale of a week that illustrated the variabil- ity of the Hα, Hβ, and HeI emission line profiles of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' From the measured radial velocity variations of the HeI 5876 Å line profile, whose narrow component traces the accretion shock, they deduced that GM Aur accretes material from its circumstel- lar disk through an inclined magnetosphere, whose axis is tilted by about 13◦ relative to the stellar rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' GM Aur in- deed harbors a strong surface magnetic field, with a mean value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 kG (Johns-Krull 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Symington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, from a recent multiwavelength X-UV-optical campaign, Espail- lat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2021) reported evidence for a transverse density strat- ification within the accretion shock at the base of the magnetic funnel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We report here the results of a new coordinated monitor- ing campaign on GM Aur that combines high-resolution optical spectroscopy and near-infrared spectropolarimetry, multiwave- length optical and near-infrared photometry, and long-term low- resolution near-infrared spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Part of the observations have been obtained simultaneously over a timescale of a few weeks, while the total duration of the campaign amounted to six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The goal of the campaign was to investigate the phys- ical processes that cause variability in GM Aur on a scale of a few stellar radii, and in particular, to constrain the structure and dynamics of the magnetospheric accretion flow from the inner disk to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We devised a long-term campaign in order to be able to probe various timescales, from days to months, and obtain a sufficiently long temporal baseline to investigate the re- lation between accretion and ejection processes on small spatial scales from the stellar surface to the inner disk regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 2 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur The campaign whose results are reported here took place in the framework of a larger project led by the ODYSSEUS team1 (see Espaillat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022), which uses the Hubble UV Legacy Library of Young Stars as Essential Standards program (ULL- YSES2, Roman-Duval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020), on HST Director’s Discre- tionary time, to monitor a sample of T Tauri stars in the UV domain, which includes GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Additional follow-up observa- tions were acquired for this project at ESO in the framework of the PENELLOPE Large Program3 (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In Section 2 we describe the observational techniques we im- plemented to perform the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In Section 3 we derive the properties of the system and analyze its photometric and spectro- scopic variability over timescales from days to months, includ- ing veiling measurements and emission line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We infer the global structure of the magnetospheric accretion flow from the observed variability and characterize associated outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In Section 4 we discuss the dynamics of the accretion and ejec- tion structure and show that short-lived episodic outflows coex- ist with a stable magnetospheric accretion pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In Section 5 we conclude on the ability of multiwavelength, multi-technique coordinated observational campaigns to unveil the physical pro- cesses at work in young stellar systems at the sub-au scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Observations In this section, we describe the acquisition and data-reduction processes of photometric, spectroscopic, and spectropolarimet- ric datasets obtained during the large-scale campaign we per- formed on the cTTS GM Aur from September 6, 2021, to March 8, 2022, using CFHT/SPIRou, OHP/SOPHIE, ESO/ExTrA, LCOGT, and ESO/REM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A summary plot of the GM Aur ob- serving campaign reported here is provided in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' LCOGT: Multiwavelength optical photometry GM Aur was observed at Las Cumbres Observatory Global Net- work (LCOGT, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2013) from September 6 to De- cember 30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We acquired 850 images in the Sloan u’g’r’i’ filters over two runs with a sub-day cadence (LCO2021B-001, PI L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Rebull;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' CLN2021B-003, PI A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bayo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The u’ images were obtained with the Sinistro 1m telescopes of the network using an exposure time of 180 seconds and reading the 2Kx2K cen- tral window of the detector with a 2x2 binning, resulting in a 13x13 arcmin field of view on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The g’r’i’ images were obtained with the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4m SBIG telescopes, offering a field of view of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2x19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 arcmin, with exposure times of 60, 20, and 20 sec- onds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We retrieved the BANZAI-reduced images from the LCOGT archive service and the noncalibrated photo- metric catalogs provided in the image headers for all detected stars in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In order to compute differential photometry, we considered two stars, HD 282625 and HD 282626, both located within 3 ar- cmin of GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The first star was used as a reference star to calibrate the differential light curve, and the second was used as a check star to assess that these are nonvariable sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These field stars have spectral types F2 and F5, respectively, and are only slightly brighter than GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We confirmed that the two stars are nonvariable from their differential light curves, and we deduced a mean rms photometric error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='025 mag in the u’g’r’ filters and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='033 mag in the i’ filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We proceeded to 1 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='edu/odysseus/ 2 https://ullyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='edu/ 3 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='com/view/cfmanara/penellope compute the differential light curve between GM Aur and HD 282625 in all four filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We adopted the mean magnitude of HD 282625 listed in the APASS and Pan-STARRS surveys, namely g’=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='331 mag, r’=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='916 mag, and i’=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='756 mag to calibrate the GM Aur light curve in the g’r’i’ filters to within an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='02 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We were not able to find an estimate of the u’-band magni- tude for the comparison stars in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Instead, we as- sumed the intrinsic (u’-g’) colors of an F2 star (Covey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Kraus & Hillenbrand 2007) for HD 282625, to which we applied interstellar reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' From the observed versus intrinsic color indices of the HD 282625 (g’-r’) and (r’-i’) bands, we de- rived AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 mag, using the R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 reddening law from Fiorucci & Munari (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This procedure yielded an estimate of the reddened (u’-g’) color of the comparison star from which we derived its u’-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The photometric calibration of the GM Aur u’-band light curve is thus relatively indirect and probably not accurate to better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 mag4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' REM: Optical and near infrared photometry Observations were performed with the 60 cm robotic REM tele- scope located at the ESO La Silla Observatory (Chile), on 15 nights from JD 2,459,497 to JD 2,459,520 (October 9 to Novem- ber 1, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' By means of a dichroic, REM simultaneously feeds two cameras at the two Nasmyth focal stations, one cam- era for the near-infrared (REMIR), and the other for the opti- cal (ROSS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The cameras have nearly the same field of view of about 10′ × 10′ and use wide-band filters (J, H, and K′ for REMIR and Sloan/SDSS g′, r′, i′, and z′ for ROSS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Exposure times were 60 s for ROSS2, which simultaneously acquires images in the four Sloan bands, and five ditherings of 3 s each were adopted for each filter of REMIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For the ROSS2 camera, we generated master flats using the twilight flat-fields taken during the observing run, which are available in the REM archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The latter were used to correct for pixel-to-pixel sensi- tivity variations, as well as for the vignetting and illumination of the field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' After subtracting the dark-frame, each scientific image was divided by the proper master-flat, depending on the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The prereduction of the REMIR images is automatically done by the AQuA pipeline (Testa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2004), and the coadded and sky-subtracted frames, resulting from five individual dither- ings, are made available to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The adopted comparison stars are reported in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 along with their griz (Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018) and JHK′ (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2003) magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Aperture photometry for all the stars listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 was performed with DAOPHOT by using the IDL5 routine Aper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For each frame and filter, we used the instrumental magnitudes of the stars listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 to generate an artifi- cial comparison, weighting them with the flux corresponding to their standard magnitude in a way similar to the ensemble pho- tometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This procedure also allowed us to evaluate a standard error based on the differences between the magnitudes calculated with different comparison stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The optical photometry gathered at REM is listed in Ta- ble A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In the common g’r’i’ bands, we found it to agree well with that obtained at LCOGT with a tighter sam- pling rate, and therefore, we did not use it further in the analysis below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The individual JHK’ measurements are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 4 The table of photometric measurements is available electronically at CDS, Strasbourg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 5 Interactive Data Language (IDL) is a registered trademark of Harris Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 3 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr REM LCOGT OHP/SOPHIE CFHT/SPIRou ESO/ExTrA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Temporal sampling of the GM Aur campaign from September 6, 2021, to March 8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: g’-band light curve from LCOGT (black dots) and from REM (magenta crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The REM g’-band magnitudes are offset in this figure by +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 mag to match the LCOGT measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The mean photometric error on both the LCOGT and REM g’-band measurements is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='025 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Vertical arrows: The vertical arrows show the dates of OHP/SOPHIE (red), CFHT/SPIRou (blue), and ESO/ExTrA (black) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The core of the campaign took place during October 2021 with contemporaneous measurements from the five instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The median errors on JHK’ measurements are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='06 mag, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, some measurements were af- fected by the nearby bright moon around JD 2,459,512, and had an error larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We chose to discard these measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We eventually derived the following values for the me- dian near-infrared magnitudes of the GM Aur system and their rms variations at the time of the observations: J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 mag, H= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='03 mag, and K’=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='07 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' OHP SOPHIE: High-resolution optical spectroscopy Observations were carried out from October 12 to 29, 2021, at Observatoire de Haute-Provence using the fiber-fed SOPHIE spectrograph (Perruchot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2008) in high-efficiency mode, which delivers a spectral resolution of R ∼ 40,000 over the wave- length range 387-694 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We obtained 15 spectra over 18 nights, with an exposure time of 3600 s, yielding a signal-to-noise ra- tio ranging from 42 to 67 at 600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The raw spectra were fully reduced at the telescope by the SOPHIE real-time pipeline (Bouchy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The data products include a resampled 1D spectrum with a constant wavelength step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='01 Å, cor- rected for barycentric radial velocity, an order-by-order estimate of the signal-to-noise ratio, and a measurement of the source ra- dial velocity, Vr, and projected rotational velocity, v sin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The latter two quantities are derived from a cross-correlation anal- ysis of nearly 7,000 spectral lines between the observed spec- trum and a K5 spectral mask template (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We list the values of these parameters in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The mean for- mal error provided by the SOPHIE pipe-line on the Vr measure- ment is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='013 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This accuracy is well suited to investigat- ing the significantly larger amplitude of photospheric line pro- file variability induced by surface spots and/or accretion flows in young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' No error is provided by the pipeline for the v sin i measurements, and we assumed that an upper limit is given by the rms deviation of the individ- ual measurements, excluding JD 2,459,512 (see below), namely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='38 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The SOPHIE spectrograph includes a second fiber that si- multaneously records the spectrum of the nearby sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Inspection of the cross-correlation function (CCF) of the sky fiber with a synthetic mask of spectral type G2 revealed that the signature of the moon becomes apparent at the expected barycentric Earth radial velocity from JD 2,459,505 onward because the growing moon approaches the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The lunar contamination culminates Article number, page 4 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' REM JHK photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Julian date J Julian date H Julian date K (2,450,000+) (mag) (2,450,000+) (mag) (2,450,000+) (mag) 9497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='73017 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 9497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='73222 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='09 9514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5918 59 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='98 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='85 9515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6107 42 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='02 9516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5321 49 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='46 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='24 † Uncertain value due to lunar contamination (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' on JD 2,459,512, as the bright moon is located about 10 degrees away from GM Aur, which explains the discrepant values mea- sured for Vr and v sin i on this date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table 2 suggests that except for JD 2,459,512, the contamination of the CCF by the moon only marginally impacts the Vr and v sin i measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' How- ever, to be conservative, we only considered the Vr and v sin i measurements obtained from the first six spectra of the observ- ing run for the subsequent analysis, from JD 2,459,499 to JD 2,459,504, where no lunar contamination is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The journal of observations is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It lists the Julian date, the signal-to-noise ratio of individual spectra at 600 nm, the radial and rotational velocities derived from each spec- trum, and the bisector span computed from the cross-correlation function (Queloz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' CFHT SPIRou: Near-infrared spectropolarimetry Near-infrared spectropolarimetric observations of GM Aur were performed at CFHT using the SPIRou near-infrared spectropo- larimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It has a spectral range covering from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='95 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 µm in a single exposure at a spectral resolution of 70,000 (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The observations were completed in the frame- work of the CFHT SPIRou Legacy Survey over four observing runs extending from September 15 to December 18, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Each monthly run was scheduled around the full moon, and we aimed at obtaining one spectrum per night during the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' An additional single spectrum was obtained on January 6, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We thus gath- ered 34 spectra, whose temporal sampling is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Each spectrum consists of four polarimetric sub-exposures6 for a total integration time of 2,200 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Individual exposures were combined to yield a single spectrum with a signal-to-noise ratio ranging from ∼60 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In one instance, on JD 2,459,503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='08, the polarimetric sequence was aborted, and the spectrum con- sists of a single sub-exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The raw data were reduced within the SPIRou consortium, using version V6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='132 of the APERO pipeline (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Spectra were cross-correlated with a K2 spectral mask template over about 6,700 spectral lines, and the radial velocity of the object was derived with sub-km s−1 accuracy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='08 km s−1 mean rms uncertainty) by fitting a Gaus- sian to the resulting CCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The median Vr amounts to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='27 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Using TWA 9A, a WTTS of spectral type K5, as a template, the veiling was derived in the JHK bands following the procedure described in Sousa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2023) The median rms er- rors on veiling measurements in the JHK bands are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='011, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='025, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='027, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The journal of observations is presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It lists the Julian date, the signal-to-noise ratio at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='16 µm, the photospheric radial velocity and its uncertainty, and the veiling in the JHK bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' ESO ExTrA: Low-resolution near-infrared spectroscopy The ExTrA facility (Bonfils et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2015), located at La Silla Ob- servatory in Chile, consists of three 60 cm telescopes and a sin- gle near-infrared (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='88 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='55 µm) fibre-fed spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We observed GM Aur on 89 nights between October 13, 2021, and March 8, 2022, using either one telescope (21 nights) or two tele- scopes simultaneously (68 nights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Five fiber units are located at the focal plane of each telescope, each consisting of two 8′′ aperture fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' One fiber is used to observe a star and the other is used to observe the nearby sky background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We observed GM Aur with one fiber unit and used another fiber unit to simultane- ously observe 2MASS J04535474+3021441 (J = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='450 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='027 mag) as a comparison star to compute differential photome- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We used the higher-resolution mode of the spectrograph (R∼200) and 300-second exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We obtained between 1 and 30 exposures per night for a total of 1898 spectra with a median signal-to-noise ratio of 105 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 µm for GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The ExTrA data were corrected for dark current, extracted using the flat- field, corrected for sky background emission, and were wave- length calibrated using custom data reduction software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Median spectra of GM Aur were computed for each night and telescope, yielding a total of 157 spectra with a median signal-to-noise ratio of 179 and a standard deviation of 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We computed differential photometry of GM Aur relative to the comparison star by integrating the individual ExTrA spectra 6 The polarimetric analysis of the dataset will be published in a com- panion paper (Zaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 5 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Journal of CFHT/SPIRou observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Julian date S/N Vr σVr rJ rH rK (2,450,000+) km s−1 September 2021 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='066 101 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='27 9475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='043 109 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='22 9476.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 † Single sub-exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' over the J-filter passband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The UKIRT-WFCAM filter transmis- sion curves were retrieved from the SVO Filter Profile Service7 (Rodrigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Rodrigo & Solano 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We multiplied each corrected individual spectrum of GM Aur and the compar- ison star by the filter transmission curve, integrated the flux, and obtained a magnitude difference from the flux ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We com- puted a differential magnitude measurement for each night as the mean and standard deviation of the individual measurements taken on that night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We then derived the J-band magnitude of GM Aur from the 2MASS magnitude of the comparison star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We obtained a median value and 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3% confidence interval of J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='417 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='061 mag for the ExTrA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The results are listed in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 of Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 7 http://svo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='es/theory/fps/ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Multicolor photometry The LCOGT u’g’r’i’ light curves of GM Aur span nearly four months and are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A TESS light curve extending over 50 days, from JD 2,459,474 to JD 2,459,524, is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We derive a mean magnitude and rms variability of u’=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 mag, g’=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='17 mag, r’=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='11 mag, and i’=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='09 mag, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The variability amplitude is much larger than the photometric errors in all bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' it amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='025 mag in the g’r’i’ bands and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='033 mag in the u’ band, and it decreases toward longer wavelengths, from u’ to i’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We ran two period-search algorithms on the light curves: a CLEAN periodogram analysis (Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1987), which is conceptu- ally similar to a Fourier transform, and the String-Length method (Dworetsky 1983), which finds the period that minimizes the av- erage distance between consecutive points in the phased light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Both yielded the same period at all wavelengths, namely P=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 days, with the uncertainty being estimated from the standard deviation of a Gaussian fitted to the periodogram peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The results of the period search are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The light curves folded in phase at the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 d period are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' They clearly show the modulation of the brightness level at this period, particularly in the u’ band, where the amplitude is largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We ascribe this low-level modulation to surface spots and the P=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 d period to stellar rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This estimate agrees within the error bars with the previously reported periods for GM Aur, namely P=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 d by Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2010) and P=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='02 d by Artemenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We therefore adopt the following ephemeris: HJD(d) = 2, 459, 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='80 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 × E, (1) which defines the rotational phase Φrot=(HJD- 2,459,460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='80)/Prot (modulo Prot), where phase zero (Φrot= 0) is chosen as the epoch of maximum optical brightness in the rotational cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Superimposed onto spot modulation, additional signs of in- trinsic variability are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This is the case, for instance, of an apparent brightness event that is centered on JD 2,459,509 and lasted for a few days, as well as more pronounced wide dips toward the end of the observing period, from JD 2,459,544 on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Interestingly, the first half of the light curve exhibits several day-long brightening events, while the last third is dominated by wide dimming events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We note that most of the brighten- ing events, with g’≤12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 mag, tend to occur at rotational phases shorter than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 or longer than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='85, that is, close to the maxi- mum brightness of the spot modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' If the photometric vari- ations of the system are modulated by the visibility of a bright accretion spot at the stellar surface, these brightening events seen around the time of maximum accretion shock visibility most likely reflect varying accretion on the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The wide dips in the last part of the light curve exhibit the same period- icity and phase as the spot modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' They reach a minimum brightness close to phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5, with an amplitude that steadily de- creases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 mag to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 mag in the g’ band over a timescale of a few weeks, from JD 2,459,550 to JD 2,459,570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 4 shows the color behavior of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As the sys- tem fades, it becomes redder, with a color slope larger than ex- pected from ISM-like extinction at least in the (u’-g’) and (g’-r’) color indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' According to Venuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2015), a large color slope at short wavelengths is characteristics of accretion-driven photometric variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, both the large scatter seen in the (u’-g’) color index at low brightness levels and the changing Article number, page 6 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' GM Aur light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: GM AUr’s u’g’r’i’ light curves from LCOGT observations that extended over nearly four months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The mean photometric error is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='025 mag in the g’r’i’ bands and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='033 mag in the u’ band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A scaled TESS light curve (black) obtained contemporaneously is overplotted onto the LCOGT r’-band light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Middle: LCOGT light curves folded in phase with a period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 days with the ephemeris of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: Same as above, with a color scale for data points that reflects the Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The amplitude of the light variations appears to increase slightly toward the end of the observing run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In all panels, the error bars on the measurements are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 7 of 30 11 12 Magnitude TESS 13 14 15 2459480 2459500 2459520 2459540 2459560 2459580 JulianDate11 12 Magnitude 13 14 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 Phase2459560 12 Magnitude 2459540 13 2459520 14 2459500 2459480 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 PhaseA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Period search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: CLEAN periodogram analysis of the u’g’r’i’ light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A peak occurs at the frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1, which corresponds to a period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: String- length analysis of the u’g’r’i’ light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A clear minimum appears for a period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Color-magnitude relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The (u’-g’), (g’-r’), and (r’-i’) colors are plotted as a function of the system brightness in the g’-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dashed lines indicate the expected ISM reddening slope computed from Fiorucci & Munari (2003) with R=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' shape of the light curve during the semester suggest that several sources of variability might be present, such as a combination of accretion and obscuration events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' High-resolution optical spectroscopy We took advantage of the wide wavelength range covered by the SOPHIE spectrograph at high spectral resolution to derive the stellar parameters of GM Aur, to measure optical veiling, and to investigate the emission line profiles and their variability across the optical range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These results are described in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Properties of the GM Aur system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Parameter Value SpT K4-K5 AV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 mag Teff 4287 ± 35 K L⋆/L⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 R⋆/R⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 M⋆/M⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='13 ˙Macc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 10−8 M⊙yr−1 EW(LiI) 420 ± 23 mÅ v sin i 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 km s−1 Prot 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 d rcor 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 R⋆(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='064 au) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Stellar parameters To derive the stellar parameters (Teff, v sin i, Vr, and Vmic), we fit synthetic ZEEMAN spectra (Landstreet 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Wade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Folsom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2012) to the average of the first six SOPHIE spec- tra, which are not contaminated by the moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Synthetic spec- tra were computed from MARCS atmosphere models (Gustafs- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2008) and the VALD line list database (Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We applied a χ2 minimization procedure based on a Levenberg-Marquart algorithm over seven wavelength windows ranging from 455 to 649 nm, excluding the regions affected by tellurics, emission, or molecular lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We set the macroturbu- lent velocity to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 km s−1 and the surface gravity log g to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0, and we assumed solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These are typical parameters for low-mass TTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' During the fitting procedure, we applied the mean veiling value derived below for the GM Aur mean opti- cal spectrum (r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='55 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2) to the synthetic spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, we averaged the results obtained from the various wavelength windows, except for one window that yielded values higher than 2σ from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We thus derived Teff = 4287 ± 35 K, Vr = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='14 km s−1, Vmic= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 km s−1, and v sin i = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We also used the ROTFIT package (Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2003, 2006) applied to the mean GM Aur’s SOPHIE spec- trum, from which we derive Teff = 4505 ± 53 K, Vr = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='26 km s−1, and v sin i = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' While all these values are consistent within 3σ, the large Teff difference derived from ZEEMAN and ROTFIT illustrates model-dependent uncertainties that are likely related to the use of different model templates (MARCs for ZEEMAN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' BTSetll for ROTFIT) and possibly to wavelength-dependent systemat- ics induced by starspots (Gangi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Flores et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For consistency with similar observing campaigns that we pre- viously performed on young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018), we adopted the results of the ZEEMAN fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This estimate also agrees better with the K6 spectral type derived by Herczeg & Hillenbrand (2014), from which they deduced Teff= 4115 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' When the Teff-SpT conversion tables from Herczeg & Hillenbrand (2014) and Pecaut & Mama- jek (2013) are used, the Teff value derived above corresponds to a K4-K5 spectral type, which agrees fairly well with previous esti- mates obtained from optical and near-infrared spectroscopy (K5- K6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Espaillat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Herczeg & Hillenbrand 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Vr and v sin i values can be compared to those derived from the uncontaminated CCF of the first six SOPHIE spectra, namely = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 km s−1 and = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='26 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The quoted uncertainties are the rms of the six mea- surements and therefore include intrinsic variability of the CCF profiles due to spot modulation, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The two estimates Article number, page 8 of 30 u g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 Frequency (1/d)20 length StringI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 Frequency (1/d)"r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 b-,r 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content="2 g'(mag)J." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur of Vr agree within the errors as well as with the median Vr value derived from the SPIRou spectra (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4), while the v sin i value derived from the CCF is significantly lower than the value deduced from spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We suspect that the discrep- ancy may arise from the color-dependent FWHM-v sin i relation used for SOPHIE CCFs, which is calibrated on main-sequence stars (Boisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2010) and may not be fully adequate for pre- main-sequence objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' From the average g’r’i’ colors reported for GM Aur in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1, the intrinsic colors of a K4-K5 dwarf listed by Kraus & Hillenbrand (2007) and Covey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2007), and using ISM extinction coefficients from Fiorucci & Munari (2003), we de- rive a visual extinction on the line of sight AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This agrees with previous determinations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Herczeg & Hil- lenbrand 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The REM photometry reported above yields a median J-band magnitude of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='11 mag, which is close to that of 2MASS (J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='34 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We dereddened the median J-band magnitude with A j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='28 × AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='084 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='084 mag and used the J-band bolometric correction listed in Pecaut & Mamajek (2013) for a K4-K5 dwarf, BCJ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='03 mag, to derive the stellar luminosity, L⋆= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 L⊙, assuming the Gaia distance of 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 pc (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We thus derive a stellar radius R⋆= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 R⊙ from Stefan’s law, and a stellar mass M⋆= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 M⊙ from the Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2000) pre-main-sequence evolution models, while CESAM models (Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2013) yield M⋆= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='12 M⊙ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alécian, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We therefore adopt M⋆= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='13 M⊙, in agreement with Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2015) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This estimate is also consistent within the errors with the dynamical mass estimate, Mdyn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='02 M⊙, reported by Guilloteau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2014), which was later revised to Mdyn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='02 M⊙ by Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table 4 summarizes the derived stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, we combined the v sin i and rotational period mea- surements with the stellar radius estimate to derive the stellar inclination sin i = Prot × v sin i / (2 π R⋆) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 with the val- ues listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Accounting for 1σ uncertainties on the stellar parameters, we derive a lower limit of i⋆ ≥ 63◦ for the stellar rotational axis onto the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This value is signif- icantly higher than the inclination inferred from high-resolution ALMA images of the outer disk of GM Aur observed at mil- limeter wavelength, which yield idisk = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 deg (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It agrees better, however, with the inclination value de- rived for the disk seen in scattered light with adaptive optics on a scale of a few dozen au, for which Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2016) ob- tained idisk = 64 ± 2 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' On the much smaller scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='013 au, Bohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2022) measured an inner-disk inclination idisk = 68+16 −28 deg from long-baseline K-band interferometry using VLTI/GRAVITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' They did not find evidence for an inner and outer disk misalignment in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As outlined by Appen- zeller & Bertout (2013), the uncertainty on the determination of stellar inclinations from rotation measurements rapidly increases at large angles and is prone to systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For GM Aur, inferring the stellar inclination from the disk inclination might therefore be more reliable than estimating it from rotation mea- surements, owing in particular to the significant uncertainty on the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Nevertheless, all independent measurements indicate a moderate to high inclination for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Veiling At optical wavelengths, accreting T Tauri stars exhibit an addi- tional source of continuum flux, which presumably arises from the accretion shock at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This optical excess Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Optical veiling measurements from the OHP/SOPHIE spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Part of a spectral window showing the template spectrum (gray), the observed spectrum (black), and the velocity broadened and veiled template spectrum (red) that fits the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: Veiling measured in several spectral windows (see text) as a function of Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The central wavelength of the spectral windows is indicated in the top left corner of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Mean optical veiling plotted as a function of rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code indicates the Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' LiI 6707 Å line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Left: The 34 line profile measurements from SOPHIE spectra are shown superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code cor- responds to successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Right: 2D periodogram across the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted horizontal line drawn at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1 indicates the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curve displays the mean line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code reflects the periodogram power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' continuum partly veils the stellar photospheric spectrum and is therefore referred to as "veiling" (Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We mea- sured the optical veiling from the high-resolution OHP/SOPHIE spectra by comparing the photospheric spectrum of GM Aur to that of the unveiled nonaccreting template V819 Tau, a WTTS with Teff= 4250 ± 50 K, Vr = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 km s−1, and v sin i Article number, page 9 of 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 2459515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 2459510 Optical 2459505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 2459500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 Phase1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 40 20 0 20 40 v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 requency Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 40 20 0 20 40 v (km/s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 No veiling nor broadening Observed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 Fit xnl Normalized 0.' metadata={'source': 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9512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 9515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 Julian Date -2,450,000 (d)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 km s−1 (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We retrieved an archival CFHT/ESPaDOnS spectrum of V819 Tau, which we resampled at the spectral resolution of OHP/SOPHIE spectra, translated into the radial velocity of GM Aur, and rotationally broadened using the rotational function from Gray (1973) to match the ro- tational velocity of GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The template spectrum was then fit to the GM Aur spectra over the same wavelength windows as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 by adjusting the veiling using the following formula: I = I0 + r 1 + r , (2) where I is the veiled spectrum, I0 is the spectrum without veil- ing, and r is the continuum veiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Rei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2018) showed that an additional line-dependent veiling component may be present in the strongest photospheric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We therefore retained only weak to moderate lines with an EW between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 Å to perform the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The veiling measured for each spectrum in each spectral window is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Systematic offsets are clearly seen between the veiling values measured in the different spectral windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These offsets may partly reflect a wavelength- dependent veiling, as veiling seems to decrease toward longer wavelengths for all but the bluest spectral window, but it may also result from systematic errors depending on the specific sam- ple of lines included in each window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We therefore computed an average veiling value, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='55, over all spectral windows for individ- ual GM Aur spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This value is listed in Table 5 with its asso- ciated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The optical veiling is moderate, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='17 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='55 at 5500 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Similar mean values were derived from ROTFIT, namely r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 at 4500, 6000, and 6500 Å, respectively, using a library of 400 spectral templates with spectral types FGKM from the OHP/ELODIE database, which yielded a best χ2 fit for spectral types ranging from K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 to K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Figure 6 shows the mean optical veiling plotted along the rota- tional phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A hint of higher veiling values towards phases 0 and 1, and minimum values around phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A pe- riodogram analysis of the mean veiling variation did not yield significant results, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The lack of significant veiling variability is confirmed by the examination of the LiI 6707 Å photospheric line profile shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The shape and depth of the line appear to remain quite stable over the duration of the observing run, and a periodogram analysis across the line profile (Giampapa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1993) revealed no signs of periodic modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This suggests that the source of optical veiling remains at least partly in view throughout the rotational cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This might arise from a high-latitude accretion shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Emission lines The main emission lines seen in the optical spectrum of GM Aur, namely Hα, Hβ, and HeI 5876 Å, are displayed in Fig- ure 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Balmer lines show a broad emission peak and a slightly blueshifted absorption component, whose depth varies from being hardly discernable to reaching below the continuum 8 The [OI] 6300 Å line is also seen in emission in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' How- ever, it is significantly contaminated by sky, which cannot be easily cor- rected for due to the different response of the object and sky fibers of the SOPHIE spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The other emission lines seen in the high signal- to-noise ratio mean SOPHIE spectrum are Ca II H&K, Hγ, Hδ, Hϵ, and He I 6678 Å, the latter being weak and affected by a deep photospheric FeI line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Optical line EW and veiling measurements from the OHP/SOPHIE spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Julian date EW Veiling Hα Hβ HeI r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='55 rms (2,450,000+) Å Å Å 9499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='51683266 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='53216217 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 † Uncertain value due to lunar contamination (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' in the Hβ profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The wide, slightly blueshifted absorption com- ponent peaks at -30 to -20 km s−1and covers a velocity range from about -90 to +40 km s−1 in the Hβ line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Additional absorption components appear at higher blueshifted velocities, peaking from -110 to -90 km s−1, and extending down to about 160 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These blueshifted absorption components cause most of the variability in the Balmer line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Hβ pro- file also exhibits significant variability over the red wing, up to velocities of +200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, none of the profiles exhibits redshifted absorption components reaching below the continuum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Owing to the complex line shapes, equivalent widths (EW) of the Hα, Hβ, and HeI 5876 Å lines were computed by directly integrating below the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The results are listed in Ta- ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The measurement accuracy is 10% or better for EW(Hα) and EW(Hβ), and about 20% for EW(HeI) due to the more un- certain continuum location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We note that on JD 2,459,512, the EWs measurements are systematically lower than during the rest of the observations, which might be due to lunar contamina- tion, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The average and rms values we obtain are EW(Hα) = 83 ± 8 Å, EW(Hβ) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 Å, and EW(HeI) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='13 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI 5876 Å line profile is roughly symmetric and con- sists of a narrow component (FWHM ∼ 30-40 km s−1) superim- posed on a broad pedestal, as previously reported by McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The peak intensity of the narrow component varies significantly, while the broad component appears relatively sta- ble (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We fit the HeI line profile with a two-component Gaussian model to extract the properties of the narrow (NC) and broad (BC) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The EWs were derived from the Gaus- sian fit of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The radial velocity, FWHM, and EW of the NC and BC, as well as their uncertainty derived from the covariance matrix of the Levenberg-Marquart fitting algorithm, are listed in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We derived the radial velocity of the narrow component and found it to be variable and redshifted by ∼5-10 km s−1 relative to the stellar velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Figure 9 shows the HeI NC radial velocity curve plotted as a function of Julian date and rotational phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As previously reported by McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020), HeI NC Vr Article number, page 10 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Optical line profile variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Series of optical line profiles Hα, Hβ, and HeI plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code corresponds to successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Middle: Same profiles, superimposed to illustrate their variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: 2D periodograms across the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code reflects the periodogram power, from zero (blue) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted horizontal red line drawn at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1 indicates the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curve is the mean line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' appears to be rotationally modulated with a full amplitude of ∼6 km s−1, as expected if the NC component of the HeI 5876 Å line profile were produced in a high-latitude accretion shock at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We fit the observed NC Vr curve with the geometrical accretion shock model described in Pouilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The model computes the variation of the HeI NC radial velocity as the combination of the rotational modulation of the accretion shock and the intrinsic inflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The free pa- rameters of the model are the inflow velocity, the latitude of the accretion shock, the phase at which it faces the observer, and the Article number, page 11 of 30 Hα day phase 9499.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 9510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 9515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 HJD (-2,450,000 d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 Vr HeI (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 Vr HeI (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='s 1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Radial velocity of the narrow component of the HeI 5876 Å line profile in the stellar rest frame plotted as a function of Julian date (top) and rotational phase computed from the ephemeris of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code corresponds to successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The fit by a geometrical accretion shock model (see text) is shown (dash-dotted curve) together with its 1σ uncertainty (gray area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' stellar inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI NC Vr curve is best reproduced with an accretion shock located at a latitude of 83 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5◦ that faces the observer at phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='08 and has a radial post-shock ve- locity of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 km s−1 in the stellar rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Because the model now includes the inflow velocity, the HeI NC radial ve- locity curve does not reach the mean velocity at the time when the spot faces the observer, as it would for the case of static stel- lar spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The stellar inclination we derive from the model is i = 64 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2◦, which is consistent with the inner disk inclination derived from K-band VLTI/GRAVITY data (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' According to the model, the accretion shock faces the observer close to the origin of the phase, which is consistent with the photometric behavior described above (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI line profile is also strongest over rotational phases ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='75 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='09 (excluding the probable accretion burst occuring at JD 2,459,510, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1), and this is also when the highest veiling values are measured in the JHK bands (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These two results further support a maximum vis- ibility of the accretion shock close to Φrot= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A bidimensional periodogram analysis (Giampapa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1993) of the Balmer and HeI 5876 Å line profiles reveals a peri- odic modulation of part of the profiles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The intensity of the narrow component of the HeI line profile is modulated at a frequency of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 d−1, corresponding to a period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 d, with a large uncertainty due to the limited temporal sampling of the spectral series, however, that translates into a poor frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As the HeI line profile NC component arises in the accretion shock (Beristain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001), it is expected to be mod- ulated at the stellar rotational period or close to it in case of lat- itudinal differential rotation at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In the Balmer line profiles, a modulation close to the stellar period also ap- pears over three distinct locations: in the highly redshifted wing over the velocity range ∼200-400 km s−1, at slightly redshifted velocities between 0 and ∼50 km s−1, and at highly blueshifted velocities from -400 to -200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' While the maximum power of the periodogram in the blue wing appears at the stellar rota- tion period, it seems to drift to longer periods, similar to that seen in the HeI NC component, toward the red wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' If the red wing modulation of Balmer line profiles is caused by the absorption of shock emission by a funnel flow crossing the line of sight, this might be an indication of differential rotation along the fun- nel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Noticeably, no sign of periodic variability is seen in the Balmer line profiles in the velocity channels in which the vari- able blueshifted absorption components arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This suggests that these components either result from sporadic ejection processes or that they vary on periods longer than ten days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices (Johns & Basri 1995) between line pro- files were computed and are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' They dis- play the degree to which temporal flux variations in a pair of spectral lines are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Matrices can be computed for a sin- gle profile (autocorrelation), for instance, Hα⋆Hα, to investigate how the different parts of the profile vary with respect to each other, or between two profiles (cross-correlation), for example, Hα⋆Hβ, to compare the intensity variations of different lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Hα⋆Hα and Hβ⋆Hβ matrices shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The blue and red wings of the profiles vary in a corre- lated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The high-velocity red wings are anticorrelated with the emission peak region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This may be a sign that high-velocity redshifted absorption components appear when the peak inten- sity is higher, although the absorptions do not reach below the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Many absorption features are superimposed on the emission line component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These features do not present a peri- odicity and are not correlated with each other, nor with the emis- sion part of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Hβ⋆Hα matrix mimics the matrices of Hα⋆Hα and the Hβ⋆Hβ, showing that both lines are formed in the same region, as expected from magnetospheric accretion models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 12 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' High-resolution near-infrared spectroscopy The main emission lines seen in the high-resolution near-infrared spectrum of GM Aur, namely HeI 10830 Å, Paβ, and Brγ, are depicted in Figures 10 and 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The near-infrared hydrogen lines exhibit broad emission profiles, with FWHM ∼200 km s−1, whose peaks are slightly blueshifted compared to the stellar ve- locity, and are located at -30 and -10 km s−1 for Paβ and Brγ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The profiles are roughly symmetric, but have pro- nounced high-velocity redshifted absorptions that extend up to +400 km s−1 and reach below the continuum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Paβ and Brγ exhibit inverse P Cygni (IPC) profiles of varying depth in most SPIRou spectra, which is in stark contrast to the optical hydro- gen line profiles, Hα and Hβ, which do not exhibit such features (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices for the near-infrared hydrogen line profiles are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Paβ⋆Paβ, Brγ⋆Brγ, and Paβ⋆Brγ matrices are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The emission part of the profiles is overall correlated, and anti-correlated with the high- velocity redshifted absorption component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This indicates that the redshifted absorption is deeper when the emission line is more intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Cross-correlation matrices between optical and near- infrared line profiles observed in the same night during the Oc- tober runs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Brγ and the Paβ emis- sion components correlate well with the main Hα and Hβ emis- sion components (-100 km s−1< v < 200 km s−1), which sug- gests that at least part of the line emission forms in the same extended region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Paβ redshifted absorption, and to a lesser extent, the Brγ redshifted absorption, correlates well with the Hα and Hβ blue and red high-velocity wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This indicates that they all form in the same region close to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The near- infrared profiles are strongly anticorrelated with the highly vari- able blueshifted absorption components that appear at velocities of about -100 km s−1 in the Balmer line profiles and presumably trace outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Except for this feature, the overall correlated vari- ations of optical and near-infrared line profiles suggest that they are all good diagnostics of the accretion flow, as previously sug- gested by Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2014), for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Detailed modeling of the line profile shapes and variability is needed, however, to ascertain the exact location from which they arise (Tessore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI 10830 Å line profile is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It is usu- ally dominated by an emission peak at low velocities that be- comes quite weak at specific times, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The profile also often displays highly redshifted absorption features, similar to the IPC components seen in the Paβ and Brγ lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Unlike the near-infrared hydrogen profiles however, the HeI line addition- ally exhibits various absorption components in the blue wing of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' At least two systems of blueshifted absorptions can be identified: a low-velocity system extending between -20 to 80 km s−1, and a high-velocity system ranging from -100 to - 250 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Figures 10 and 11 show that the two systems are quite variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These two absorption systems are reminiscent of those observed in the Hα and Hβ lines, although they occur at slightly bluer velocities in the HeI profile, with an offset of about 40 km s−1 compared to the optical profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices involving the HeI 10830 Å line are dis- played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI autocorrelation matrix shows sev- 9 Paγ and Paδ also appear in SPIRou spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Their shape and variabil- ity behavior is similar to that of Paβ and Brγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Brδ is also included in the SPIRou wavelength range, but lies at the intersection of spectral orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In all SPIRou spectra, we also detect a weak H2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='12 µm line, with EW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 Å, and FWHM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 Å (≃ 20 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This line is located at the stellar rest velocity (Vr= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='75 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' eral correlated components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Over the velocity channels extend- ing from -100 km s−1 up to 200 km s−1, the line flux varies in a correlated fashion, but this region of the line does not correlate with the rest of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Similarly, the redshifted absorption region, which extends from 200 km s−1 to 400 km s−1, varies as a whole and does not correlate with the rest of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The blue wing of the profile presents many short-duration absorp- tion components superimposed on the emission component, and, probably due to this, each velocity bin varies independently of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Comparison of the HeI 10830 Å line profile to optical and near-infrared hydrogen profiles indicates that the emission core and peak intensity are correlated, as are the high-velocity redshifted absorptions seen in the HeI, Paβ, and Brγ line pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, the HeI blue wing shows a strong anticorrelation with the core of the Hβ profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The periodogram analysis of the line profiles is shown in Fig- ure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In the three lines, the IPC components appear to be peri- odically modulated at the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As the SPIRou dataset extends over four months, this suggests that a quite sta- ble structure, presumably the accretion funnel flow, gives rise to this spectral feature10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The redshifted absorption component is deeper when the emission line is more intense, which agrees with the assumption that the two components trace the densest part of the accretion column close to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The low-velocity red wing of the HeI line profile also shows a periodic modula- tion at the same period, which might trace the visibility of the accretion shock, as seen in the optical HeI line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The pe- riodogram power in the blue wing of the lines is weaker, ex- cept perhaps over the high-velocity channels of the Paβ line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In particular, the variable blueshifted absorption systems seen in the HeI line profile are not periodically modulated on this long timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We performed a similar analysis of each SPIRou run individually, and the results are shown in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 reveal that the most conspicuous high-velocity redshifted absorptions in the HeI, Paβ, and Brγ line profiles occur prefer- entially around Φrot=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' During the SPIRou October run, which was simultaneous with the OHP/SOPHIE observations, we re- alized that the highly blueshifted velocity channels of the near- infrared lines appear to be modulated at the stellar rotation pe- riod (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Hence, while the modulation of high-velocity blueshifted absorptions seen in the HeI profile disappear on a timescale of several months, the modulation may survive over a few rotational periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, similar to what is seen in the Balmer line profiles over a much shorter time span (see Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3), there is a marginal indication from the periodograms of the near-infrared lines that the modulation period drifts from the blue to the red wing of the profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This might be a sign of differential rotation in the source of the variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The EW of the HeI 10830 Å, Paβ, and Brγ lines was com- puted on the residual profiles using two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The first method consists of adjusting a Gaussian fit to the line profile, and the second method is integrating below the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The two methods were applied to the Paβ and Brγ lines that most often exhibit a strong Gaussian-like emission and, at times, a pronounced redshifted absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The difference between the EW derived from the Gaussian fit and from the profile integra- tion then provides an estimate of the strength of the redshifted absorption component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI line exhibits a complex profile, with pronounced absorptions appearing in both the blue and red wings, and it cannot be fit by a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We report here HeI 10 The long time coverage of the SPIRou observations enabled us to explore periods up to 100 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, we did not find significant periods longer than those reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 13 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Near-infrared line profiles: HeI (left), Paβ (center), and Brγ (right), plotted with arbitrary offsets as a function of Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Each color represents a SPIRou run, namely September (blue), October (orange), November (green), December 2021 (red), and January 2022 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The October SPIRou run is contemporaneous to the OHP/SOPHIE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 14 of 30 Hel Paβ Bry day day day 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 9475.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 一 一 500 0 500 500 0 500 500 0 500 v (km/s) v (km/s) v (km/s)J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Near-infrared line profile variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Series of residual near-infrared line profiles HeI (left), Paβ (center), and Brγ (right), plotted superimposed to illustrate their variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Each color represents a SPIRou run as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: 2D periodograms across the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code reflects the periodogram power from zero (blue) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted red horizontal line drawn at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1 indicates the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curve displays the mean line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EWs measured through profile integration only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW measure- ments are usually accurate to within 5% because of the well- defined adjacent stellar continuum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The results are listed in Table 7, and the evolution of EW during the observing campaign is illustrated on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' While the EW variations of the three lines are usually correlated, there are notable exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For instance, during the first SPIRou run around JD 2,459,478, the HeI line goes into absorption, driven by a broad redshifted absorption component that reaches half the continuum value, while the Paβ and Brγ lines reach a local inten- sity maximum, even though they also exhibit a pronounced red- shifted absorption (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In contrast, during the second SPIRou run, the variations of the three lines are well correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Line EWs and near-infrared veiling measurements (listed in Ta- ble 3) are both shown as a function of Julian date and rotational phase in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 (Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Both quantities appear to be ro- tationally modulated, with maximum values occurring close to Φrot=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, the photospheric radial velocity curve derived from SPIRou spectra is shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Vr is found to vary between 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='96 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 km s−1, with a median value of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='65 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A CLEAN periodogram analysis reveals a period P=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='11 d, and the String-Length method yields P = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='12 d, both consistent with the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 d photometric period derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The photospheric radial velocity curve folded in phase at the stellar rotation period is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Vr exhibits a roughly sinusoidal variations in rotational phase, which suggests that it is modulated by a surface spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It reaches the median velocity going blueward around Φrot= 0, as expected from a stellar spot facing the observer at this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Be- cause this is also the phase of maximum brightness of the system (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1), this suggests that a hot spot modulates the pho- tospheric Vr curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Vr curve derived from the near-infrared photospheric lines (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 13) is inverted compared to the Vr curve derived for the HeI 5876 Å NC line profile (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Similar antiphase radial velocity variations between absorption Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW of near-infrared lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: EW of the HeI, Paβ, and Brγ lines measured on SPIRou spectra plotted as a function of Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The measurement uncertainties are about the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: Dif- ference between the EWs measured by direct line profile integration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', including the redshifted absorption component) and the EW de- rived from the Gaussian fitting of the emission component only plot- ted as a function of Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This differential quantity measures the strength of the redshifted absorption component in the Paβ and Brγ line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The more negative the differential quantity, the deeper the red- shifted absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' and emission lines have been reported for other accreting T Tauri stars and were interpreted as caused by the modulation of the line profiles by a hot spot at the stellar surface (Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Gahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alternatively, the photospheric radial velocity variations may also be produced by a cool spot that co- exists with the accretion shock at the same location on the stellar Article number, page 15 of 30 Hel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 400 200 0 200 400 v (km/s)Paβ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 400 200 0 200 400 v (km/s)Bry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 xn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 400 200 0 200 400 v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 (1/d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 (1/d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 Frequency ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)20 Hel PaB BrG 3 10 EW C 9480 9500 9520 9540 9560 9580 JulianDate-2,450,000PaB BrG dEW 9480 9500 9520 9540 9560 9580 JulianDate-2,450,000A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Radial velocity variations measured in the near-infrared photo- spheric lines of the SPIRou spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Radial velocity as a function of Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: Radial velocity curve folded in phase at the stellar rotational period P=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code reflects the Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Near-infrared line EW measurements from the CFHT/SPIRou spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Julian date EW (Å) (2,450,000+) HeIint Paβg Paβint Paβdif f Brγg Brγint Brγdif f 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='066 2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='17 9566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='953 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='08 9585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='941 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='31 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='67 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='47 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='02 Note: EWg is obtained from a Gaussian fit to the line profile, while EWint is measured from line profile integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EWdif f is the difference between EWint and EWg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' surface, as Doppler images of accreting T Tauri stars suggest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2010, 2011, 2019, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Low-resolution near-infrared spectroscopy The median nightly spectra obtained from the ExTrA-T2 and ExTrA-T3 telescopes are shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We computed the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 Relative intensity T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 Relative intensity T3 HeI Paβ Paγ Paδ Paϵ Paζ Paη Paθ 9500 9525 9550 9575 9600 9625 9650 BJD - 2,450,000 0 10 20 EW(Paβ) [˚A] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Median spectrum for each night from the ExTrA-T2 telescope (lower part, 88 spectra) and ExTrA-T3 telescope (upper part, 69 spec- tra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The main emission lines are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color is proportional to the EW of the Paβ line shown in the top panel as a function of Julian date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW of the HeI 10830 Å, Paβ, Paγ, and Paδ lines from the ExTrA spectra using specutils11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We analyzed each telescope inde- pendently because the point spread function (and therefore the resolution) of the spectrograph depends on the position on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' First, we fit a Gaussian to each line on the median spec- tra to measure the line center and full width, the latter we defined as amounting to six times the standard deviation of the Gaussian fit in order to isolate the line profile from the nearby continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The local continuum around each line was modeled with a third- degree polynomial, adjusted on three line widths centered on the line, but excluding the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Then, for each line in each of the 1898 individual spectra, we computed the EW by integrating the flux over the full line width using the parameters and the nor- malization region defined from the median spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, we computed the median of the individual measurements for each night, regardless of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The median is less affected by outliers than the mean, and the differences between the mean and the median values are within the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 in Appendix C lists the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In order to estimate the reliability of the procedure, we com- pared EWs derived from ExTrA and from SPIRou spectra for the Paβ line using 20 measurements obtained on each instru- ment less than one day apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The results show an excellent cor- relation, with a slight tendency for the EW measured from Ex- TrA to exceed those measured from SPIRou, with a mean dif- ference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A similar result is obtained for the HeI line, with a mean difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='31 Å and an rms of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 Å be- tween ExTrA and SPIRou estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The comparison with high- resolution measurements thus validates the EWs obtained from low-resolution spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The line variability is illustrated on Figure 14, where the me- dian nightly spectra are superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The median and extreme 11 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='com/astropy/specutils Article number, page 16 of 30 9580 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9560 (km/s) 9540 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 D Vrad 9520 9500 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9480 9480 9500 9520 9540 9560 9580 JulianDate-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0009580 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9560 (km/s) 9540 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 D Vrad 9520 9500 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9480 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 PhaseJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW variability of the near-infrared line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Left: EW measurements plotted as a function of Julian date for the HeI, Paβ, Paγ, and Paδ lines from the ExTrA spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Right: GLS periodogram of the EW measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The period on the x-axis is displayed on a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EWs of near-infrared line profiles and J-band magnitude vari- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: EW of the near-infrared HeI and Paβ lines folded in phase at the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW(HeI) is offset by -10 Å for clar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: J-band light curve, deduced from ExTrA spectra, folded in phase at the stellar period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The brightness modulation is similar to that observed at optical wavelengths (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Mid-term variation of EW(Paβ) (red) compared to the system u’-band light curve (blue) for measurements taken less than one day apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' To facilitate the comparison, the EW measurements are plotted on a magnitude scale and are offset, namely -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 log EW(Paβ) + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Minimum, median, and maximum near-infrared line EW mea- surements from the ExTrA spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW (Å) HeI Paβ Paγ Paδ Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 values of EWs measured for the HeI, Paβ, Paγ, and Paδ lines are listed in Table 8, and the night-by-night measurements are listed in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We note that the HeI line at times appears to be in absorption at this low spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Figure 15 shows the EW measurements plotted as a function of time and its generalized Lomb-Scargle periodogram (GLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Zechmeister & Kürster 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The EW of the four lines is found to be modulated with a period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='087 days, consistent with the stellar rotation period, where the error es- timate is the standard deviation of a Gaussian fit to the peri- odogram peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 16 shows the HeI and Paβ line EWs folded in phase at the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The modulation of the line strength clearly appears, with maximum flux around phase zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As shown in the same figure, these variations follow the modu- lated brightness level of the system in the J band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Longer-term EW variations of higher amplitudes are also clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' These variations seem to be correlated with the multicolor photometry presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1, in par- ticular, during the brightness event centered on JD 2,459,509 and the wide dip around JD 2,459,549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In Figure 17, a clear correla- tion appears between the photometric variations in the u’ band and the Paβ line variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' If the brightening of the system in the u’ band is linked to accretion, this correlation suggests that most of the emission line flux is connected to the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Mass-accretion rate The combination of optical and near-infrared spectroscopy of- fers a number of emission lines from which we can estimate the mass accretion rate onto the star, using the empirical re- lations between line luminosity and accretion luminosity pro- posed by Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Combining the range of Hα and Hβ EWs reported above with the nearby continuum fluxes com- puted from the r’ and g’ magnitudes corrected for extinction, we obtain the line fluxes and luminosities as follows: Fline = Article number, page 17 of 30 20 Hel Paβ Pa Pad 15 A 10 EW 5 0 9500 9520 9540 9560 9580 9600 9620 9640 BJD - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 Normalized power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 2 3 4567 10 20 30 4060 Period [d]20 EW(PaB) EW(Hel)-10 10 3 2459600 2459550 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 Phase9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 2459600 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 (mag 2459550 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 Phase11 PaB 12 M 14 15 9500 9520 9540 9560 9580 Juliandate-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='000A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fo λ × EW(line) × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4(mλ−Aλ) and Lline = 4πd2Fline, where Fo λ is the flux of a zero-magnitude star12, mλ and Aλ are the magnitude and extinction in the photometric band of interest, and d is the distance to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The accretion luminosity was derived from the line luminosity using the relation reported by Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2017), and ˙Macc was deduced from Lacc assuming a magneto- spheric radius of 5 R⋆ (see Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2017, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Taking the uncertainties on all involved quantities into account, we thus derive ˙Macc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 × 10−8 M⊙yr−1 from the Hα and Hβ line fluxes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We performed a similar analysis using the extensive mea- surements of Paβ EWs obtained from the ExTrA spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' From the extreme values, namely EW(Paβ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 –18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 Å, and the mean REM J-band magnitude of the system during the observ- ing period, J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10, we derive ˙Macc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 × 10−8 M⊙yr−1, with a median value of ˙Macc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 × 10−8 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, we derived additional estimates of ˙Macc from the Brγ line flux, using the median and extreme values of EW(Brγ) mea- sured on SPIRou spectra, namely 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 Å, which we calibrated with the mean REM K’-band magnitude of the system, K’= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Using the relations reported by Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2017) between line and accretion luminosity, we thus derived ˙Macc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 × 10−8 M⊙yr−1, with a median value of ˙Macc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 ×10−8 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dispersion in the ˙Macc estimates partly results from in- trinsic ˙Macc variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For instance, the SOPHIE spectra were obtained during a brightening of the system that occurred around JD 2,459, 512, at a time at which all optical and infrared lines were relatively strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We thus derive a relatively high value of ˙Macc from these spectra compared, for example, to the smaller ˙Macc estimate derived from the Brγ line in the SPIRou spec- tra that were obtained during more quiescent phases of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We cannot exclude, however, that some of the dispersion may also arise from systematic uncertainties in the empirical line-to-accretion luminosity relations over the optical and near- infrared wavelength ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In any case, the various estimates agree globally, with ˙Macc typically varying between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 ×10−8 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Discussion The monitoring campaign we performed on GM Aur reveals significant but relatively low-level temporal variability over a timescale of six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The photometric variations are mild, as might be expected for this moderately accreting young system ( ˙Macc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 × 10−8 M⊙yr−1), with amplitudes ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 mag in the u’ band to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 mag in the i’ band, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 mag in the J band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The brightness of the system varies smoothly because it is modulated by the visibility of surface spots at the stellar rota- tional period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Except for the HeI 10830 Å line profile, the spectral appearance of the system does not change drasti- cally on a timescale of months, and the veiling is low and stable, amounting to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 in the optical range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The strength of the main emission lines (Hα, Hβ, HeI 5876Å, Paβ, and Brγ) varies by a factor of 2 to 3 over the course of the semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In contrast, the HeI 10830 Å line profile exhibits extreme variability and is sometimes barely noticeable in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The line profile shapes are strongly variable on a timescale of days, and the develop- ment of both blueshifted and redshifted absorption components is superimposed onto a broad emission component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Blueshifted 12 Fo λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='43 10−9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='27 10−9 erg s−1 cm−2 Å−1 in the r’ and g’ bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' absorption components indicate outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Kwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2007), and redshifted components probe fun- nel flows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Re- markably, the redshifted absorption features that reach below the continuum level, that is, inverse P Cygni profiles (IPC) (see Cal- vet & Hartmann 1992), which are seen in the high-resolution near infrared line profiles (HeI 10830 Å, Paβ, Brγ), are steadily modulated by stellar rotation over an extended observing time span of 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Rotational modulation is also clearly detected in the strength of the near-infrared lines and is continuously ob- served at low resolution for more than five months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Assuming that most of the line flux arises from the magnetospheric accre- tion region, as suggested by the periodic appearance of the IPCs and the correlation between line flux and u’-band excess, this in- dicates a stable large-scale accretion structure on this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In contrast, high-velocity blueshifted absorption components are neither periodic nor stable on this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' While they are ubiquitous in the optical line profiles, most notably Hα and Hβ, and are also quite conspicuous in the HeI 10830 Å line profile, their signature evolves on a timescale of a few days, sometimes drifting in velocity before disappearing altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' As an exam- ple, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3 shows the evolution of the blueshifted absorption components in the HeI line profile over the course of the SPIRou November run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' On JD 9537, a high-velocity blueshifted feature appears in the profile and drifts toward lower velocities over the next several days, from -200 km s−1 to -120 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Soon after, on JD 9540, a new high-velocity component appears and follows the same trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A similar behavior is seen in the high-velocity blueshifted component appearing in the HeI 10830 Å line pro- file during the SPIRou September run, which drifted from -240 km s−1 to -150 km s−1 on a timescale of five days, from JD 9475 to JD 9480 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This suggests episodic outflows lasting for a few days only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We see no sign of a steady, constant veloc- ity wind in the HeI line profiles over the observing period, nor do we find evidence for a rotational modulation of the outflow signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The only stable outflow signature seen in the optical and near-infrared line profiles consists of a narrow, low-velocity blueshifted absorption in the Hα profile that peaks at -20 km s−1 and remains visible over more than two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Observations thus suggest that we witness a globally stable accretion structure and a succession of short-lived episodic out- flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The contrasting behavior of accretion and outflow diag- nostics observed on a timescale of months thus raises the ques- tion whether the two processes are physically connected on the scale of a few stellar radii that we probe here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' To examine this is- sue, we investigated the aftermath of the brightening event GM Aur underwent around JD 2,459,509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' On this date, the system exhibited a significant brightening at optical and near-infrared wavelengths, most notably in the u’ band (∼1 mag), as well as some of the strongest line fluxes and highest optical and near- infrared veiling values measured during the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A simul- taneous TESS light curve of the system recorded the brightening event (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2), which started on JD 2,459,508 and ended on JD 2,459,511, and exhibited a flat peak lasting for two days with a 20% flux increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We therefore interpret this episode as an accretion burst that occurred around the rotational phase Φrot=0, corresponding to the maximum visibility of the accretion shock, and lasted for several days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' From the measured continuum level and Balmer line EW during the burst, we derive an increase of a factor of 2 in the accretion rate, reaching ∼2×10−8 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Inspection of the optical and near-infrared line profiles during and after this event reveals high-velocity blueshifted absorption components that appear in the Balmer and HeI 10830 Å line pro- Article number, page 18 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Magnetospheric ejection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Development of a magnetospheric ejection computed from star-disk interaction MHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The snapshots shown here are extracted from model 3 of Pantolmos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020), where the magnetospheric truncation radius amounts to 54% of the corotation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The three snapshots are shown at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 Prot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curves indicate expanding magnetic field lines that give rise to the ejection of plasmoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color scale indicates density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The green arrows show the velocity field of the stellar and disk winds and of MEs at their interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Magnetospheric ejection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Velocity of the gas in the ejected plasmoid at a distance of 10 (magenta), 20 (blue), and 30 (cyan) stellar radii as a function of time in units of the stellar rotational pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Successive ejections of plasmoids are featured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The vertical lines correspond to the snapshots shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' On JD 2,459,509 a new blueshifted absorption component appears in the HeI infrared line at a velocity of -280 km s−1and extends down to -350 km s−1 before it reaches the continuum level again, and it drifts to lower velocities (∼ -150 km s−1) over the next few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The closest optical spectrum was recorded only toward the end of the accretion burst, on JD 2,459,510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In this spectrum, the Hα and Hβ profiles feature a new blueshifted absorption component at a velocity of -110 km s−1 that lasts for a few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The contemporaneous occurrence of an accretion burst rapidly followed by outflow signatures in the line profiles therefore suggests that the accretion and ejection processes are physically connected on small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We propose that the most likely scenario that accounts for these episodic events is magnetospheric ejections, possibly trig- gered by magnetic reconnections in the accreting magneto- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Magnetospheric ejections (Zanni & Ferreira 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sauty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2022), or nonstationary conical winds (Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2009), are caused by the expansion and reconnection of the field lines that connect the star with the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The inflation process is the result of the star-disk differential rotation and the conse- quent build-up of toroidal magnetic field pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Good- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Quasi-periodic ejections of plasmoids are pre- dicted to occur throughout the magnetospheric inflation cycle on a timescale of about the rotational period (Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The speed and variability of the outflows likely depend on var- ious parameters, such as the magnetic field strength and topol- ogy, the thermal disk pressure, nonideal MHD effects, and the interaction of the magnetospheric-ejection region with the sur- rounding outflows, that is, stellar and disk winds (Miller & Stone 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Zanni & Ferreira 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Previous monitoring campaigns on young stellar objects have reported ev- idence for magnetospheric inflation cycles (Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We show a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5D MHD simulation of the interaction between an inner accretion disk and a dipolar mag- netosphere from Pantolmos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The figure illustrates the ejection of plasmoids along expanding field lines at the distance of a few stellar radii from the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The timescale for successive ejections is about that of the stellar ro- tation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The outflow speed, shown in Fig 19, reaches more than 200 km s−1 in the early phases of the ejection, then deceler- ates to about 150 km s−1 on a timescale of days (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3×Prot), and finally vanishes altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' A direct comparison of the model to observations is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The terminal speed we de- rive from observations is higher than predicted by the simula- tion, and its temporal evolution on a timescale of a few days might be dominated by projection effects as the system rotates and does not trace the evolution of the plasmoid velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Moreover, the sporadic ejections we observe do not appear to have the quasi-periodic character of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Both the wind speed and ejection timescale may, however, depend on numerical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In any case, the behavior of the magnetospheric ejection model qualitatively matches the dynamics of the high-velocity blueshifted absorptions seen in the line profiles of GM Aur on a timescale of days to weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Moreover, in the scenario of a mag- netospheric inflation cycle, magnetic reconnection leads to an accretion burst and simultaneously triggers an ejection episode Article number, page 19 of 30 9- 5 3 2 1 4 0 1 Log10(p/p*) t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2P* t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5P* t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8P* t= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1P* 326 km/s 30 20 R/R* 10 十0 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 R/R* R/R* R/R* R/R*250 200 Speed [km/s] S 150 10R* 20R* 30R* 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 Time [Prot]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Goodson & Winglee 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This is quite reminiscent in- deed of what is suggested by the variability of GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We also noted a change in the GM Aur light curve whose first part is dominated by successive low-level brightening events, up to the major accretion burst described above, while it exhibits luminosity dips toward the end of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It is unclear whether the contrasting behavior of the system luminosity ob- served before and after the main accretion burst is a consequence of the burst itself, perhaps inducing a structural change in the star-disk interaction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It is conceivable that the rearrange- ment of the magnetic topology after the inflation or reconnection event that led to the burst has impacted the large-scale geometry of the star-disk interaction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Either a modest increase in the inner disk scale-height (Nagel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2017) or a reduction of the extent of the truncation radius is prone to trigger a periodic obscuration of the central star by circumstellar dust, that is, a dip- per phenomenon (Cody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2014), especially in young systems seen at high inclination (McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bodman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The magnetic topology and the mass accretion rate may thus have slightly evolved over the six-month time span of the campaign, as suggested by the long-term variations of the emis- sion line EWs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, the results we report here clearly indicate that the large-scale geometry of the star- disk interaction was not drastically modified over this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This is evidenced by the phase stability of the modulated light curve, the smooth variability of the emission line profiles around their mean shape, and the strictly periodic appearance of IPC profiles over at least three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' All these accretion diagnos- tics support a globally stable magnetospheric accretion structure during the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Finally, it is interesting to compare the line profile shape and variability reported here to those reported by McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020), which were obtained in 2011, that is, ten years prior to our observing campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The shapes of the Hα and Hβ profiles are quite different in the two studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020), these are pure emission profiles with a triangular shape, with- out any significant absorption components, and they exhibit lit- tle variability over the timescale of a week (see their Figures S5, S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Here, the same profiles appear to be much more struc- tured, with highly variable absorption components on the same timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The HeI 5876Å line profile variability also differs be- tween the two studies, but in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In both stud- ies, the line profile consists of a broad and a narrow component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' However, in McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020), the intensity of the broad component clearly varies, especially on the blue wing, while we found it to be quite stable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It seems that the behavior of the system was different between the two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Hα and Hβ line EWs are indeed systematically higher in McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' (2020) and were comparable to the highest values we mea- sured here during the JD 2,459,509 accretion burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It is therefore likely that GM Aur was in a state of more active accretion during the 2011 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This is consistent with the more triangular shape and lack of structure of the Balmer emission line profiles, which are predicted to become more optical thick as the funnel flow density increases (Muzerolle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' It is also consis- tent with the higher level of variability seen in the blue wing of the broad component of the HeI 5876 Å line profile that may betray the existence of a hot accretion-driven wind at times of enhanced accretion (Beristain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Long-term changes in the system behavior driven by a varying mass accretion rate and/or a change in the magnetic topology are therefore likely to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Over the six-month span of our observing campaign, the significant variation observed in the EW of the Paβ line profile, which by the end of the campaign reaches similar levels to those measured during the JD 2,459,509 accretion burst (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 15), suggests that such changes may occur on a timescale of a few weeks to months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Conclusion By combining optical and near-infrared high-resolution spectro- scopic time series, seconded by a long-term monitoring of the photometric variability of the system and low-resolution near- infrared spectrophotometry, we were able to characterize the ac- cretion and ejection process occurring in the young system GM Aur on a timescale ranging from days to months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We report a stable accretion pattern according to which the large-scale mag- netic field of the star controls the accretion of gas from the in- ner disk onto the central star along funnel flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The appear- ance of inverse P Cygni profiles that signal the crossing of fun- nel flows on the line of sight is remarkably periodic at the stel- lar rotation period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Similarly, the photometric and line flux variations, both driven by the visibility of the accre- tion shock located at the foot of the main accretion column, are modulated at the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' While the amplitude varies, the phase of variability of all these accretion diagnostics remains stable over the 30 rotational periods covered by the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This suggests that the underlying magnetic topology that con- trols the non-axisymmetric accretion flow, presumably an in- clined dipole on the large scale, did not undergo major changes over a timescale of six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' In stark contrast, high-velocity blueshifted absorption components that indicate outflows appear at random times in the emission line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' They are not rota- tionally modulated, and their signatures last for a few days only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We argue that these transient outflows associated with a stable accretion pattern are best accounted for by magnetospheric ejec- tion models, as predicted by MHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Thus, by prob- ing the dynamics of the star-disk interaction region, these results show that the physical connection between accretion and ejec- tion processes that has long been established on large scales also appears to be valid on the much smaller sub-au scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We thank the referee for a prompt and detailed report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This study is based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The observations at the CFHT were performed with care and respect from the sum- mit of Maunakea which is a significant cultural and historic site;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' based on ob- servations made at Observatoire de Haute Provence (CNRS), France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' based on data collected under the ExTrA project at the ESO La Silla Paranal Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' ExTrA is a project of Institut de Planétologie et d’Astrophysique de Grenoble (IPAG/CNRS/UGA), funded by the European Research Council under the ERC Grant Agreement n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 337591-ExTrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We thank Ágnes Kóspál for providing a reduced TESS light curve of GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Funding for the TESS mission is pro- vided by NASA’s Science Mission directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This project has received fund- ing from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 742095;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' SPIDI: Star-Planets-Inner Disk-Interactions, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='spidi-eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We ac- knowledge funding from the French National Research Agency (ANR) under contract number ANR-18-CE31-0019 (SPlaSH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' SHPA acknowledges financial support from CNPq, CAPES and Fapemig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' JFD acknowledges funding from the European Research Council (ERC) under the H2020 research & innovation pro- gramme (grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 740651 NewWorlds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' AF acknowledges support by the PRIN-INAF 2019 STRADE (Spectroscopically TRAcing the Disk dis- persal Evolution) and by the Large Grant INAF YODA (YSOs Outflow, Disks and Accretion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' JFG was supported by fundação para a Ciência e Tecnologia (FCT) through the research grants UIDB/04434/2020 and UIDP/04434/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' This work benefited from discussions with the ODYSSEUS (HST AR-16129) and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2020b, MNRAS, 498, 5684 Donati, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Skelly, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Bouvier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Covino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2003, A&A, 405, 149 Frasca, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Guillout, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Marilli, E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Petrov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', & Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2013, A&A, 560, A57 Gaia Collaboration, Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Bouvier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Herbst, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', & Shevchenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2007, A&A, 461, 183 Gravity Collaboration, Garcia Lopez, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Venuti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2021, A&A, 650, A196 Marques, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Goupil, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} 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A&A, 374, 265 Woitke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Kamp, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', Antonellini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2019, PASP, 131, 064301 Zanni, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' & Ferreira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2013, A&A, 550, A99 Zechmeister, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' & Kürster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2009, A&A, 496, 577 Article number, page 21 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Appendix A: REM comparison stars and optical photometry Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 lists the comparison stars we used to calibrate the REM optical photometry (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 lists the derived g’r’i’z’ magnitudes for GM Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Appendix B: Line profile correlation matrices This section provides correlation matrices between line profiles (Johns & Basri 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation coefficients are computed across the line profiles for every pair of velocity channels com- mon to the two profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 present the correlation matrices of the optical and near-infrared line profiles studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Appendix C: EWs and J-band measurements from the ExTrA spectra We provide the HeI 10830 Å, Paβ, Paγ, and Paδ line EWs and the J-band photometry measured from the ExTrA spectra in Ta- ble C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' For each night, the table lists the mean observation time, the median EW measurement and its standard deviation for each spectral line, and the J-band magnitude and its error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Appendix D: nIR veiling and line EWs We present in Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 the evolution of near-infrared veiling in the JHK bands and of the HeI, Paβ, and Brγ line EWs over the course of the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Appendix E: Line profile variability on successive SPIRou runs Figures E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 show the line profile variability of the HeI 10830 Å, Paβ, and Brγ line profiles for the successive SPIRou runs in September, October, November, and December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The figures also include 2D periodograms for each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' We note, however, that only the October SPIRou run is long enough, ex- tending over 14 days, to yield significant results when search- ing for a modulation of the line profiles around the rotational period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='04 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The September, November, and December runs lasted for only 9, 6, and 9 days, respectively, which is too short to reliably investigate periods longer than 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 22 of 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Literature data for comparison stars in the field of GM Aur on the REM cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Id Name 2MASS g r i z J H K′ (mag) (mag) (mag) (mag) (mag) (mag) (mag) 2 HD 282625 J04551650+3022369 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='310 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='914 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='772 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='707 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='781 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='554 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='487 3 TIC 96533048 J04551015+3021333 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='361 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='399 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='936 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='668 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='295 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='716 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='563 4 HD 282626 J04550536+3020382 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} 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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' J04551400+3020168 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='279 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='637 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='312 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='109 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} 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+page_content='288 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='604 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='135 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='564 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='828 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='608 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' J04550253+3019228 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='529 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='494 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='892 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='496 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='994 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='264 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='086 Notes: griz magnitudes from Pan-STARRS (Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' JHK′ magnitudes from 2MASS (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices for hydrogen optical line profiles computed from the 15 OHP/SOPHIE spectra obtained during the campaign: Hα⋆Hα (left), Hβ⋆Hβ (center), and Hα⋆Hβ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices for the hydrogen near-infrared line profiles computed from the 34 CFHT/SPIRou spectra obtained during the campaign : Paβ⋆Paβ (left), Brγ⋆Brγ (center), and Paβ⋆Brγ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 23 of 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 200 v(km/s)- ha 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 400 200 0 200 400 v (km/s) - brgA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices between optical and near-infrared hydrogen line profiles computed from ten OHP/SOPHIE and ten CFHT/SPIRou spectra obtained over the same nights during the October runs: Hα⋆Paβ (left), Hβ⋆Paβ (center left), Hα⋆Brγ (center right), and Hβ⋆Brγ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Correlation matrices for the HeI 10830 Å and hydrogen lines: HeI⋆HeI (left), Paβ⋆HeI (center), and Hβ⋆HeI (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The Brγ⋆HeI and Hα⋆HeI matrices are not shown as they are 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200 400 y (km/s) - heiir400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 200 v (km/s) - hb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 400 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='001 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='011 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='008 Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' EW measurements and J-band photometry from the ExTrA spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Julian date EW(HeI) EW(Paβ) EW(Paγ) EW(Paδ) J errJ (2,450,000+) (Å) (Å) (Å) (Å) (mag) (mag) 9501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='89138 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='80 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='28 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='028 † No J-band photometry could be obtained for that night due to poor seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 25 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Veiling measured on SPIRou spectra in the JHK bands (tri- angles) and the EW of the HeI, Paβ, and Brγ lines (dots) plotted as a function of Julian date (top) and rotational phase (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 26 of 30 EW(Hel) 10 EW(PaB) EW(BrG) EWs and △ rh rk IR veiling near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 9480 9500 9520 9540 9560 9580 Juliandate-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content="000EW(Hel) 10 EW(PaB) g and Ew's (A) EW(BrG) rij △ rh 1 rk IRveiling AA near 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 A A A 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 PhaseJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over nine days during the September 2021 SPIRou run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The colors represent successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: 2D periodograms across the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted horizontal red line drawn at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1 indicates the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curve displays the mean line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code reflects the periodogram power from zero (blue) to 1 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 27 of 30 day phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='03 9475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 9480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 948 482 500 500-500 0 500 v (km/s) v (km/s)day phase 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='03 947 9480 9481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 9482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 500 0 500-500 o 500 v (km/s) v (km/s)day phase 9473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='03 94/5 9477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 9478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 9480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 9481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='68 9482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='85 500 500-500 0 500 v (km/s) v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 Frequency (1/d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 45342corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over 14 days during the October 2021 SPIRou run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The colors represent successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: 2D periodograms across the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted horizontal red line drawn at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1 indicates the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curve displays the mean line profile.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='82 9513 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='83 9514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' 9515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='95 9516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='0 500 0 500-500 0 500 v (km/s) v (km/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='40 Frequency (1/d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} 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+page_content='40 Frequency (1/d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='10 400 200 0 200 400 v (km/s)J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bouvier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Sousa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Pouilly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' : GMAur Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over six days during the November 2021 SPIRou run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The colors represent successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: 2D periodograms across the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted horizontal red line drawn at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='166 day−1 indicates the stellar rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The white curve displays the mean line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The color code reflects the periodogram power from zero (blue) to 1 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Article number, page 29 of 30 day phase 500 500-500 0 500 v (km/s) v (km/s)day phase 9535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content='1 11.' metadata={'source': 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+page_content=' Near-infrared HeI (left), Paβ (center), and Brγ (right) line profiles obtained over nine days during the December 2021 SPIRou run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Top: Line profiles are plotted as a function of Julian date (left subpanels) and rotational phase (right subpanels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The colors represent successive rotational cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' Bottom: 2D periodograms across the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf'} +page_content=' The dotted horizontal red line drawn at a frequency of 0.' metadata={'source': 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file mode 100644 index 0000000000000000000000000000000000000000..8146e5fadc843e089f0b5ed9916df55648c48e5c --- /dev/null +++ b/u9E3T4oBgHgl3EQf-QtO/content/tmp_files/2301.04824v1.pdf.txt @@ -0,0 +1,3502 @@ +A Network Science perspective of Graph Convolutional Networks: A survey +MINGSHAN JIA, BOGDAN GABRYS, and KATARZYNA MUSIAL, University of Technology Sydney, +Australia +The mining and exploitation of graph structural information have been the focal points in the study of complex networks. Traditional +structural measures in Network Science focus on the analysis and modelling of complex networks from the perspective of network +structure, such as the centrality measures, the clustering coefficient, and motifs and graphlets, and they have become basic tools for +studying and understanding graphs. In comparison, graph neural networks, especially graph convolutional networks (GCNs), are +particularly effective at integrating node features into graph structures via neighbourhood aggregation and message passing, and +have been shown to significantly improve the performances in a variety of learning tasks. These two classes of methods are, however, +typically treated separately with limited references to each other. In this work, aiming to establish relationships between them, we +provide a network science perspective of GCNs. Our novel taxonomy classifies GCNs from three structural information angles, i.e., the +layer-wise message aggregation scope, the message content, and the overall learning scope. Moreover, as a prerequisite for reviewing +GCNs via a network science perspective, we also summarise traditional structural measures and propose a new taxonomy for them. +Finally and most importantly, we draw connections between traditional structural approaches and graph convolutional networks, and +discuss potential directions for future research. +CCS Concepts: • Networks → Topology analysis and generation; Network structure; • Computing methodologies → Ma- +chine learning. +Additional Key Words and Phrases: Graph Convolutional Networks, Network Science, graph structural measures +1 +INTRODUCTION +Networks or graphs are a general language for modelling and analysing complex systems that are abstracted as entities +and their connections [13, 151]. In the representation of networks, domain data is no longer only being a set of isolated +data points but also contains important information about the relationships among them. The entities are related to +each other according to the structure of the network, and modelling these relational structures allows us to build more +accurate models of the domain data. Various types of real-world data can naturally be modelled as networks, such as +social networks representing social actors and their relationships [149], molecular graphs representing chemical atoms +and their bonds [90], transportation networks representing infrastructures and traffic flow [18], control flow graphs +representing code blocks and their executions [143], etc. +Although networks are very powerful at modelling relational data, processing them is significantly more difficult, +mainly due to their intricate topological structures. Compared to other common data formats such as images or text, +network data does not have a starting or an ending point that can be defined in Euclidean space, nor the essential notion +of spacial locality and proximity. Therefore, understanding and exploiting graph structure has always been a core theme +in analysing complex networks. Traditional network science approaches are mostly structure-related heuristics, such as +various types of node centralities [127] for node-level analysis, common neighbours similarity and its variants [138] for +link-level analysis, and motifs [142] and significance profile [141] for graph-level analysis. These methods, along with +others, have become the standard tools for analysing graphs and have been used in all kinds of networks. Certainly, +these approaches have their limitations. First is their applicability — each is effective for examining specific properties +but falls short of capturing other structural aspects. Another drawback is that most heuristic approaches focus on graph +structures while overlooking the rich information that could be contained in nodes or on edges [126]. +1 +arXiv:2301.04824v1 [cs.SI] 12 Jan 2023 + +Graph Convolutional Networks +Layer-wise Message Aggregation Scope +(Where a node aggregates message +from at each layer) +Message Content +(What message is gathered +and passed on) +Learning Scope +(Where GNNs are +trained on) +Fig. 1. Taxonomy of graph convolutional networks from structural perspectives. +Another mainstream class of methods is grounded in deep learning on graphs, especially the recently emerging and +quickly gaining in popularity graph convolutional networks (GCNs) [189]. GCNs are generalised from the notion of +Convolutional Neural Networks (CNNs) [8], redefining them for non-Euclidean graph data. GCNs ingeniously combine +graph structure and node/edge features via neighbourhood message aggregation and a structure-based propagation +scheme. Being a rapidly evolving area of research, a large number of graph convolutional network approaches have +emerged in recent years, aiming to improve its expressivity, scalability, or targeting specific tasks or types of networks +[1]. However, there are still many challenges and opportunities in this field. Some of the key open problems include +developing more powerful and efficient GCN architectures, extending these models to handle temporal, multi-layered, +or other more complex graph data, and improving the interpretability and transparency of GCN models. +Traditional structural measures of Network Science are direct and efficient tools for analysing and understanding +complex networks, while graph convolutional networks are deep learning models designed specifically for graph data +in order to address various learning problems. As discussed, the two classes of methods have their own strengths and +weaknesses. Surprisingly, they are very often treated separately with relatively limited references to each other. Network +science researchers may be sceptical about the lack of explainability of deep learning approaches, while deep learning +researchers tend to overlook the advance in traditional non-learning approaches. We believe, however, that with the +established foundations of traditional structural measures in Network Science, and GCNs emerging as a new powerful +class of methods, there would be great benefits to be realised from a closer integration and awareness of the two +communities. On the one hand, GCNs gracefully incorporate node features, which are largely overlooked in traditional +structural measures, into the structure of graphs, and achieve state-of-the-art performances in various tasks. On the +other hand, traditional network science notions, being the foundations of understanding and characterising complex +networks, are also indispensable in studying GCNs. Different types of structural measures are being exploited in the +recent advance of GCNs as well [27, 100, 117, 192]. Therefore, in this work, we aim to link the two classes of methods +together by comprehensively reviewing each of them, proposing new taxonomies and discussing their connections. +Along with the phenomenal development of GCNs, many survey articles appeared to summarise and review them. +Some have a broad scope that covers graph representation learning [79] or graph deep learning [32, 189, 216] in general. +Some others are focused on specific aspects, such as the design pipeline or the composition modules of graph neural +networks [219], the dynamic mechanisms [170], or the learning on limited labelled samples [190]. However, there still +lacks an examination that focuses on how graph structure information (which is the main focus of traditional network +science approaches), is exploited in graph convolutional networks. Thus, in this work, we propose new taxonomies +of GCNs from the perspective of graph local structure, and at the same time, review the latest works that improve +graph neural networks through exploiting local structural information. Specifically, we propose to summarise graph +2 + +Subgraph Count Based Approaches +Message Passing Based Approaches +Subgraph Formation Based Approaches +Mixed Approaches +Global Path Based Approaches +Structural Measures +on Graphs +Fig. 2. Structural measures on graphs. +convolutional networks from three structural angles, i.e., the scope of layer-wise message aggregation, the content of +the message being passed on, and the overall scope of learning on graphs (Figure 1). +Moreover, a systematic understanding of traditional graph structural approaches is the prerequisite for thoroughly +reviewing GCNs via a network science perspective. Therefore, before jumping into the sphere of graph neural networks, +we first summarise and classify non-learning graph structural measures. The study of graph structures is so ubiquitous +that they often appear in different terms, such as the big family of centrality measures [47, 127, 164], the popular +notion of motifs [142] and graphlets [140] and the set of subgraph formation measurements such as the clustering +coefficient [184], the closure coefficient [196], the square clustering coefficient [124], etc. Existing surveys on structure +measurements only cover one or two sets of those notions, and fail to unite them from an overarching perspective +or to draw connections and comparisons between them. Therefore, in this work with a focus on graph structure, we +also propose a new taxonomy that brings all these concepts together. Specifically, we group existing graph structural +measures into five categories: subgraph count based measures, subgraph formation based measures, global path based +measures, message passing based measures, and hybrid measures ( Figure 2). More importantly, through summarising +both the traditional structural measures and the graph convolutional network approaches, we could draw connections +between the two, strengthen the understanding and analysis of GCNs and lead to insightful discussions about potential +research avenues. +To summarise, the main contributions of this survey are as follows: +• We propose a new taxonomy that brings together various types of traditional structure-based approaches. We +make a clearer distinction between the concepts of local and global, and we first introduce and summarise the +category of subgraph formation based approaches. +• We propose a novel taxonomy of graph convolutional networks, with a focus on the exploitation of graph +structural information. The taxonomy categorises GCNs from three structural information angles, i.e., the +layer-wise message aggregation scope, the message content, and the overall learning scope. We review and +summarise the latest GCN approaches with a structural focus, and provide a thorough analysis of the time and +space complexities. +• We draw connections between the graph convolutional networks and the traditional structure-based approaches, +and discuss three potential future research avenues in the joint area. +The rest of this survey is organised as follows: In Section 2, we introduce and compare two pairs of concepts, i.e., +local and global, and motifs and graphlets. In Section 3, we present the five categories of graph structural measures +3 + +and discuss four open problems. In Section 4, we introduce the novel taxonomy of graph convolutional networks, and +discuss their time and space complexities. In Section 5, we discuss the connections between the traditional structural +measures and the graph convolutional networks, and present some potential research directions. Finally, we conclude +the article in Section 6. +2 +PRELIMINARIES AND BACKGROUND +This section introduces preliminary concepts that are helpful for understanding the proposed taxonomies. +2.1 +Local vs. Global +When discussing graph structural measures, we need first to distinguish what is local and what is global. Previous works +[56, 92, 133, 138] either only focus on where the measures are defined by dividing them into two or three categories: +(i) the "local", "micro" or "individual" level; (ii) the "global", "macro" or "aggregate" level; and (iii) sometimes at the +"mesoscopic", "quasi-local" or "subnetwork" level; or they are defined solely based on the scope of information involved +in their computation. This, however, leads to some confusion. For example, the betweenness centrality is defined for +nodes (at the node-level) but requires global information to compute. Should it be termed a local measure or a global +measure? Similarly, the average clustering coefficient is defined at the network-level, but only needs local information +at each node — calculating the local clustering coefficient at each node, then averaging over all nodes. +Therefore, we propose the following terms to distinguish both at what level the measures are defined and the scope +of information that is needed to calculate them: +• Local-level measure is a measurement defined on a node-level or link-level (the link here also includes the +non-existing or potential link which is often used in a link prediction task). Thus, it can be further divided into a +node-level measure and a link-level measure. +• Network-level measure is a measurement defined for the entire network. +• Local structural measure is a measurement whose computation only involves the nearby neighbourhood of a +node, i.e., within a range of k-hop away from a node. In most cases, k is less than or equal to 4. Many traditional +measures only care about the immediate neighbourhood around a node, and we name them as Strict-local +structural measures. +• Global structural measure is a measurement that involves the global information in computation. This type of +measurement often involves the computation of paths between nodes in the network. +Now, when we revisit the betweenness centrality, it is both a local-level and a global structural measure. The average +clustering coefficient, on the other hand, is both a network-level measure and a local structural measure. Notice that the +average clustering coefficient involves the extra step of averaging over all nodes. Indeed, it is 𝑛 times the complexity +of computing the local clustering coefficient at a single node. However, any local-level measure can easily have an +extended definition at the network-level through averaging over all nodes or edges. Moreover, in the practice of network +analysis, local-level measures are often calculated for the entire network, looping over all nodes or all edges. Therefore, +when defining local or global structural measures, we choose to exclude this aggregation or averaging step. +To summarise, we use the terms “local-level” and “global-level” to distinguish where the measure is defined, and +we use the terms “local structural” and “global structural” to distinguish the scope of information involved in the +computation, before the aggregation/averaging step. +4 + +Table 1. Some 3-node and 4-node motifs in directed networks[142]. Motifs containing bidirectional edges are not included. +Motif +Designation +Type of network +3-node feed-forward loop +Gene regulation network +Neural network +Electronic circuits (forward logic chips) +3-chain +Food webs +3-node feedback loop +Gene regulation network +Neural network +Electronic circuits (forward logic chips) +Bi-fan +Gene regulation network +Neural network +Electronic circuits (forward logic chips) +Electronic circuits II +Bi-parallel +Neural network +Food webs +Electronic circuits (forward logic chips) +4-node feedback loop +Electronic circuits II +2.2 +Motifs vs. Graphlets +Next, we distinguish three similar concepts that are later used in our taxonomies, i.e., subgraphs, motifs and graphlets. +A subgraph, as the name implies, is a smaller graph whose node set and edge set are subsets of those of the original +graph. We then recap the notions of motifs [142] and graphlets [140] according to the papers that proposed them. +Network motifs [142] are subgraphs that recur much more frequently in the real network than in an ensemble of +randomised networks. They are defined at the network-level, in order to uncover the basic building blocks of directed +networks across domains. Subgraphs having a 𝑝-value less than 0.01 are deemed as motifs, where 𝑝 is the probability of +the subgraph appearing more times in randomised networks than in the real network. The statistical significance of a +motif can also be captured by the Z-score, which is calculated as follows: +𝑍𝑖 = +� +𝑁 real +𝑖 +− ¯𝑁 rand +𝑖 +� +/std +� +𝑁 rand +𝑖 +� +, +where 𝑁 real +𝑖 +is the number of subgraphs of type 𝑖 in the real network, and 𝑁 rand +𝑖 +is the number of subgraphs of type +𝑖 in a randomised network. A natural downside of this approach, however, is that it needs to generate a large number of +5 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +G0 +G1 +G2 +G3 +G4 +G5 +G6 +G7 +G8 +19 +20 +18 +21 +15 +16 +17 +G9 +G10 +Fig. 3. Graphlets and their orbits [140] +random networks (e.g. 100s or 1000s) using a certain configuration model. The original work only focuses on 3-node +and 4-node directed subgraphs, finding that particular subgraphs such as 3-node feed-forward loop, 3-node feedback +loop, bi-fan, bi-parallel, and 4-node feedback loop are significant building blocks in several different types of directed +networks (Table 1). +Graphlets [140], are nonisomorphic induced subgraphs around a focal node. In the original work, it is defined for +undirected networks. A key difference between motifs and graphlets is that graphlets are defined at node-level. The +term automorphism orbits, or orbits for short, are used to distinguish different positions of the focal node in a subgraph. +Therefore, when subgraph size is limited to a range of 2 to 5 nodes, there are 73 different orbits on 30 different subgraphs. +We recap graphlets with the orbits in Figure 3 (in order to save some space, the majority of 5-node graphlets are omitted). +It is worth mentioning that the idea of counting induced subgraphs is also extended to the link-level, leading to the +notion of edge orbits [82]. Taking graphlet 𝐺1 in Figure 3 for example, there exist two (node) orbits denoted ’1’ and ’2’, +respectively. In contrast, there is only one edge orbit in it since the two edges are structurally equivalent. +To summarise, motifs and graphlets are both small induced subgraphs, but they are different in the following aspects +(Figure 4): motifs are defined at the network-level while graphlets are defined at the node-level; motifs are proposed for +directed networks while graphlets are for undirected networks; motifs are discovered from comparing real networks to +randomised networks with the same degree sequence while graphlets are calculated on the network itself; lastly, motifs +contain 3 - 4 nodes while graphlets have 2 -5 nodes. Notice that most of the analyses stop at 4 or 5 nodes because a +subgraph containing more than 5 nodes would become too complicated for us to enumerate and interpret all possible +subgraphs or orbits. For example, a 6-node induced subgraph leads up to 112 different types of subgraphs and 407 +different orbits. Taking link directions into consideration, there are 1, 530, 843 subgraphs and 9, 031, 113 orbits [162]. +3 +GRAPH STRUCTURAL MEASURES +In order to set up the context of reviewing graph convolutional networks from a Network Science perspective, we first +summarise traditional graph structural measures and propose a novel taxonomy for them, which will later be used in +our categorisation and analysis of GCNs. Specifically, We divide existing structural measures into five categories (see +Figure 2): +• Subgraph Count Based Approaches. These measures are defined based on the number of a particular subgraph or +subgraphs. +6 + +Small +induced +nonisomorphic +subgraph +Network-level +Node-level +Directed network +Undirected network +Subgraphs of high frequency +Subgraphs of any frequency +Compare to random networks +Calculate on itself +Subgraph size: 3 - 4 +Subgraph size: 2 - 5 +Motifs +Graphlets +Fig. 4. Motifs vs. Graphlets +• Subgraph Formation Based Approaches. In this category, the measures are defined by the ratio of the numbers of +two subgraphs: one contains fewer edges (or nodes) and is viewed as the formation base of another. +• Global Path Based Approaches. As the name implies, these measures are based on unbounded paths. They involve +the calculation of shortest paths or all paths originating from a node to any node in the entire graph. +• Message Passing Based Approaches. Unlike previous categories, message passing-based approaches utilise graph +structural information in an implicit manner: every node is initialised with an importance score. Then iteratively, +each node updates its score through aggregating the scores of its neighbours. Graph Neural Network approaches +(see more in Section 4) can be viewed as transforming this traditional message passing approach into a learnable +process. +• Hybrid Approaches. These measures are simply some combinations of the previous four categories. +We now explain the logic behind our taxonomy. The first two categories both cover a local area of the whole network +(within a certain distance from the focal node, or containing a limited number of nodes and edges). The first category — +subgraph count based approaches — is built from counting the number of particular local structures. For example, the +number of neighbours, local paths or subgraphs. The second category — subgraph formation based approaches — is +uniquely defined based on the ratio of two subgraphs and thus bears the meaning of measuring the formation of certain +local structures. To have both of them in the taxonomy instead of combining them into one category is to stress their +differences. +Then, the third category expands its scope to the entire network. We name it global path based approaches instead +of just global approaches. This is because all global approaches involve either the calculation of shortest paths or all +paths originating from a node to any other node in the entire graph. Notice here that a path is also a particular type +of subgraph. However, a local path or bounded path, such as a 2-path or 3-path, belongs to the category of subgraph +count based measures, whereas a global path or unbounded path is in this category. We choose to differentiate the third +category from the previous two categories from the perspective of the covered scope. +Next, the fourth category — message passing based approaches — is based on the idea of propagating information +along the edges. It is a different form to the abovementioned three categories because it does not calculate any type +of subgraphs or global paths. Instead, the structure is utilised in an implicit way. Every node is initialised with an +importance score. Then iteratively, each node updates its score through aggregating the scores of its neighbours. +Although these four categories are largely different from each other, there are many approaches that combine them +together, which are naturally put into the fifth category — mixed approaches. +7 + +3.1 +Subgraph Count Based Approaches +Subgraph count based measures are based on the number of a particular subgraph or subgraphs. We further divide +them into three subclasses, i.e., measures defined on 1-hop neighbours, measures defined on k-hop neighbours/local +paths, and measures defined on multi-subgraphs. Figure 5 gives the detailed categorisation. The colour of the block +differentiates where the approach is defined: grey is on the node-level, blue is on the link-level, and orange is on the +network-level. +Subgraph +Count +1-hop neighbs. +Degree cent. +Local cent. +Graphlet Degree +Subgraph centrality +Triad Signif. Profile/Subgraph Ratio Profile +Multi-subgraphs +ℎ-index/𝑔-index +local paths\ +k-hop neighbs. +𝑘-core +𝑘-truss/CN +Local path index +Collective influence +𝜅-path cent. +𝜅-path edge cent. +𝑘-betweenness cent. +Potential theory/Quad motifs index +Fig. 5. Subgraph count based measures. +3.1.1 +1-hop neighbours. As the name implies, the calculations within this category only require the immediate +neighbourhood around a node or a link. +– Degree centrality. Through calculating the number of nodes directly connected to a node, the degree centrality is +an easy and straightforward way to assess the importance or influence of the node[65]. In order to render it within +the range of (0,1], it is often normalised by the size of the network minus one. Mathematically, the normalised degree +centrality of node 𝑖 is defined as: +Θ𝐷 (𝑖) = +𝑑𝑖 +𝑛 − 1. +(1) +Despite being so simple, the degree centrality has been widely applied in various domains. For example, in customer +networks, the degree centrality is used to find opinion leaders [163], and in biomedical semantic networks, it is +effective in selecting crucial information for summarising a disease treatment [208]. Some interesting extensions +of the degree centrality include the in-degree/out-degree centrality in directed networks, the strength centrality +and weighted strength centrality in weighted networks [35] and the cross-layer degree centrality in multi-layered +networks [31]. +– ℎ-index/𝑔-index. ℎ-index is proposed to evaluate the impact of an individual’s research output: A researcher has an +index of ℎ if ℎ of his or her papers have at least ℎ citations [81]. It is then used as a centrality measure in networks, +and named as lobby index or 𝑙-index [110]. The 𝑙-index of a node is the largest integer 𝑘 such that the node has at +least 𝑘 neighbours with a degree of at least 𝑘. Egghe argued that the influence of highly cited papers is underplayed +in the ℎ-index, and proposed a 𝑔-index to overcome this disadvantage [58]. After ranking a researcher’s papers +according to their citations, the 𝑔-index is defined as the highest rank 𝑔 such that the top 𝑔 papers together have at +least 𝑔2 citations. From its definition, the 𝑔-index is always greater than or equal to the ℎ-index. To address the same +8 + +issue, an 𝑒-index is proposed to complement the ℎ-index for excess citations[207]. Recently, a local ℎ-index centrality +is proposed to identify influential spreaders by simultaneously considering the ℎ-index values of the node and its +neighbours [125]: Θ𝐿𝐻 (𝑖) = ℎ(𝑖) + � +𝑗 ∈𝑁𝑖 ℎ(𝑗). +– 𝑘-core [107]. Instead of only calculating the number of 1-hop neighbours at one node (as in the degree centrality) +or at both the node and its neighbours (as in the ℎ-index), a 𝑘-core or coreness takes into account the number of +neighbours at every node. Specifically, the 𝑘-core is defined as a subgraph in which all nodes of a degree smaller +than k have been removed and the remaining nodes have a degree of at least 𝑘. A node located in a higher 𝑘-core is +deemed more important than a node in a lower 𝑘-core. The 𝑘-core is calculated through the 𝑘-shell decomposition +[37] which incrementally (from 1 to 𝑘) removes nodes with degree less than 𝑘 (which in turn results in lowering the +degree of remaining nodes) until no more nodes need to be removed. Given that the degree centrality, the ℎ-index and +the coreness are all based on the number of 1-hop neighbours, Lü et al. further revealed their relationships through +proposing the high-order ℎ-indices [130]. Bae et al. further propose a neighbourhood coreness that considers both +the degree of a node and the coreness of its neighbours [12]: +Θ𝑁𝐶 (𝑖) = +∑︁ +𝑗 ∈𝑁 (𝑖) +𝑘𝑠(𝑗). +(2) +The assumption is that a node having more connections to the neighbours located in the core of the network is more +influential. +– 𝑘-truss/Common neighbours. A 𝑘-truss is a subgraph where every edge is contained in at least 𝑘 − 2 triangles[45, +180]. It is found through counting the number of common neighbours of a pair of nodes that forms an edge, i.e., the +number of triangles that the edge participates in. The 𝑘-truss is also a (𝑘 + 1)-core. Counting common neighbours +around a pair of nodes that have not formed an edge (a non-edge) is also a basic approach in a link prediction task +[123]. There is a big family of similar approaches based on the number of neighbours around non-edges, such as the +Adamic-Adar index, the resource allocation index, the preferential attachment index, among others [138]. Notice that +both 𝑘-truss and Common Neighbours-like approaches are defined at the link-level. The block colour is therefore +blue in Figure 5. +3.1.2 +local paths/k-hop neighbours. The group of methods in this category requires the calculation of local paths or +k-hop neighbours. +– 𝑘-betweenness centrality [26]. The 𝑘-betweenness centrality or bounded-distance betweenness centrality is a +variation of the well-known betweenness centrality that limits the length of shortest paths to a predefined value 𝑘. +Specifically, the 𝑘-betweenness centrality of any node 𝑖 is calculated by: +Θ𝐵𝑘 (𝑖) = +∑︁ +𝑠,𝑡 ∈𝑉 +𝜎𝑘 (𝑠,𝑡 | 𝑖) +𝜎𝑘 (𝑠,𝑡) , +(3) +where 𝜎𝑘 (𝑠,𝑡) is the number of shortest paths of length at most 𝑘 between a node pair 𝑠 and 𝑡, and 𝜎𝑘 (𝑠,𝑡 | 𝑖) is the +number of those paths that pass through node 𝑖. The reason for proposing a bounded-distance betweenness centrality +is that in some networks, long paths are rarely used for the propagation of influence. +9 + +– 𝜅-path centrality [7]. Instead of limiting the length of shortest paths between node pairs, the 𝜅-path centrality +assumes that message traversals are along random simple paths of length at most 𝑘, and proposes to calculate the +sum of the probability that a message originating from any possible node goes through the focal node. The 𝜅-path +centrality of node 𝑖 is defined as: +Θ𝑃𝑘 (𝑖) = +∑︁ +𝑠 ∈𝑉 +𝜎𝑘 (𝑠 | 𝑖) +𝜎𝑘 (𝑠) , +(4) +where 𝑠 are all the possible source nodes, 𝜎𝑘 (𝑠 | 𝑖) is the number of 𝑘-paths originating from 𝑠 and passing through +𝑖, and 𝜎𝑘 (𝑠) is the overall number of 𝑘-paths originating from 𝑠. In order to calculate it more efficiently in large +networks, a randomised approximation algorithm called RA-𝜅path is also proposed. [7] +– 𝜅-path edge centrality [51]. Moving the 𝜅-path centrality definition to link-level, we then have the 𝜅-path edge +centrality. The 𝑘-path edge centrality of any given edge 𝑒 is defined as the sum of the frequency with which a message +originated from any possible node traverses 𝑒, assuming that the message traversals are along random simple paths +of length at most 𝑘: +Θ𝑃𝑘 (𝑒) = +∑︁ +𝑠 ∈𝑉 +𝜎𝑘 (𝑠 | 𝑒) +𝜎𝑘 (𝑠) +. +(5) +Quite similar to Equation 5, only here 𝜎𝜅𝑠 (𝑒) is the number of 𝜅-paths originating from 𝑠 that go over the edge 𝑒. The +original 𝜅-path edge centrality is very expensive to compute in large networks with a big 𝑘, therefore two randomised +approximations have been further proposed, i.e., ERW-𝜅path and WERW-𝜅path [51]. +– Local centrality [40]. Local centrality, sometimes summarised as LocalRank [127] utilises the information within a +node’s 4-hop neighbourhood. Concretely, the local centrality of node 𝑖 is defined as: +Θ𝐿𝑅(𝑖) = +∑︁ +𝑗 ∈𝑁 (𝑖) +𝑄(𝑗), +𝑄(𝑗) = +∑︁ +𝑘 ∈𝑁 (𝑗) +𝑅(𝑘), +(6) +where 𝑁 (𝑖) and 𝑁 (𝑗) are the set of 1-hop neighbours of node 𝑖 and 𝑗, and 𝑅(𝑘) is the number of both 1-hop and +2-hop neighbours of node 𝑘. It is said to perform better than betweenness centrality and almost as well as closeness +centrality to identify influential nodes under the setting of a SIR model, with only a time complexity of 𝑂(𝑛⟨𝑘⟩2). +– Collective influence [145]. Collective influence (CI) is another interesting method that takes higher-order neigh- +bourhoods into consideration. The idea is to find those nodes that will cause the biggest drop in the “energy function” +when removed. Specifically, the level 𝑘 collective influence of a node 𝑖 is defined as: +Θ𝐶𝐼𝑘 (𝑖) = (𝑑𝑖 − 1) +∑︁ +𝑗 ∈𝑁𝑘 (𝑖) +(𝑑𝑗 − 1), +(7) +where 𝑁𝑘 (𝑖) is 𝑘-hop neighbours of a node 𝑖. After applying the collective influence score, the paper finds that a +large number of previously neglected weakly connected nodes (nodes of lower degree) emerge among the optimal +influencers [145]. +3.1.3 +Multi-subgraphs. Methods of this category involve the count of multiple different subgraphs. They can be at the +node level, the link level or the network level. +– Graphlet degree [140]. As discussed in Section 2.2, graphlets are nonisomorphic induced subgraphs around a node. +Graphlet degree is a 73-dimensional vector formed by all different orbits in the subgraphs of size 2-5 nodes. The paper +10 + +discovers that in PPI networks, nodes grouped together under this measure belong to the same protein complexes, +perform the same biological functions and have the same tissue expressions. Some interesting extensions of graphlets +include the dynamic graphlets for temporal networks[91], the directed graphlets for directed networks[11], the +coloured graphlets for heterogeneous networks[75], and the typed-edge graphlets for edge-labelled networks [97]. +– Subgraph centrality [61]. Subgraph centrality focuses on subgraphs captured by closed walks of different lengths +around a given node. For example, when the walk length is 4, three types of subgraphs are covered, which are +2-cliques, 2-paths, and 4-cycles. The number of closed walks of length 𝑘 around node 𝑖 can be calculated from the 𝑖th +diagonal entry of the 𝑘th power of the adjacency matrix. When the walk becomes unbounded, the subgraph centrality +of node 𝑖 is calculated by: +Θ𝑆 (𝑖) = +∞ +∑︁ +𝑘=0 +𝜇𝑘 (𝑖) +𝑘! , +(8) +where 𝜇𝑘 (𝑖) = +� +A𝑘� +𝑖𝑖. It is shown to be more discriminative than many popular centrality measures such as the +degree centrality, the betweenness and the eigenvector centrality. +– Local path index [128]. Extended from common neighbours, the local path index counts both the number of 2- +paths and 3-paths between a pair of nodes. The approach is proposed for link prediction, and therefore focuses on +non-connected node pairs. Concretely, the local path index of a node pair 𝑖 and 𝑗 is defined as: +Θ𝐿𝑃 (𝑖, 𝑗) = 𝐴2 +𝑖𝑗 + 𝜖𝐴3 +𝑖𝑗, +(9) +where 𝜖 is a weigh parameter for 3-paths. The paper finds out that the local path index remarkably outperforms +common neighbours and can reach a competitive accuracy to the Katz index where all paths are considered. Some +other works compare 3-paths approaches against 2-paths approaches in link prediction and find out that 3-path +approaches perform better in PPI networks and food webs [111, 148, 220]. +– Potential theory/Quad motifs index. The potential theory aims to predict links in directed networks. By counting +the numbers of 4 different directed 2-paths and 8 different directed 3-paths around a pair of nodes, the paper finds out +that a link has a higher probability of appearing if it could generate more bi-fan subgraphs [213]. Very similar to the +idea of potential theory, the quad motifs index proposes to count particularly three types of directed open-quadriad +(3-paths) subgraphs: two of them are the bases for bi-parallel subgraphs and the other one is for bi-fan [87]. Specifically, +the quad motifs index of a pair of nodes 𝑖 and 𝑗 is defined as: +Θ𝑄𝑀 (𝑖, 𝑗) = 𝛼 × 𝑠𝐹 (𝑖, 𝑗) + (1 − 𝛼) +2 +(𝑠𝑃1(𝑖, 𝑗) + 𝑠𝑃2(𝑖, 𝑗)) , +(10) +where 𝑠𝐹 (𝑖, 𝑗) is the contribution from the bi-fan base while 𝑠𝑃1(𝑖, 𝑗) and 𝑠𝑃2(𝑖, 𝑗) are the contributions from two +bi-parallel bases. Together with the local path index, it is interesting to see that 3-path subgraphs are of particular +importance in link prediction. +– Triad significance profile/Subgraph ratio profile [141]. Extended from network motifs [142], the triad signifi- +cance profile (TSP) is constructed from normalised 𝑍 scores of 13 different directed 3-node subgraphs. +𝑇𝑆𝑃 = {𝑆𝑃1,𝑆𝑃2, ...,𝑆𝑃13}, +SP𝑖 = 𝑍𝑖/(Σ𝑍 2 +𝑖 )1/2. +(11) +11 + +𝑍𝑖 is in turn calculated from comparing with an ensemble of randomised networks with the same degree sequence, +i.e., 𝑍𝑖 = +� +𝑁 real +𝑖 +− ¯𝑁 rand +𝑖 +� +/std +� +𝑁 rand +𝑖 +� +. Subgraph ratio profile (SRP), on the other hand, is built from 6 undirected +4-node subgraphs (𝐺3 to 𝐺8 in Figure 3) : +𝑆𝑅𝑃 = {𝑆𝑅𝑃1,𝑆𝑅𝑃2, ...,𝑆𝑅𝑃6}, +SRP𝑖 = Δ𝑖/(ΣΔ𝑖 2)1/2. +(12) +Unlike TSP, SRP uses the abundance of each subgraph relative to random networks, i.e., Δ𝑖 = +𝑁 real𝑖−⟨𝑁rand𝑖 ⟩ +𝑁 real𝑖+⟨𝑁 rand𝑖 ⟩+𝜀 . +Previously seemingly unrelated networks are found to belong to several superfamilies with very similar significance +profiles. Notice also that these two approaches are defined on network-level, not on node or link-level as we have +seen often. +3.2 +Subgraph Formation Based Approaches +Subgraph formation based measures view a subgraph being built from another less complex subgraph, i.e., with one +link, multiple links, or one node less. We further divide them into three categories according to the size of the subgraph, +3-node, 4-node and 4-node plus (Figure 6). Most of these approaches are defined at node-level, except that the edge +clustering coefficient is at link-level and the interest clustering coefficient is at network-level. +Subgraph +Formation +3-node +4-node +4-node + +Clustering coef. +Closure coef. +Quadrangle coef. +Square clustering coef. +HO closure coef. +Grid coef. +Edge clustering coef. +HO clustering coef. F +HO clustering coef. Y +Interest clustering coef. +Weighted degree cent. +Fig. 6. Subgraph formation based measures. +3.2.1 +3-node subgraph. The 3-node subgraph is the simplest yet most important category in the taxonomy. +– Clustering coefficient [184]. The clustering coefficient is the first and most influential measure in this category. It +measures the extent to which the neighbours of a node connect to each other. From a structural formation perspective, +it measures the formation of triangles upon open-triads (also called wedges). Specifically, the clustering coefficient of +node 𝑖 is defined as the ratio between the number of triangles containing node 𝑖 (denoted 𝑇 (𝑖)) and the number of +open triads (denoted 𝑂𝑇 (𝑖)): +C𝐶 (𝑖) = 𝑇 (𝑖) +𝑂𝑇 (𝑖) = +1 +2 +� +𝑗 ∈𝑁 (𝑖) |𝑁 (𝑖) ∩ 𝑁 (𝑗)| +1 +2𝑑𝑖 (𝑑𝑖 − 1) +. +(13) +Due to its significance and simplicity in definition, the clustering coefficient has been widely applied in studying +complex networks [33, 160, 166] and extended to directed networks [6, 62], weighted networks [15, 156, 206] and +signed networks [46, 115]. +– Closure coefficient [196]. The closure coefficient measures the formation of triangles from a new perspective, i.e., +with the focal node located at the end of an open-triad. Different from the ordinary centre node perspective in +clustering coefficient (orbit 2 of 𝐺1 in Figure 3, denoted as 𝐺 (2) +1 +), the focal node in closure coefficient serves as the +end node of an open triad (orbit type 𝐺 (1) +1 ). The closure coefficient of node 𝑖 is calculated as the fraction of open +12 + +triads (𝑂𝑇𝐸 (𝑖)), where 𝑖 serves as the end node, that actually forms triangles: +C𝐸 (𝑖) = 2 · 𝑇 (𝑖) +𝑂𝑇𝐸 (𝑖) = +� +𝑗 ∈𝑁 (𝑖) |𝑁 (𝑖) ∩ 𝑁 (𝑗)| +� +𝑗 ∈𝑁 (𝑖) (𝑑𝑗 − 1) +. +(14) +Despite this subtle difference in definition, the closure coefficient has very different properties compared to the +clustering coefficient. It has been extended to directed networks [94, 197] and weighted networks [95]. +– Edge clustering coefficient [181]. Defined on link-level, the edge clustering coefficient (ECC) evaluates to what +extent nodes cluster around the focal edge. From a structure formation view, it measures the formation of triangles +upon this link. The edge clustering coefficient of an edge 𝑒𝑖𝑗 is defined as: +C𝑒 (𝑖, 𝑗) = +𝑇 (𝑖, 𝑗) +min �𝑑𝑖 − 1,𝑑𝑗 − 1� , +(15) +where 𝑇 (𝑖, 𝑗) is the number of triangles that 𝑒𝑖𝑗 participates, and min �𝑑𝑖 − 1,𝑑𝑗 − 1� is the number of triangles that +edge could possibly form. Based on ECC, a node centrality measure is then defined as the sum of the edge clustering +coefficients of all edges connected to it, i.e., C𝑁 (𝑖) = � +𝑗 ∈𝑁𝑖 C𝑒 (𝑖, 𝑗). This measure has been proven to be more +efficient for identifying essential proteins than many other centrality measures. +– Weighted degree centrality [173]. Weighted degree centrality (WDC) is also proposed to identify essential proteins. +Although this name seems to suggest a close relation to the degree centrality, it is in fact an extension of the edge +clustering coefficient. This approach is different in that it takes into account not only the PPI graph data but also the +gene expression data. Specifically, a weight of an interaction is calculated as: +C𝑤(𝑖, 𝑗) = C𝑒 (𝑖, 𝑗) + 𝑟 (𝑖′, 𝑗 ′), +(16) +where C𝑒 (𝑖, 𝑗) is the edge clustering coefficient from the graph data, and 𝑟 (𝑖′, 𝑗 ′) is the Pearson correlation coefficient +calculated from the gene expression data. Similarly, the weighted degree centrality of a node is then defined as: +Θ𝑊 (𝑖) = � +𝑗 ∈𝑁𝑖 C𝑤(𝑖, 𝑗). This approach essentially integrates node features when analysing networks. +3.2.2 +4-node subgraph. 4-node subgraphs are much more complicated than the 3-node subgraphs. There are in total 6 +different subgraphs and 11 different orbits in 4-node subgraphs (Figure 3). +– Quadrangle coefficients [96]. Many real networks (such as PPI networks, neural networks and food webs) are +naturally rich in quadrangles. Quadrangle coefficients, or i-quad coefficient and o-quad coefficient, are thus proposed +to measure the formation of quadrangles upon open-quadriads (3-paths). As there are two orbits in an open-quadriad +(𝐺 (5) +3 +and 𝐺 (4) +3 ), i-quad coefficient has the focal node at 𝐺 (5) +3 +while o-quad coefficient has the focal node at 𝐺 (4) +3 . +Specifically, the quadrangle coefficients of node 𝑖 are defined as: +C𝐼 (𝑖) = 2𝑄(𝑖) +𝑂𝑄𝐼 (𝑖) , +C𝑂 (𝑖) = +2𝑄(𝑖) +𝑂𝑄𝑂(𝑖) , +(17) +where 𝑄(𝑖) is the number of quadrangles; 𝑂𝑄𝐼 (𝑖) and 𝑂𝑄𝐼 (𝑖) are number of open-quadriads with 𝑖 as the inner node +and outer node respectively. They are found to be more efficient than 3-node measures in classifying networks and +predicting links. +13 + +– Grid coefficients [34]. Grid coefficients, including the primary grid coefficient and the secondary grid coefficient, +also aim to measure the formation of 4-cycles. The primary grid coefficient measures the formation of “primary +quadrilaterals” upon a node and three of its 1-hop neighbours, which is essentially the formation of chordal cycles +(𝐺7) from tailed-triangles (orbit 𝐺 (11) +6 +). Concretely, the primary grid coefficient of node 𝑖 is defined as: +C𝐺𝑝 (𝑖) = +𝑄𝑝 (𝑖) +𝑑𝑖 (𝑑𝑖 − 1)(𝑑𝑖 − 2)/2, +(18) +where 𝐺𝑝 (𝑖) is the number of chordal-cycles containing 𝑖 and the denominator is the number of possible chordal- +cycles built from a node and its three neighbours. The secondary coefficient measures the formation of “secondary +quadrilaterals” from a node, two of its 1-hop neighbours and one of its 2-hop neighbours: +C𝐺𝑠 (𝑖) = +𝑄𝑠 (𝑖) +𝑑𝑖,2𝑛𝑑𝑑𝑖 (𝑑𝑖 − 1)/2. +(19) +Notice, however, in this definition the 2-hop neighbour could be at orbit 𝐺 (4) +3 +or at orbit 𝐺 (20) +10 +. The latter essentially +involves 5 nodes in total. +– Square clustering coefficient. As triangles (3-cycles) are absent in bipartite networks, the square clustering coeffi- +cient is proposed to measure the formation of 4-cycles in the context of bipartite networks [124]. What is unusual +about this approach is that it views 4-cycles being built from node overlapping instead of node connection. Specifically, +the square coefficient of node 𝑖, with a pair of its neighbours 𝑚 and 𝑛, is calculated as: +C𝑆 (𝑖|𝑚,𝑛) = +𝑄𝑖𝑚𝑛 +(𝑑𝑚 − 𝜂𝑖𝑚𝑛)(𝑑𝑛 − 𝜂𝑖𝑚𝑛) + 𝑄𝑖𝑚𝑛 +, +(20) +where 𝑄𝑖𝑚𝑛 is the number of 4-cycles containing nodes 𝑖, 𝑚, 𝑛; and 𝜂𝑖𝑚𝑛 = 1 + 𝑞𝑖𝑚𝑛 if 𝑚 and 𝑛 are not connected (or +𝜂𝑖𝑚𝑛 = 2 + 𝑞𝑖𝑚𝑛 if 𝑚 and 𝑛 are connected). Zhang et al. [212] later proposed a modified version of square clustering +coefficient: C𝑆𝑍 (𝑖|𝑚,𝑛) = +𝑄𝑖𝑚𝑛 +(𝑑𝑚−𝜂𝑖𝑚𝑛)+(𝑑𝑛−𝜂𝑖𝑚𝑛)+𝑄𝑖𝑚𝑛 . With this minor change at the denominator, 4-cycles are now +built from connecting nodes. It is mainly applied in community detection. +– Interest clustering coefficient [176]. An interest clustering coefficient is introduced to measure the “clustering +of interest links” in directed social networks. It argues that the best way of defining a relationship between two +individuals is through common interests, i.e., two individuals having links towards a common neighbour will have +a higher chance to follow other common neighbours. From a structural view, the interest clustering coefficient +essentially measures the formation of bi-fan subgraphs (Table 1) upon open bi-fans: +C𝐼 = +4 · # bifan +# open-bifan. +(21) +Note that this metric is defined at network-level. The paper finds out that the interest clustering coefficient of Twitter +is higher than the traditional directed clustering coefficient, and further proves its usage in a link recommendation +task. +3.2.3 +Beyond 4-node subgraph. Some approaches are introduced with a variable subgraph size. In actual usage, however, +due to high complexity, they seldom go beyond the size of 6 nodes. +– Higher-order clustering coefficientsF [67]. Fronczak et al. propose the higher clustering coefficients to evaluate +the probabilities that the shortest paths between any two neighbours of node 𝑖 equals 𝑘, when all paths passing +14 + +Table 2. Metrics for 3-node and 4-node subgraph formation. +3-node/4-node +subgraph formation +Undirected networks +Directed networks +Weighted networks +clustering coef.[184] +directed clustering coef.[6, 62] +wgted. clustering coef.[15, 156, 206] +wgted. signed clustering coef.[46, 115] +wgted. directed clustering coef.[62] +closure coef.[196] +directed closure coef. [94, 197] +weighted closure coef. [95] +edge clustering coef.[181] +higher-order clustering +coef. (Fronczak)[67] +higher-order clustering +coef. (Abdo)[2] +None +None +square clustering coef. +(Lind [124], Zhang [212]) +i-quad coef. [96] +primary grid coef. [34] +None +None +o-quad coef. [96] +None +weighted o-quad coef. [96] +higher-order clustering +coef. (Yin)[195] +None +None +higher-order closure +coef. (Yin)[196] +None +None +through node 𝑖 are neglected. Particularly, a clustering coefficient of order 𝑘 for node 𝑖 is calculated as: +C𝐻𝐹 (𝑖 | 𝑘) = 2𝐸(𝑖 | 𝑘) +𝑑𝑖 (𝑑𝑖 − 1) , +(22) +where 𝐸(𝑖 | 𝑘) denotes the number of shortest paths of length 𝑘 between 𝑖’s neighbours. When 𝑘 equals 1, it degrades +to the standard clustering coefficient, and when 𝑘 equals 2, it measures the formation of 4-cycles. Note that each pair +of neighbours could have multiple shortest paths of the same length, and only one of them should be counted so that +the value of higher-order clustering coefficients is bounded by 1. +15 + +– Higher-order clustering coefficientY [195]. The higher-order clustering coefficient proposed by Yin et al. is another +generalisation of the traditional clustering coefficient. It aims to measure the formation of higher-order cliques. +Specifically, a 𝑘th-order clustering coefficient of node 𝑖 is defined as the probability that a 𝑘-clique plus an edge +incident to 𝑖 (termed as 𝑘-wedge) forms a (𝑘 + 1)-clique: +C𝐻𝑌 (𝑖 | 𝑘) = 𝑘 · |𝐶𝑘+1(𝑖)| +|𝑊𝑘 (𝑖)| += +𝑘 · |𝐶𝑘+1(𝑖)| +(𝑑𝑖 − 𝑘 + 1)|𝐶𝑘 (𝑖)|, +(23) +where 𝐶𝑘+1(𝑖) is the set of (𝑘 + 1)-cliques containing node 𝑖, and 𝑊𝑘 (𝑖) is the set of 𝑘-wedges with 𝑖 as the centre +node. The properties of higher-order clustering coefficient in random graph and the small-world model have also +been thoroughly investigated [195]. +– Higher-order closure coefficient [196]. Higher-order closure coefficient measures the formation of higher-order +cliques from a different perspective, i.e., the focal node being the end-node of a 𝑘-wedge (instead of the centre-node). +The 𝑘th-order closure coefficient of node 𝑖 is thus defined as the fraction of end-node based 𝑘-wedges that are closed +(a closed 𝑘-wedge is a (𝑘 + 1)-clique): +C𝐻𝐸 (𝑖 | 𝑘) = 𝑘 · |𝐶𝑘+1(𝑖)| +|𝑊 ′ +𝑘 (𝑖)| += +𝑘 · |𝐶𝑘+1(𝑖)| +� +𝑗 ∈𝑁 (𝑖) [𝐶𝑘 (𝑗) − (𝑘 − 1)𝐶𝑘 (𝑖)] , +(24) +where 𝐶𝑘+1(𝑖) is the set of (𝑘 + 1)-cliques containing node 𝑖, and𝑊𝑘 (𝑖)′ is the set of 𝑘-wedges with 𝑖 as the end-node. +Higher-order closure coefficient is proven to be useful in finding seeds for personalised PageRank community +detection. +An illustrative summary for most abovementioned approaches is given in Table 2. +3.3 +Global Path Based Approaches +Global path based approaches require structural information across the whole network in the form of unbounded paths +between nodes. One set of methods is based on the paths from one node to all other nodes, such as the well known +closeness centrality and Katz index; another set of methods is based on paths between all node pairs, represented by the +betweenness centrality (Figure 7). +Global +Path +Betweenness cent. +Katz index +Closeness cent. +Heatmap cent. +Reaching cent. +Edge Btw. cent. +Flow btw. cent./Communicability btw. cent. +Random-walk btw. cent. +Gravity cent./Gravity model +1-to-all +all-to-all +Fig. 7. Global path based measures. +3.3.1 +One-to-all. The approaches of this type involve the paths from one node to all other nodes. They are also referred +to as radial measures. +16 + +– Closeness centrality [65]. Being one of the most classic centrality measures, closeness centrality is defined as the +reciprocal of the average shortest path distance from a focal node 𝑖 to all other nodes: +Θ𝐶 (𝑖) = +|𝑉 | − 1 +� +𝑗 ∈𝑉,𝑗≠𝑖 𝑑(𝑖, 𝑗) . +(25) +Obviously, the original definition is not suitable for graphs with more than one connected component. To address +this problem, a modified version of the closeness centrality is defined as [183]: +Θ𝐶′(𝑖) = 𝑛 − 1 +|𝑉 | − 1 +𝑛 − 1 +�𝑛−1 +𝑗=1 𝑑(𝑖, 𝑗) +, +(26) +where 𝑛 is the number of nodes in one connected component. Due to its intuitiveness in definition, the closeness +centrality keeps being applied and extended in various fields. Some recent works include the neighbourhood closeness +centrality in predicting essential proteins [116], and the backward/forward closeness in studying global value chains +[76]. +– Katz index [102]. Unlike the closeness centrality that focuses on shortest paths, Katz centrality of a node considers +all paths reaching other nodes with longer paths contributing less. Concretely, the Katz centrality of a node 𝑖 is +calculated as: +Θ𝐾 (𝑖) = +∑︁ +𝑗 +∞ +∑︁ +𝑘=1 +𝛽𝑘A𝑘 +𝑖𝑗, +(27) +where 𝑘 is a path length and 𝛽 is an attenuation parameter in a range (0, 1 +𝜆 ), 𝜆 being the largest eigenvalue of A. +Further, the overall matrix M = �∞ +𝑘=1(𝛽 · A)𝑘 is an weighted ensemble of all paths. Thus, M𝑖𝑗 represents the weighted +sum of paths from 𝑖 to 𝑗 in all possible hops. Note that this definition is naturally suitable in directed networks and a +recent work proposes to generate node embedding of a directed graph by performing a singular value decomposition +on the Katz index matrix [158]. +– Gravity model [121] /Gravity centrality [131] . Inspired by Newton’s gravity law formula, a gravity model is +proposed by viewing the degree of a node as its mass and the shortest path length between two nodes as their +distance: +Θ𝐺 (𝑖) = +∑︁ +𝑗 ∈𝑉,𝑗≠𝑖 +𝑑𝑖 · 𝑑𝑗 +𝑑(𝑖, 𝑗)2 . +(28) +In order to make it easier to compute in large networks, a modified version limits the radius from the entire network +to a certain length. Adopting a similar idea, the gravity centrality is introduced by regarding the coreness of a node +as its mass, and the shortest path length between nodes as their distance: +Θ′ +𝐺 (𝑖) = +∑︁ +𝑗 ∈𝑁𝑘 (𝑖) +𝑘𝑠(𝑖) · 𝑘𝑠(𝑗) +𝑑(𝑖, 𝑗)2 +, +(29) +where 𝑁𝑘 (𝑖) is the neighbourhood of node 𝑖 within 𝑘-hops, and 𝑘𝑠(𝑖) is the coreness of node 𝑖. The two approaches +are shown to be effective in identifying influential spreaders through analyses of the SIR model on real networks. +– Heatmap centrality [57]. Heatmap centrality measures the influence of a node by comparing the farness of the +node with the average farness of its neighbours. Farness, the reciprocal of closeness, is defined as the sum of the +lengths of shortest paths from a node to all other nodes, i.e., 𝑓 (𝑖) = � +𝑗 ∈𝑉,𝑗≠𝑖 𝑑(𝑖, 𝑗). Therefore, the heatmap centrality +17 + +of node 𝑖 is quantified as: +Θ𝐻𝑀 (𝑖) = 𝑓 (𝑖) − +� +𝑗 ∈𝑁 (𝑖) 𝑓 (𝑗) +|𝑁 (𝑖)| +. +(30) +The intuition of this metric is that if a node has a smaller farness than its neighbours, the probability of information +passing through it is higher. Note that according to heatmap centrality, a top-ranked node of influence should have +the most negative value. Although the definition of heatmap centrality is more related to the closeness centrality, it is +revealed that it is highly correlated with the betweenness centrality. +– Reaching centrality [144]. Reaching centrality aims to rank the influence of a node in directed networks. Intuitively, +the reaching centrality of node 𝑖 is quantified as the proportion of nodes that can be reached by the node via outgoing +edges, i.e., the number of nodes with a directed distance from 𝑖, divided by |𝑉 | −1. Further, a global reaching centrality +is then defined as: +𝐺𝑅𝐶 = +� +𝑖 ∈𝑉 [Θ𝑚𝑎𝑥 +𝑅 +− Θ𝑅(𝑖)] +|𝑉 | − 1 +, +(31) +where Θ𝑚𝑎𝑥 +𝑅 +is the largest reaching centrality of all nodes. The meaning of 𝐺𝑅𝐶 is the difference between the +maximum reaching centrality and the average reaching centrality. Global reaching centrality is used as a hierarchy +measure for directed networks and is shown to be capable of capturing the degree of hierarchy in both synthetic and +real networks. +3.3.2 +All-to-all. The approaches here involve the count of paths between all node pairs, and among them the ones that +pass through a focal node or edge. They are also referred to as medial measures. +– Betweenness centrality [64]. Betweenness centrality, or more precisely, the shortest-path betweenness centrality +is one of the best-known centrality measures. The betweenness centrality of node 𝑖 is quantified as the sum of the +fraction of all-pairs shortest paths going through 𝑖: +Θ𝐵(𝑖) = +∑︁ +𝑠,𝑡 ∈𝑉 +𝜎(𝑠,𝑡 | 𝑖) +𝜎(𝑠,𝑡) , +(32) +where 𝜎(𝑠,𝑡 | 𝑖) is the number of shortest paths between node pair 𝑠 and 𝑡 that pass through node 𝑖, and 𝜎(𝑠,𝑡) is the +number of all shortest paths between 𝑠 and 𝑡. It is often normalised by ( |𝑉 |−1) ( |𝑉 |−2) +2 +, in order to be compared in +different networks. The betweenness centrality has also been generalised to directed networks[186] and weighted +networks [157]. +– Edge betweenness centrality [70]. With a small modification on the original betweenness centrality, Girvan and +Newman propose an edge betweenness centrality in order to detect a community structure in complex networks. The +edge betweenness centrality of an edge 𝑒 is quantified as the sum of the fraction of all-pairs shortest paths passing +through 𝑒: +Θ𝐵(𝑒) = +∑︁ +𝑠,𝑡 ∈𝑉 +𝜎(𝑠,𝑡 | 𝑒) +𝜎(𝑠,𝑡) , +(33) +According to the definition, edges which lie between communities will have large edge betweenness. Therefore, +the underlying communities of the network would be uncovered by removing edges of high edge betweenness +centrality. It has been widely applied in a community detection task, and some recent applications include the study +of anti-vaccination sentiment on Facebook [83] and the analysis of microbial diversity in marine sediment [85]. +18 + +– Flow betweenness centrality [66]/ Communicability betweenness centrality [60]. A major limitation of the +betweenness centrality is that it exclusively focuses on the shortest paths. In real situations, however, information +often takes a more circuitous path randomly or intentionally [171]. The flow betweenness addresses this issue by +considering all paths between nodes. Specifically, the flow betweenness centrality of a node 𝑖 is defined as: +Θ𝐹 (𝑖) = +∑︁ +𝑠,𝑡 ∈𝑉 +𝜙(𝑠,𝑡 | 𝑖) +𝜙(𝑠,𝑡) , +(34) +where 𝜙(𝑠,𝑡 | 𝑖) is the maximum flow between 𝑠 and 𝑡 that passes through 𝑖, and 𝜙(𝑠,𝑡) is the total flow between 𝑠 +and 𝑡. The maximum flow is in turn calculated by the minimum cut capacity [63]. Having established the notion of +“capacity ” on links, the flow betweenness centrality is naturally suitable for weighted networks. Instead of treating +each path equally, the communicability betweenness centrality proposes to reduce the weight for longer paths: +2 +(𝑛 − 1)(𝑛 − 2) +∑︁ +𝑠,𝑡 ∈𝑉 +�∞ +𝑘=0 +1 +𝑘!𝜇𝑘 (𝑠,𝑡 | 𝑖) +�∞ +𝑘=0 +1 +𝑘!𝜇𝑘 (𝑠,𝑡) +, +(35) +where 𝜇𝑘 (𝑠,𝑡 | 𝑖) is the number of paths between 𝑠 and 𝑡 passing 𝑖 with length 𝑘, and 𝜇𝑘 (𝑠,𝑡) is the number of paths +between 𝑠 and 𝑡 with a length 𝑘. +– Random-walk betweenness centrality [152]. A random-walk betweenness centrality, also known as a current- +flow betweenness centrality, is another popular variant of the betweenness centrality. It models information spreading +in a network analogous to an electrical current flow in a circuit. Concretely, the current-flow betweenness centrality +of node 𝑖 is defined as the amount of current flowing through 𝑖, averaged over all node pairs: +Θ𝐶𝐹 (𝑖) = +� +𝑠,𝑡 ∈𝑉 𝐼 (𝑠,𝑡 | 𝑖) +(1/2)𝑛(𝑛 − 1) , +(36) +where 𝐼 (𝑠,𝑡 | 𝑖) is the current flowing from 𝑠 to 𝑡 that passes 𝑖. The paper then proves that a message spreading along +random walks is equivalent to the above definition. +3.4 +Message Passing Based Approaches +The above mentioned approaches depend solely on the topological information of a network, such as the number of +particular subgraphs, the ratio between two subgraphs, the length of shortest paths, or the number of paths. Message +passing based approaches further consider the information contained in each node. From a microscopic point of view, +in one iteration, only local information is needed at each node. It is worth noticing that the popular graph convolutional +network is also based on this idea, i.e, iteratively gathering information from nearby nodes. +Message +Passing +PageRank +LeaderRank +Eigenvector cent. +HITS +Nonbacktracking cent. +Alpha cent. +Fig. 8. Message passing based approaches. +19 + +– Eigenvector centrality [24]. The eigenvector centrality is another classic centrality measure. The idea is that a +node’s centrality depends on the centralities of its neighbours: +𝑥(𝑖) = 𝑐 +∑︁ +𝑗 ∈𝑁 (𝑖) +𝑥(𝑗), +(37) +where 𝑐 is a normalisation constant. The equation is recursive and computed by starting with a set of initial influence +scores and iterating the computation until it converges. In a vectorised form, i.e., �𝑥 = 𝑐A�𝑥, �𝑥 is found to converge to +the dominant eigenvector of A and 𝑐 converges to the reciprocal of the dominant eigenvalue of A. The eigenvector +centrality has some problems in very sparse networks, i.e., the leading eigenvector is localised around nodes of the +highest degree and diminishes the effectiveness of quantifying nodes’ importance [113]. +– Nonbacktracking centrality [137]. The nonbacktracking centrality is proposed to address the above mentioned +localisation issue. The same as in the eigenvector centrality, a node’s centrality is the sum of its neighbours’ centralities, +but now the neighbours’ centralities are calculated without the influence of this node. This is achieved by using the +nonbacktracking matrix [80]. The nonbacktracking matrix B, is a 2𝑚 ×2𝑚 matrix, defined on the directed edges of the +graph (undirected edges are converted to bidirectional edges), and elements B𝑖→𝑗,𝑘→𝑙 = 𝛿𝑖,𝑙 (1 − 𝛿𝑗𝑘), where 𝛿 is the +Kronecker delta. Then, 𝑒𝑗→𝑖 of the leading eigenvector of B gives the centrality of node 𝑗 ignoring the contribution +of 𝑖. Finally, the nonbacktracking centrality of node 𝑖 is 𝑥(𝑖) = � +𝑗 A𝑗𝑖𝑒𝑗→𝑖. The eigenvalues of the nonbacktracking +matrix are also found to be useful in a community detection task [114]. +– Alpha centrality [25]. When the eigenvector centrality is applied in directed networks, a node’s centrality is +determined by those who pointed at it. Thus, the vector form becomes: �𝑥 = 1 +𝜆 A𝑇 �𝑥. The problem is that nodes with +no incoming edges would have zero centrality value. The alpha centrality proposes to solve this problem by taking +into account the "external status characteristics". The equation then becomes: +�𝑥 = 𝛼A𝑇 �𝑥 + �𝑒, +(38) +where �𝑒 is a vector of the exogenous sources of characteristics and 𝛼 is a parameter which reflects the relative +importance of a topological structure versus exogenous factors. +– PageRank [30]. PageRank, a popular variation of the eigenvector centrality, is proposed to rank the importance of +web pages. Web pages and the links among them are modelled as a directed network, and a page should have a high +rank if the sum of the ranks of pages that point to it is high. Specifically, the rank of page 𝑖 is calculated as: +𝑟 (𝑖) = 𝑐 +∑︁ +𝑗 ∈𝑁 𝑖𝑛 +𝑖 +𝑟 (𝑗) +𝑑𝑜𝑢𝑡 +𝑗 +, +(39) +where 𝑁𝑖𝑛 +𝑖 +is the set of pages pointing to 𝑖 (𝑖’s in-neighbours), and 𝑑𝑜𝑢𝑡 +𝑗 +is out-degree of page 𝑗. In order to deal +with the “rank sink” problem, where several pages form a loop without other outgoing links, a source of the rank is +introduced over all pages (also viewed as a random jumping factor), denoted as a vector �𝑒. Therefore, the rank of page +𝑖 becomes: 𝑟 (𝑖) = 𝑐(� +𝑗 ∈𝑁 𝑖𝑛 +𝑖 +𝑟 (𝑗) +𝑑𝑜𝑢𝑡 +𝑗 ++ 𝑒(𝑖)), and the corresponding vector form is �𝑟 = 𝑐(A𝑇 + �𝑒 × 1)�𝑟. The PageRank has +also been extended to weighted networks [191], on nonbacktracking matrix [9], and applied to many different areas +[71]. +20 + +– HITS [109]. Unlike the PageRank which focuses on pages having many incoming links, HITS, abbreviated from a +hyperlink induced topic search, proposes to distinguish two roles in the hyperlink structure, i.e., authorities and hubs. +Authorities are reliable information sources, and hubs are the websites pointing to them. Based on the intuition that +an authority should be pointed to by hubs and a hub should point to authorities, an authority weight and a hub +weight of page 𝑖 are thus defined in a mutually dependent manner: +𝑎(𝑖) = +∑︁ +𝑗 ∈𝑁 𝑖𝑛 +𝑖 +ℎ(𝑗) +ℎ(𝑖) = +∑︁ +𝑗 ∈𝑁 𝑜𝑢𝑡 +𝑖 +𝑎(𝑗). +(40) +The corresponding vector forms are: �𝑎 = A𝑇 �ℎ, and �ℎ = A�𝑎. �𝑎 and �ℎ are updated iteratively, and it is proven that �𝑎 +converges to the leading eigenvector of A𝑇 A, and �ℎ converges to the leading eigenvector of AA𝑇 . Based on HITS, ARC +(Automatic Resource Compilation) later proposes to incorporate textual information around the link by assigning +each link a weight [38], and Co-HITS proposes to extend the idea to bipartite networks [53]. +– LeaderRank [129]. In order to solve the above mentioned rank sink problem, the LeaderRank proposes to add a +ground node that connects to other nodes via bidirectional links. In the beginning, each node other than the ground +node is initialised by one unit of score, and the ground node is initialised to zero. Then, the same as the PageRank, at +each iteration, the score of node 𝑖 is calculated as: 𝑠(𝑖)(𝑡) = 𝑐 � +𝑗 ∈𝑁 𝑖𝑛 +𝑖 +𝑠 (𝑗) (𝑡−1) +𝑑𝑜𝑢𝑡 +𝑗 +. After the scores of all nodes reach a +steady state, the score of the ground node will be distributed evenly to other nodes, and thus the final score of node 𝑖 +is: +𝑠(𝑖) = 𝑠(𝑖)𝑐 + 𝑠(𝑔)𝑐 +|𝑉 | , +(41) +where 𝑠(𝑖)𝑐 is the steady score of node 𝑖, and 𝑠(𝑔)𝑐 is the steady score of the ground node. A major advantage of the +LeaderRank is that it has no additional parameter that needs to be optimised. Some interesting extensions of the +LeaderRank include the weighted LeaderRank that assigns degree-dependent weights onto links associated with the +ground node [119] and the adaptive LeaderRank that introduces H-index into the weighted mechanism [194]. +3.5 +Hybrid Approaches +The methods in the fifth and final category are combinations of previously introduced approaches. +Hybrid +ClusterRank +HybridRank +BridgeRank +Local structural cent. +Local triangle struc. cent. +CCPA +Hybrid degree cent. +Fig. 9. Hybrid Approaches. +– ClusterRank [41]. Previous studies have shown that a large clustering coefficient may slow the spreading process +of disease in the entire network [59, 221]. A ClusterRank thus proposes to consider not only the number of a node’s +neighbours, but also the negative effect of local clustering when identifying influential nodes. The ClusterRank score +of node 𝑖 is defined as: +Θ𝐶𝑅(𝑖) = 𝑓 (𝑐𝑖) +∑︁ +𝑗 ∈𝑁 𝑜𝑢𝑡 +𝑖 +(𝑑𝑜𝑢𝑡 +𝑗 ++ 1), +(42) +21 + +where 𝑐𝑖 = +� +𝑗∈𝑁𝑜𝑢𝑡 +𝑖 +|𝑁 𝑜𝑢𝑡 (𝑖)∩𝑁 (𝑗) | +𝑑𝑜𝑢𝑡 +𝑖 +(𝑑𝑜𝑢𝑡 +𝑖 +−1) +is a modified version of clustering coefficient in directed networks. 𝑓 (𝑐𝑖) is a +function that is negatively correlated with 𝑐𝑖, for example an exponential function 𝑓 (𝑐𝑖) = 10−𝑐𝑖 . Although the +ClusterRank is proposed for directed networks, it can be easily extended to undirected networks [41] and weighted +networks[182]. Experiments on several real networks demonstrate that the ClusterRank score outperforms the +PageRank and the LeaderRank while being more efficient in computation. +– Local structural Centrality [69]. Aiming to evaluate the spreading ability of nodes, a local structural centrality +essentially extends the local centrality (section 3.1.2) by further considering the connections between higher-order +neighbours. The idea is that a node has a better spreading ability when its neighbours are better connected because +a neighbour node can be affected directly by the source node or indirectly by another neighbour node. The local +structural centrality of node 𝑖 is defined as: +Θ𝐿𝑆 (𝑖) = +∑︁ +𝑗 ∈𝑁𝑖 +(𝛼|𝑁 1,2 +𝑗 +| + (1 − 𝛼) +∑︁ +𝑘 ∈𝑁 1,2 +𝑗 +𝑐(𝑘)), +(43) +where 𝑁 1,2 +𝑗 +is the node set of 1-hop and 2-hop neighbours of node 𝑗, and 𝑐(𝑘) is the clustering coefficient of node 𝑘. +𝛼 is a tunable parameter between 0 and 1, balancing a direct and indirect spreading contribution. Notice that the part +of the clustering coefficient is multiplied in the ClusterRank when evaluating spreading speed, but added up here +when measuring the spreading ability. +– Local triangle structure centrality [134]. A local triangle structure centrality (LTSC) proposes to include the +triangle proportion of a node, instead of its clustering coefficient when evaluating a node’s spreading ability. The +triangle proportion is able to indicate the location of a node, whether it is located in a denser or sparser part of a +network. LTSC partitions the spreading ability into two parts, i.e., inner spreading ability and outer spreading ability. +Specifically, the local triangle structural centrality of node 𝑖 is defined as: +Θ𝑇𝑆 (𝑖) = +∑︁ +𝑗 ∈𝑁𝑖 +(𝑑𝑗 (1 +𝑇𝑃(𝑗)) + ( +∑︁ +𝑘 ∈𝑁𝑗 +𝑑𝑘 − 𝑑𝑗)), +(44) +where 𝑇𝑃(𝑗) is the triangle proportion of node 𝑗, calculated by the number of triangles containing 𝑗 divided by +the total number of triangles in the network. For each neighbour 𝑗 of a given node 𝑖, the part of 𝑑𝑗 (1 +𝑇𝑃(𝑗) is to +measure its inner spreading ability, and the part of � +𝑘 ∈𝑁𝑗 𝑑𝑘 − 𝑑𝑗 is to measure its outer spreading ability. Finally, +the local triangle structure centrality of node 𝑖 is the sum of the spreading abilities of its neighbours. +– Hybrid degree centrality [132]. The spreading probabilities of networks describing diseases, opinions, and rumours +should obviously differ. Most existing centrality measures, however, fail to take that into consideration. The per- +formance of centrality measures is sensitive to the spreading probability. The degree centrality, for example, works +best with modest spreading probabilities, while the local centrality (section 3.1.2) works better with higher ones [69]. +In order to alleviate the sensitivity to different spreading probabilities, a hybrid degree centrality is introduced by +integrating the degree centrality and a modified local centrality. The hybrid degree centrality of node 𝑖 is defined as: +Θ𝐻𝐷 (𝑖) = (𝛽 − 𝑝) · 𝛼 · Θ𝐷 (𝑖) + 𝑝 · Θ′ +𝐿𝑅(𝑖), +(45) +22 + +where Θ′ +𝐿𝑅(𝑖) = Θ𝐿𝑅(𝑖) − 2 � +𝑗 ∈𝑁𝑖 |𝑁𝑗 | is the modified local centrality, 𝑝 is the spreading probability, 𝛼 and 𝛽 are +two tunable parameters. The part contributed by the degree centrality is viewed as a near-source influence, and the +part of modified local centrality is a distant influence. +– HybridRank [4]. A HybridRank proposes to identify influential spreaders by combining the neighbourhood coreness +centrality (section 3.1.1) and the eigenvector centrality. The reason for integrating these two measures is that they +both regard a node as influential if the node is connected to other influential nodes. The hybrid centrality of node 𝑖 is +defined as: +Θ𝐻𝑅(𝑖) = Θ𝑁𝐶 (𝑖) × Θ𝐸 (𝑖), +(46) +where Θ𝑁𝐶 (𝑖) = � +𝑗 ∈𝑁𝑖 𝑘𝑠(𝑗) is the neighbourhood coreness of 𝑖, and Θ𝐸 (𝑖) is the eigenvector centrality of node 𝑖. +The HybridRank algorithm further suggests that when selecting influential spreaders, the neighbours of selected +ones should be neglected in order to maximise the spreading range. The effectiveness of the HybridRank has also +been tested in real networks using a SIR model. +– BridgeRank [167]. In order to lower the time complexity of the closeness centrality while keeping comparable +performance, a BridgeRank proposes to compute the shortest paths to just a few core nodes in the network. In +the BridgeRank algorithm, at first, communities are identified by the Louvain algorithm [21]. Then, core nodes are +discovered through calculating the betweenness centralities within each community (one core node per community). +Finally, the BridgeRank centrality of each node is defined as a filtered closeness centrality to these core nodes: +Θ𝐵𝑅(𝑖) = +1 +� +𝑗 ∈C 𝑑(𝑖, 𝑗) , +(47) +where C is the set of identified core nodes in each community. The time complexity is therefore reduced from 𝑂(|𝑉 |3) +to 𝑂(|𝑉 |𝑙𝑜𝑔|𝑉 |). A modified version that allows multiple core nodes being selected in a community is also introduced +[167]. Other community structure based methods include 𝑘-medoid that uses information transfer probabilities +between any node pairs [215], and the influence maximization algorithm based on label propagation [218]. +– CCPA [5]. A common neighbour and centrality based parameterised algorithm, or CCPA, is an approach for a link +prediction. It aims to bring together two essential properties of nodes, i.e., the common neighbours and the closeness +centrality. The similarity score between a pair of nodes 𝑖 and 𝑗 is defined as: +𝑠(𝑖, 𝑗) = 𝛼 · (|𝑁𝑖 ∩ 𝑁𝑗 |) + (1 − 𝛼) · +|𝑉 | +𝑑(𝑖, 𝑗) . +(48) +|𝑁𝑖 ∩ 𝑁𝑗 | is obviously the part of common neighbours. +|𝑉 | +𝑑 (𝑖,𝑗) , reciprocal of the normalised distance between two +nodes, is deemed as the closeness centrality of them, since it has a similar form as the classic node closeness centrality. +𝛼 ∈ [0, 1] is a user-defined parameter controlling the weight of the two parts. Experiments on real-world datasets +suggest that the change in performance (measured in average AUC) caused by the change of 𝛼 is not significant. +3.6 +Discussion and Outlook +To end this section, we further discuss graph structural measures in different types of networks and highlight some +research avenues for future studies. We then briefly talk about the importance and role of reviewing traditional structural +measures in the following part of the survey on GCNs. +23 + +Dynamic Networks. Most approaches covered in the survey assume that networks are static or time-independent. +Many real-world networks, however, are in fact dynamic, nodes and edges appearing and disappearing over time +[84, 122]. In telecommunication networks, the connection between agents is often bursty and fluctuates across time; in +social networks, relationships among people are typically intermittent and recurrent; in transportation networks, the +frequency of public transport service is usually higher in rush hours. This extra dimension of time adds richness and +complexity to the graph representation of a system, necessitating the development of more advanced approaches that +can leverage temporal information. Many studies have generalised the classic graph structural measures to dynamic +networks, including temporal degree centrality[104], temporal clustering coefficient [153], temporal closeness and +betweenness centrality [103], temporal eigenvector centrality [174], temporal Katz centrality [153], temporal motifs +[112, 159] and temporal graphlets [91]. Despite the large number of structural measures proposed for dynamic networks, +there are still many open questions to be tackled. For example, what is the impact of the temporal network’s structure on +the dynamics of processes that occur on it; how to apply temporal measures in inferring spreading chains in incomplete +temporal networks, etc. +Multilayer Networks. Sometimes, systems are so complicated that multiple-layered networks are needed to better +represent and study them [20, 22, 48, 108]. For example, a multilayer social network incorporates both friendship and +financial relationships among individuals; a multilayer brain network contains both the anatomical brain layer and +functional brain layer, and a multilayer transportation network integrates all sorts of transportation. Since interlayer +connections cause new structural and dynamic correlations between components, neglecting them or simply aggregating +over layers will alter the original topological properties. Therefore, it is desirable to develop structural measures taking +interlayer relationships into consideration. Not surprisingly, fundamental single-layer approaches have been largely +generalised to multilayer networks, such as multilayer degree, clustering coefficient, closeness and betweenness +centralities, [22, 48, 55], multilayer motifs and graphlets [17, 54], multilayer eigenvector, PageRank and HITS centralities +[49, 50, 77]. Some tailor-made approaches for multilayer networks are also recently introduced, for example, the +minimal-layers power community index [16], and the singular vector of tensor centrality [179]. The study of multilayer +structures, however, is still in an early stage. There is still much room for developing new cross-layer structural +approaches that better model inter-layer spreading processes [168] and captures multiplex dynamics, and controllability +[98]. +Node/edge attributes. Network data, besides the pure topological presence, are often accompanied by rich information +on node attributes and/or edge attributes, and they are also referred to as labelled networks or attributed networks. Most +structural measures, as the name suggests, focus solely on capturing the part of topological properties. Theoretically, +message passing approaches are able to include numeric node attributes, such as the initial rank and source of rank +in the PageRank [30], or the endogenous and exogenous status in the alpha centrality [25]. In practice though, these +features are usually set to identical values for all nodes, for example, all ones for the initial rank and 0.15 for the source +of rank in the PageRank. Multidimensional features are not supported in message passing approaches either. There have +also been attempts to integrate node/edge attributes with other graph structural measures. For instance, the degree +and betweenness centralities are combined with node attributes in studying criminal networks [29]; nodes’ attributes +are used as a threshold in LRIC index [10]; and node/edge attributes are fused into graphlets [93, 165]. We believe +there is still great potential for developing novel structural approaches that integrate rich information on nodes and/or +edges. It is also worth mentioning that one reason for the popularity of graph neural networks is that it naturally +enables integrating node attributes and some recent works also propose to take edge attributes into account in GNNs +[42, 72, 99]. +24 + +Finally, we discuss how the traditional structure-based approaches are linked to GCNs. The importance and role of +reviewing traditional structural measures in the survey of GCNs are Multifaceted. First, traditional structural approaches, +the outcome of decades of Network Science studies, are the precursors and foundations of graph neural networks. +For example, the key idea of neighbourhood aggregation and message passing in GCNs can trace back to 1972 when +Bonacich proposed the eigenvector centrality [23]. Basic network science notions such as the clustering coefficient, +motifs and graphlets are utilised in GCNs as well. Second, the taxonomy of traditional approaches from the perspective +of structure information inspired us to develop a new taxonomy for GCNs. We will see later how the taxonomy of +GCNs from a layer-wise message aggregation scope is similar to that of subgraph count based measures in Section 3.1. +Third and last, a comprehensive review of traditional structural measures not only helps in revealing their connections +to GCN approaches but also benefits the discovery of knowledge gaps. We will see that some GCN approaches are +inspired by the traditional message passing based approaches, and that many subgraph count based approaches find +their usages in GCNs. However, the connections between GCNs and subgraph formation based or global path based +approaches are still largely left undiscovered. +4 +STRUCTURE INFORMATION IN GRAPH CONVOLUTIONAL NETWORKS +After summarising the traditional Network Science structural measures, we are set to review the graph convolutional +networks from a novel perspective of graph structural information. +In recent years, graph neural networks, especially graph convolutional networks, have become one of the most +prominent research areas in the study of complex networks. It extends the traditional convolutional neural networks to +graph data and enables an effective combination of the rich node features information and graph topological structure. +Graph convolutional networks have been successfully applied in different types of graph learning tasks, including +node classification, link prediction, graph classification and graph clustering. Amongst the large family of graph deep +learning approaches [135, 216], we particularly focus on graph convolutional networks not only because they have +a wider range of applicability, but also because they are the bases of many other graph deep learning approaches, +including graph autoencoders, graph reinforcement learning, graph adversarial methods, etc. +There exist several comprehensive surveys on graph neural networks. Bronstein et al. [32] provide a thorough +review of geometric deep learning, which presents its problems, difficulties, solutions and applications. Hamilton et al. +[79] develop a unified encoder-decoder framework for graph representation learning approaches, bringing together +matrix factorisation-based methods, random-walk-based algorithms and graph neural networks. Chami et al. [39] later +extend the framework by including more recent advancements in the area. Zhang et al. [214] propose a comprehensive +review specifically on graph convolutional networks. Zhou et al. [219] introduce a detailed taxonomy after dividing +GNNs into several modules, including the propagation module, the sampling module and the pooling module. Wu et al. +[189] propose to divide GNNs into four categories, i.e., recurrent GNNs, convolutional GNNs, graph autoencoders and +spatial-temporal GNNs. +These reviews, when introducing convolutional neural networks, usually focus on the domain of convolutional +operations and propose a dichotomy, i.e., the spectral-based methods and the spatial-based methods. However, the line +between the two is sometimes blurred. For example, GCN is an approximation of spectral graph convolutions, but it +operates directly on graphs — applying filters acting on the k-hop neighbourhood of the graph in the spatial domain +[32]. Another recent work also proves that spectral convolutional graph neural networks can be viewed as a particular +case of spatial convolutional neural networks [147]. +25 + +Different from existing reviews, in this survey we primarily, but not exclusively, focus on how local structure plays +its role in graph convolutional networks. we propose to categorise GCN approaches from three different perspectives, +which are the layer-wise message aggregation scope, the message content, and the overall learning scope. +• Layer-wise message aggregation scope. Analogous to convolutional neural networks, multilayer architecture is +one of the key features in graph convolutional networks. Taking the vanilla GCN for example, at each layer, a node +gathers information from its 1-hop neighbours. Then from stacking 𝑘 layers, the node would convolve its 𝑘𝑡ℎ- +order neighbourhood. Thereafter, many other approaches propose to apply different scope at each layer, including +2-hop neighbourhood, k-hop neighbourhood, local-random-walk neighbourhood, subgraph neighbourhood, etc. +This first structural perspective in GCN design can be summarised into the following question: From where a +node aggregates message at each layer? The detailed taxonomy of GCNs from the perspective of layer-wise +message aggregation scope and related approaches are given in Subsection 4.1. +• Message content. Compared to traditional deep learning models such as CNNs and RNNs, the strength of GCNs +comes from the ingenious integration of graph structure and node features — node features are passed through +the edges of the graph. In many cases, the feature of nodes is independent of graph structure, such as numerical +ratings, word vectors generated from sentences, positional gene sets, immunological signatures, and more. +Meanwhile, there are emerging works that include other structural information as part of node features, from the +simplest node degree to more complicated distance or subgraph information [27, 78, 117]. This second structural +perspective in GCN design can be summarised into the question: What structural information is included in +the node feature when initialising or running the message passing scheme? The detailed taxonomy from the +message content perspective and the related approaches are given in Subsection 4.2. +• Learning scope. Traditional graph representation learning approaches are generally based on matrix factorisation, +which thus requires the fixed whole graph. Although the original GCN approach also takes the whole graph’s +adjacency matrix as input, it has soon been extended to various settings, such as subgraphs, localised subgraphs, +and more. To put in a question format, the third structural perspective in a GNN design is: Where GCNs are +trained on? or What is/are the input graph/graphs in GCNs? The detailed taxonomy of GCNs from the learning +scope perspective and the related approaches are given in Subsection 4.3. +4.1 +Layer-wise message scope +To begin with, we discuss in detail the first structural perspective in a GCN design, i.e., a layer-wise message scope. By +answering the question of where a node aggregates message from at each layer, we divide existing GCN approaches +into four categories, which are 1-hop neighbourhood approaches, k-hop neighbourhood approaches, local-random-walk +neighbourhood approaches, and subgraph neighbourhood approaches. The taxonomy and representative approaches +are given in Figure 10. The colour of the block indicates what task the approach is proposed for: grey is the most +common node classification task, orange is a network classification, and blue is the link prediction task which will +appear later in Section 4.2. Notice that a graph representation can be readily obtained via graph pooling, so approaches +proposed for a node classification can potentially be applied in a network classification task. Likewise, some approaches +proposed for the network classification also generate node representations, making them possible to be used in the +node classification. +4.1.1 +1-hop neighbourhood approaches. Many influential GCN approaches adopt the 1-hop neighbourhood aggregation +strategy, where a node’s representation is iteratively updated through aggregating representations of its neighbours. +26 + +Layer-wise Message +Aggregation Scope +(Where a node aggregates +message from) +k-hop neighbourhood +Random-walk +neighbourhood +Subgraph +neighbourhood +DGCN +MixHop +k-hop GNN +PinSage +GraLSP +k-GNN +GCN +GraphSAGE +GAT +GIN +Adapt +FastGCN +PATCHY-SAN +DGCNN +GRAPE +GraphSNN +SGC +1-hop neighbourhood +DGP +Fig. 10. Taxonomy from the Layer-wise Message Aggregation Scope perspective. +One iteration happens at one convolutional layer, and after stacking multiple layers, the node’s representation is able to +capture a wider range of neighbourhoods. +GCN. Motivated by the first-order approximation of localised spectral filters on a graph [52], GCN proposes the +following layer-wise propagation rule operating directly on graphs: +𝐻 (𝑙) = 𝜎 +� +ˆ𝐴𝐻 (𝑙−1)𝑊 (𝑙)� +(49) +, where ˆ𝐴 = ˜𝐷− 1 +2 ˜𝐴 ˜𝐷− 1 +2 , ˜𝐴 is adjacency matrix with added self-connections, and ˜𝐷 is degree matrix of ˜𝐴. 𝐻 (𝑙) is the +representations at the 𝑙𝑡ℎ layer, and 𝑊 (𝑙) is the learnable weight matrix at 𝑙𝑡ℎ layer. 𝜎 denotes a nonlinear activation +function such as ReLU. The multiplication of the normalised self-connection added adjacency matrix ˆ𝐴 and the nodes’ +representation matrix 𝐻 represents a normalised sum of neighbouring nodes’ (and self node’s) representation. From a +microscopic point of view, the representation of node 𝑣 at layer 𝑙 is calculated as: +ℎ(𝑙) +𝑣 += 𝜎 �� +� +∑︁ +𝑢∈N(𝑣) +1 +𝑐𝑣𝑢 +ℎ(𝑙−1) +𝑢 +𝑊 (𝑙)�� +� +, +(50) +where N (𝑣) is the set of node 𝑣’s one-hop neighbours (with added self-loops to each node), 𝑐𝑣𝑢 = +√︁ +|N (𝑣)| +√︁ +|N (𝑢)| +is the normalization constant based on the node degree. The loss is then computed as: L = − � +𝑙 ∈Y𝐿 +�𝐹 +𝑓 =1 𝑌𝑙 ln𝑍𝑙𝑓 , +where Y𝐿 is the set of labelled nodes, 𝑍 is the output embedding and 𝐹 are the feature maps. Successfully bringing +convolutional operations on graphs, GCN has become one of the most popular graph representation learning approaches. +It is worth mentioning that the Iterative Classification Algorithm (ICA) also uses neighbourhood information to train a +model [19, 150]. The model is then used to iteratively update the labels of nodes in the test set. Obviously, without +a multi-layer convolutional network, the scope of ICA in the training stage is strictly limited within the immediate +neighbourhood. +GraphSAGE. Hamilton et al. later proposed the GraphSAGE framework, which extends the GCN to a more general +setting that supports a mini-batch approach and different aggregation functions [78]. Specifically, the representation of +27 + +node 𝑣 at layer 𝑙 is given by: +h𝑙 +N(𝑣) ← AGGREGATE 𝑙 +�� +h𝑙−1 +𝑢 , ∀𝑢 ∈ N (𝑣) +�� +, +h𝑙 +𝑣 ← 𝜎 +� +W𝑙 · CONCAT +� +h𝑙−1 +𝑣 +, h𝑙 +N(𝑣) +�� +(51) +The framework thus gives us the flexibility to choose different aggregator functions, such as mean aggregator +(equivalent to GCN), LSTM aggregator and pooling aggregator. Further, unlike GCN which requires full batch gradient +descent, GraphSAGE enables mini-batch setting and therefore can also be applied to unseen nodes (also known as +inductive learning). In addition, GraphSAGE proposes to sample a fixed-size of neighbours around each node in their +aggregation scheme, instead of using all neighbours, which helps to keep the computational cost of each batch fixed. +GIN. Although GCN and GraphSAGE have achieved excellent performances in graph learning tasks, especially +in node classification tasks, they are unable to distinguish some simple graph structures due to their limits in the +neighbourhood aggregation scheme. Graph Isomorphism Network architecture (GIN) [193] is proposed to overcome +this shortcoming and is proven to be as powerful as the Weisfeiler-Lehman graph isomorphism test [185]. Specifically, +in order to achieve the same discriminative power as the Weisfeiler-Lehman test, the representation of node 𝑣 at layer 𝑙 +should be as: +ℎ(𝑙) +𝑣 += 𝜙 (𝑙) � +ℎ(𝑙−1) +𝑣 +, 𝑓 (𝑙−1) �� +ℎ(𝑙−1) +𝑢 +: 𝑢 ∈ N (𝑣) +��� +, +(52) +where 𝑓 (𝑙−1) is a function operating on multisets and 𝜙 (𝑙) is an injective function. The choice of multiset on neighbour- +hood information aggregation, instead of mean pooling in GCN or max pooling in GraphSAGE, enables it to better +preserve neighbourhood structural information. The above representation is then proven to be equivalent to: +ℎ(𝑙) +𝑣 += MLP(𝑙) �� +� +� +1 + 𝜖 (𝑙)� +· ℎ(𝑙−1) +𝑣 ++ +∑︁ +𝑢∈N(𝑣) +ℎ(𝑙−1) +𝑢 +�� +� +, +(53) +where 𝜖 (𝑙) is a scalar representing the importance of the focal node, and MLP is used to model the composition of the +function 𝑓 and 𝜙. +GAT. Although in the aggregation scheme of the GCN, nodes from the same neighbourhood are assigned different +weights by introducing the normalisation term (𝑐𝑣𝑢 in Equation 50), the approach lacks the flexibility of introducing +other weight mechanisms. To overcome this shortcoming, GAT proposes to use a masked self-attentional layer on +graphs. “Masked” means that only 1-hop neighbours, rather than all other nodes, of a given node, are included in the +attention scheme. Specifically, the attention coefficient of an edge 𝑒𝑣𝑢 at layer 𝑙 is given by: +𝛼 (𝑙) +𝑣𝑢 = +exp +� +𝑒 (𝑙) +𝑣𝑢 +� +� +𝑤∈N(𝑣) exp +� +𝑒 (𝑙) +𝑣𝑤 +� ,𝑒 (𝑙) +𝑣𝑢 = LeakyReLU +� +�𝑎(𝑙) [W(𝑙)ℎ(𝑙) +𝑣 ∥W(𝑙)ℎ(𝑙) +𝑢 ] +� +, +(54) +where 𝑎(𝑙) is a shared feedforward neural network parameterised by a weight vector �𝑎, and W(𝑙) is a shared linear +transformation of input or hidden features. Then the representation of node 𝑣 at layer (𝑙 + 1), is obtained through +applying the attention coefficients on 𝑣’s neighbour nodes: +ℎ(𝑙+1) +𝑣 += 𝜎 �� +� +∑︁ +𝑗 ∈N(𝑖) +𝛼 (𝑙) +𝑣𝑢 W(𝑙)ℎ(𝑙) +𝑢 �� +� +(55) +FastGCN. One issue of the GCN’s neighbourhood aggregation scheme is the quick neighbourhood expansion across +layers, which largely limits its scalability in large and dense graphs. To address this problem, FastGCN [43] proposes to +sample a fixed number of nodes at each layer while applying neighbourhood aggregation, so the number of involved +28 + +nodes is up-bounded by the sample size. Concretely, the representation of nodes at layer 𝑙 is given by: +𝐻 (𝑙+1) (𝑣, :) = 𝜎 �� +� +𝑛 +𝑠 +𝑠∑︁ +𝑗=1 +ˆ𝐴 +� +𝑣,𝑢 (𝑙) +𝑗 +� +𝐻 (𝑙) � +𝑢 (𝑙) +𝑗 , : +� +𝑊 (𝑙)�� +� +, +(56) +where 𝑠 is the sample size and 𝑛 is the total number of nodes in a graph. Compared to the node-wise sampling strategy +proposed by GraphSAGE, this layer-wise sampling method further improves the computational efficiency of the model. +For example, in a 2-layer setup, when 10 nodes are sampled from a node’s neighbourhood, there will be a total of +102 = 100 nodes involved. In contrast, when 10 nodes are sampled at each layer, the total number of involved nodes is +at most 10 ∗ 2 = 20. +GraphSNN. A common feature of the above-mentioned approaches is that each node gathers information from +its neighbours, that is to say treating the neighbourhood as a 1-hop subtree. A recent work argues that this scheme +ignores the rich structure information among the neighbour nodes, and therefore proposes a model named GraphSNN +to treat the neighbourhood as a 1-hop subgraph by including the connections among neighbours [187]. Concretely, +the work first defines “structure coefficients” for each node and its neighbours and generate a weighted adjacency +matrix 𝐴𝑣𝑢 = 𝑤 (𝑆𝑣,𝑆𝑣𝑢), where 𝑆𝑣 is 1-hop neighbourhood subgraph of node 𝑣, and 𝑆𝑣𝑢 is overlap subgraphs of node 𝑣 +and 𝑢. 𝑤 is a function on 𝑆𝑣 and 𝑆𝑣𝑢 exhibiting properties of local closeness and local denseness, which is designed as +|𝐸𝑣𝑢 | +|𝑉𝑣𝑢 |·|𝑉𝑣𝑢−1| |𝑉𝑣𝑢|𝜆 in the paper. 𝜆 is a positive value chosen by users. Then, the representation of node 𝑣 at layer 𝑙 is +generated by: +ℎ(𝑙) +𝑣 += MLP(𝑙) �� +� +𝛾 (𝑙−1) �� +� +∑︁ +𝑢∈N(𝑣) +˜𝐴𝑣𝑢 + 1�� +� +ℎ(𝑙−1) +𝑣 ++ +∑︁ +𝑢∈N(𝑣) +� +˜𝐴𝑣𝑢 + 1 +� +ℎ(𝑙−1) +𝑢 +�� +� +, +(57) +where 𝛾 (𝑙−1) is a learnable scalar parameter, and ˜𝐴𝑣𝑢 is the normalised weighted adjacency matrix. The part before +ℎ(𝑙−1) +𝑣 +signifies the focal node’s self-importance while the part � +𝑢∈N(𝑣) +� +˜𝐴𝑣𝑢 + 1 +� +before ℎ(𝑙−1) +𝑢 +is to apply different +weights on different neighbours based on the overlap subgraph between the focal node and the neighbour node. From +this perspective, GraphSNN is also an attention-like scheme that takes the 1-hop subgraph structure into account. +DGCNN. In order to apply a GCN on graph-level learning tasks, Deep Graph Convolutional Neural Network (DGCNN) +proposes to sort and pool the nodes’ representations from multiple graph convolutional layers, then pass them to a +traditional CNN architecture, i.e., a one-dimensional convolutional layer followed by dense layers before the final +softmax output layer [210]. As the GCN can be viewed as “a differentiable and parameterised generalisation of the +1-dim Weisfeiler-Lehman algorithm” [105], each node’s representation can be viewed as a “continuous colour” at that +layer. The order of nodes in DGCNN is thus calculated according to the nodes’ representations, i.e., nodes’ colours, +at the graph convolutional layers (first comparing the representations at the last layer, then the representations at +the second-to-last layer when some nodes have the same representation, and so on). Next, in order to fit into the +following CNN architecture, the sorted nodes’ representation needs to be truncated or extended, which is done by +deleting excessive rows or adding zero rows. This bridge layer between GCN and CNN is also known as SortPooling. +4.1.2 +k-hop neighbourhood approaches. A natural idea to improve the performance of the GCN is to expand its message +aggregation scope at each layer. This leads us to the second subcategory, i.e., k-hop neighbourhood approaches. +MixHop. The layer-wise message passing scope of the vanilla GCN is limited to 1-hop neighbours and therefore lacks +the ability to mix latent information from neighbours at different distances. MixHop is proposed to address this issue +through a higher-order message passing scheme that aggregates information from further neighbours [3]. Concretely, +the convolutional layer is defined as: +29 + +𝐻 (𝑖+1) = 𝜎 +�����𝑗 ∈𝐾 +� +𝐴𝑗𝐻 (𝑖)𝑊 (𝑖) +𝑗 +� +, +(58) +where 𝐾 is a set of integers representing the scope, and ∥ denotes column-wise concatenation. When 𝐾 = 1, the +operation degrades to the vanilla GCN. The paper also proves theoretically that the vanilla GCN cannot recover a 2-hop +delta operator and thus cannot represent a general layer-wise neighbourhood mixing. In contrast, MixHop is able to +learn a general mixing of information from neighbours at various distances. Their experiments on a synthetic dataset +show that MixHop performs significantly better than several baselines on graphs of low levels of homophily. +k-hop GNN. A simple example of the limitation of the 1-hop neighbourhood aggregation is that it cannot distinguish +regular graphs of the same size and degree. In order to improve the expressivity of the vanilla GCN, k-hop GCN also +proposes to take k-hop neighbours into consideration in the layer-wise aggregation scheme [155]. The general model is +presented as: +𝑎(𝑙) +𝑣 += AGGREGATE(𝑙) �� +ℎ(𝑙−1) +𝑢 +| 𝑢 ∈ N𝑘 (𝑣) +�� +, +ℎ(𝑙) +𝑣 += MERGE(𝑙) � +ℎ(𝑙−1) +𝑣 +,𝑎(𝑙) +𝑣 +� +, +(59) +where N𝑘 (𝑣) denotes the k-hop neighbourhood of node 𝑣. Specifically, it adopts an outside-to-inside updating scheme in +the aggregation part: gradually updating neighbouring nodes from the furthest to the immediate ones. Each neighbour +node 𝑢 at a distance 𝑑 from the focal node 𝑣 goes through two update functions successively: +𝑥𝑢 = UPDATE(𝑙) +𝑑,𝑎𝑐𝑟𝑜𝑠𝑠 (𝑢, N1(𝑢) ∩ 𝑅𝑑+1(𝑣)), +𝑥𝑢 = UPDATE(𝑙) +𝑑,𝑤𝑖𝑡ℎ𝑖𝑛(𝑢, N1(𝑢) ∩ 𝑅𝑑 (𝑣)), +(60) +where 𝑅𝑑+1(𝑣) or 𝑅𝑑 (𝑣) denote the set of nodes that are at a distance 𝑑 + 1 or 𝑑 from node 𝑣. The first function learns +representation from node u’s neighbours that are (𝑑 + 1)-hop away from node 𝑣; and the second function learns +from node u’s neighbours that are 𝑑-hop away from node 𝑣. The update functions are defined as: UPDATE(𝑢,𝑆) = +MLP1 (MLP2 (𝑥𝑢) + � +𝑤∈𝑆 MLP3 (𝑥𝑤)). Finally, the representation of a node𝑣 is calculated as:ℎ(𝑙) +𝑣 += UPDATE(𝑙) +0,𝑎𝑐𝑟𝑜𝑠𝑠 (𝑣, N1(𝑣)). +Although this model can capture structural information from the k-hop neighbourhood at a single layer, it requires up +to 2𝑘 update functions and the aggregation scheme is much more complicated and computationally expensive. +Adapt. Similar to the FastGCN, Adaptive Sampling GCN (abbreviated as Adapt) adopts the layer-wise sampling +strategy in order to accelerate the training of the GCN [89]. Lower layer sampling is conditioned on the higher layer. +Compared to node-wise sampling, layer-wise sampling not only has a fixed number of nodes at each layer but also +preserves the connections between lower-layer neighbours and higher-layer parent nodes. Furthermore, the approach +proposes to aggregate information from distant nodes via skip connections, i.e., connecting layer 𝑙 + 1 with layer 𝑙 − 1. +Specifically, the skip-connection representation of node 𝑣 at layer (𝑙 + 1) is formulated as: +ℎ(𝑙+1) +𝑣skip = +∑︁ +𝑠 ∈V (𝑙−1) +ˆ𝑎𝑠𝑘𝑖𝑝 (𝑣,𝑠) ℎ(𝑙−1) +𝑠 +𝑊 (𝑙−1) +skip , +(61) +where 𝑠 denotes sampled nodes at layer (𝑙 − 1), ˆ𝑎𝑠𝑘𝑖𝑝 (𝑣,𝑠) = � +𝑢∈V (𝑙) ˆ𝑎 (𝑣,𝑢) ˆ𝑎 (𝑢,𝑠), and 𝑊 (𝑙−1) +skip += 𝑊 (𝑙−1)𝑊 (𝑙). The +part of skip-connection is then added to the classic GCN layer before a nonlinear transformation. Therefore, the overall +representation of node 𝑣 is: +ℎ(𝑙+1) +𝑣 += 𝜎 �� +� +∑︁ +𝑢∈V (𝑙) +ˆ𝑎 (𝑣,𝑢) ℎ(𝑙) +𝑢 𝑊 (𝑙) + ℎ(𝑙+1) +𝑣skip +�� +� +. +(62) +30 + +That is to say, each node gathers information from both its 1-hop neighbours and 2-hop neighbours. Their experiments +on the Cora dataset show that although skip connection does not lead to significant improvement in accuracy, it helps +to speed up the convergence. +DGP. Aiming to improve the performance of zero-shot learning tasks on knowledge graphs (directed graphs), Dense +Graph Propagation (DGP) proposes to adopt a two-phase propagation scheme on two separate connectivity patterns +(one having nodes connected to their ancestors and the other having nodes connected to their descendants) [101]. +Furthermore, at each phase, DGP introduces a weighting scheme to include the contributions from distant nodes. +Concretely, the overall representation is formulated as: +𝐻 = 𝜎 +� 𝐾 +∑︁ +𝑘=0 +𝛼𝑎 +𝑘 ˆ𝐴𝑎 +𝑘𝜎 +� 𝐾 +∑︁ +𝑘=0 +𝛼𝑑 +𝑘 ˆ𝐴𝑑 +𝑘𝑋𝑊𝑑 +� +𝑊𝑎 +� +, +(63) +where ˆ𝐴𝑎 +𝑘 and ˆ𝐴𝑑 +𝑘 denote the normalised adjacency matrices containing k-hop connections to ancestors and to descen- +dants, respectively. 𝛼𝑎 +𝑘 and 𝛼𝑑 +𝑘 are learnable weights denoting contributions from nodes that are k-hop away from a +given node. We see from the above equation that DGP can be viewed as consisting of two convolutional layers where +the inner layer aggregates information from 1 to k-hop out-neighbours, and the outer layer aggregates information +from 1 to k-hop in-neighbours. +DGCN. Directed Graph Convolutional Networks (DGCN) is another attempt to extend the GCN to directed graphs +[175]. It proposes to expand the receptive field of convolutional operation by considering the first- and second-order +proximities. Specifically, they first define the notions of second-order in-degree proximity matrix and second-order +out-degree proximity matrix as: +𝐴𝑆in (𝑢, 𝑣) = +∑︁ +𝑤 +𝐴𝑤,𝑢𝐴𝑤,𝑣 +� +𝑥 𝐴𝑤,𝑥 +, +𝐴𝑆out (𝑢, 𝑣) = +∑︁ +𝑤 +𝐴𝑢,𝑤𝐴𝑣,𝑤 +� +𝑥 𝐴𝑥,𝑤 +. +(64) +The idea is that if two nodes (a given node and its 2-hop neighbour) share many common in-neighbours (or out- +neighbours), they have higher second-order in-degree (or out-degree) proximity. When capturing first-order proximity, +they choose to make the adjacency matrix symmetric by ignoring link directions. Then the overall representation at +layer 𝑙 is formulated as: +H(𝑙) = 𝐶𝑜𝑛𝑐𝑎𝑐𝑡 +� +𝜎 +� +ˆAFH(𝑙−1)Θ(𝑙−1)� +, 𝜎 +� +ˆASinH(𝑙−1)Θ(𝑙−1)� +, 𝜎 +� +ˆASout H(𝑙−1)Θ(𝑙−1)�� +, +(65) +where ˆAF, ˆASin and ˆASout are normalised first-order proximity matrix, normalised second-order in-degree proximity +matrix and normalised second-order out-degree proximity matrix, respectively. Notice that although DGP also considers +2-hop neighbours in directed graphs when 𝑘 equals 2, DGCN and DGP have different definitions of directed 2-hops. +SGC. In order to improve the efficiency and scalability of GCN, Simple Graph Convolution (SGC) proposes to +remove the nonlinear transformation between layers [188]. They argue that the main advantage of GCN lies in its +neighbourhood aggregation scheme, not the nonlinearity between convolutional layers. After removing all nonlinear +activations, the final output of the SGC model is represented as follows: +ˆ𝑌 = softmax +� +ˆ𝐴 . . . ˆ𝐴 ˆ𝐴𝐻 (0)𝑊 (1)𝑊 (2) . . .𝑊 (𝐿)� += softmax +� +ˆ𝐴𝐿𝐻 (0)𝑊 +� +, +(66) +where ˆ𝐴 is the normalised self-connection added adjacency matrix, 𝐻 (0) is the input node feature matrix, and 𝑊 is a +single weight matrix. This output representation thus only requires learning a single weight matrix, and the term ˆ𝐴𝐿𝐻 (0) +can be computed directly. Note that the meaning of ˆ𝐴𝐿𝐻 (0) is the sum of features from k-hop neighbouring nodes. +31 + +Therefore, the SGC model is actually equivalent to a single convolutional layer where nodes aggregate information +from their k-hop neighbours. +PATCHY-SAN [154]. Traditional image-based convolutional networks can be viewed as traversing a node sequence, +i.e., a receptive field moving from left to right and from top to bottom. In order to employ the convolutional architecture +to graphs where spatial order is missing, PATCHY-SAN proposes to first impose an order on nodes according to a +certain ranking algorithm, then construct receptive fields from a fixed number of neighbour nodes for each node in +a preselected node sequence. Note here the neighbour nodes are selected by performing a breadth-first search, so it +can go beyond 1-hop neighbours. The receptive fields, after being normalised, will then be fed into a one-dimensional +convolutional layer and other dense layers. Comparing this CNN-like approach with the GCN, we see that it requires +an extra procedure to rank nodes, and there are more hyper parameters to tune, such as the length of node sequence, +the stride and the number of neighbour nodes in the receptive field. +4.1.3 +Random-walk neighbourhood approaches. Instead of defining neighbourhood based on the distance to the focal +node, some GCN approaches adopt a random-walk based neighbourhood, which might enable them to capture random +processes on certain types of graphs. +PinSage. In order to apply GCN to web-scale recommender systems, PinSage proposes to construct neighbourhoods +via random walks, also referred to as importance-based neighbourhoods [198]. The convolutional operation is similar +to that of GraphSAGE: +h𝑙 +N𝑟 (𝑣) ← 𝛾 +�� +𝜎 +� +Qlℎ𝑢 +� +| 𝑢 ∈ N𝑟 (𝑣) +� +, 𝜶 +� +, +h𝑙 +𝑣 ← 𝜎 +� +W𝑙 · CONCAT +� +h𝑙−1 +𝑣 +, h𝑙 +N𝑟 (𝑣) +�� +, +(67) +where N𝑟 (𝑣) is a random-walk neighbourhood, 𝛾 is an aggregation function, ℎ𝑢 is a set of embeddings of nodes in +the neighbourhood, 𝜶 is a set of weights on nodes in the neighbourhood, Ql and Wl are learnable model parameters. +Specifically, N𝑟 (𝑣) comes from simulating a random walk starting from the focal node and calculating the 𝐿1-normalised +count of visited nodes, then the top𝑇 nodes with the highest counts are selected as the neighbourhood in the layer-wise +message passing. There are two benefits in this neighbourhood definition: first, the number of nodes involved in the +aggregation is fixed, so the cost of the algorithm is predictable; second, the normalised visit counts can be directly used +as weights to represent the importance of each node in the neighbourhood. PinSage also introduces some strategies to +improve the model’s scalability, such as the producer-consumer minibatch construction and a MapReduce pipeline. +Notice that PinSage is originally designed for recommender systems which are bipartite networks. +GraLSP. Based on the idea that anonymous walks can capture structures through reconstructing local subgraphs +[139], GraLSP proposes to adopt random anonymous walks into the neighbourhood aggregation scheme [100]. It also +combines some other techniques to enhance the model performance, such as adaptive receptive radius, attention and +channel-wise amplification. Specifically, the convolutional layer is formulated as: +a(𝑘) +𝑣 += MEANwk∈W (𝑖),𝑝 ∈[1,𝑟wk] +� +𝜆(𝑘) +𝑣,wk +� +q(𝑘) +𝑣,wk ⊙ h(𝑘−1) +wk𝑝 +�� +, +h(𝑘) +𝑣 += ReLU +� +W(𝑘)h(𝑘−1) +𝑣 ++ U(𝑘)a(𝑘) +𝑣 +� +, +(68) +where 𝑤𝑘 denotes a walk from the set of random walk sequence W (𝑖), 𝑤𝑘𝑝 is the 𝑝-th node in walk 𝑤𝑘. 𝑟𝑤𝑘, 𝜆𝑣,wk +and q𝑣,wk denote receptive radius, attention coefficient and amplification coefficient, respectively. Adaptive radius is +introduced in order to regulate the scope of walks so that nodes that are too far away in the constructed subgraph are +excluded while nodes in clustered subgraphs are included. Concretely, it is defined as 𝑟𝑤𝑘 = +� +2𝑙 +𝐶𝑤𝑘 +� +, where 𝑙 is walk +length and 𝐶𝑤𝑘 is the number of distinct nodes visited by the walk. Finally, the attention coefficient is introduced to +32 + +assign different importance to visited nodes, and the channel-wise amplification is used to model the selection of node +features. +4.1.4 +Subgraph neighbourhood approaches. In addition to the fixed-hop neighbourhood and random-walk neighbour- +hood definition in layer-wise message aggregation, some approaches view the neighbourhood as k-node tuples or +subgraphs. +k-GNN. As the ability of the GCN to distinguish nonisomorphic graphs is equivalent to that of the 1-dimensional +Weisfeiler-Leman algorithm (1-WL), a k-GNN is proposed to achieve a higher expressivity as that of a k-WL [146]. +Different from the vanilla GCN where each node gathers information from a defined neighbourhood, the k-GNN works on +the level of node tuple. Accordingly, the neighbourhood of a k-tuple is defined as other k-tuples containing one node that +is not in the focal k-tuple. Specifically, the neighbourhood of k-tuple 𝑠 is defined as: 𝑁𝑘 (𝑠) = +� +𝑡 ∈ [𝑉 ]𝑘 ||𝑠 ∩ 𝑡 |= 𝑘 − 1 +� +, +where [𝑉 ]𝑘 is a set of all k-tuples in a given graph. The convolutional operation at layer 𝑙 is then defined as: +ℎ(𝑙) (𝑠) = 𝜎 �� +� +ℎ(𝑙−1) (𝑠) ·𝑊 (𝑙) +1 ++ +∑︁ +𝑢∈𝑁𝑘 (𝑠) +ℎ(𝑙−1) (𝑢) ·𝑊 (𝑙) +2 +�� +� +. +(69) +At the beginning, ℎ(0) (𝑠) is set as ℎ𝑖𝑠𝑜 (𝑠), which is a one-hot encoding of the isomorphism type of induced subgraph of +𝑠. To improve the model’s scalability and avoid overfitting, a more restricted k-tuple neighbourhood 𝐿𝑁𝑘 (𝑠), named +local neighbourhood, is defined as the tuples in 𝑁𝑘 (𝑠) also satisfying (𝑢, 𝑣) ∈ 𝐸 for 𝑢 ∈ 𝑠\𝑡 and 𝑣 ∈ 𝑠\𝑡. In other words, +the non-overlapped nodes in a given k-tuple and a neighbouring k-tuple needs to be connected so that the neighbouring +k-tuple is a local neighbourhood. Notice that as k-GNN is defined on the k-tuple level, it is unsuitable for node-level +tasks. +GRAPE. In order to improve GCN’s ability to discriminate graph isomorphism, GRAPE proposes to consider specific +subgraph patterns in its layer-wise neighbourhood aggregation [192]. First, nodes of a given subgraph pattern are +grouped into different sets according to their egocentric automorphic equivalences, abbreviated as the Ego-AE set. +For example, in a triangle subgraph, the focal node is in one set, and the other two nodes are in another set. Then, +different weights are learned for each Ego-AE set. Concretely, node 𝑣’s Ego-AE sets in a given subgraph 𝑆 are denoted +as: +� +AE𝑆,1 (𝑣) , . . . , AE𝑆,𝑖 (𝑣) , . . . , AE𝑆,𝑚 (𝑣) +�, where 𝑚 is the total number of AE-set. The convolutional operation +for subgraph 𝑆 is then formulated as: +ℎ𝑙 +𝑆 (𝑣) = MLP �� +� +∑︁ +𝑖 +𝛽𝑆,𝑖 · +∑︁ +𝑢∈AE𝑆,𝑖 (𝑣) +ℎ𝑙−1 +𝑆 +(𝑢)�� +� +, +(70) +where 𝛽𝑆,𝑖 are learnable weights representing the importance of each set AE𝑆,𝑖. Note that the focal node 𝑣 also belongs +to an Ego-AE set. The final embedding of node 𝑣 at layer 𝑙 is then achieved through combining embeddings from a +set of different subgraph patterns: ℎ𝑙 (𝑣) = � +𝑆 ∈Ω 𝛼𝑙 +𝑆 · ℎ𝑙 +𝑆 (𝑣), where Ω denotes the set of subgraph patterns, and 𝛼𝑙 +𝑆 is +learnable weight on a given subgraph pattern. This way GRAPE is able to differentiate neighbouring nodes according to +their structural roles captured by Ego-AE. Certainly, the approach involves an extra step of choosing subgraph patterns. +33 + +Approach +Layer-wise aggregation scope +Aggregator +Task +Batch size +Type of graph +GCN +1-hop neighbourhood +Sum/Mean +Node & Graph level +full-batch +General +GraphSAGE +1-hop neighbourhood with sampling +Flexible choice +Node & Graph level +mini-batch +General +GIN +1-hop neighbourhood +Multiset +Node & Graph level +mini-batch +General +GAT +1-hop neighbourhood with attention scheme +Weighted sum/mean +Node & Graph level +mini-batch +General +FastGCN +1-hop neighbourhood with +layer-wise sampling +Sum/Mean +Node & Graph level +mini-batch +General +GraphSNN +1-hop neighbourhood with +structural coefficients +Sum/Mean +Node & Graph level +mini-batch +General +DGCNN +1-hop neighbourhood +Sum/Mean +Graph level +full-batch +Directed +MixHop +k-hop neighbourhood +Sum/Mean +Node & Graph level +full-batch +General +k-hop GNN +k-hop neighbourhood +Outside-to-insides scheme +Node & Graph level +mini-batch +General +Adapt +2-hop neighbourhood +Sum/Mean +Node & Graph level +mini-batch +General +DGP +k-hop neighbourhood +Sum/Mean +Node & Graph level +full-batch +Directed +DGCN +2-hop neighbourhood +Sum/Mean +Node & Graph level +full-batch +Directed +SGC +k-hop neighbourhood +Sum/Mean +Node & Graph level +mini-batch +General +Patchy-SAN +fixed number of nodes +CNN-like (weighted sum) +Graph-level +mini-batch +General +PinSage +random-walk neighbourhood +Weighted sum/mean +Node & Graph level +mini-batch +Bipartite +GraLSP +random-walk neighbourhood +Sum/Mean +Node & Graph level +mini-batch +General +k-GNN +k-tuple neighbourhood +Sum +Graph-level +mini-batch +General +GRAPE +subgraph neighbourhood +Weighted sum +Node & Graph level +mini-batch +General +Table 3. Summary of approaches in the first category. +34 + +Message Content +(What message is +gathered and passed on) +Count of subgraphs + X +GSN +ℱ-MPNN +SMP +Distance encoding + X +Other information + X +rGIN +ID-GNN +P-GNN +DE-GNN +Fig. 11. Taxonomy from the message content perspective. +4.2 +Message Content +The superior performance of the GCN lies in its ingenious combination of node attributes and a graph structure +with node attributes used as initial representations and then subsequently propagated on the graph through certain +convolutional operations. In contrast, some learning approaches only exploit structural information, such as the matrix +decomposition based methods [36, 158] and the random walk based methods [73, 161]. In situations where no node +attributes are provided, a simple structural metric like the node degree is often used as initial representations in the +GCNs [78]. Another group of approaches further propose to improve the GCN’s distinguishability through injecting +more complicated structural features into the node representations, such as the count of graphlets, distance-based +information, etc. The taxonomy and representative approaches are given in Figure 11. +4.2.1 +Count of subgraphs. The number of certain subgraphs or substructures is often used as a node feature in +traditional network studies [140]. Some approaches thus propose to include this type of structural information as part +of a node representation in the GCN’s message passing scheme. +GSN. Graph Substructure Network (GSN) proposes to capture structural features by counting the appearance of +particular graphlet orbits and include them as part of node features in the convolutional operation [27]. Specifically, +node 𝑣’s representation at layer 𝑙 is defined as: +h𝑙+1(𝑣) = MLP1 �� +� +ℎ𝑙 (𝑣), +∑︁ +𝑢∈N(𝑣) +𝑀𝐿𝑃2 +�h𝑡 (𝑣), h𝑡 (𝑢), x𝑉 (𝑣), x𝑉 (𝑢), e(𝑢, 𝑣)��� +� +, +(71) +where x𝑉 (𝑣) and x𝑉 (𝑢) are structure features of nodes 𝑣 and 𝑢, respectively. e(𝑢, 𝑣) is an edge feature if provided. The +structural feature is a vector containing the counts of node orbits. For example, if subgraphs 2-path and 3-clique are con- +sidered (𝐺1 and𝐺2 in Figure 4), the counts of three node orbits will be included in the vector. GSN further introduces a ver- +sion based on edge orbits, which is formulated as: h𝑙+1(𝑣) = MLP1 +� +ℎ𝑙 (𝑣), � +𝑢∈N(𝑣) 𝑀𝐿𝑃2 +�h𝑡 (𝑣), h𝑡 (𝑢), x𝐸 (𝑢, 𝑣), e(𝑢, 𝑣)�� +, +where e(𝑢, 𝑣) denotes the edge structural feature, i.e., a vector containing the count of edge orbits. The GSN has been +proven to be strictly more powerful than the 1-WL test when the chosen subgraphs are not star graphs. Certainly, the +choice of subgraphs is the core of this approach, and a larger subgraph will lead to higher computational complexity in +a preprocessing step. +F -MPNN. A local graph parameter enabled GNN (F -MPNN) also proposes to include a subgraph count into the GCN +[14]. F = {𝑃𝑟 +1, ..., 𝑃𝑟 +𝑘} is a set of pre-selected subgraph patterns with 𝑟 referring to a node. The “homomorphism count” +of each pattern 𝑃𝑟 +𝑖 for node 𝑣 in the original graph 𝐺 is denoted as ℎ𝑜𝑚(𝑃𝑟 +𝑖 ,𝐺𝑣), which is actually equivalent to the +count of a given node orbit (see Section 2.2). Then a structural feature vector (ℎ𝑜𝑚(𝑃𝑟 +1,𝐺𝑣), ...,ℎ𝑜𝑚(𝑃𝑟 +𝑘,𝐺𝑣)) is added to +35 + +the one-hot encoding of node 𝑣’s label, serving as 𝑣’s initial feature vector. Concretely, the framework is formulated as: +h(0) +𝑣 +:= +� +𝑥𝑣, hom �𝑃𝑟 +1,𝐺𝑣� , . . . , hom +� +𝑃𝑘 +ℓ ,𝐺𝑣�� +, +h(𝑙) +𝑣 +:= MERGE +� +x(𝑙−1) +𝑣 +, AGGREGATE +��� +x(𝑙−1) +𝑢 +| 𝑢 ∈ 𝑁 (𝑣) +���� +, +(72) +where 𝑥𝑣 is the one-hot encoding of node 𝑣’s label, MERGE and AGGREGATE are two MLPs. Since this structural +feature is only applied to enhance the initial feature of nodes, it can be used as an add-on to any GCN architecture. +Similar to the GSN, the choice of subgraph patterns is the core of F -MPNN. Cycles of length smaller than 10 and cliques +of size smaller than 5 are used as subgraph patterns in the experiment. +ID-GNN. Identity-aware GNN (ID-GNN) proposes to improve the expressivity of the GCN through distinguishing +the root node of the extracted computation graphs from other nodes in its message passing scheme [200]. It contains +two major steps: the first step, named inductive identity colouring, is to uniquely colour the root node in its k-hop +ego network; then in the second step, a heterogeneous message passing is applied to all the extracted ego networks. +Specifically, the representation of any node 𝑣 in an extracted computation graph 𝐺𝑟 (rooted at node 𝑟) is formulated as: +m(𝑙) +𝑢 += MSG(𝑙) +1[𝑢=𝑟 ] +� +h(𝑙−1) +𝑢 +� +, +h(𝑙) +𝑣 += AGG(𝑙) �� +m(𝑙) +𝑢 ,𝑢 ∈ N (𝑣) +� +, h(𝑙−1) +𝑣 +� +. +(73) +MSG(𝑙) +1[𝑢=𝑟 ] (·) means that MSG(𝑙) +1 (·) is applied to the root node while MSG(𝑙) +0 (·) is applied to other nodes. In this way, +the representation of the root node is different from that of other nodes and will help distinguish other nodes when +propagated to later layers. The approach is inductive since the colouring is based on the extracted computation graphs +instead of the original graph. Further, in order to avoid the overhead of extracting ego-networks, ID-GNN-Fast proposes +to use the count of cycles as an augmented node feature. Therefore the input node feature is built from concatenating +the original node feature and the augmented node feature. +4.2.2 +Distance information. Distance measures such as shortest paths between nodes are widely used in traditional +network studies [28]. Naturally, some approaches propose to enhance the performance of the GCN through including +distance information in their message passing scheme or as an additional initial node feature. +P-GNN. Position-aware graph neural network (P-GNN) proposes to let each node aggregate information from +several randomly chosen subsets of nodes, instead of its own 1-hop neighbours [201]. As every node shares the same +neighbourhood in P-GNN, distance information is included to indicate the relative position of each node to those +subsets. Specifically, given 𝑘 randomly sampled subsets, 𝑆𝑖 denoting the 𝑖th subset, the representation of node 𝑣 at layer +𝑙 is formulated as: +h𝑙 +𝑣 = AGG(𝑙) � +M𝑙−1 +𝑖 +, ∀𝑖 ∈ [1,𝑘] +� +, +M𝑙−1 +𝑖 += {𝐹 (𝑑𝑢𝑣,ℎ𝑙−1 +𝑢 ,ℎ𝑙−1 +𝑣 +), ∀𝑢 ∈ 𝑆𝑖}. +(74) +𝐹 is a message computation function accounting for both distance information and feature information of a pair of +nodes. The output at the last layer is constructed with M𝑖 being the 𝑖th embedding dimension, thus making the final +representation “position aware”. Note that the subsets are resampled at each convolution layer, so that each node can +aggregate information from different sets of nodes at each layer. +DE-GNN. Distance-Encoding GNN (DE-GNN) also proposes to improve the GCN’s expressivity through adding +distance information [117]. Intuitively, for any given node set 𝑆 whose representation is to be learnt, other nodes are +encoded with their distances to each node of 𝑆. Formally, DE of node 𝑢 with regard to the target node set 𝑆 is defined as: +𝜁 (𝑢 | 𝑆) = AGG({𝜁 (𝑢 | 𝑣) | 𝑣 ∈ 𝑆}), +𝜁 (𝑢 | 𝑣) = 𝑓 +�� +(𝑀)𝑢𝑣, +� +𝑀2� +𝑢𝑣 , . . . , +� +𝑀𝑘� +𝑢𝑣 , . . . +�� +, +(75) +36 + +where 𝑀 = 𝐴𝐷−1 is a matrix of landing probabilities through random walks,𝑓 can be a heuristic function or a learnable +neural network. Different distance measures can be captured by the above equation such as the shortest path distance or +the generalised PageRank score. DE is denoted as DE-|𝑆| according to the size of set 𝑆. For example, DE-2 when |𝑆| = 2. +One way of improving the GCN through distance encoding is to use it as an extra node feature:ℎ(0) +𝑣 += 𝐶𝑂𝑁𝐶𝐴𝑇 (𝑥𝑣,𝜁 (𝑣 | +𝑆)). Another approach is to use DE-1 in the layer-wise aggregation: ℎ(𝑙+1) +𝑣 += 𝑓1 +� +ℎ(𝑙) +𝑣 , AGG +�� +(𝑓2 +� +ℎ(𝑙) +𝑢 +� +,𝜁 (𝑢 | 𝑣)) +� +𝑢∈𝑉 +�� +. +Although DE-GNN adopts minibatch training, the distance information needs to be computed for every node set and +for all nodes in its extracted L-hop ego-network, leading to a higher computational cost. Also note that DE-GNN is +flexible for tasks on different levels: DE-1 for node-level tasks, DE-2 for link prediction tasks and DE-3 for triangle +prediction tasks. To highlight that the DE-GNN is suitable for both node and link-level tasks, we use two colours in its +block in Figure 11. +4.2.3 +Other approaches. Apart from the count of certain subgraphs and the distance information, some other infor- +mation such as the “local context matrix” or even random features are also used to enhance the performance of the +GCN. +SMP. In order to improve the GCN’s performances on structure-related tasks, Structural Message Passing (SMP) +proposes to maintain a “local context matrix” at each node, instead of a feature vector as in the vanilla GCN. Specifically, +each node is initialised as a one-hot encoding 𝑴 (0) +𝑖 += 1𝑖 ∈ R𝑛×1, and the additional node feature 𝑥𝑖 of 𝑣𝑖 is appended +at the 𝑖th row: 𝑀 (0) +𝑖 +[𝑖, :] = [1,𝑥𝑖] ∈ R1+𝑐𝑋 . Then the local context matrix of node 𝑣𝑖 at layer 𝑙 is formulated as: +𝑀 (𝑙) +𝑖 += 𝑀𝐿𝑃 (𝑙−1) +1 +� +𝑀 (𝑙−1) +𝑖 +,𝐴𝐺𝐺 +�� +𝑀𝐿𝑃 (𝑙−1) +2 +� +𝑀 (𝑙−1) +𝑖 +, 𝑀 (𝑙−1) +𝑗 +,𝑒𝑖𝑗 +�� +𝑣𝑗 ∈𝑁𝑖 +�� +. +(76) +𝐴𝐺𝐺 is an aggregation function which is by default a normalised sum aggregator: � +𝑣𝑗 ∈𝑁𝑖 𝑀𝐿𝑃 (𝑙−1) +2 +� +𝑴 (𝑙) +𝑖 , 𝑴 (𝑙) +𝑗 , 𝒆𝑖𝑗 +� +/𝑑avg +� +. +In this way, the 𝑗th row in 𝑀𝑖 is the representation node 𝑣𝑖 has of node 𝑣𝑗. Finally, the vector form representation +of node 𝑣𝑖 is obtained through applying an equivariant neural network for sets on the rows of its context matrix. +Although node ordering is needed when constructing the local context matrix, the learned representation is proven to +be order-invariant when 𝑀𝐿𝑃1, 𝑀𝐿𝑃2 and 𝐴𝐺𝐺 are permutation equivariant. SMP is shown to excel in various tasks, +such as the detection of structural properties including distance, eccentricity connectivity, diameter, etc. +rGIN. Apart from various types of explicit structural features that are being added to the GCN, another work (termed +rGIN) proves that the expressive power of the GCN can be enhanced by just adding random features to each node [169]. +Specifically, rGIN first assigns a random value 𝑟𝑣 to each node and concatenates it with the original node feature 𝑥𝑣, +then performs GIN’s convolutional operations: +ℎ(0) +𝑣 += MLP(0) (𝐶𝑂𝑁𝐶𝐴𝑇 (𝑥𝑣,𝑟𝑣)) , +ℎ(𝑙) +𝑣 += MLP(𝑙) �� +� +� +1 + 𝜀 (𝑙)� +ℎ(𝑙−1) +𝑣 ++ +∑︁ +𝑢∈N(𝑣) +ℎ(𝑙−1) +𝑢 +�� +� +. +(77) +With this simple modification on the initial node feature, rGIN is proven to be able to distinguish any local structure +with high probability. The idea of injecting random features into nodes is that the GCN fails to distinguish graphs with +identical node features. For example, a GCN with the node degree as an input feature cannot distinguish a node in a +3-cycle graph from a node in a 6-cycle graph. rGIN is shown to perform well on structure-related tasks such as learning +the existence of triangles, learning the local clustering coefficient, and learning the algorithm for the MDS (Minimum +Dominating Set) problem. +37 + +Learning Scope +(Input graph) +Cluster-GCN +G-Meta +Subgraphs +GraphSAINT +SEAL +Local subgraphs +LGCN +Other graphs +DiffPool +Shadow-GNN +NGNN +AM-GCN/kNN-GCN +GNN-AK +Fig. 12. Taxonomy from the learning scope perspective. +4.3 +Learning scope +The third structural perspective on GCNs is regarding the learning scope or the input graph. Whether it is full-batch +or mini-batch training, most GCNs still have the whole graph as an input, i.e., in an L-layer GCN, each node has the +scope of its L-hop neighbourhood in the original graph. A higher number of layers leads to a neighbourhood explosion +and thus higher computational cost. To address this issue, many approaches limit the scope to subgraphs or localised +subgraphs while some other methods propose to run the GCN on particular types of generated graphs. The taxonomy +and related approaches are given in Figure 12. Again, the block’s colour indicates the task the approach is proposed for: +grey represents a node classification, blue represents a link prediction, and orange represents a network classification. +4.3.1 +Subgraphs. An intuitive idea is to limit the training scope to several selected subgraphs instead of the original +whole graph, so the neighbourhood is restricted within the sphere of subgraphs no matter how many layers are stacked. +GraphSAINT. In order to enhance the scalability of the GCN, GraphSAINT proposes to train a GCN model iteratively +on several sampled subgraphs. Each sampled subgraph 𝐺𝑠 ∈ G is a mini-batch. The representation of node 𝑣 in a +sampled subgraph 𝐺𝑠 is formulated as: +ℎ(𝑙+1) +𝑣 += +∑︁ +𝑢∈𝑁𝑣 |𝐺𝑠 +˜𝐴𝑣,𝑢 +𝛼𝑢,𝑣 +𝑊 (𝑙)ℎ(𝑙) +𝑢 , +(78) +where 𝑢 is 𝑣’s neighbour in 𝐺𝑠, and 𝛼𝑢,𝑣 is a coefficient to offset the biases from the sampler. Specifically, 𝛼𝑢,𝑣 is defined +as the probability of edge (𝑢, 𝑣) being sampled, divided by the probability of node 𝑣 being sampled. Given a set of +pre-sampled subgraphs G, 𝛼𝑢,𝑣 = 𝐶𝑢,𝑣 +𝐶𝑣 , where 𝐶𝑢,𝑣 and 𝐶𝑣 are the number of times edge (𝑢, 𝑣) and node 𝑣 appear in G, +respectively. Finally, the batch loss is calculated as: 𝐿batch = +1 +|G| +� +𝐺𝑠 ∈G +� +𝑣 +𝐿𝑣 +𝜆𝑣 , where 𝐿𝑣 is the loss on node 𝑣 in the +GCN’s output layer, and 𝜆𝑣 is a loss normalisation term computed by the number of node 𝑣 appearing in G divided +by the total number of nodes in the original graph. Different samplers are integrated within the framework, such as +random node sampler, random edge sampler and random walk based sampler. According to the experiment, the random +walk based sampler tends to have the best performance. +Cluster-GCN. Also to address the issue of neighbourhood explosion in large graphs, a Cluster-GCN proposes to first +partition the whole graph into several clusters according to certain clustering algorithms, then run the GCN on those +clusters. Given 𝑐 clusters, the original adjacency matrix 𝐴 is approximated as a list of submatrices 𝐴11, 𝐴11, ..., 𝐴𝑐𝑐 at +diagonal. The representation of nodes at layer 𝑙 in the 𝑡th cluster is thus formulated as: +𝐻 (𝑙) +𝑡 += ˆ𝐴𝑡𝑡𝐻 (𝑙−1) +𝑡 +𝑊 (𝑙−1), +(79) +where ˆ𝐴𝑡𝑡 is the normalised version of 𝐴𝑡𝑡. The loss is then calculated as: 𝐿𝑡 = +1 +|𝑉𝑡 | +� +𝑖 ∈𝑉𝑡 loss +� +𝑦𝑖,ℎ(𝐿) +𝑖 +� +. At each iteration, +the model weights are updated based on the loss of the cluster. This way, no matter how many convolutional layers are +38 + +involved, the neighbourhood scope is restricted to one cluster. In order to offset the bias of clustering algorithms, a +better version of the Cluster-GCN proposes to randomly form a subgraph with several randomly chosen clusters, then +at each iteration, run GCN on one subgraph. Experiments on very large datasets show that the Cluster-GCN is able to +train a deeper GCN without time and space overhead and achieves advanced performance. +LGCN. A learnable graph convolutional network (LGCN) proposes to transform graph data into a grid-like data +structure and apply the traditional convolutional operation on it [68]. As traditional CNN requires a fixed number of +ordered units in the receptive fields, the LGCN proposes to sort features at each dimension and select the k-largest +ones to form a grid structure. The transformed data is then fed into a one-dimensional CNN to generate the final +representation of the focal node. Specifically, the nodes’ representation at layer 𝑙 is formulated as: +𝐻 (𝑙+1) = 𝑐(𝑔(𝐻 (𝑙),𝐴,𝑘)), +(80) +where 𝑘 is a hyper parameter, 𝑔(·) is the function that performs k-largest selection to transfer the original graph data +into grid data, and 𝑐(·) is a one-dimensional CNN. Furthermore, as Cluster-GCN and GraphSAINT, the LGCN also +proposes to train the neural network on subgraphs. Each subgraph is built from randomly selecting a few initial nodes +and then expanding adjacent nodes into it using a breadth-first-search algorithm. At each training iteration, multiple +subgraphs can be included in a mini-batch. The subgraph training strategy is shown to be more time and space efficient, +with only negligible loss in performance. +4.3.2 +Local subgraphs. Another popular idea to address the computational overhead is training the GCN on local +subgraphs. Note that local subgraphs are different from subgraphs in that they are extracted around each node or link. +In contrast, subgraphs, as we have discussed earlier, have in general a wider range, without focusing on a node or a link. +SEAL. SEAL is a GCN based framework specially designed for a link prediction task [209]. Motivated by the fact +that many successful link prediction heuristics, such as the common neighbour, Adamic-Adar and resource allocation, +only involve the 1-hop or 2-hop neighbours around a node pair, SEAL proposes to train a GCN on the local subgraphs +extracted around each target link. Specifically, the local subgraph is the induced graph from each target node pair and +their k-hop neighbours. After having constructed the training data, it further introduces a node labelling procedure to +give the target node pair special weights as well as to distinguish the neighbouring nodes in a given local subgraph. +Specifically, it labels each node in the target pair as “1”, and assigns larger labels to other nodes according to their +distances to the target pair. The assigned labels are then concatenated with other features to construct the feature matrix +of the local subgraph. In the final step, a GCN is trained on the local subgraphs and their label-enhanced feature matrices. +In the experimental implementation, SEAL chooses to use DGCNN, a GCN model designed for graph classification (see +Section 4.1.1), as the default GCN model. Essentially, the link existence problem in the original graph is modelled as a +graph classification problem on the extracted local subgraphs. +G-Meta. Motivated by the idea that local subgraphs may contain transferable knowledge that can be adapted to +unseen tasks, G-Meta proposes to leverage local subgraph information in few-shot graph meta-learning [88]. For the +node classification task, local subgraphs are constructed as induced graphs from each node and its k-hop neighbours; +and when it comes to link prediction, local subgraphs are built as in SEAL. Then a typical GCN is used on these local +subgraphs to generate graph embeddings. At last, a prototypical loss and Model-Agnostic Meta-Learning (MAML) +algorithm are used to update the GCN’s parameters. Specifically, the prototype 𝑡𝑙 of label 𝑙 is calculated through +averaging over subgraph embeddings in the support set: 𝑡𝑙 = +1 +𝑁𝑙 +� +𝑦𝑗=𝑙 h𝑗. Then for each local subgraph 𝑆𝑢 in both +support and query set, a class distribution vector p is calculated as: p𝑙 = +exp(−∥h𝑆𝑢 −t𝑙 ∥) +� +ˆ𝑙 exp(−∥h𝑆𝑢 −tˆ𝑙 ∥) . Finally, the cross-entropy +39 + +loss is formulated as: L(p, y) = � +𝑗 y𝑗 log p𝑗. Experiments on synthetic and real networks show that local subgraphs are +vital for few-shot graph learning. It is worth mentioning that the best performance is yielded when 2-hop neighbours +are included. +Shadow-GNN. From the perspective of decoupling the scope (i.e., a receptive field) and the depth (i.e., a number +of layers) of the GCN, a Shadow-GNN also proposes to adopt local subgraph as an input [204]. Typically on a full +graph, the scope of the GCN increases with the number of layers — an L-layer GCN means an L-hop neighbourhood +scope. As the GCN model is also viewed as a form of Laplacian smoothing that mixes the feature of a node and its +neighbours, when the scope becomes too large, node features may be oversmoothed [118]. To address this problem, the +Shadow-GNN proposes to train the GCN on local subgraphs, so that the scope is bounded by the range of the local +subgraphs, regardless of the number of layers. In this setting, the depth can be larger than the scope. It means that +nodes in the subgraphs may exchange information multiple times, which could lead to better expressivity. Different +subgraph extractors can be selected, such as an L-hop neighbourhood extractor or a random-walk-based extractor. In +actual implementation, the scope is set as a 2- or 3-hop neighbourhood while the depth is deeper (3 or 5 layers). +NGNN. A nested graph neural network (NGNN) proposes to apply the local subgraph training strategy on a graph +classification task [211]. The extracted local subgraphs, termed rooted subgraphs, are also induced subgraphs from each +node and its k-hop neighbours. First, a base GCN is applied on all rooted subgraphs. Taking root node 𝑣 for example, at +layer 𝑙, any node 𝑢 in its k-hop rooted subgraph 𝐺𝑘𝑣 is formulated as: +ℎ(𝑙) +𝑢,𝐺𝑘𝑣 = 𝑈𝑃𝐷𝐴𝑇𝐸(𝑙−1) ��� +� +ℎ(𝑙−1) +𝑢,𝐺𝑘𝑣 , +∑︁ +𝑤∈𝑁 (𝑢 |𝐺𝑘𝑣 ) +𝑀𝑆𝐺 (𝑙−1) +� +ℎ(𝑙−1) +𝑢,𝐺𝑘𝑣 ,ℎ(𝑙−1) +𝑤,𝐺𝑘𝑣 ,𝑒𝑢𝑤 +���� +� +. +(81) +Then, the final representation of root node 𝑣 at layer 𝐿 is set to be equal to its rooted subgraph representation obtained +from applying a subgraph pooling on all nodes in the subgraph: ℎ𝑣 = ℎ𝐺𝑘𝑣 = 𝑃𝑂𝑂𝐿1 +�� +ℎ(𝐿) +𝑢,𝐺𝑘𝑣 | 𝑢 ∈ 𝐺𝑘𝑣 +�� +. With the +same base GCN applied on all nodes’ rooted subgraphs, the representation of each node can be obtained, and the graph +representation can be generated from applying another GCN, termed outer GCN, on those updated node representations. +To make it simple, the outer GCN can be just a graph pooling layer: ℎ𝐺 = 𝑃𝑂𝑂𝐿2(ℎ𝑣 | 𝑣 ∈ 𝐺). The work theoretically +proves that a proper NGNN can discriminate almost all 𝑟-regular graphs where the vanilla GCN cannot. +GNN-AK. GNN-As Kernel (GNN-AK) is another local subgraph based approach for a graph classification problem +[217]. Different from the NGNN which directly uses the rooted subgraph embedding to represent each node, the +GNN-AK proposes to construct node representation from concatenating three types of embedding, i.e., subgraph +embedding, centroid embedding, and context embedding. Centroid embedding is simply the root node representation +in its own subgraph, while context embedding is built from the representation of this node in other nodes’ rooted +subgraphs. It is argued that these two additional embeddings contain information which is not captured in the subgraph +embedding. Formally, the representation of node 𝑣 at layer 𝑙 is: +ℎ(𝑙) +𝑣 += 𝐶𝑂𝑁𝐶𝐴𝑇 (ℎ(𝑙) +𝑣,centroid,ℎ(𝑙) +𝑣,subgraph,ℎ(𝑙) +𝑣,context), +(82) +with ℎ(𝑙) +𝑣,centroid = ℎ𝑣|𝐺𝑘𝑣 , ℎ(𝑙) +𝑣,subgraph = 𝑃𝑂𝑂𝐿1({ℎ𝑖 |𝐺𝑘𝑣 | 𝑖 ∈ N𝑘 (𝑣)}), and ℎ(𝑙) +𝑣,context = 𝑃𝑂𝑂𝐿2({ℎ𝑣|𝐺𝑘 +𝑗 ∀𝑗 s.t. 𝑣 ∈ N𝑘 (𝑗)}). +ℎ𝑣|𝐺𝑘𝑢 denotes the representation of node 𝑣 in node 𝑢’s rooted subgraph. Then, the final graph representation is obtained +from another pooling at the output layer: ℎ𝐺 = 𝑃𝑂𝑂𝐿3({ℎ𝐿𝑣 | 𝑣 ∈ 𝑉 }). It is worth mentioning that a subgraph drop +strategy is further introduced to improve the scalability of GNN-AK so that the number of local subgraphs can be much +smaller than the number of nodes in the original graph. +40 + +4.3.3 +Other types of graphs. Subgraphs or local subgraphs are still part of the original graphs. In the third subcategory, +we see approaches that use differently constructed graphs, such as the coarsened graph and the feature graph. +DiffPool. Analogous to the idea of spatial pooling in a traditional CNN, a DiffPool proposes to learn a graph +representation in a hierarchical manner. Nodes at layer 𝑙 will be collapsed into higher-level cluster nodes at layer 𝑙 + 1 +via a learned assignment matrix, and after stacking several hierarchical layers, the singular node’s embedding at the +final layer is viewed as the representation for the whole graph. Concretely, node embedding matrices 𝑍 (𝑙) are learned +from a GCN (called an embedding GNN), and an assignment matrix 𝑆 (𝑙) is learned from another GCN, called a pooling +GNN: +𝑍 (𝑙) = 𝐺𝑁𝑁 (𝑙) +embed(𝐴(𝑙),𝑋 (𝑙)), +𝑆 (𝑙) = softmax(𝐺𝑁𝑁 (𝑙) +pool(𝐴(𝑙),𝑋 (𝑙))), +(83) +where 𝐴(𝑙) and 𝑋 (𝑙) are the coarsened adjacency matrix and the cluster nodes feature matrix at layer 𝑙, respectively. +The dimension of assignment matrix 𝑆 (𝑙) is 𝑛𝑙 × 𝑛𝑙+1, so that each role is one of the 𝑛𝑙 nodes at layer 𝑙 and each column +is one of the cluster nodes at layer 𝑙 + 1. Then, 𝐴(𝑙+1) and 𝑋 (𝑙+1) which are used as the next layer’s inputs are generated +as: +𝑋 (𝑙+1) = 𝑆 (𝑙)𝑇𝑍 (𝑙), +𝐴(𝑙+1) = 𝑆 (𝑙)𝑇𝐴(𝑙)𝑆 (𝑙). +(84) +The assignment matrix 𝑆 (𝐿−1) at the penultimate layer is set to be a vector of 1’s, so that all nodes will collapse into a +single cluster node at the final layer, and the corresponding node embedding is viewed as the representation for the +original graph. Note that the number of clusters is a predefined hyperparameter, which is usually set as a percentage of +the number of nodes at the previous layer. +AM-GCN. An Adaptive Multi-channel Graph Conventional Network (AM-GCN) proposes to not only run the GCN +on the original (topological) graph, but also on a feature graph constructed from a feature similarity matrix. Specifically, +the similarity matrix is computed using cosine similarity or heat kernel. Then, edges will be added between each node +and 𝑘 other nodes of top similarity scores. The generated feature graph 𝐺𝑓 = (A𝑓 , X) is also called the k-nearest +neighbour (kNN) graph. Therefore, the embeddings on the feature graph are formulated as follows: +H(𝑙) +𝑓 += ReLU +� +ˆA𝑓 H(𝑙−1) +𝑓 +W(𝑙) +𝑓 +� +, +(85) +where ˆA𝑓 is the normalised feature graph adjacency matrix. Another GCN is used to generate node embeddings on +the original graph 𝐺 = (A, X): H(𝑙) +𝑡 += ReLU +� +ˆAH(𝑙−1) +𝑡 +W(𝑙) +𝑡 +� +. Further, in order to capture the correlation between a +topological space and a feature space, a common convolutional module is introduced as: H(𝑙) +𝑐𝑡 = ReLU +� +ˆAH(𝑙−1) +𝑐𝑡 +W(𝑙) +𝑐 +� +; +H(𝑙) +𝑐𝑓 = ReLU +� +ˆA𝑓 H(𝑙−1) +𝑐𝑓 +W(𝑙) +𝑐 +� +. Note that the same weight matrix W(𝑙) +𝑐 +is shared in H(𝑙) +𝑐𝑡 and H(𝑙) +𝑐𝑓 . Under this setting, node +features are propagated not only in a topological space but also in a feature space. The final representation is then obtained +through combining the above four embeddings with an attention scheme: Z = 𝛼𝑡 · H(𝐿) +𝑡 ++ 𝛼𝑓 · H(𝐿) +𝑓 ++ 𝛼𝑐 · ( +H(𝐿) +𝑐𝑡 +H(𝐿) +𝑐𝑓 +2 +), +where 𝛼𝑡, 𝛼𝑓 and 𝛼𝑐 are attention vectors. +4.4 +Discussion +4.4.1 +Differences between layer-wise scope and overall learning scope. Here we emphasize the differences between +layer-wise scope and overall learning scope. Layer-wise message aggregation scope, or a receptive field, is where a +node receives the message from. It can be a 1-hop neighbourhood, k-hop neighbourhood, random-walk neighbourhood, +or subgraph neighbourhood according to our taxonomy. Although the receptive field is usually small, distant nodes +41 + +Neighbourhood definition (representative approach) +Time complexity +Space complexity +1-hop neighbourhood (GCN) +𝑂 �𝐿 �|𝑉 |2𝐶 + |𝑉 |𝐶2�� +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +1-hop neighbourhood (GraphSAGE) +𝑂(𝐿|𝑉 |(𝑠𝐶 + 𝐶2)) +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +h-hop neighbourhood (MixHop) +𝑂 �𝐿 �|𝑉 |2𝐶ℎ + |𝑉 |𝐶2�� +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +h-hop neighbourhood (k-hop GNN) +𝑂(𝐿|𝑉 |(𝑘ℎ𝑚𝑎𝑥𝐶 + 𝐶2)) +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +Random-walk neighbourhood (PinSage) +𝑂(𝐿|𝑉 |(𝑤𝑙 + 𝑣𝑙𝑜𝑔𝑣 + 𝑠𝐶 + 𝐶2)) +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +k-node subgraph neighbourhood (k-GNN) +𝑂(𝐿�|𝑉 | +𝑘 +�(𝑘|𝑉 |𝐶 + 𝐶2)) +𝑂(𝐿�|𝑉 | +𝑘 +�𝐶 + 𝐶2) +Table 4. Time and space complexity from the perspective of a layer-wise message scope. |𝑉 | is the number of nodes in the graph, 𝐶 +are node feature channels (assuming the number of features is fixed for all layers), 𝐿 is the number of convolutional layers, 𝑠 is the +number of sampled nodes, ℎ is the number of hops away from a focal node, 𝑘𝑚𝑎𝑥 is maximum node degree, 𝑤 is the number of +random walks, 𝑙 is the length of a random walk, 𝑣 is the number of visited nodes, and 𝑘 is the number of nodes in a subgraph. +can exchange messages after stacking multiple GCN layers, causing the well-known neighbourhood explosion issue. +Obviously, with a large enough number of layers, a node can exchange information with any other node in the entire +graph. Overall learning scope, in contrast, is determined by the input graph, which can be the entire original graph, +extracted subgraphs or local subgraphs, or coarsened graphs according to our taxonomy. Taking an extracted local +subgraph for example, no matter how large the receptive field is or how many layers are stacked, a node can only +exchange messages with other nodes in the same subgraph. This naturally solves the neighbourhood explosion issue. A +large number of layers on a relatively small subgraph also means that nodes may exchange information multiple times, +which is argued to help the GCN “better absorb and embed information” [204]. +4.4.2 +Time and space complexity analysis. In the discussion about complexity, we focus on how the different definitions +of the neighbourhood in a convolutional layer influence the cost of computation (corresponding to the layer-wise +message scope taxonomy in Figure 10). The time and space complexities of each category are listed in Table 4. +First, according to the propagation rule of the vanilla GCN (Equation 49), which is essentially the multiplication of +three matrices 𝐴 ∈ R|𝑉 |×|𝑉 |, 𝐻 ∈ R|𝑉 |×𝐶, and 𝑊 ∈ R𝐶×𝐶, the time complexity at each layer is 𝑂 �|𝑉 |2𝐶 + |𝑉 |𝐶2�, and +thus the overall complexity is 𝑂 �𝐿 �|𝑉 |2𝐶 + |𝑉 |𝐶2��. Certainly, when |𝑉 | >> 𝐶, and when the sparsity of adjacency +matrix is exploited (for instance through the compressed sparse row format), its time complexity is sometimes expressed +as 𝑂(𝐿|𝐸|𝐶) [44, 178]. As the GCN’s space complexity is concerned, we need to store the embeddings of all nodes plus +the weight matrix at each layer, which is 𝑂(𝐿|𝑉 |𝐶 +𝐿𝐶2), or 𝑂(𝐿|𝑉 |𝐶) when |𝑉 | >> 𝐶. GraphSAGE illustrates the same +propagation procedure from a microscopic view, with a fixed number, denoted 𝑠, of sampled neighbours involved in +the convolutional operation (Equation 51). The overall time complexity of GraphSAGE is, therefore: 𝑂(𝐿|𝑉 |(𝑠𝐶 + 𝐶2)). +Notice that when 𝑠 equals |𝑉 |, the time complexity of GraphSAGE is the same as the vanilla GCN. +Then, when each node aggregates messages from its higher-order neighbours, denoted h-hop neighbours here, the +propagation rule can be put as: 𝐻 (𝑙) = 𝜎 +� +ˆ𝐴ℎ𝐻 (𝑙−1)𝑊 (𝑙)� +. A typical representative is MixHop [3] (refer to Section 4.1.2). +Thus, the time complexity at each layer is: 𝑂(|𝑉 |2𝐶ℎ + |𝑉 |𝐶2) or 𝑂(|𝑉 |2𝐶ℎ) when |𝑉 |ℎ >> 𝐶, and the space complexity +stays unchanged. From a microscopic view, represented by the approach k-hop GNN [155], the time complexity +of involving h-hop neighbours in convolutional operation would be 𝑂(𝐿|𝑉 |(𝑘ℎ𝑚𝑎𝑥𝐶 + 𝐶2)), or 𝑂(𝐿|𝑉 |𝑘ℎ𝑚𝑎𝑥𝐶) when +𝑘ℎ𝑚𝑎𝑥 >> 𝐶. Clearly, the time complexities of both macroscopic and microscopic algorithms grow with ℎ, and when ℎ +equals one, they degrade to the versions of 1-hop neighbourhood algorithms, i.e., the vanilla GCN and GraphSAGE. +42 + +Message (representative approach) +Time complexity +Space complexity +Node feature X (GCN) +𝑂 �𝐿 �|𝑉 |2𝐶 + |𝑉 |𝐶2�� +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +Count of graphlets + X (GSN) +𝑂(|𝑉 |𝑘 |𝑆 |−1 +𝑚𝑎𝑥 + 𝐿(|𝑉 |2(𝐶 + 𝑜) + |𝑉 |(𝐶 + 𝑜)2) +𝑂 �L|𝑉 |(𝐶 + 𝑜) + 𝐿(𝐶 + 𝑜)2� +Distance information + X (P-GNN) +𝑂(|𝑉 |3 + 𝐿|𝑉 |(𝑇𝑛 + 𝐶2)) +𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� +Random feature + X (rGIN) +𝑂(𝐿(|𝑉 |2(𝐶 + 𝑟) + |𝑉 |(𝐶 + 𝑟)2) +𝑂 �L|𝑉 |(𝐶 + 𝑟) + 𝐿(𝐶 + 𝑟)2� +Table 5. Time and space complexity from the perspective of a message content. |𝑉 | is the number of nodes in the graph, 𝐶 is node +feature channels (assuming the number of features is fixed for all layers), 𝐿 is the number of convolutional layers, |𝑆 | is the maximum +size of a set of graphlets, 𝑜 is the number of orbits in graphlets, 𝑘𝑚𝑎𝑥 is maximum node degree, 𝑇 is the number of anchor sets, 𝑛 is +the maximum number of nodes in an anchor set, and 𝑟 is the length of the random feature vector. +Thirdly, approaches with neighbourhood defined on random walks typically include the following steps (represented +by PinSage [198]): performing 𝑤 times random walks of length 𝑙, ranking the visited 𝑣 nodes based on the visited times, +aggregating messages from the top 𝑠 nodes, and finally applying weight matrix on node representations. Therefore, +the overall time complexity is termed as: 𝑂(𝐿|𝑉 |(𝑤𝑙 + 𝑣𝑙𝑜𝑔𝑣 + 𝑠𝐶 + 𝐶2)). Normally, there is no need to record all the +random walks, so the space complexity is still 𝑂(𝐿|𝑉 |𝐶 + 𝐿𝐶2). Comparing the time complexity of PinSage with that of +GraphSAGE, we see that with the extra step of performing random walks and ranking visited nodes, i.e., the term 𝑤𝑙 +and the term 𝑣𝑙𝑜𝑔𝑣, PinSage is more expensive in computation. +In the fourth subcategory, we take k-GNN [146] as an example to analyse the complexity of having k-node subgraphs +as neighbours. The approach aims to learn embeddings for k-node tuples, and the neighbours of each k-tuple are defined +as other k-tuples containing one node that is not in the focal k-tuple(refer to Section 4.1.4). Each k-tuple aggregates +messages from all its k-tuple neighbours, with a time complexity of 𝑂(𝑘|𝑉 |𝐶). Therefore, on all �|𝑉 | +𝑘 +� k-tuples and 𝐿 +layers, the overall time complexity is: 𝑂(𝐿�|𝑉 | +𝑘 +�(𝑘|𝑉 |𝐶 + 𝐶2)). To store the embeddings of �|𝑉 | +𝑘 +� node tuples and the +weight matrices at all layers, it requires 𝑂(𝐿�|𝑉 | +𝑘 +�𝐶 +𝐶2)) space. This approach is essentially different from the previous +ones, in that it is to generate embeddings for k-tuples instead of for each node, resulting in the term �|𝑉 | +𝑘 +� appearing in +both its time and space complexities. Clearly, its complexity grows combinatorially with 𝑘, and easily surpasses the +complexities of all other algorithms when 𝑘 is relatively large. In practice, however, the value of 𝑘 generally does not +exceed 3. +In addition, Table 5 lists the time and space complexities of approaches that include extra node features in the +GNNs (corresponding to the message content taxonomy in Figure 11). First, when the count of graphlets, or more +specifically, the count of node orbits is added to the node features (represented by the approach GSN [27]), it requires a +preprocessing step to count the number of each node orbit, then performing the general convolutional operation. The +cost of counting orbits depends on the size of graphlet |𝑆| and the maximum degree of nodes 𝑘𝑚𝑎𝑥. Another difference +from the vanilla GCN is that the node feature dimension will increase by the number of orbits, denoted 𝑜. Therefore, its +time complexity is 𝑂(|𝑉 |𝑘 |𝑆 |−1 +𝑚𝑎𝑥 + 𝐿(|𝑉 |2(𝐶 + 𝑜) + |𝑉 |(𝐶 + 𝑜)2), and its space complexity is 𝑂 �L|𝑉 |(𝐶 + 𝑜) + 𝐿(𝐶 + 𝑜)2�. +Second, when distance information is included, as in the approach P-GNN [201], it requires first calculating the shortest +path distances between all nodes (𝑂(|𝑉 |3) in the typical Floyd-Warshall algorithm), then aggregating message from +a number of anchor sets (𝑇 anchor sets and each containing at most 𝑛 nodes). Therefore the time complexity would +be 𝑂(|𝑉 |3 + 𝐿|𝑉 |(𝑇𝑛 + 𝐶2)). Another less expensive version is to calculate a limited-hop, e.g., h-hop, shortest path +distance in the preprocessing step, whose time complexity is 𝑂(|𝑉 |𝑘ℎ𝑚𝑎𝑥). Third and lastly, when random features are +43 + +included, represented by the approach rGIN [169], the impact on time complexity is mainly due to the increase in +feature dimension. This is because the cost of generating random features is generally negligible. +We finally discuss the complexity of approaches that have different learning scopes (corresponding to the taxonomy +in Figure 10). For GCNs running on subgraphs, represented by the GraphSAINT [205], the cost includes two steps, i.e., +the subgraph sampling and the training. Given the cost of sampling𝑇𝑠 and a set sampled subgraphs G (maximum number +of nodes in sampled subgraphs denoted |𝑉𝑠 |), its complexity is: 𝑂(𝑇𝑠 + |G|𝐿(|𝑉𝑠 |2𝐶 + |𝑉𝑠 |𝐶2)). The cost 𝑇𝑠, depending +on the choice of the sampler, is normally less expensive than the training. The key term is, therefore, |G|𝐿|𝑉𝑠 |2𝐶. When +|𝑉𝑠 | << |𝑉 |, subgraph-based approaches significantly reduce the training cost of the GCNs. Similarly, for GCNs running +on local subgraphs, exemplified by the Shadow-GNN [204], the two steps are extracting local subgraphs (extraction +cost is denoted as 𝑇𝑒, the maximum number of nodes in extracted local subgraphs is denoted as |𝑉𝑙 |), and training the +GCN on them. Therefore, the time complexity is 𝑂(𝑇𝑒 + |𝑉 |𝐿(|𝑉𝑙 |2𝐶 + |𝑉𝑙 |𝐶2)). Note that local subgraphs are usually +extracted at each node, so the number of extracted subgraphs equals the number of nodes |𝑉 |. Given that |𝑉𝑙 | << |𝑉 | +(we should also have |𝑉𝑙 | << |𝑉𝑠 |), local subgraph based GCNs are generally much faster in training than full graph or +subgraph based GCNs. +5 +DISCUSSION AND OUTLOOK +After reviewing the traditional structural measures and the graph convolutional networks, we are set to answer the +following research question: How are these two classes of methods related, especially how traditional structural +measures of Network Science can inform GCN methods? In this section, we first briefly discuss the performance of +GCNs in major learning tasks, then move on to drawing connections between GCNs and traditional structure based +approaches, and finally introduce three future directions. +5.1 +GCN’s performance in learning tasks +Convolutional Neural Networks have been shown to be state-of-the-art in various tasks in the area of image processing, +including image classification, object detection, and semantic segmentation [74, 120]. GCNs have also achieved promising +performances in various graph-related tasks. As an extension of CNNs in graph data, GCNs, since their appearance, +have received a lot of attention and are viewed as state-of-the-art by default. However, there are works showing that +simple heuristics from traditional network science achieve a comparative performance of GCNs [203, 209], or even beat +them in link prediction and network reconstruction tasks [136]. A recent paper shows that simply feeding heuristics +derived from nodes similarity scores in a logistic regression model can achieve the best performance in link prediction +among many deep learning approaches, including GCNs [136]. In addition, Katz index is the top performer in the +network reconstruction task, followed by VGAE which uses GCN as the graph encoder [106]. +Although the majority of GCN approaches focus on node classification and graph classification tasks, they rarely +include structural heuristic-based methods as baselines in the experiment. This overlook could hinder a comprehensive +evaluation of the performance of graph convolutional networks. Additionally, comparing GCN approaches with +traditional heuristic-based methods could help to better understand the strengths and limitations of GCNs. We believe a +closer integration of graph deep learning approaches and traditional network science approaches would immensely +benefit both communities, and revealing the connections between the two classes of methods lays the foundation of +this integration. +44 + +5.2 +Connections between traditional network science approaches and GCNs +Based on the current literature, the connections between GCNs and traditional structure based approaches are observed +via the following four aspects. The first aspect covers the foundations of GCNs in traditional Network Science; the +second aspect focuses on their similarities in dealing with directed networks; the third and final aspect cover two typical +applications of traditional structural information in GCNs: (i) number of graphlets and (ii) distance information. +5.2.1 +Message passing based approaches and GCN. As we have seen in message passing based approaches (Section 3.4), +a node’s influential score or centrality is calculated through iteratively aggregating the scores of its neighbours until it +converges. Taking the eigenvector centrality, for example, the centrality of node 𝑖, denoted 𝑥(𝑖), is formulated as: +𝑥(𝑖) = 𝑐 +∑︁ +𝑗 ∈𝑁 (𝑖) +𝑥(𝑗). +𝑥, a vector of all nodes’ centralities, is found to converge to the dominant eigenvector of the adjacency matrix 𝐴, and 𝑐 +converges to the reciprocal of the dominant eigenvalue of 𝐴. +Interestingly, graph convolutional networks adopt the same idea of neighbourhood aggregation, and the iteration +process is implemented through the usage of multiple layers. Taking the vanilla GCN for example, we have the following +convolutional operation: +ℎ(𝑙) +𝑣 += 𝜎 �� +� +∑︁ +𝑢∈N(𝑣) +1 +𝑐𝑣𝑢 +ℎ(𝑙−1) +𝑢 +𝑊 (𝑙)�� +� +. +Comparing the above two expressions, one major difference is obviously the appearance of weight matrices: in +eigenvector centrality, the influential score is directly calculated from forward propagation (e.g., a power iteration), +while in the GCN, weight matrices are updated in the backward propagation with the help of labelled samples. Another +subtle yet significant difference is that GCNs allow rich node features (n-dimensional vector for each node), while +traditional message passing approaches, such as the eigenvector centrality, alpha centrality or PageRank, only support +using a numeric value that represents the node’s importance or influence. These two points are also the main reasons why +GCNs have quickly gained popularity — the learnable setting makes GCNs suitable for various types of tasks, and the +support of rich node features makes them appropriate for different types of real-world data. Despite the advancements +and popularity of graph convolutional networks, traditional network science approaches remain important in the field. +They have a strong theoretical foundation, which can provide insights into the underlying mechanisms of networked +systems. Furthermore, traditional approaches are often more computationally efficient than deep learning approaches, +making them more practical for certain types of tasks or data. Overall, the continued use and development of traditional +network science approaches alongside newer methods, such as GCNs, can help to deepen our understanding of complex +networked systems and advance the field as a whole. +5.2.2 +Dealing with link direction. When directions of links are considered, we observe interesting connections between +the traditional message passing approach HITS [109] and the recent graph convolutional approach DGP [101]. HITS +proposes to distinguish two roles in webpages, i.e., authorities and hubs. Authorities, being reliable information sources, +are pointed by hubs (based on incoming edges to the node), while hubs, acting as a home page or library, point to +authorities (based on outgoing edges from the node). An authority score and a hub score are defined in a mutually +dependent way: +𝑎(𝑖) = +∑︁ +𝑗 ∈𝑁 𝑖𝑛 +𝑖 +ℎ(𝑗), +ℎ(𝑖) = +∑︁ +𝑗 ∈𝑁 𝑜𝑢𝑡 +𝑖 +𝑎(𝑗). +45 + +Interestingly, DGP, as a graph convolutional approach, proposes to distinguish link direction through a two-phase +propagation scheme, i.e., one phase capturing outgoing connections and the other capturing incoming connections +(find more in Section 4.1.2): +𝐻 = 𝜎 +� 𝐾 +∑︁ +𝑘=0 +𝛼𝑎 +𝑘 ˆ𝐴𝑎 +𝑘𝜎 +� 𝐾 +∑︁ +𝑘=0 +𝛼𝑑 +𝑘 ˆ𝐴𝑑 +𝑘𝑋𝑊𝑑 +� +𝑊𝑎 +� +. +Clearly, the major difference here is that in DGP one type of connection is stacked on top of another, and therefore +only one representation is learnt, instead of two scores as in HITS. Besides, k-hop outgoing/incoming connections are +included at once in one convolutional layer. Another GCN approach that applies exactly the same idea of distinguishing +outgoing edges and incoming edges is Asymmetric GNN, or AGNN [172]. It proposes a one-way message passing that +only operates on the outgoing or incoming edges of a graph. Two embeddings are then generated for each node to +model their roles of sending and receiving information. It is also possible to design a one-way GCN at particular layers, +while still considering both types of edges in other layers, which could allow the model to focus on different aspects of +the graph structure at different stages of processing. +5.2.3 +Number of graphlets. The number of graphlets, or more specifically, node orbits or edge orbits are important +topological features around individual nodes or edges (find more in Section 2.2). In a traditional non-learning setting, a +vector composed of the counts of a chosen set of node orbits is used to distinguish the roles of nodes [97, 140]. Weights +of the orbits, when introduced, are calculated from hand-coded function. In graph convolutional networks, the count of +graphlets is added as additional features in the message passing scheme, as we have seen in GSN [27], F -MPNN [14], +and ID-GNN [200]. Taken GSN for example, node orbits 𝑥𝑉 (𝑢), 𝑥𝑉 (𝑣) or edge orbits 𝑥𝐸 (𝑢, 𝑣) are introduced as follow: +h𝑙+1(𝑣) = MLP1 �� +� +ℎ𝑙 (𝑣), +∑︁ +𝑢∈N(𝑣) +𝑀𝐿𝑃2 +�h𝑡 (𝑣), h𝑡 (𝑢), x𝑉 (𝑣), x𝑉 (𝑢), e(𝑢, 𝑣)��� +� +, +h𝑙+1(𝑣) = MLP1 �� +� +ℎ𝑙 (𝑣), +∑︁ +𝑢∈N(𝑣) +𝑀𝐿𝑃2 +�h𝑡 (𝑣), h𝑡 (𝑢), x𝐸 (𝑢, 𝑣), e(𝑢, 𝑣)��� +� +. +Obviously, in a learning setting, the weights on all types of features, including the count of graphlets, are learned in the +training stage. Another interesting difference between non-learning approaches and GCN approaches is that the former +chooses to include all node or edge orbits within a given size, while the latter tends to focus on specific substructures +like cycles or cliques. One open problem in using graphlets or orbits in GCNs is determining which ones to choose. +Existing approaches have focused on using cliques and/or cycles within a specific range [14, 27, 200], without providing +much rationale for this choice. While these types of graphlets and orbits are crucial in some contexts, it is likely that +other types could also be exploited to improve the performance of GCN models. There is still much to be explored in +terms of the utility of different graphlets and orbits in GCN models, and further research in this area could lead to +advances in the field. +5.2.4 +Distance information. The path related information is largely used in traditional structural measures, such as in +closeness centrality, betweenness centrality, 𝜅-path centrality, etc. Taking the closeness centrality, for example, it is +defined as the reciprocal of the average shortest path from the focal node 𝑖 to all other nodes: +Θ𝐶 (𝑖) = +|𝑉 | − 1 +� +𝑗 ∈𝑉,𝑗≠𝑖 𝑑(𝑖, 𝑗) . +46 + +The value of a node’s closeness centrality is directly used to describe the node’s capacity of spreading information on +the graph. Unsurprisingly, the distance information is also made of use in graph convolutional networks, as we have +seen in P-GNN [201] and DE-GNN [117]. In P-GNN, for example, the distance between a node and several anchor sets +is included in the convolutional operation: +h𝑙 +𝑣 = AGG(𝑙) � +M𝑙−1 +𝑖 +, ∀𝑖 ∈ [1,𝑘] +� +, +M𝑙−1 +𝑖 += {𝐹 (𝑑𝑢𝑣,ℎ𝑙−1 +𝑢 ,ℎ𝑙−1 +𝑣 +), ∀𝑢 ∈ 𝑆𝑖}. +In DE-GNN, the distance information between node 𝑣 and a target node set 𝑆 is used as an extra initial node feature: +ℎ(0) +𝑣 += 𝐶𝑂𝑁𝐶𝐴𝑇 (𝑥𝑣,𝜁 (𝑣 | 𝑆)). Recall that this idea of including extra structural features as additional initial node +features is also found in F -MPNN [14], ID-GNN [200], and rGIN [169]. +5.3 +Future directions +Although recent years have witnessed the great success of graph convolutional networks in various domains, there are +still many open problems to be solved and a lot of room for further exploration [32, 216, 219]. Except for the frequently +mentioned directions, such as proposing GCNs for more complicated types of networks or to further increase the +expressivity or scalability of GCNs, we would like to point out three potential directions which combine the traditional +graph analysis approaches and GCN approaches. +Exploring the applicability of more structural measures in GCNs. We have seen appearances of various structural +measures in GCNs, from the simplest node degree [78] to the much more complicated distance information [201] and +graphlet orbits [27]. However, there are many other traditional structural measures that have yet to be fully explored in +the context of GCNs. For example, subgraph formation based measures, such as the clustering coefficient [184] and +the closure coefficient [94, 196], could be incorporated as node-level features or used to weight the edges of the graph +[177]. Global path based measures, such as the closeness or betweenness centrality measures, can be used to guide the +sampling of nodes, edges or subgraphs when constructing the training set for a GCN [219]. For example, we could use +closeness centrality to select the nodes that are most influential in the graph and build subgraphs based on these nodes +as the input to the GCN. It would be interesting to see how these and other structural measures could be utilised in +GCNs to improve performance on certain tasks or in particular types of networks. +Improving the explainability of GCNs / Guiding the choice of GCNs via traditional structural measures. When it comes +to the explainability of GCN models, existing methods, represented by perturbation-based methods, mostly focus +on generating explanations for a trained GCN [199, 202]. There are, however, still many questions to be answered, +such as how different GCNs perform differently on different types of networks, and what are the reasons for these +differences. An analysis from a structural information perspective can provide more insights into how different GCN +models extract and utilise graph structural information, and how the information may differ across different GCN +models and graph types. This can help to better understand the strengths and limitations of different GCN models and +how to effectively apply them in different scenarios. Moreover, in view of the large collection of GCN models and their +composition modules, it is difficult to decide which one to choose and how to set it up for the targeted dataset and task +[86]. Traditional structural measures could be used as indices for selecting the appropriate GCN model and the related +modules. For example, for graphs that are rich in triangles, a particular GCN would be a better choice, while for graphs +where quadrangles are overrepresented, another GCN model should be selected. +Integrating edge features in GCNs. While the vanilla GCN primarily focuses on aggregating and passing information +from neighbouring nodes, it is important to consider the role of edge attributes in many real-world networks. For +example, in consumer review networks, the ratings of products are often labelled on the edges, and in social networks, +47 + +the type and frequency of interactions are labelled on the edges. Integrating edge features into GCNs could not only +enhance the applicability of the model but also increase the accuracy and relevance of its predictions. There are works +that naively include edge features in GCN or propose a tailor-made model to encompass them [27, 72]. However, there +is still much to be learned about the utility of traditional edge-level structural measures in GCN models, such as the +edge orbits [82], the edge clustering coefficient [181], the local path index [128], etc. Further research in this area is +likely to yield valuable insights and improvements to the performance of GCN models. +6 +CONCLUSION +The complexity of graph data mainly comes from its intricate topological structures. Mining and exploiting graph +structural information have always been one of the focal points in the study of graphs. A large amount of work in +traditional network science proposes various types of structural measures, especially local structural measures, to +characterise and study complex networks. When more nodes or edges are involved, such approaches, however, become +infeasibly complicated. Graph convolutional networks, on the other hand, are proposed to automatically extract relevant +features from nodes’ neighbourhoods, and in this manner, avoid choosing and manually calculating structural metrics. +In order to reveal the connections between the two classes of methods, especially how traditional structural measures +can inform GCNs, in this paper, we first reviewed the traditional structure-based approaches in Network Science and +proposed a new taxonomy encompassing many seemingly unrelated concepts from a structural perspective. With +this prerequisite knowledge, we then extend the scope to the prominent and powerful graph convolutional networks, +and provide a Network Science perspective on them — review and classify GCNs from three structural angles, which +are the layer-wise message aggregation scope, the message content, and the overall learning scope. Furthermore, we +extensively discuss the connections between the traditional structural approaches and the graph convolutional networks +and suggest three future research directions in the joint research area. We believe that the well-established foundations +of traditional structure-based approaches in Network Science not only form the basis for GCNs but also could, and +probably should, serve as a driving force for their future advances. +ACKNOWLEDGMENTS +The authors thank Yu-Xuan Qiu, Joakim Skarding and Xiaohan Zhang for their helpful comments and discussions. This +work was supported by the Australian Research Council, Grant No. DP190101087: “Dynamics and Control of Complex +Social Networks”. +REFERENCES +[1] Sergi Abadal, Akshay Jain, Robert Guirado, Jorge López-Alonso, and Eduard Alarcón. 2021. Computing graph neural networks: A survey from +algorithms to accelerators. ACM Computing Surveys (CSUR) 54, 9 (2021), 1–38. +[2] Alexandre H Abdo and APS de Moura. 2006. 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Maximal planar networks with large clustering coefficient and power-law degree distribution. +Physical Review E 71, 4 (2005), 046141. +55 + diff --git a/u9E3T4oBgHgl3EQf-QtO/content/tmp_files/load_file.txt b/u9E3T4oBgHgl3EQf-QtO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..24271bcb5420cb7efa44fab08b351eaea50bbd02 --- /dev/null +++ b/u9E3T4oBgHgl3EQf-QtO/content/tmp_files/load_file.txt @@ -0,0 +1,2512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf,len=2511 +page_content='A Network Science perspective of Graph Convolutional Networks: A survey MINGSHAN JIA, BOGDAN GABRYS, and KATARZYNA MUSIAL, University of Technology Sydney, Australia The mining and exploitation of graph structural information have been the focal points in the study of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Traditional structural measures in Network Science focus on the analysis and modelling of complex networks from the perspective of network structure, such as the centrality measures, the clustering coefficient, and motifs and graphlets, and they have become basic tools for studying and understanding graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In comparison, graph neural networks, especially graph convolutional networks (GCNs), are particularly effective at integrating node features into graph structures via neighbourhood aggregation and message passing, and have been shown to significantly improve the performances in a variety of learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' These two classes of methods are, however, typically treated separately with limited references to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In this work, aiming to establish relationships between them, we provide a network science perspective of GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Our novel taxonomy classifies GCNs from three structural information angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the layer-wise message aggregation scope, the message content, and the overall learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Moreover, as a prerequisite for reviewing GCNs via a network science perspective, we also summarise traditional structural measures and propose a new taxonomy for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally and most importantly, we draw connections between traditional structural approaches and graph convolutional networks, and discuss potential directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' CCS Concepts: • Networks → Topology analysis and generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Network structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' • Computing methodologies → Ma- chine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Additional Key Words and Phrases: Graph Convolutional Networks, Network Science, graph structural measures 1 INTRODUCTION Networks or graphs are a general language for modelling and analysing complex systems that are abstracted as entities and their connections [13, 151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the representation of networks, domain data is no longer only being a set of isolated data points but also contains important information about the relationships among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The entities are related to each other according to the structure of the network, and modelling these relational structures allows us to build more accurate models of the domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Various types of real-world data can naturally be modelled as networks, such as social networks representing social actors and their relationships [149], molecular graphs representing chemical atoms and their bonds [90], transportation networks representing infrastructures and traffic flow [18], control flow graphs representing code blocks and their executions [143], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although networks are very powerful at modelling relational data, processing them is significantly more difficult, mainly due to their intricate topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Compared to other common data formats such as images or text, network data does not have a starting or an ending point that can be defined in Euclidean space, nor the essential notion of spacial locality and proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, understanding and exploiting graph structure has always been a core theme in analysing complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Traditional network science approaches are mostly structure-related heuristics, such as various types of node centralities [127] for node-level analysis, common neighbours similarity and its variants [138] for link-level analysis, and motifs [142] and significance profile [141] for graph-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' These methods, along with others, have become the standard tools for analysing graphs and have been used in all kinds of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Certainly, these approaches have their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' First is their applicability — each is effective for examining specific properties but falls short of capturing other structural aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another drawback is that most heuristic approaches focus on graph structures while overlooking the rich information that could be contained in nodes or on edges [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='04824v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='SI] 12 Jan 2023 Graph Convolutional Networks Layer-wise Message Aggregation Scope (Where a node aggregates message from at each layer) Message Content (What message is gathered and passed on) Learning Scope (Where GNNs are trained on) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taxonomy of graph convolutional networks from structural perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another mainstream class of methods is grounded in deep learning on graphs, especially the recently emerging and quickly gaining in popularity graph convolutional networks (GCNs) [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GCNs are generalised from the notion of Convolutional Neural Networks (CNNs) [8], redefining them for non-Euclidean graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GCNs ingeniously combine graph structure and node/edge features via neighbourhood message aggregation and a structure-based propagation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Being a rapidly evolving area of research, a large number of graph convolutional network approaches have emerged in recent years, aiming to improve its expressivity, scalability, or targeting specific tasks or types of networks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, there are still many challenges and opportunities in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some of the key open problems include developing more powerful and efficient GCN architectures, extending these models to handle temporal, multi-layered, or other more complex graph data, and improving the interpretability and transparency of GCN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Traditional structural measures of Network Science are direct and efficient tools for analysing and understanding complex networks, while graph convolutional networks are deep learning models designed specifically for graph data in order to address various learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As discussed, the two classes of methods have their own strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Surprisingly, they are very often treated separately with relatively limited references to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Network science researchers may be sceptical about the lack of explainability of deep learning approaches, while deep learning researchers tend to overlook the advance in traditional non-learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We believe, however, that with the established foundations of traditional structural measures in Network Science, and GCNs emerging as a new powerful class of methods, there would be great benefits to be realised from a closer integration and awareness of the two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' On the one hand, GCNs gracefully incorporate node features, which are largely overlooked in traditional structural measures, into the structure of graphs, and achieve state-of-the-art performances in various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' On the other hand, traditional network science notions, being the foundations of understanding and characterising complex networks, are also indispensable in studying GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different types of structural measures are being exploited in the recent advance of GCNs as well [27, 100, 117, 192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, in this work, we aim to link the two classes of methods together by comprehensively reviewing each of them, proposing new taxonomies and discussing their connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Along with the phenomenal development of GCNs, many survey articles appeared to summarise and review them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some have a broad scope that covers graph representation learning [79] or graph deep learning [32, 189, 216] in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some others are focused on specific aspects, such as the design pipeline or the composition modules of graph neural networks [219], the dynamic mechanisms [170], or the learning on limited labelled samples [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, there still lacks an examination that focuses on how graph structure information (which is the main focus of traditional network science approaches), is exploited in graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thus, in this work, we propose new taxonomies of GCNs from the perspective of graph local structure, and at the same time, review the latest works that improve graph neural networks through exploiting local structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, we propose to summarise graph 2 Subgraph Count Based Approaches Message Passing Based Approaches Subgraph Formation Based Approaches Mixed Approaches Global Path Based Approaches Structural Measures on Graphs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Structural measures on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' convolutional networks from three structural angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the scope of layer-wise message aggregation, the content of the message being passed on, and the overall scope of learning on graphs (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Moreover, a systematic understanding of traditional graph structural approaches is the prerequisite for thoroughly reviewing GCNs via a network science perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, before jumping into the sphere of graph neural networks, we first summarise and classify non-learning graph structural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The study of graph structures is so ubiquitous that they often appear in different terms, such as the big family of centrality measures [47, 127, 164], the popular notion of motifs [142] and graphlets [140] and the set of subgraph formation measurements such as the clustering coefficient [184], the closure coefficient [196], the square clustering coefficient [124], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Existing surveys on structure measurements only cover one or two sets of those notions, and fail to unite them from an overarching perspective or to draw connections and comparisons between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, in this work with a focus on graph structure, we also propose a new taxonomy that brings all these concepts together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, we group existing graph structural measures into five categories: subgraph count based measures, subgraph formation based measures, global path based measures, message passing based measures, and hybrid measures ( Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' More importantly, through summarising both the traditional structural measures and the graph convolutional network approaches, we could draw connections between the two, strengthen the understanding and analysis of GCNs and lead to insightful discussions about potential research avenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To summarise, the main contributions of this survey are as follows: We propose a new taxonomy that brings together various types of traditional structure-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We make a clearer distinction between the concepts of local and global, and we first introduce and summarise the category of subgraph formation based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We propose a novel taxonomy of graph convolutional networks, with a focus on the exploitation of graph structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The taxonomy categorises GCNs from three structural information angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the layer-wise message aggregation scope, the message content, and the overall learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We review and summarise the latest GCN approaches with a structural focus, and provide a thorough analysis of the time and space complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We draw connections between the graph convolutional networks and the traditional structure-based approaches, and discuss three potential future research avenues in the joint area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The rest of this survey is organised as follows: In Section 2, we introduce and compare two pairs of concepts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', local and global, and motifs and graphlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In Section 3, we present the five categories of graph structural measures 3 and discuss four open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In Section 4, we introduce the novel taxonomy of graph convolutional networks, and discuss their time and space complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In Section 5, we discuss the connections between the traditional structural measures and the graph convolutional networks, and present some potential research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, we conclude the article in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2 PRELIMINARIES AND BACKGROUND This section introduces preliminary concepts that are helpful for understanding the proposed taxonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Local vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global When discussing graph structural measures, we need first to distinguish what is local and what is global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Previous works [56, 92, 133, 138] either only focus on where the measures are defined by dividing them into two or three categories: (i) the "local", "micro" or "individual" level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (ii) the "global", "macro" or "aggregate" level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' and (iii) sometimes at the "mesoscopic", "quasi-local" or "subnetwork" level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' or they are defined solely based on the scope of information involved in their computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This, however, leads to some confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, the betweenness centrality is defined for nodes (at the node-level) but requires global information to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Should it be termed a local measure or a global measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Similarly, the average clustering coefficient is defined at the network-level, but only needs local information at each node — calculating the local clustering coefficient at each node, then averaging over all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, we propose the following terms to distinguish both at what level the measures are defined and the scope of information that is needed to calculate them: Local-level measure is a measurement defined on a node-level or link-level (the link here also includes the non-existing or potential link which is often used in a link prediction task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thus, it can be further divided into a node-level measure and a link-level measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Network-level measure is a measurement defined for the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Local structural measure is a measurement whose computation only involves the nearby neighbourhood of a node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', within a range of k-hop away from a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In most cases, k is less than or equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Many traditional measures only care about the immediate neighbourhood around a node, and we name them as Strict-local structural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global structural measure is a measurement that involves the global information in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This type of measurement often involves the computation of paths between nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Now, when we revisit the betweenness centrality, it is both a local-level and a global structural measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The average clustering coefficient, on the other hand, is both a network-level measure and a local structural measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that the average clustering coefficient involves the extra step of averaging over all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Indeed, it is 𝑛 times the complexity of computing the local clustering coefficient at a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, any local-level measure can easily have an extended definition at the network-level through averaging over all nodes or edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Moreover, in the practice of network analysis, local-level measures are often calculated for the entire network, looping over all nodes or all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, when defining local or global structural measures, we choose to exclude this aggregation or averaging step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To summarise, we use the terms “local-level” and “global-level” to distinguish where the measure is defined, and we use the terms “local structural” and “global structural” to distinguish the scope of information involved in the computation, before the aggregation/averaging step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some 3-node and 4-node motifs in directed networks[142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Motifs containing bidirectional edges are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Motif ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Designation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Type of network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3-node feed-forward loop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Gene regulation network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Neural network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Electronic circuits (forward logic chips) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3-chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Food webs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3-node feedback loop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Gene regulation network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Neural network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Electronic circuits (forward logic chips) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Bi-fan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Gene regulation network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Neural network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Electronic circuits (forward logic chips) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Electronic circuits II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Bi-parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Neural network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Food webs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Electronic circuits (forward logic chips) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4-node feedback loop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Electronic circuits II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Motifs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graphlets Next, we distinguish three similar concepts that are later used in our taxonomies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', subgraphs, motifs and graphlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A subgraph, as the name implies, is a smaller graph whose node set and edge set are subsets of those of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We then recap the notions of motifs [142] and graphlets [140] according to the papers that proposed them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Network motifs [142] are subgraphs that recur much more frequently in the real network than in an ensemble of randomised networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They are defined at the network-level, in order to uncover the basic building blocks of directed networks across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraphs having a 𝑝-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='01 are deemed as motifs, where 𝑝 is the probability of the subgraph appearing more times in randomised networks than in the real network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The statistical significance of a motif can also be captured by the Z-score, which is calculated as follows: 𝑍𝑖 = � 𝑁 real 𝑖 − ¯𝑁 rand 𝑖 � /std � 𝑁 rand 𝑖 � , where 𝑁 real 𝑖 is the number of subgraphs of type 𝑖 in the real network, and 𝑁 rand 𝑖 is the number of subgraphs of type 𝑖 in a randomised network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A natural downside of this approach, however, is that it needs to generate a large number of 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 G0 G1 G2 G3 G4 G5 G6 G7 G8 19 20 18 21 15 16 17 G9 G10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graphlets and their orbits [140] random networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 100s or 1000s) using a certain configuration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The original work only focuses on 3-node and 4-node directed subgraphs, finding that particular subgraphs such as 3-node feed-forward loop, 3-node feedback loop, bi-fan, bi-parallel, and 4-node feedback loop are significant building blocks in several different types of directed networks (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graphlets [140], are nonisomorphic induced subgraphs around a focal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the original work, it is defined for undirected networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A key difference between motifs and graphlets is that graphlets are defined at node-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The term automorphism orbits, or orbits for short, are used to distinguish different positions of the focal node in a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, when subgraph size is limited to a range of 2 to 5 nodes, there are 73 different orbits on 30 different subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We recap graphlets with the orbits in Figure 3 (in order to save some space, the majority of 5-node graphlets are omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is worth mentioning that the idea of counting induced subgraphs is also extended to the link-level, leading to the notion of edge orbits [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking graphlet 𝐺1 in Figure 3 for example, there exist two (node) orbits denoted ’1’ and ’2’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In contrast, there is only one edge orbit in it since the two edges are structurally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To summarise, motifs and graphlets are both small induced subgraphs, but they are different in the following aspects (Figure 4): motifs are defined at the network-level while graphlets are defined at the node-level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' motifs are proposed for directed networks while graphlets are for undirected networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' motifs are discovered from comparing real networks to randomised networks with the same degree sequence while graphlets are calculated on the network itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' lastly, motifs contain 3 - 4 nodes while graphlets have 2 -5 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that most of the analyses stop at 4 or 5 nodes because a subgraph containing more than 5 nodes would become too complicated for us to enumerate and interpret all possible subgraphs or orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, a 6-node induced subgraph leads up to 112 different types of subgraphs and 407 different orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking link directions into consideration, there are 1, 530, 843 subgraphs and 9, 031, 113 orbits [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3 GRAPH STRUCTURAL MEASURES In order to set up the context of reviewing graph convolutional networks from a Network Science perspective, we first summarise traditional graph structural measures and propose a novel taxonomy for them, which will later be used in our categorisation and analysis of GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, We divide existing structural measures into five categories (see Figure 2): Subgraph Count Based Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' These measures are defined based on the number of a particular subgraph or subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 6 Small induced nonisomorphic subgraph Network-level Node-level Directed network Undirected network Subgraphs of high frequency Subgraphs of any frequency Compare to random networks Calculate on itself Subgraph size: 3 - 4 Subgraph size: 2 - 5 Motifs Graphlets Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Motifs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graphlets Subgraph Formation Based Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In this category, the measures are defined by the ratio of the numbers of two subgraphs: one contains fewer edges (or nodes) and is viewed as the formation base of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global Path Based Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As the name implies, these measures are based on unbounded paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They involve the calculation of shortest paths or all paths originating from a node to any node in the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Message Passing Based Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Unlike previous categories, message passing-based approaches utilise graph structural information in an implicit manner: every node is initialised with an importance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then iteratively, each node updates its score through aggregating the scores of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graph Neural Network approaches (see more in Section 4) can be viewed as transforming this traditional message passing approach into a learnable process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Hybrid Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' These measures are simply some combinations of the previous four categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We now explain the logic behind our taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The first two categories both cover a local area of the whole network (within a certain distance from the focal node, or containing a limited number of nodes and edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The first category — subgraph count based approaches — is built from counting the number of particular local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, the number of neighbours, local paths or subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The second category — subgraph formation based approaches — is uniquely defined based on the ratio of two subgraphs and thus bears the meaning of measuring the formation of certain local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To have both of them in the taxonomy instead of combining them into one category is to stress their differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, the third category expands its scope to the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We name it global path based approaches instead of just global approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This is because all global approaches involve either the calculation of shortest paths or all paths originating from a node to any other node in the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice here that a path is also a particular type of subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, a local path or bounded path, such as a 2-path or 3-path, belongs to the category of subgraph count based measures, whereas a global path or unbounded path is in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We choose to differentiate the third category from the previous two categories from the perspective of the covered scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Next, the fourth category — message passing based approaches — is based on the idea of propagating information along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is a different form to the abovementioned three categories because it does not calculate any type of subgraphs or global paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Instead, the structure is utilised in an implicit way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Every node is initialised with an importance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then iteratively, each node updates its score through aggregating the scores of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although these four categories are largely different from each other, there are many approaches that combine them together, which are naturally put into the fifth category — mixed approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Subgraph Count Based Approaches Subgraph count based measures are based on the number of a particular subgraph or subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We further divide them into three subclasses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', measures defined on 1-hop neighbours, measures defined on k-hop neighbours/local paths, and measures defined on multi-subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Figure 5 gives the detailed categorisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The colour of the block differentiates where the approach is defined: grey is on the node-level, blue is on the link-level, and orange is on the network-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraph Count 1-hop neighbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Degree cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Local cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graphlet Degree Subgraph centrality Triad Signif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Profile/Subgraph Ratio Profile Multi-subgraphs ℎ-index/𝑔-index local paths\\ k-hop neighbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑘-core 𝑘-truss/CN Local path index Collective influence 𝜅-path cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝜅-path edge cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑘-betweenness cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Potential theory/Quad motifs index Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraph count based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 1-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As the name implies, the calculations within this category only require the immediate neighbourhood around a node or a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Degree centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Through calculating the number of nodes directly connected to a node, the degree centrality is an easy and straightforward way to assess the importance or influence of the node[65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to render it within the range of (0,1], it is often normalised by the size of the network minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Mathematically, the normalised degree centrality of node 𝑖 is defined as: Θ𝐷 (𝑖) = 𝑑𝑖 𝑛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (1) Despite being so simple, the degree centrality has been widely applied in various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, in customer networks, the degree centrality is used to find opinion leaders [163], and in biomedical semantic networks, it is effective in selecting crucial information for summarising a disease treatment [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some interesting extensions of the degree centrality include the in-degree/out-degree centrality in directed networks, the strength centrality and weighted strength centrality in weighted networks [35] and the cross-layer degree centrality in multi-layered networks [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – ℎ-index/𝑔-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ℎ-index is proposed to evaluate the impact of an individual’s research output: A researcher has an index of ℎ if ℎ of his or her papers have at least ℎ citations [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is then used as a centrality measure in networks, and named as lobby index or 𝑙-index [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝑙-index of a node is the largest integer 𝑘 such that the node has at least 𝑘 neighbours with a degree of at least 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Egghe argued that the influence of highly cited papers is underplayed in the ℎ-index, and proposed a 𝑔-index to overcome this disadvantage [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' After ranking a researcher’s papers according to their citations, the 𝑔-index is defined as the highest rank 𝑔 such that the top 𝑔 papers together have at least 𝑔2 citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From its definition, the 𝑔-index is always greater than or equal to the ℎ-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To address the same 8 issue, an 𝑒-index is proposed to complement the ℎ-index for excess citations[207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Recently, a local ℎ-index centrality is proposed to identify influential spreaders by simultaneously considering the ℎ-index values of the node and its neighbours [125]: Θ𝐿𝐻 (𝑖) = ℎ(𝑖) + � 𝑗 ∈𝑁𝑖 ℎ(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – 𝑘-core [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Instead of only calculating the number of 1-hop neighbours at one node (as in the degree centrality) or at both the node and its neighbours (as in the ℎ-index), a 𝑘-core or coreness takes into account the number of neighbours at every node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the 𝑘-core is defined as a subgraph in which all nodes of a degree smaller than k have been removed and the remaining nodes have a degree of at least 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A node located in a higher 𝑘-core is deemed more important than a node in a lower 𝑘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝑘-core is calculated through the 𝑘-shell decomposition [37] which incrementally (from 1 to 𝑘) removes nodes with degree less than 𝑘 (which in turn results in lowering the degree of remaining nodes) until no more nodes need to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Given that the degree centrality, the ℎ-index and the coreness are all based on the number of 1-hop neighbours, Lü et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' further revealed their relationships through proposing the high-order ℎ-indices [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' further propose a neighbourhood coreness that considers both the degree of a node and the coreness of its neighbours [12]: Θ𝑁𝐶 (𝑖) = ∑︁ 𝑗 ∈𝑁 (𝑖) 𝑘𝑠(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (2) The assumption is that a node having more connections to the neighbours located in the core of the network is more influential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – 𝑘-truss/Common neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A 𝑘-truss is a subgraph where every edge is contained in at least 𝑘 − 2 triangles[45, 180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is found through counting the number of common neighbours of a pair of nodes that forms an edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the number of triangles that the edge participates in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝑘-truss is also a (𝑘 + 1)-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Counting common neighbours around a pair of nodes that have not formed an edge (a non-edge) is also a basic approach in a link prediction task [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There is a big family of similar approaches based on the number of neighbours around non-edges, such as the Adamic-Adar index, the resource allocation index, the preferential attachment index, among others [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that both 𝑘-truss and Common Neighbours-like approaches are defined at the link-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The block colour is therefore blue in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 local paths/k-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The group of methods in this category requires the calculation of local paths or k-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – 𝑘-betweenness centrality [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝑘-betweenness centrality or bounded-distance betweenness centrality is a variation of the well-known betweenness centrality that limits the length of shortest paths to a predefined value 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the 𝑘-betweenness centrality of any node 𝑖 is calculated by: Θ𝐵𝑘 (𝑖) = ∑︁ 𝑠,𝑡 ∈𝑉 𝜎𝑘 (𝑠,𝑡 | 𝑖) 𝜎𝑘 (𝑠,𝑡) , (3) where 𝜎𝑘 (𝑠,𝑡) is the number of shortest paths of length at most 𝑘 between a node pair 𝑠 and 𝑡, and 𝜎𝑘 (𝑠,𝑡 | 𝑖) is the number of those paths that pass through node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The reason for proposing a bounded-distance betweenness centrality is that in some networks, long paths are rarely used for the propagation of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 9 – 𝜅-path centrality [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Instead of limiting the length of shortest paths between node pairs, the 𝜅-path centrality assumes that message traversals are along random simple paths of length at most 𝑘, and proposes to calculate the sum of the probability that a message originating from any possible node goes through the focal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝜅-path centrality of node 𝑖 is defined as: Θ𝑃𝑘 (𝑖) = ∑︁ 𝑠 ∈𝑉 𝜎𝑘 (𝑠 | 𝑖) 𝜎𝑘 (𝑠) , (4) where 𝑠 are all the possible source nodes, 𝜎𝑘 (𝑠 | 𝑖) is the number of 𝑘-paths originating from 𝑠 and passing through 𝑖, and 𝜎𝑘 (𝑠) is the overall number of 𝑘-paths originating from 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to calculate it more efficiently in large networks, a randomised approximation algorithm called RA-𝜅path is also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [7] – 𝜅-path edge centrality [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Moving the 𝜅-path centrality definition to link-level, we then have the 𝜅-path edge centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝑘-path edge centrality of any given edge 𝑒 is defined as the sum of the frequency with which a message originated from any possible node traverses 𝑒, assuming that the message traversals are along random simple paths of length at most 𝑘: Θ𝑃𝑘 (𝑒) = ∑︁ 𝑠 ∈𝑉 𝜎𝑘 (𝑠 | 𝑒) 𝜎𝑘 (𝑠) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (5) Quite similar to Equation 5, only here 𝜎𝜅𝑠 (𝑒) is the number of 𝜅-paths originating from 𝑠 that go over the edge 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The original 𝜅-path edge centrality is very expensive to compute in large networks with a big 𝑘, therefore two randomised approximations have been further proposed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', ERW-𝜅path and WERW-𝜅path [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Local centrality [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Local centrality, sometimes summarised as LocalRank [127] utilises the information within a node’s 4-hop neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the local centrality of node 𝑖 is defined as: Θ𝐿𝑅(𝑖) = ∑︁ 𝑗 ∈𝑁 (𝑖) 𝑄(𝑗), 𝑄(𝑗) = ∑︁ 𝑘 ∈𝑁 (𝑗) 𝑅(𝑘), (6) where 𝑁 (𝑖) and 𝑁 (𝑗) are the set of 1-hop neighbours of node 𝑖 and 𝑗, and 𝑅(𝑘) is the number of both 1-hop and 2-hop neighbours of node 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is said to perform better than betweenness centrality and almost as well as closeness centrality to identify influential nodes under the setting of a SIR model, with only a time complexity of 𝑂(𝑛⟨𝑘⟩2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Collective influence [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Collective influence (CI) is another interesting method that takes higher-order neigh- bourhoods into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The idea is to find those nodes that will cause the biggest drop in the “energy function” when removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the level 𝑘 collective influence of a node 𝑖 is defined as: Θ𝐶𝐼𝑘 (𝑖) = (𝑑𝑖 − 1) ∑︁ 𝑗 ∈𝑁𝑘 (𝑖) (𝑑𝑗 − 1), (7) where 𝑁𝑘 (𝑖) is 𝑘-hop neighbours of a node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' After applying the collective influence score, the paper finds that a large number of previously neglected weakly connected nodes (nodes of lower degree) emerge among the optimal influencers [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Multi-subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Methods of this category involve the count of multiple different subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They can be at the node level, the link level or the network level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Graphlet degree [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2, graphlets are nonisomorphic induced subgraphs around a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graphlet degree is a 73-dimensional vector formed by all different orbits in the subgraphs of size 2-5 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The paper 10 discovers that in PPI networks, nodes grouped together under this measure belong to the same protein complexes, perform the same biological functions and have the same tissue expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some interesting extensions of graphlets include the dynamic graphlets for temporal networks[91], the directed graphlets for directed networks[11], the coloured graphlets for heterogeneous networks[75], and the typed-edge graphlets for edge-labelled networks [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Subgraph centrality [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraph centrality focuses on subgraphs captured by closed walks of different lengths around a given node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, when the walk length is 4, three types of subgraphs are covered, which are 2-cliques, 2-paths, and 4-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The number of closed walks of length 𝑘 around node 𝑖 can be calculated from the 𝑖th diagonal entry of the 𝑘th power of the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When the walk becomes unbounded, the subgraph centrality of node 𝑖 is calculated by: Θ𝑆 (𝑖) = ∞ ∑︁ 𝑘=0 𝜇𝑘 (𝑖) 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' , (8) where 𝜇𝑘 (𝑖) = � A𝑘� 𝑖𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is shown to be more discriminative than many popular centrality measures such as the degree centrality, the betweenness and the eigenvector centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Local path index [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Extended from common neighbours, the local path index counts both the number of 2- paths and 3-paths between a pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The approach is proposed for link prediction, and therefore focuses on non-connected node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the local path index of a node pair 𝑖 and 𝑗 is defined as: Θ𝐿𝑃 (𝑖, 𝑗) = 𝐴2 𝑖𝑗 + 𝜖𝐴3 𝑖𝑗, (9) where 𝜖 is a weigh parameter for 3-paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The paper finds out that the local path index remarkably outperforms common neighbours and can reach a competitive accuracy to the Katz index where all paths are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some other works compare 3-paths approaches against 2-paths approaches in link prediction and find out that 3-path approaches perform better in PPI networks and food webs [111, 148, 220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Potential theory/Quad motifs index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The potential theory aims to predict links in directed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' By counting the numbers of 4 different directed 2-paths and 8 different directed 3-paths around a pair of nodes, the paper finds out that a link has a higher probability of appearing if it could generate more bi-fan subgraphs [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Very similar to the idea of potential theory, the quad motifs index proposes to count particularly three types of directed open-quadriad (3-paths) subgraphs: two of them are the bases for bi-parallel subgraphs and the other one is for bi-fan [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the quad motifs index of a pair of nodes 𝑖 and 𝑗 is defined as: Θ𝑄𝑀 (𝑖, 𝑗) = 𝛼 × 𝑠𝐹 (𝑖, 𝑗) + (1 − 𝛼) 2 (𝑠𝑃1(𝑖, 𝑗) + 𝑠𝑃2(𝑖, 𝑗)) , (10) where 𝑠𝐹 (𝑖, 𝑗) is the contribution from the bi-fan base while 𝑠𝑃1(𝑖, 𝑗) and 𝑠𝑃2(𝑖, 𝑗) are the contributions from two bi-parallel bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Together with the local path index, it is interesting to see that 3-path subgraphs are of particular importance in link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Triad significance profile/Subgraph ratio profile [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Extended from network motifs [142], the triad signifi- cance profile (TSP) is constructed from normalised 𝑍 scores of 13 different directed 3-node subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑇𝑆𝑃 = {𝑆𝑃1,𝑆𝑃2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=',𝑆𝑃13}, SP𝑖 = 𝑍𝑖/(Σ𝑍 2 𝑖 )1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (11) 11 𝑍𝑖 is in turn calculated from comparing with an ensemble of randomised networks with the same degree sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', 𝑍𝑖 = � 𝑁 real 𝑖 − ¯𝑁 rand 𝑖 � /std � 𝑁 rand 𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraph ratio profile (SRP), on the other hand, is built from 6 undirected 4-node subgraphs (𝐺3 to 𝐺8 in Figure 3) : 𝑆𝑅𝑃 = {𝑆𝑅𝑃1,𝑆𝑅𝑃2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=',𝑆𝑅𝑃6}, SRP𝑖 = Δ𝑖/(ΣΔ𝑖 2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (12) Unlike TSP, SRP uses the abundance of each subgraph relative to random networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', Δ𝑖 = 𝑁 real𝑖−⟨𝑁rand𝑖 ⟩ 𝑁 real𝑖+⟨𝑁 rand𝑖 ⟩+𝜀 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Previously seemingly unrelated networks are found to belong to several superfamilies with very similar significance profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice also that these two approaches are defined on network-level, not on node or link-level as we have seen often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Subgraph Formation Based Approaches Subgraph formation based measures view a subgraph being built from another less complex subgraph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', with one link, multiple links, or one node less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We further divide them into three categories according to the size of the subgraph, 3-node, 4-node and 4-node plus (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Most of these approaches are defined at node-level, except that the edge clustering coefficient is at link-level and the interest clustering coefficient is at network-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraph Formation 3-node 4-node 4-node + Clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Closure coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Quadrangle coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Square clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' HO closure coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Grid coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Edge clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' HO clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' F HO clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Y Interest clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Weighted degree cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraph formation based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 3-node subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 3-node subgraph is the simplest yet most important category in the taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Clustering coefficient [184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The clustering coefficient is the first and most influential measure in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It measures the extent to which the neighbours of a node connect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From a structural formation perspective, it measures the formation of triangles upon open-triads (also called wedges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the clustering coefficient of node 𝑖 is defined as the ratio between the number of triangles containing node 𝑖 (denoted 𝑇 (𝑖)) and the number of open triads (denoted 𝑂𝑇 (𝑖)): C𝐶 (𝑖) = 𝑇 (𝑖) 𝑂𝑇 (𝑖) = 1 2 � 𝑗 ∈𝑁 (𝑖) |𝑁 (𝑖) ∩ 𝑁 (𝑗)| 1 2𝑑𝑖 (𝑑𝑖 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (13) Due to its significance and simplicity in definition, the clustering coefficient has been widely applied in studying complex networks [33, 160, 166] and extended to directed networks [6, 62], weighted networks [15, 156, 206] and signed networks [46, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Closure coefficient [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The closure coefficient measures the formation of triangles from a new perspective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', with the focal node located at the end of an open-triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different from the ordinary centre node perspective in clustering coefficient (orbit 2 of 𝐺1 in Figure 3, denoted as 𝐺 (2) 1 ), the focal node in closure coefficient serves as the end node of an open triad (orbit type 𝐺 (1) 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The closure coefficient of node 𝑖 is calculated as the fraction of open 12 triads (𝑂𝑇𝐸 (𝑖)), where 𝑖 serves as the end node, that actually forms triangles: C𝐸 (𝑖) = 2 · 𝑇 (𝑖) 𝑂𝑇𝐸 (𝑖) = � 𝑗 ∈𝑁 (𝑖) |𝑁 (𝑖) ∩ 𝑁 (𝑗)| � 𝑗 ∈𝑁 (𝑖) (𝑑𝑗 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (14) Despite this subtle difference in definition, the closure coefficient has very different properties compared to the clustering coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It has been extended to directed networks [94, 197] and weighted networks [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Edge clustering coefficient [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Defined on link-level, the edge clustering coefficient (ECC) evaluates to what extent nodes cluster around the focal edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From a structure formation view, it measures the formation of triangles upon this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The edge clustering coefficient of an edge 𝑒𝑖𝑗 is defined as: C𝑒 (𝑖, 𝑗) = 𝑇 (𝑖, 𝑗) min �𝑑𝑖 − 1,𝑑𝑗 − 1� , (15) where 𝑇 (𝑖, 𝑗) is the number of triangles that 𝑒𝑖𝑗 participates, and min �𝑑𝑖 − 1,𝑑𝑗 − 1� is the number of triangles that edge could possibly form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Based on ECC, a node centrality measure is then defined as the sum of the edge clustering coefficients of all edges connected to it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', C𝑁 (𝑖) = � 𝑗 ∈𝑁𝑖 C𝑒 (𝑖, 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This measure has been proven to be more efficient for identifying essential proteins than many other centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Weighted degree centrality [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Weighted degree centrality (WDC) is also proposed to identify essential proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although this name seems to suggest a close relation to the degree centrality, it is in fact an extension of the edge clustering coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This approach is different in that it takes into account not only the PPI graph data but also the gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, a weight of an interaction is calculated as: C𝑤(𝑖, 𝑗) = C𝑒 (𝑖, 𝑗) + 𝑟 (𝑖′, 𝑗 ′), (16) where C𝑒 (𝑖, 𝑗) is the edge clustering coefficient from the graph data, and 𝑟 (𝑖′, 𝑗 ′) is the Pearson correlation coefficient calculated from the gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Similarly, the weighted degree centrality of a node is then defined as: Θ𝑊 (𝑖) = � 𝑗 ∈𝑁𝑖 C𝑤(𝑖, 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This approach essentially integrates node features when analysing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 4-node subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4-node subgraphs are much more complicated than the 3-node subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There are in total 6 different subgraphs and 11 different orbits in 4-node subgraphs (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Quadrangle coefficients [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Many real networks (such as PPI networks, neural networks and food webs) are naturally rich in quadrangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Quadrangle coefficients, or i-quad coefficient and o-quad coefficient, are thus proposed to measure the formation of quadrangles upon open-quadriads (3-paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As there are two orbits in an open-quadriad (𝐺 (5) 3 and 𝐺 (4) 3 ), i-quad coefficient has the focal node at 𝐺 (5) 3 while o-quad coefficient has the focal node at 𝐺 (4) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the quadrangle coefficients of node 𝑖 are defined as: C𝐼 (𝑖) = 2𝑄(𝑖) 𝑂𝑄𝐼 (𝑖) , C𝑂 (𝑖) = 2𝑄(𝑖) 𝑂𝑄𝑂(𝑖) , (17) where 𝑄(𝑖) is the number of quadrangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑂𝑄𝐼 (𝑖) and 𝑂𝑄𝐼 (𝑖) are number of open-quadriads with 𝑖 as the inner node and outer node respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They are found to be more efficient than 3-node measures in classifying networks and predicting links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 13 – Grid coefficients [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Grid coefficients, including the primary grid coefficient and the secondary grid coefficient, also aim to measure the formation of 4-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The primary grid coefficient measures the formation of “primary quadrilaterals” upon a node and three of its 1-hop neighbours, which is essentially the formation of chordal cycles (𝐺7) from tailed-triangles (orbit 𝐺 (11) 6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the primary grid coefficient of node 𝑖 is defined as: C𝐺𝑝 (𝑖) = 𝑄𝑝 (𝑖) 𝑑𝑖 (𝑑𝑖 − 1)(𝑑𝑖 − 2)/2, (18) where 𝐺𝑝 (𝑖) is the number of chordal-cycles containing 𝑖 and the denominator is the number of possible chordal- cycles built from a node and its three neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The secondary coefficient measures the formation of “secondary quadrilaterals” from a node, two of its 1-hop neighbours and one of its 2-hop neighbours: C𝐺𝑠 (𝑖) = 𝑄𝑠 (𝑖) 𝑑𝑖,2𝑛𝑑𝑑𝑖 (𝑑𝑖 − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (19) Notice, however, in this definition the 2-hop neighbour could be at orbit 𝐺 (4) 3 or at orbit 𝐺 (20) 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The latter essentially involves 5 nodes in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Square clustering coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As triangles (3-cycles) are absent in bipartite networks, the square clustering coeffi- cient is proposed to measure the formation of 4-cycles in the context of bipartite networks [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' What is unusual about this approach is that it views 4-cycles being built from node overlapping instead of node connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the square coefficient of node 𝑖, with a pair of its neighbours 𝑚 and 𝑛, is calculated as: C𝑆 (𝑖|𝑚,𝑛) = 𝑄𝑖𝑚𝑛 (𝑑𝑚 − 𝜂𝑖𝑚𝑛)(𝑑𝑛 − 𝜂𝑖𝑚𝑛) + 𝑄𝑖𝑚𝑛 , (20) where 𝑄𝑖𝑚𝑛 is the number of 4-cycles containing nodes 𝑖, 𝑚, 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' and 𝜂𝑖𝑚𝑛 = 1 + 𝑞𝑖𝑚𝑛 if 𝑚 and 𝑛 are not connected (or 𝜂𝑖𝑚𝑛 = 2 + 𝑞𝑖𝑚𝑛 if 𝑚 and 𝑛 are connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [212] later proposed a modified version of square clustering coefficient: C𝑆𝑍 (𝑖|𝑚,𝑛) = 𝑄𝑖𝑚𝑛 (𝑑𝑚−𝜂𝑖𝑚𝑛)+(𝑑𝑛−𝜂𝑖𝑚𝑛)+𝑄𝑖𝑚𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' With this minor change at the denominator, 4-cycles are now built from connecting nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is mainly applied in community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Interest clustering coefficient [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An interest clustering coefficient is introduced to measure the “clustering of interest links” in directed social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It argues that the best way of defining a relationship between two individuals is through common interests, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', two individuals having links towards a common neighbour will have a higher chance to follow other common neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From a structural view, the interest clustering coefficient essentially measures the formation of bi-fan subgraphs (Table 1) upon open bi-fans: C𝐼 = 4 · # bifan # open-bifan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (21) Note that this metric is defined at network-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The paper finds out that the interest clustering coefficient of Twitter is higher than the traditional directed clustering coefficient, and further proves its usage in a link recommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Beyond 4-node subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some approaches are introduced with a variable subgraph size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In actual usage, however, due to high complexity, they seldom go beyond the size of 6 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Higher-order clustering coefficientsF [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Fronczak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' propose the higher clustering coefficients to evaluate the probabilities that the shortest paths between any two neighbours of node 𝑖 equals 𝑘, when all paths passing 14 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Metrics for 3-node and 4-node subgraph formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3-node/4-node subgraph formation Undirected networks Directed networks Weighted networks clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [184] directed clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [6, 62] wgted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [15, 156, 206] wgted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' signed clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [46, 115] wgted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' directed clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [62] closure coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [196] directed closure coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [94, 197] weighted closure coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [95] edge clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [181] higher-order clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (Fronczak)[67] higher-order clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (Abdo)[2] None None square clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (Lind [124], Zhang [212]) i-quad coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [96] primary grid coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [34] None None o-quad coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [96] None weighted o-quad coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [96] higher-order clustering coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (Yin)[195] None None higher-order closure coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (Yin)[196] None None through node 𝑖 are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Particularly, a clustering coefficient of order 𝑘 for node 𝑖 is calculated as: C𝐻𝐹 (𝑖 | 𝑘) = 2𝐸(𝑖 | 𝑘) 𝑑𝑖 (𝑑𝑖 − 1) , (22) where 𝐸(𝑖 | 𝑘) denotes the number of shortest paths of length 𝑘 between 𝑖’s neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When 𝑘 equals 1, it degrades to the standard clustering coefficient, and when 𝑘 equals 2, it measures the formation of 4-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that each pair of neighbours could have multiple shortest paths of the same length, and only one of them should be counted so that the value of higher-order clustering coefficients is bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 15 – Higher-order clustering coefficientY [195].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The higher-order clustering coefficient proposed by Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' is another generalisation of the traditional clustering coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It aims to measure the formation of higher-order cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, a 𝑘th-order clustering coefficient of node 𝑖 is defined as the probability that a 𝑘-clique plus an edge incident to 𝑖 (termed as 𝑘-wedge) forms a (𝑘 + 1)-clique: C𝐻𝑌 (𝑖 | 𝑘) = 𝑘 · |𝐶𝑘+1(𝑖)| |𝑊𝑘 (𝑖)| = 𝑘 · |𝐶𝑘+1(𝑖)| (𝑑𝑖 − 𝑘 + 1)|𝐶𝑘 (𝑖)|, (23) where 𝐶𝑘+1(𝑖) is the set of (𝑘 + 1)-cliques containing node 𝑖, and 𝑊𝑘 (𝑖) is the set of 𝑘-wedges with 𝑖 as the centre node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The properties of higher-order clustering coefficient in random graph and the small-world model have also been thoroughly investigated [195].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Higher-order closure coefficient [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Higher-order closure coefficient measures the formation of higher-order cliques from a different perspective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the focal node being the end-node of a 𝑘-wedge (instead of the centre-node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The 𝑘th-order closure coefficient of node 𝑖 is thus defined as the fraction of end-node based 𝑘-wedges that are closed (a closed 𝑘-wedge is a (𝑘 + 1)-clique): C𝐻𝐸 (𝑖 | 𝑘) = 𝑘 · |𝐶𝑘+1(𝑖)| |𝑊 ′ 𝑘 (𝑖)| = 𝑘 · |𝐶𝑘+1(𝑖)| � 𝑗 ∈𝑁 (𝑖) [𝐶𝑘 (𝑗) − (𝑘 − 1)𝐶𝑘 (𝑖)] , (24) where 𝐶𝑘+1(𝑖) is the set of (𝑘 + 1)-cliques containing node 𝑖, and𝑊𝑘 (𝑖)′ is the set of 𝑘-wedges with 𝑖 as the end-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Higher-order closure coefficient is proven to be useful in finding seeds for personalised PageRank community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An illustrative summary for most abovementioned approaches is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Global Path Based Approaches Global path based approaches require structural information across the whole network in the form of unbounded paths between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' One set of methods is based on the paths from one node to all other nodes, such as the well known closeness centrality and Katz index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' another set of methods is based on paths between all node pairs, represented by the betweenness centrality (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global Path Betweenness cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Katz index Closeness cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Heatmap cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Reaching cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Edge Btw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Flow btw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='/Communicability btw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Random-walk btw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Gravity cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='/Gravity model 1-to-all all-to-all Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global path based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 One-to-all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The approaches of this type involve the paths from one node to all other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They are also referred to as radial measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 16 – Closeness centrality [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Being one of the most classic centrality measures, closeness centrality is defined as the reciprocal of the average shortest path distance from a focal node 𝑖 to all other nodes: Θ𝐶 (𝑖) = |𝑉 | − 1 � 𝑗 ∈𝑉,𝑗≠𝑖 𝑑(𝑖, 𝑗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (25) Obviously, the original definition is not suitable for graphs with more than one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To address this problem, a modified version of the closeness centrality is defined as [183]: Θ𝐶′(𝑖) = 𝑛 − 1 |𝑉 | − 1 𝑛 − 1 �𝑛−1 𝑗=1 𝑑(𝑖, 𝑗) , (26) where 𝑛 is the number of nodes in one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Due to its intuitiveness in definition, the closeness centrality keeps being applied and extended in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some recent works include the neighbourhood closeness centrality in predicting essential proteins [116], and the backward/forward closeness in studying global value chains [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Katz index [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Unlike the closeness centrality that focuses on shortest paths, Katz centrality of a node considers all paths reaching other nodes with longer paths contributing less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the Katz centrality of a node 𝑖 is calculated as: Θ𝐾 (𝑖) = ∑︁ 𝑗 ∞ ∑︁ 𝑘=1 𝛽𝑘A𝑘 𝑖𝑗, (27) where 𝑘 is a path length and 𝛽 is an attenuation parameter in a range (0, 1 𝜆 ), 𝜆 being the largest eigenvalue of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Further, the overall matrix M = �∞ 𝑘=1(𝛽 · A)𝑘 is an weighted ensemble of all paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thus, M𝑖𝑗 represents the weighted sum of paths from 𝑖 to 𝑗 in all possible hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that this definition is naturally suitable in directed networks and a recent work proposes to generate node embedding of a directed graph by performing a singular value decomposition on the Katz index matrix [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Gravity model [121] /Gravity centrality [131] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Inspired by Newton’s gravity law formula, a gravity model is proposed by viewing the degree of a node as its mass and the shortest path length between two nodes as their distance: Θ𝐺 (𝑖) = ∑︁ 𝑗 ∈𝑉,𝑗≠𝑖 𝑑𝑖 · 𝑑𝑗 𝑑(𝑖, 𝑗)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (28) In order to make it easier to compute in large networks, a modified version limits the radius from the entire network to a certain length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Adopting a similar idea, the gravity centrality is introduced by regarding the coreness of a node as its mass, and the shortest path length between nodes as their distance: Θ′ 𝐺 (𝑖) = ∑︁ 𝑗 ∈𝑁𝑘 (𝑖) 𝑘𝑠(𝑖) · 𝑘𝑠(𝑗) 𝑑(𝑖, 𝑗)2 , (29) where 𝑁𝑘 (𝑖) is the neighbourhood of node 𝑖 within 𝑘-hops, and 𝑘𝑠(𝑖) is the coreness of node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The two approaches are shown to be effective in identifying influential spreaders through analyses of the SIR model on real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Heatmap centrality [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Heatmap centrality measures the influence of a node by comparing the farness of the node with the average farness of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Farness, the reciprocal of closeness, is defined as the sum of the lengths of shortest paths from a node to all other nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', 𝑓 (𝑖) = � 𝑗 ∈𝑉,𝑗≠𝑖 𝑑(𝑖, 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the heatmap centrality 17 of node 𝑖 is quantified as: Θ𝐻𝑀 (𝑖) = 𝑓 (𝑖) − � 𝑗 ∈𝑁 (𝑖) 𝑓 (𝑗) |𝑁 (𝑖)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (30) The intuition of this metric is that if a node has a smaller farness than its neighbours, the probability of information passing through it is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that according to heatmap centrality, a top-ranked node of influence should have the most negative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although the definition of heatmap centrality is more related to the closeness centrality, it is revealed that it is highly correlated with the betweenness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Reaching centrality [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Reaching centrality aims to rank the influence of a node in directed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Intuitively, the reaching centrality of node 𝑖 is quantified as the proportion of nodes that can be reached by the node via outgoing edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the number of nodes with a directed distance from 𝑖, divided by |𝑉 | −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Further, a global reaching centrality is then defined as: 𝐺𝑅𝐶 = � 𝑖 ∈𝑉 [Θ𝑚𝑎𝑥 𝑅 − Θ𝑅(𝑖)] |𝑉 | − 1 , (31) where Θ𝑚𝑎𝑥 𝑅 is the largest reaching centrality of all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The meaning of 𝐺𝑅𝐶 is the difference between the maximum reaching centrality and the average reaching centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global reaching centrality is used as a hierarchy measure for directed networks and is shown to be capable of capturing the degree of hierarchy in both synthetic and real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 All-to-all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The approaches here involve the count of paths between all node pairs, and among them the ones that pass through a focal node or edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They are also referred to as medial measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Betweenness centrality [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Betweenness centrality, or more precisely, the shortest-path betweenness centrality is one of the best-known centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The betweenness centrality of node 𝑖 is quantified as the sum of the fraction of all-pairs shortest paths going through 𝑖: Θ𝐵(𝑖) = ∑︁ 𝑠,𝑡 ∈𝑉 𝜎(𝑠,𝑡 | 𝑖) 𝜎(𝑠,𝑡) , (32) where 𝜎(𝑠,𝑡 | 𝑖) is the number of shortest paths between node pair 𝑠 and 𝑡 that pass through node 𝑖, and 𝜎(𝑠,𝑡) is the number of all shortest paths between 𝑠 and 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is often normalised by ( |𝑉 |−1) ( |𝑉 |−2) 2 , in order to be compared in different networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The betweenness centrality has also been generalised to directed networks[186] and weighted networks [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Edge betweenness centrality [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' With a small modification on the original betweenness centrality, Girvan and Newman propose an edge betweenness centrality in order to detect a community structure in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The edge betweenness centrality of an edge 𝑒 is quantified as the sum of the fraction of all-pairs shortest paths passing through 𝑒: Θ𝐵(𝑒) = ∑︁ 𝑠,𝑡 ∈𝑉 𝜎(𝑠,𝑡 | 𝑒) 𝜎(𝑠,𝑡) , (33) According to the definition, edges which lie between communities will have large edge betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the underlying communities of the network would be uncovered by removing edges of high edge betweenness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It has been widely applied in a community detection task, and some recent applications include the study of anti-vaccination sentiment on Facebook [83] and the analysis of microbial diversity in marine sediment [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 18 – Flow betweenness centrality [66]/ Communicability betweenness centrality [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A major limitation of the betweenness centrality is that it exclusively focuses on the shortest paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In real situations, however, information often takes a more circuitous path randomly or intentionally [171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The flow betweenness addresses this issue by considering all paths between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the flow betweenness centrality of a node 𝑖 is defined as: Θ𝐹 (𝑖) = ∑︁ 𝑠,𝑡 ∈𝑉 𝜙(𝑠,𝑡 | 𝑖) 𝜙(𝑠,𝑡) , (34) where 𝜙(𝑠,𝑡 | 𝑖) is the maximum flow between 𝑠 and 𝑡 that passes through 𝑖, and 𝜙(𝑠,𝑡) is the total flow between 𝑠 and 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The maximum flow is in turn calculated by the minimum cut capacity [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Having established the notion of “capacity ” on links, the flow betweenness centrality is naturally suitable for weighted networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Instead of treating each path equally, the communicability betweenness centrality proposes to reduce the weight for longer paths: 2 (𝑛 − 1)(𝑛 − 2) ∑︁ 𝑠,𝑡 ∈𝑉 �∞ 𝑘=0 1 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝜇𝑘 (𝑠,𝑡 | 𝑖) �∞ 𝑘=0 1 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝜇𝑘 (𝑠,𝑡) , (35) where 𝜇𝑘 (𝑠,𝑡 | 𝑖) is the number of paths between 𝑠 and 𝑡 passing 𝑖 with length 𝑘, and 𝜇𝑘 (𝑠,𝑡) is the number of paths between 𝑠 and 𝑡 with a length 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Random-walk betweenness centrality [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A random-walk betweenness centrality, also known as a current- flow betweenness centrality, is another popular variant of the betweenness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It models information spreading in a network analogous to an electrical current flow in a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the current-flow betweenness centrality of node 𝑖 is defined as the amount of current flowing through 𝑖, averaged over all node pairs: Θ𝐶𝐹 (𝑖) = � 𝑠,𝑡 ∈𝑉 𝐼 (𝑠,𝑡 | 𝑖) (1/2)𝑛(𝑛 − 1) , (36) where 𝐼 (𝑠,𝑡 | 𝑖) is the current flowing from 𝑠 to 𝑡 that passes 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The paper then proves that a message spreading along random walks is equivalent to the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4 Message Passing Based Approaches The above mentioned approaches depend solely on the topological information of a network, such as the number of particular subgraphs, the ratio between two subgraphs, the length of shortest paths, or the number of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Message passing based approaches further consider the information contained in each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From a microscopic point of view, in one iteration, only local information is needed at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is worth noticing that the popular graph convolutional network is also based on this idea, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e, iteratively gathering information from nearby nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Message Passing PageRank LeaderRank Eigenvector cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' HITS Nonbacktracking cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Alpha cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Message passing based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 19 – Eigenvector centrality [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The eigenvector centrality is another classic centrality measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The idea is that a node’s centrality depends on the centralities of its neighbours: 𝑥(𝑖) = 𝑐 ∑︁ 𝑗 ∈𝑁 (𝑖) 𝑥(𝑗), (37) where 𝑐 is a normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The equation is recursive and computed by starting with a set of initial influence scores and iterating the computation until it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In a vectorised form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', �𝑥 = 𝑐A�𝑥, �𝑥 is found to converge to the dominant eigenvector of A and 𝑐 converges to the reciprocal of the dominant eigenvalue of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The eigenvector centrality has some problems in very sparse networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the leading eigenvector is localised around nodes of the highest degree and diminishes the effectiveness of quantifying nodes’ importance [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Nonbacktracking centrality [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The nonbacktracking centrality is proposed to address the above mentioned localisation issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The same as in the eigenvector centrality, a node’s centrality is the sum of its neighbours’ centralities, but now the neighbours’ centralities are calculated without the influence of this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This is achieved by using the nonbacktracking matrix [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The nonbacktracking matrix B, is a 2𝑚 ×2𝑚 matrix, defined on the directed edges of the graph (undirected edges are converted to bidirectional edges), and elements B𝑖→𝑗,𝑘→𝑙 = 𝛿𝑖,𝑙 (1 − 𝛿𝑗𝑘), where 𝛿 is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, 𝑒𝑗→𝑖 of the leading eigenvector of B gives the centrality of node 𝑗 ignoring the contribution of 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the nonbacktracking centrality of node 𝑖 is 𝑥(𝑖) = � 𝑗 A𝑗𝑖𝑒𝑗→𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The eigenvalues of the nonbacktracking matrix are also found to be useful in a community detection task [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Alpha centrality [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When the eigenvector centrality is applied in directed networks, a node’s centrality is determined by those who pointed at it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thus, the vector form becomes: �𝑥 = 1 𝜆 A𝑇 �𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The problem is that nodes with no incoming edges would have zero centrality value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The alpha centrality proposes to solve this problem by taking into account the "external status characteristics".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The equation then becomes: �𝑥 = 𝛼A𝑇 �𝑥 + �𝑒, (38) where �𝑒 is a vector of the exogenous sources of characteristics and 𝛼 is a parameter which reflects the relative importance of a topological structure versus exogenous factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – PageRank [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PageRank, a popular variation of the eigenvector centrality, is proposed to rank the importance of web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Web pages and the links among them are modelled as a directed network, and a page should have a high rank if the sum of the ranks of pages that point to it is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the rank of page 𝑖 is calculated as: 𝑟 (𝑖) = 𝑐 ∑︁ 𝑗 ∈𝑁 𝑖𝑛 𝑖 𝑟 (𝑗) 𝑑𝑜𝑢𝑡 𝑗 , (39) where 𝑁𝑖𝑛 𝑖 is the set of pages pointing to 𝑖 (𝑖’s in-neighbours), and 𝑑𝑜𝑢𝑡 𝑗 is out-degree of page 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to deal with the “rank sink” problem, where several pages form a loop without other outgoing links, a source of the rank is introduced over all pages (also viewed as a random jumping factor), denoted as a vector �𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the rank of page 𝑖 becomes: 𝑟 (𝑖) = 𝑐(� 𝑗 ∈𝑁 𝑖𝑛 𝑖 𝑟 (𝑗) 𝑑𝑜𝑢𝑡 𝑗 + 𝑒(𝑖)), and the corresponding vector form is �𝑟 = 𝑐(A𝑇 + �𝑒 × 1)�𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The PageRank has also been extended to weighted networks [191], on nonbacktracking matrix [9], and applied to many different areas [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 20 – HITS [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Unlike the PageRank which focuses on pages having many incoming links, HITS, abbreviated from a hyperlink induced topic search, proposes to distinguish two roles in the hyperlink structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', authorities and hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Authorities are reliable information sources, and hubs are the websites pointing to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Based on the intuition that an authority should be pointed to by hubs and a hub should point to authorities, an authority weight and a hub weight of page 𝑖 are thus defined in a mutually dependent manner: 𝑎(𝑖) = ∑︁ 𝑗 ∈𝑁 𝑖𝑛 𝑖 ℎ(𝑗) ℎ(𝑖) = ∑︁ 𝑗 ∈𝑁 𝑜𝑢𝑡 𝑖 𝑎(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (40) The corresponding vector forms are: �𝑎 = A𝑇 �ℎ, and �ℎ = A�𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' �𝑎 and �ℎ are updated iteratively, and it is proven that �𝑎 converges to the leading eigenvector of A𝑇 A, and �ℎ converges to the leading eigenvector of AA𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Based on HITS, ARC (Automatic Resource Compilation) later proposes to incorporate textual information around the link by assigning each link a weight [38], and Co-HITS proposes to extend the idea to bipartite networks [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – LeaderRank [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to solve the above mentioned rank sink problem, the LeaderRank proposes to add a ground node that connects to other nodes via bidirectional links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the beginning, each node other than the ground node is initialised by one unit of score, and the ground node is initialised to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, the same as the PageRank, at each iteration, the score of node 𝑖 is calculated as: 𝑠(𝑖)(𝑡) = 𝑐 � 𝑗 ∈𝑁 𝑖𝑛 𝑖 𝑠 (𝑗) (𝑡−1) 𝑑𝑜𝑢𝑡 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' After the scores of all nodes reach a steady state, the score of the ground node will be distributed evenly to other nodes, and thus the final score of node 𝑖 is: 𝑠(𝑖) = 𝑠(𝑖)𝑐 + 𝑠(𝑔)𝑐 |𝑉 | , (41) where 𝑠(𝑖)𝑐 is the steady score of node 𝑖, and 𝑠(𝑔)𝑐 is the steady score of the ground node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A major advantage of the LeaderRank is that it has no additional parameter that needs to be optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some interesting extensions of the LeaderRank include the weighted LeaderRank that assigns degree-dependent weights onto links associated with the ground node [119] and the adaptive LeaderRank that introduces H-index into the weighted mechanism [194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='5 Hybrid Approaches The methods in the fifth and final category are combinations of previously introduced approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Hybrid ClusterRank HybridRank BridgeRank Local structural cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Local triangle struc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' CCPA Hybrid degree cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Hybrid Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – ClusterRank [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Previous studies have shown that a large clustering coefficient may slow the spreading process of disease in the entire network [59, 221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A ClusterRank thus proposes to consider not only the number of a node’s neighbours, but also the negative effect of local clustering when identifying influential nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The ClusterRank score of node 𝑖 is defined as: Θ𝐶𝑅(𝑖) = 𝑓 (𝑐𝑖) ∑︁ 𝑗 ∈𝑁 𝑜𝑢𝑡 𝑖 (𝑑𝑜𝑢𝑡 𝑗 + 1), (42) 21 where 𝑐𝑖 = � 𝑗∈𝑁𝑜𝑢𝑡 𝑖 |𝑁 𝑜𝑢𝑡 (𝑖)∩𝑁 (𝑗) | 𝑑𝑜𝑢𝑡 𝑖 (𝑑𝑜𝑢𝑡 𝑖 −1) is a modified version of clustering coefficient in directed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑓 (𝑐𝑖) is a function that is negatively correlated with 𝑐𝑖, for example an exponential function 𝑓 (𝑐𝑖) = 10−𝑐𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although the ClusterRank is proposed for directed networks, it can be easily extended to undirected networks [41] and weighted networks[182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Experiments on several real networks demonstrate that the ClusterRank score outperforms the PageRank and the LeaderRank while being more efficient in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Local structural Centrality [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Aiming to evaluate the spreading ability of nodes, a local structural centrality essentially extends the local centrality (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2) by further considering the connections between higher-order neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The idea is that a node has a better spreading ability when its neighbours are better connected because a neighbour node can be affected directly by the source node or indirectly by another neighbour node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The local structural centrality of node 𝑖 is defined as: Θ𝐿𝑆 (𝑖) = ∑︁ 𝑗 ∈𝑁𝑖 (𝛼|𝑁 1,2 𝑗 | + (1 − 𝛼) ∑︁ 𝑘 ∈𝑁 1,2 𝑗 𝑐(𝑘)), (43) where 𝑁 1,2 𝑗 is the node set of 1-hop and 2-hop neighbours of node 𝑗, and 𝑐(𝑘) is the clustering coefficient of node 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝛼 is a tunable parameter between 0 and 1, balancing a direct and indirect spreading contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that the part of the clustering coefficient is multiplied in the ClusterRank when evaluating spreading speed, but added up here when measuring the spreading ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Local triangle structure centrality [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A local triangle structure centrality (LTSC) proposes to include the triangle proportion of a node, instead of its clustering coefficient when evaluating a node’s spreading ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The triangle proportion is able to indicate the location of a node, whether it is located in a denser or sparser part of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' LTSC partitions the spreading ability into two parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', inner spreading ability and outer spreading ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the local triangle structural centrality of node 𝑖 is defined as: Θ𝑇𝑆 (𝑖) = ∑︁ 𝑗 ∈𝑁𝑖 (𝑑𝑗 (1 +𝑇𝑃(𝑗)) + ( ∑︁ 𝑘 ∈𝑁𝑗 𝑑𝑘 − 𝑑𝑗)), (44) where 𝑇𝑃(𝑗) is the triangle proportion of node 𝑗, calculated by the number of triangles containing 𝑗 divided by the total number of triangles in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For each neighbour 𝑗 of a given node 𝑖, the part of 𝑑𝑗 (1 +𝑇𝑃(𝑗) is to measure its inner spreading ability, and the part of � 𝑘 ∈𝑁𝑗 𝑑𝑘 − 𝑑𝑗 is to measure its outer spreading ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the local triangle structure centrality of node 𝑖 is the sum of the spreading abilities of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – Hybrid degree centrality [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The spreading probabilities of networks describing diseases, opinions, and rumours should obviously differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Most existing centrality measures, however, fail to take that into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The per- formance of centrality measures is sensitive to the spreading probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The degree centrality, for example, works best with modest spreading probabilities, while the local centrality (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2) works better with higher ones [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to alleviate the sensitivity to different spreading probabilities, a hybrid degree centrality is introduced by integrating the degree centrality and a modified local centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The hybrid degree centrality of node 𝑖 is defined as: Θ𝐻𝐷 (𝑖) = (𝛽 − 𝑝) · 𝛼 · Θ𝐷 (𝑖) + 𝑝 · Θ′ 𝐿𝑅(𝑖), (45) 22 where Θ′ 𝐿𝑅(𝑖) = Θ𝐿𝑅(𝑖) − 2 � 𝑗 ∈𝑁𝑖 |𝑁𝑗 | is the modified local centrality, 𝑝 is the spreading probability, 𝛼 and 𝛽 are two tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The part contributed by the degree centrality is viewed as a near-source influence, and the part of modified local centrality is a distant influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – HybridRank [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A HybridRank proposes to identify influential spreaders by combining the neighbourhood coreness centrality (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1) and the eigenvector centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The reason for integrating these two measures is that they both regard a node as influential if the node is connected to other influential nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The hybrid centrality of node 𝑖 is defined as: Θ𝐻𝑅(𝑖) = Θ𝑁𝐶 (𝑖) × Θ𝐸 (𝑖), (46) where Θ𝑁𝐶 (𝑖) = � 𝑗 ∈𝑁𝑖 𝑘𝑠(𝑗) is the neighbourhood coreness of 𝑖, and Θ𝐸 (𝑖) is the eigenvector centrality of node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The HybridRank algorithm further suggests that when selecting influential spreaders, the neighbours of selected ones should be neglected in order to maximise the spreading range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The effectiveness of the HybridRank has also been tested in real networks using a SIR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – BridgeRank [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to lower the time complexity of the closeness centrality while keeping comparable performance, a BridgeRank proposes to compute the shortest paths to just a few core nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the BridgeRank algorithm, at first, communities are identified by the Louvain algorithm [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, core nodes are discovered through calculating the betweenness centralities within each community (one core node per community).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the BridgeRank centrality of each node is defined as a filtered closeness centrality to these core nodes: Θ𝐵𝑅(𝑖) = 1 � 𝑗 ∈C 𝑑(𝑖, 𝑗) , (47) where C is the set of identified core nodes in each community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The time complexity is therefore reduced from 𝑂(|𝑉 |3) to 𝑂(|𝑉 |𝑙𝑜𝑔|𝑉 |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A modified version that allows multiple core nodes being selected in a community is also introduced [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Other community structure based methods include 𝑘-medoid that uses information transfer probabilities between any node pairs [215], and the influence maximization algorithm based on label propagation [218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' – CCPA [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A common neighbour and centrality based parameterised algorithm, or CCPA, is an approach for a link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It aims to bring together two essential properties of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the common neighbours and the closeness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The similarity score between a pair of nodes 𝑖 and 𝑗 is defined as: 𝑠(𝑖, 𝑗) = 𝛼 · (|𝑁𝑖 ∩ 𝑁𝑗 |) + (1 − 𝛼) · |𝑉 | 𝑑(𝑖, 𝑗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (48) |𝑁𝑖 ∩ 𝑁𝑗 | is obviously the part of common neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' |𝑉 | 𝑑 (𝑖,𝑗) , reciprocal of the normalised distance between two nodes, is deemed as the closeness centrality of them, since it has a similar form as the classic node closeness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝛼 ∈ [0, 1] is a user-defined parameter controlling the weight of the two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Experiments on real-world datasets suggest that the change in performance (measured in average AUC) caused by the change of 𝛼 is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='6 Discussion and Outlook To end this section, we further discuss graph structural measures in different types of networks and highlight some research avenues for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We then briefly talk about the importance and role of reviewing traditional structural measures in the following part of the survey on GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 23 Dynamic Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Most approaches covered in the survey assume that networks are static or time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Many real-world networks, however, are in fact dynamic, nodes and edges appearing and disappearing over time [84, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In telecommunication networks, the connection between agents is often bursty and fluctuates across time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' in social networks, relationships among people are typically intermittent and recurrent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' in transportation networks, the frequency of public transport service is usually higher in rush hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This extra dimension of time adds richness and complexity to the graph representation of a system, necessitating the development of more advanced approaches that can leverage temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Many studies have generalised the classic graph structural measures to dynamic networks, including temporal degree centrality[104], temporal clustering coefficient [153], temporal closeness and betweenness centrality [103], temporal eigenvector centrality [174], temporal Katz centrality [153], temporal motifs [112, 159] and temporal graphlets [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Despite the large number of structural measures proposed for dynamic networks, there are still many open questions to be tackled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, what is the impact of the temporal network’s structure on the dynamics of processes that occur on it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' how to apply temporal measures in inferring spreading chains in incomplete temporal networks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Multilayer Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Sometimes, systems are so complicated that multiple-layered networks are needed to better represent and study them [20, 22, 48, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, a multilayer social network incorporates both friendship and financial relationships among individuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' a multilayer brain network contains both the anatomical brain layer and functional brain layer, and a multilayer transportation network integrates all sorts of transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Since interlayer connections cause new structural and dynamic correlations between components, neglecting them or simply aggregating over layers will alter the original topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, it is desirable to develop structural measures taking interlayer relationships into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Not surprisingly, fundamental single-layer approaches have been largely generalised to multilayer networks, such as multilayer degree, clustering coefficient, closeness and betweenness centralities, [22, 48, 55], multilayer motifs and graphlets [17, 54], multilayer eigenvector, PageRank and HITS centralities [49, 50, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some tailor-made approaches for multilayer networks are also recently introduced, for example, the minimal-layers power community index [16], and the singular vector of tensor centrality [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The study of multilayer structures, however, is still in an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There is still much room for developing new cross-layer structural approaches that better model inter-layer spreading processes [168] and captures multiplex dynamics, and controllability [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Node/edge attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Network data, besides the pure topological presence, are often accompanied by rich information on node attributes and/or edge attributes, and they are also referred to as labelled networks or attributed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Most structural measures, as the name suggests, focus solely on capturing the part of topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Theoretically, message passing approaches are able to include numeric node attributes, such as the initial rank and source of rank in the PageRank [30], or the endogenous and exogenous status in the alpha centrality [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In practice though, these features are usually set to identical values for all nodes, for example, all ones for the initial rank and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='15 for the source of rank in the PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Multidimensional features are not supported in message passing approaches either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There have also been attempts to integrate node/edge attributes with other graph structural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For instance, the degree and betweenness centralities are combined with node attributes in studying criminal networks [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' nodes’ attributes are used as a threshold in LRIC index [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' and node/edge attributes are fused into graphlets [93, 165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We believe there is still great potential for developing novel structural approaches that integrate rich information on nodes and/or edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is also worth mentioning that one reason for the popularity of graph neural networks is that it naturally enables integrating node attributes and some recent works also propose to take edge attributes into account in GNNs [42, 72, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 24 Finally, we discuss how the traditional structure-based approaches are linked to GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The importance and role of reviewing traditional structural measures in the survey of GCNs are Multifaceted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' First, traditional structural approaches, the outcome of decades of Network Science studies, are the precursors and foundations of graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, the key idea of neighbourhood aggregation and message passing in GCNs can trace back to 1972 when Bonacich proposed the eigenvector centrality [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Basic network science notions such as the clustering coefficient, motifs and graphlets are utilised in GCNs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Second, the taxonomy of traditional approaches from the perspective of structure information inspired us to develop a new taxonomy for GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We will see later how the taxonomy of GCNs from a layer-wise message aggregation scope is similar to that of subgraph count based measures in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Third and last, a comprehensive review of traditional structural measures not only helps in revealing their connections to GCN approaches but also benefits the discovery of knowledge gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We will see that some GCN approaches are inspired by the traditional message passing based approaches, and that many subgraph count based approaches find their usages in GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, the connections between GCNs and subgraph formation based or global path based approaches are still largely left undiscovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4 STRUCTURE INFORMATION IN GRAPH CONVOLUTIONAL NETWORKS After summarising the traditional Network Science structural measures, we are set to review the graph convolutional networks from a novel perspective of graph structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In recent years, graph neural networks, especially graph convolutional networks, have become one of the most prominent research areas in the study of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It extends the traditional convolutional neural networks to graph data and enables an effective combination of the rich node features information and graph topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graph convolutional networks have been successfully applied in different types of graph learning tasks, including node classification, link prediction, graph classification and graph clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Amongst the large family of graph deep learning approaches [135, 216], we particularly focus on graph convolutional networks not only because they have a wider range of applicability, but also because they are the bases of many other graph deep learning approaches, including graph autoencoders, graph reinforcement learning, graph adversarial methods, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There exist several comprehensive surveys on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Bronstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [32] provide a thorough review of geometric deep learning, which presents its problems, difficulties, solutions and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [79] develop a unified encoder-decoder framework for graph representation learning approaches, bringing together matrix factorisation-based methods, random-walk-based algorithms and graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Chami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [39] later extend the framework by including more recent advancements in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [214] propose a comprehensive review specifically on graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [219] introduce a detailed taxonomy after dividing GNNs into several modules, including the propagation module, the sampling module and the pooling module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [189] propose to divide GNNs into four categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', recurrent GNNs, convolutional GNNs, graph autoencoders and spatial-temporal GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' These reviews, when introducing convolutional neural networks, usually focus on the domain of convolutional operations and propose a dichotomy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the spectral-based methods and the spatial-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, the line between the two is sometimes blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, GCN is an approximation of spectral graph convolutions, but it operates directly on graphs — applying filters acting on the k-hop neighbourhood of the graph in the spatial domain [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another recent work also proves that spectral convolutional graph neural networks can be viewed as a particular case of spatial convolutional neural networks [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 25 Different from existing reviews, in this survey we primarily, but not exclusively, focus on how local structure plays its role in graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' we propose to categorise GCN approaches from three different perspectives, which are the layer-wise message aggregation scope, the message content, and the overall learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Layer-wise message aggregation scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Analogous to convolutional neural networks, multilayer architecture is one of the key features in graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking the vanilla GCN for example, at each layer, a node gathers information from its 1-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then from stacking 𝑘 layers, the node would convolve its 𝑘𝑡ℎ- order neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thereafter, many other approaches propose to apply different scope at each layer, including 2-hop neighbourhood, k-hop neighbourhood, local-random-walk neighbourhood, subgraph neighbourhood, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This first structural perspective in GCN design can be summarised into the following question: From where a node aggregates message at each layer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The detailed taxonomy of GCNs from the perspective of layer-wise message aggregation scope and related approaches are given in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Message content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Compared to traditional deep learning models such as CNNs and RNNs, the strength of GCNs comes from the ingenious integration of graph structure and node features — node features are passed through the edges of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In many cases, the feature of nodes is independent of graph structure, such as numerical ratings, word vectors generated from sentences, positional gene sets, immunological signatures, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Meanwhile, there are emerging works that include other structural information as part of node features, from the simplest node degree to more complicated distance or subgraph information [27, 78, 117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This second structural perspective in GCN design can be summarised into the question: What structural information is included in the node feature when initialising or running the message passing scheme?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The detailed taxonomy from the message content perspective and the related approaches are given in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Traditional graph representation learning approaches are generally based on matrix factorisation, which thus requires the fixed whole graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although the original GCN approach also takes the whole graph’s adjacency matrix as input, it has soon been extended to various settings, such as subgraphs, localised subgraphs, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To put in a question format, the third structural perspective in a GNN design is: Where GCNs are trained on?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' or What is/are the input graph/graphs in GCNs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The detailed taxonomy of GCNs from the learning scope perspective and the related approaches are given in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Layer-wise message scope To begin with, we discuss in detail the first structural perspective in a GCN design, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a layer-wise message scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' By answering the question of where a node aggregates message from at each layer, we divide existing GCN approaches into four categories, which are 1-hop neighbourhood approaches, k-hop neighbourhood approaches, local-random-walk neighbourhood approaches, and subgraph neighbourhood approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The taxonomy and representative approaches are given in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The colour of the block indicates what task the approach is proposed for: grey is the most common node classification task, orange is a network classification, and blue is the link prediction task which will appear later in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that a graph representation can be readily obtained via graph pooling, so approaches proposed for a node classification can potentially be applied in a network classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Likewise, some approaches proposed for the network classification also generate node representations, making them possible to be used in the node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 1-hop neighbourhood approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Many influential GCN approaches adopt the 1-hop neighbourhood aggregation strategy, where a node’s representation is iteratively updated through aggregating representations of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 26 Layer-wise Message Aggregation Scope (Where a node aggregates message from) k-hop neighbourhood Random-walk neighbourhood Subgraph neighbourhood DGCN MixHop k-hop GNN PinSage GraLSP k-GNN GCN GraphSAGE GAT GIN Adapt FastGCN PATCHY-SAN DGCNN GRAPE GraphSNN SGC 1-hop neighbourhood DGP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taxonomy from the Layer-wise Message Aggregation Scope perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' One iteration happens at one convolutional layer, and after stacking multiple layers, the node’s representation is able to capture a wider range of neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Motivated by the first-order approximation of localised spectral filters on a graph [52], GCN proposes the following layer-wise propagation rule operating directly on graphs: 𝐻 (𝑙) = 𝜎 � ˆ𝐴𝐻 (𝑙−1)𝑊 (𝑙)� (49) , where ˆ𝐴 = ˜𝐷− 1 2 ˜𝐴 ˜𝐷− 1 2 , ˜𝐴 is adjacency matrix with added self-connections, and ˜𝐷 is degree matrix of ˜𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝐻 (𝑙) is the representations at the 𝑙𝑡ℎ layer, and 𝑊 (𝑙) is the learnable weight matrix at 𝑙𝑡ℎ layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝜎 denotes a nonlinear activation function such as ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The multiplication of the normalised self-connection added adjacency matrix ˆ𝐴 and the nodes’ representation matrix 𝐻 represents a normalised sum of neighbouring nodes’ (and self node’s) representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From a microscopic point of view, the representation of node 𝑣 at layer 𝑙 is calculated as: ℎ(𝑙) 𝑣 = 𝜎 �� � ∑︁ 𝑢∈N(𝑣) 1 𝑐𝑣𝑢 ℎ(𝑙−1) 𝑢 𝑊 (𝑙)�� � , (50) where N (𝑣) is the set of node 𝑣’s one-hop neighbours (with added self-loops to each node), 𝑐𝑣𝑢 = √︁ |N (𝑣)| √︁ |N (𝑢)| is the normalization constant based on the node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The loss is then computed as: L = − � 𝑙 ∈Y𝐿 �𝐹 𝑓 =1 𝑌𝑙 ln𝑍𝑙𝑓 , where Y𝐿 is the set of labelled nodes, 𝑍 is the output embedding and 𝐹 are the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Successfully bringing convolutional operations on graphs, GCN has become one of the most popular graph representation learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is worth mentioning that the Iterative Classification Algorithm (ICA) also uses neighbourhood information to train a model [19, 150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The model is then used to iteratively update the labels of nodes in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Obviously, without a multi-layer convolutional network, the scope of ICA in the training stage is strictly limited within the immediate neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GraphSAGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' later proposed the GraphSAGE framework, which extends the GCN to a more general setting that supports a mini-batch approach and different aggregation functions [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the representation of 27 node 𝑣 at layer 𝑙 is given by: h𝑙 N(𝑣) ← AGGREGATE 𝑙 �� h𝑙−1 𝑢 , ∀𝑢 ∈ N (𝑣) �� , h𝑙 𝑣 ← 𝜎 � W𝑙 · CONCAT � h𝑙−1 𝑣 , h𝑙 N(𝑣) �� (51) The framework thus gives us the flexibility to choose different aggregator functions, such as mean aggregator (equivalent to GCN), LSTM aggregator and pooling aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Further, unlike GCN which requires full batch gradient descent, GraphSAGE enables mini-batch setting and therefore can also be applied to unseen nodes (also known as inductive learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In addition, GraphSAGE proposes to sample a fixed-size of neighbours around each node in their aggregation scheme, instead of using all neighbours, which helps to keep the computational cost of each batch fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although GCN and GraphSAGE have achieved excellent performances in graph learning tasks, especially in node classification tasks, they are unable to distinguish some simple graph structures due to their limits in the neighbourhood aggregation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graph Isomorphism Network architecture (GIN) [193] is proposed to overcome this shortcoming and is proven to be as powerful as the Weisfeiler-Lehman graph isomorphism test [185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, in order to achieve the same discriminative power as the Weisfeiler-Lehman test, the representation of node 𝑣 at layer 𝑙 should be as: ℎ(𝑙) 𝑣 = 𝜙 (𝑙) � ℎ(𝑙−1) 𝑣 , 𝑓 (𝑙−1) �� ℎ(𝑙−1) 𝑢 : 𝑢 ∈ N (𝑣) ��� , (52) where 𝑓 (𝑙−1) is a function operating on multisets and 𝜙 (𝑙) is an injective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The choice of multiset on neighbour- hood information aggregation, instead of mean pooling in GCN or max pooling in GraphSAGE, enables it to better preserve neighbourhood structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The above representation is then proven to be equivalent to: ℎ(𝑙) 𝑣 = MLP(𝑙) �� � � 1 + 𝜖 (𝑙)� ℎ(𝑙−1) 𝑣 + ∑︁ 𝑢∈N(𝑣) ℎ(𝑙−1) 𝑢 �� � , (53) where 𝜖 (𝑙) is a scalar representing the importance of the focal node, and MLP is used to model the composition of the function 𝑓 and 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although in the aggregation scheme of the GCN, nodes from the same neighbourhood are assigned different weights by introducing the normalisation term (𝑐𝑣𝑢 in Equation 50), the approach lacks the flexibility of introducing other weight mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To overcome this shortcoming, GAT proposes to use a masked self-attentional layer on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' “Masked” means that only 1-hop neighbours, rather than all other nodes, of a given node, are included in the attention scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the attention coefficient of an edge 𝑒𝑣𝑢 at layer 𝑙 is given by: 𝛼 (𝑙) 𝑣𝑢 = exp � 𝑒 (𝑙) 𝑣𝑢 � � 𝑤∈N(𝑣) exp � 𝑒 (𝑙) 𝑣𝑤 � ,𝑒 (𝑙) 𝑣𝑢 = LeakyReLU � �𝑎(𝑙) [W(𝑙)ℎ(𝑙) 𝑣 ∥W(𝑙)ℎ(𝑙) 𝑢 ] � , (54) where 𝑎(𝑙) is a shared feedforward neural network parameterised by a weight vector �𝑎, and W(𝑙) is a shared linear transformation of input or hidden features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then the representation of node 𝑣 at layer (𝑙 + 1), is obtained through applying the attention coefficients on 𝑣’s neighbour nodes: ℎ(𝑙+1) 𝑣 = 𝜎 �� � ∑︁ 𝑗 ∈N(𝑖) 𝛼 (𝑙) 𝑣𝑢 W(𝑙)ℎ(𝑙) 𝑢 �� � (55) FastGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' One issue of the GCN’s neighbourhood aggregation scheme is the quick neighbourhood expansion across layers, which largely limits its scalability in large and dense graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To address this problem, FastGCN [43] proposes to sample a fixed number of nodes at each layer while applying neighbourhood aggregation, so the number of involved 28 nodes is up-bounded by the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the representation of nodes at layer 𝑙 is given by: 𝐻 (𝑙+1) (𝑣, :) = 𝜎 �� � 𝑛 𝑠 𝑠∑︁ 𝑗=1 ˆ𝐴 � 𝑣,𝑢 (𝑙) 𝑗 � 𝐻 (𝑙) � 𝑢 (𝑙) 𝑗 , : � 𝑊 (𝑙)�� � , (56) where 𝑠 is the sample size and 𝑛 is the total number of nodes in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Compared to the node-wise sampling strategy proposed by GraphSAGE, this layer-wise sampling method further improves the computational efficiency of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, in a 2-layer setup, when 10 nodes are sampled from a node’s neighbourhood, there will be a total of 102 = 100 nodes involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In contrast, when 10 nodes are sampled at each layer, the total number of involved nodes is at most 10 ∗ 2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GraphSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A common feature of the above-mentioned approaches is that each node gathers information from its neighbours, that is to say treating the neighbourhood as a 1-hop subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A recent work argues that this scheme ignores the rich structure information among the neighbour nodes, and therefore proposes a model named GraphSNN to treat the neighbourhood as a 1-hop subgraph by including the connections among neighbours [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the work first defines “structure coefficients” for each node and its neighbours and generate a weighted adjacency matrix 𝐴𝑣𝑢 = 𝑤 (𝑆𝑣,𝑆𝑣𝑢), where 𝑆𝑣 is 1-hop neighbourhood subgraph of node 𝑣, and 𝑆𝑣𝑢 is overlap subgraphs of node 𝑣 and 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑤 is a function on 𝑆𝑣 and 𝑆𝑣𝑢 exhibiting properties of local closeness and local denseness, which is designed as |𝐸𝑣𝑢 | |𝑉𝑣𝑢 |·|𝑉𝑣𝑢−1| |𝑉𝑣𝑢|𝜆 in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝜆 is a positive value chosen by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, the representation of node 𝑣 at layer 𝑙 is generated by: ℎ(𝑙) 𝑣 = MLP(𝑙) �� � 𝛾 (𝑙−1) �� � ∑︁ 𝑢∈N(𝑣) ˜𝐴𝑣𝑢 + 1�� � ℎ(𝑙−1) 𝑣 + ∑︁ 𝑢∈N(𝑣) � ˜𝐴𝑣𝑢 + 1 � ℎ(𝑙−1) 𝑢 �� � , (57) where 𝛾 (𝑙−1) is a learnable scalar parameter, and ˜𝐴𝑣𝑢 is the normalised weighted adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The part before ℎ(𝑙−1) 𝑣 signifies the focal node’s self-importance while the part � 𝑢∈N(𝑣) � ˜𝐴𝑣𝑢 + 1 � before ℎ(𝑙−1) 𝑢 is to apply different weights on different neighbours based on the overlap subgraph between the focal node and the neighbour node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From this perspective, GraphSNN is also an attention-like scheme that takes the 1-hop subgraph structure into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DGCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to apply a GCN on graph-level learning tasks, Deep Graph Convolutional Neural Network (DGCNN) proposes to sort and pool the nodes’ representations from multiple graph convolutional layers, then pass them to a traditional CNN architecture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a one-dimensional convolutional layer followed by dense layers before the final softmax output layer [210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As the GCN can be viewed as “a differentiable and parameterised generalisation of the 1-dim Weisfeiler-Lehman algorithm” [105], each node’s representation can be viewed as a “continuous colour” at that layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The order of nodes in DGCNN is thus calculated according to the nodes’ representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', nodes’ colours, at the graph convolutional layers (first comparing the representations at the last layer, then the representations at the second-to-last layer when some nodes have the same representation, and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Next, in order to fit into the following CNN architecture, the sorted nodes’ representation needs to be truncated or extended, which is done by deleting excessive rows or adding zero rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This bridge layer between GCN and CNN is also known as SortPooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 k-hop neighbourhood approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A natural idea to improve the performance of the GCN is to expand its message aggregation scope at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This leads us to the second subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', k-hop neighbourhood approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' MixHop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The layer-wise message passing scope of the vanilla GCN is limited to 1-hop neighbours and therefore lacks the ability to mix latent information from neighbours at different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' MixHop is proposed to address this issue through a higher-order message passing scheme that aggregates information from further neighbours [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the convolutional layer is defined as: 29 𝐻 (𝑖+1) = 𝜎 �����𝑗 ∈𝐾 � 𝐴𝑗𝐻 (𝑖)𝑊 (𝑖) 𝑗 � , (58) where 𝐾 is a set of integers representing the scope, and ∥ denotes column-wise concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When 𝐾 = 1, the operation degrades to the vanilla GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The paper also proves theoretically that the vanilla GCN cannot recover a 2-hop delta operator and thus cannot represent a general layer-wise neighbourhood mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In contrast, MixHop is able to learn a general mixing of information from neighbours at various distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Their experiments on a synthetic dataset show that MixHop performs significantly better than several baselines on graphs of low levels of homophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' k-hop GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A simple example of the limitation of the 1-hop neighbourhood aggregation is that it cannot distinguish regular graphs of the same size and degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to improve the expressivity of the vanilla GCN, k-hop GCN also proposes to take k-hop neighbours into consideration in the layer-wise aggregation scheme [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The general model is presented as: 𝑎(𝑙) 𝑣 = AGGREGATE(𝑙) �� ℎ(𝑙−1) 𝑢 | 𝑢 ∈ N𝑘 (𝑣) �� , ℎ(𝑙) 𝑣 = MERGE(𝑙) � ℎ(𝑙−1) 𝑣 ,𝑎(𝑙) 𝑣 � , (59) where N𝑘 (𝑣) denotes the k-hop neighbourhood of node 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, it adopts an outside-to-inside updating scheme in the aggregation part: gradually updating neighbouring nodes from the furthest to the immediate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Each neighbour node 𝑢 at a distance 𝑑 from the focal node 𝑣 goes through two update functions successively: 𝑥𝑢 = UPDATE(𝑙) 𝑑,𝑎𝑐𝑟𝑜𝑠𝑠 (𝑢, N1(𝑢) ∩ 𝑅𝑑+1(𝑣)), 𝑥𝑢 = UPDATE(𝑙) 𝑑,𝑤𝑖𝑡ℎ𝑖𝑛(𝑢, N1(𝑢) ∩ 𝑅𝑑 (𝑣)), (60) where 𝑅𝑑+1(𝑣) or 𝑅𝑑 (𝑣) denote the set of nodes that are at a distance 𝑑 + 1 or 𝑑 from node 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The first function learns representation from node u’s neighbours that are (𝑑 + 1)-hop away from node 𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' and the second function learns from node u’s neighbours that are 𝑑-hop away from node 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The update functions are defined as: UPDATE(𝑢,𝑆) = MLP1 (MLP2 (𝑥𝑢) + � 𝑤∈𝑆 MLP3 (𝑥𝑤)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the representation of a node𝑣 is calculated as:ℎ(𝑙) 𝑣 = UPDATE(𝑙) 0,𝑎𝑐𝑟𝑜𝑠𝑠 (𝑣, N1(𝑣)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although this model can capture structural information from the k-hop neighbourhood at a single layer, it requires up to 2𝑘 update functions and the aggregation scheme is much more complicated and computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Similar to the FastGCN, Adaptive Sampling GCN (abbreviated as Adapt) adopts the layer-wise sampling strategy in order to accelerate the training of the GCN [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Lower layer sampling is conditioned on the higher layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Compared to node-wise sampling, layer-wise sampling not only has a fixed number of nodes at each layer but also preserves the connections between lower-layer neighbours and higher-layer parent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Furthermore, the approach proposes to aggregate information from distant nodes via skip connections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', connecting layer 𝑙 + 1 with layer 𝑙 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the skip-connection representation of node 𝑣 at layer (𝑙 + 1) is formulated as: ℎ(𝑙+1) 𝑣skip = ∑︁ 𝑠 ∈V (𝑙−1) ˆ𝑎𝑠𝑘𝑖𝑝 (𝑣,𝑠) ℎ(𝑙−1) 𝑠 𝑊 (𝑙−1) skip , (61) where 𝑠 denotes sampled nodes at layer (𝑙 − 1), ˆ𝑎𝑠𝑘𝑖𝑝 (𝑣,𝑠) = � 𝑢∈V (𝑙) ˆ𝑎 (𝑣,𝑢) ˆ𝑎 (𝑢,𝑠), and 𝑊 (𝑙−1) skip = 𝑊 (𝑙−1)𝑊 (𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The part of skip-connection is then added to the classic GCN layer before a nonlinear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the overall representation of node 𝑣 is: ℎ(𝑙+1) 𝑣 = 𝜎 �� � ∑︁ 𝑢∈V (𝑙) ˆ𝑎 (𝑣,𝑢) ℎ(𝑙) 𝑢 𝑊 (𝑙) + ℎ(𝑙+1) 𝑣skip �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (62) 30 That is to say, each node gathers information from both its 1-hop neighbours and 2-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Their experiments on the Cora dataset show that although skip connection does not lead to significant improvement in accuracy, it helps to speed up the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Aiming to improve the performance of zero-shot learning tasks on knowledge graphs (directed graphs), Dense Graph Propagation (DGP) proposes to adopt a two-phase propagation scheme on two separate connectivity patterns (one having nodes connected to their ancestors and the other having nodes connected to their descendants) [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Furthermore, at each phase, DGP introduces a weighting scheme to include the contributions from distant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the overall representation is formulated as: 𝐻 = 𝜎 � 𝐾 ∑︁ 𝑘=0 𝛼𝑎 𝑘 ˆ𝐴𝑎 𝑘𝜎 � 𝐾 ∑︁ 𝑘=0 𝛼𝑑 𝑘 ˆ𝐴𝑑 𝑘𝑋𝑊𝑑 � 𝑊𝑎 � , (63) where ˆ𝐴𝑎 𝑘 and ˆ𝐴𝑑 𝑘 denote the normalised adjacency matrices containing k-hop connections to ancestors and to descen- dants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝛼𝑎 𝑘 and 𝛼𝑑 𝑘 are learnable weights denoting contributions from nodes that are k-hop away from a given node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We see from the above equation that DGP can be viewed as consisting of two convolutional layers where the inner layer aggregates information from 1 to k-hop out-neighbours, and the outer layer aggregates information from 1 to k-hop in-neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Directed Graph Convolutional Networks (DGCN) is another attempt to extend the GCN to directed graphs [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It proposes to expand the receptive field of convolutional operation by considering the first- and second-order proximities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, they first define the notions of second-order in-degree proximity matrix and second-order out-degree proximity matrix as: 𝐴𝑆in (𝑢, 𝑣) = ∑︁ 𝑤 𝐴𝑤,𝑢𝐴𝑤,𝑣 � 𝑥 𝐴𝑤,𝑥 , 𝐴𝑆out (𝑢, 𝑣) = ∑︁ 𝑤 𝐴𝑢,𝑤𝐴𝑣,𝑤 � 𝑥 𝐴𝑥,𝑤 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (64) The idea is that if two nodes (a given node and its 2-hop neighbour) share many common in-neighbours (or out- neighbours), they have higher second-order in-degree (or out-degree) proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When capturing first-order proximity, they choose to make the adjacency matrix symmetric by ignoring link directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then the overall representation at layer 𝑙 is formulated as: H(𝑙) = 𝐶𝑜𝑛𝑐𝑎𝑐𝑡 � 𝜎 � ˆAFH(𝑙−1)Θ(𝑙−1)� , 𝜎 � ˆASinH(𝑙−1)Θ(𝑙−1)� , 𝜎 � ˆASout H(𝑙−1)Θ(𝑙−1)�� , (65) where ˆAF, ˆASin and ˆASout are normalised first-order proximity matrix, normalised second-order in-degree proximity matrix and normalised second-order out-degree proximity matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that although DGP also considers 2-hop neighbours in directed graphs when 𝑘 equals 2, DGCN and DGP have different definitions of directed 2-hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' SGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to improve the efficiency and scalability of GCN, Simple Graph Convolution (SGC) proposes to remove the nonlinear transformation between layers [188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They argue that the main advantage of GCN lies in its neighbourhood aggregation scheme, not the nonlinearity between convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' After removing all nonlinear activations, the final output of the SGC model is represented as follows: ˆ𝑌 = softmax � ˆ𝐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ˆ𝐴 ˆ𝐴𝐻 (0)𝑊 (1)𝑊 (2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑊 (𝐿)� = softmax � ˆ𝐴𝐿𝐻 (0)𝑊 � , (66) where ˆ𝐴 is the normalised self-connection added adjacency matrix, 𝐻 (0) is the input node feature matrix, and 𝑊 is a single weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This output representation thus only requires learning a single weight matrix, and the term ˆ𝐴𝐿𝐻 (0) can be computed directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that the meaning of ˆ𝐴𝐿𝐻 (0) is the sum of features from k-hop neighbouring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 31 Therefore, the SGC model is actually equivalent to a single convolutional layer where nodes aggregate information from their k-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PATCHY-SAN [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Traditional image-based convolutional networks can be viewed as traversing a node sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a receptive field moving from left to right and from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to employ the convolutional architecture to graphs where spatial order is missing, PATCHY-SAN proposes to first impose an order on nodes according to a certain ranking algorithm, then construct receptive fields from a fixed number of neighbour nodes for each node in a preselected node sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note here the neighbour nodes are selected by performing a breadth-first search, so it can go beyond 1-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The receptive fields, after being normalised, will then be fed into a one-dimensional convolutional layer and other dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Comparing this CNN-like approach with the GCN, we see that it requires an extra procedure to rank nodes, and there are more hyper parameters to tune, such as the length of node sequence, the stride and the number of neighbour nodes in the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Random-walk neighbourhood approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Instead of defining neighbourhood based on the distance to the focal node, some GCN approaches adopt a random-walk based neighbourhood, which might enable them to capture random processes on certain types of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PinSage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to apply GCN to web-scale recommender systems, PinSage proposes to construct neighbourhoods via random walks, also referred to as importance-based neighbourhoods [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The convolutional operation is similar to that of GraphSAGE: h𝑙 N𝑟 (𝑣) ← 𝛾 �� 𝜎 � Qlℎ𝑢 � | 𝑢 ∈ N𝑟 (𝑣) � , 𝜶 � , h𝑙 𝑣 ← 𝜎 � W𝑙 · CONCAT � h𝑙−1 𝑣 , h𝑙 N𝑟 (𝑣) �� , (67) where N𝑟 (𝑣) is a random-walk neighbourhood, 𝛾 is an aggregation function, ℎ𝑢 is a set of embeddings of nodes in the neighbourhood, 𝜶 is a set of weights on nodes in the neighbourhood, Ql and Wl are learnable model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, N𝑟 (𝑣) comes from simulating a random walk starting from the focal node and calculating the 𝐿1-normalised count of visited nodes, then the top𝑇 nodes with the highest counts are selected as the neighbourhood in the layer-wise message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There are two benefits in this neighbourhood definition: first, the number of nodes involved in the aggregation is fixed, so the cost of the algorithm is predictable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' second, the normalised visit counts can be directly used as weights to represent the importance of each node in the neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PinSage also introduces some strategies to improve the model’s scalability, such as the producer-consumer minibatch construction and a MapReduce pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that PinSage is originally designed for recommender systems which are bipartite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GraLSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Based on the idea that anonymous walks can capture structures through reconstructing local subgraphs [139], GraLSP proposes to adopt random anonymous walks into the neighbourhood aggregation scheme [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It also combines some other techniques to enhance the model performance, such as adaptive receptive radius, attention and channel-wise amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the convolutional layer is formulated as: a(𝑘) 𝑣 = MEANwk∈W (𝑖),𝑝 ∈[1,𝑟wk] � 𝜆(𝑘) 𝑣,wk � q(𝑘) 𝑣,wk ⊙ h(𝑘−1) wk𝑝 �� , h(𝑘) 𝑣 = ReLU � W(𝑘)h(𝑘−1) 𝑣 + U(𝑘)a(𝑘) 𝑣 � , (68) where 𝑤𝑘 denotes a walk from the set of random walk sequence W (𝑖), 𝑤𝑘𝑝 is the 𝑝-th node in walk 𝑤𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑟𝑤𝑘, 𝜆𝑣,wk and q𝑣,wk denote receptive radius, attention coefficient and amplification coefficient, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Adaptive radius is introduced in order to regulate the scope of walks so that nodes that are too far away in the constructed subgraph are excluded while nodes in clustered subgraphs are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, it is defined as 𝑟𝑤𝑘 = � 2𝑙 𝐶𝑤𝑘 � , where 𝑙 is walk length and 𝐶𝑤𝑘 is the number of distinct nodes visited by the walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the attention coefficient is introduced to 32 assign different importance to visited nodes, and the channel-wise amplification is used to model the selection of node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4 Subgraph neighbourhood approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In addition to the fixed-hop neighbourhood and random-walk neighbour- hood definition in layer-wise message aggregation, some approaches view the neighbourhood as k-node tuples or subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' k-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As the ability of the GCN to distinguish nonisomorphic graphs is equivalent to that of the 1-dimensional Weisfeiler-Leman algorithm (1-WL), a k-GNN is proposed to achieve a higher expressivity as that of a k-WL [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different from the vanilla GCN where each node gathers information from a defined neighbourhood, the k-GNN works on the level of node tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Accordingly, the neighbourhood of a k-tuple is defined as other k-tuples containing one node that is not in the focal k-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the neighbourhood of k-tuple 𝑠 is defined as: 𝑁𝑘 (𝑠) = � 𝑡 ∈ [𝑉 ]𝑘 ||𝑠 ∩ 𝑡 |= 𝑘 − 1 � , where [𝑉 ]𝑘 is a set of all k-tuples in a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The convolutional operation at layer 𝑙 is then defined as: ℎ(𝑙) (𝑠) = 𝜎 �� � ℎ(𝑙−1) (𝑠) ·𝑊 (𝑙) 1 + ∑︁ 𝑢∈𝑁𝑘 (𝑠) ℎ(𝑙−1) (𝑢) ·𝑊 (𝑙) 2 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (69) At the beginning, ℎ(0) (𝑠) is set as ℎ𝑖𝑠𝑜 (𝑠), which is a one-hot encoding of the isomorphism type of induced subgraph of 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To improve the model’s scalability and avoid overfitting, a more restricted k-tuple neighbourhood 𝐿𝑁𝑘 (𝑠), named local neighbourhood, is defined as the tuples in 𝑁𝑘 (𝑠) also satisfying (𝑢, 𝑣) ∈ 𝐸 for 𝑢 ∈ 𝑠\\𝑡 and 𝑣 ∈ 𝑠\\𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In other words, the non-overlapped nodes in a given k-tuple and a neighbouring k-tuple needs to be connected so that the neighbouring k-tuple is a local neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that as k-GNN is defined on the k-tuple level, it is unsuitable for node-level tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GRAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to improve GCN’s ability to discriminate graph isomorphism, GRAPE proposes to consider specific subgraph patterns in its layer-wise neighbourhood aggregation [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' First, nodes of a given subgraph pattern are grouped into different sets according to their egocentric automorphic equivalences, abbreviated as the Ego-AE set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, in a triangle subgraph, the focal node is in one set, and the other two nodes are in another set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, different weights are learned for each Ego-AE set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, node 𝑣’s Ego-AE sets in a given subgraph 𝑆 are denoted as: � AE𝑆,1 (𝑣) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' , AE𝑆,𝑖 (𝑣) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' , AE𝑆,𝑚 (𝑣) �, where 𝑚 is the total number of AE-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The convolutional operation for subgraph 𝑆 is then formulated as: ℎ𝑙 𝑆 (𝑣) = MLP �� � ∑︁ 𝑖 𝛽𝑆,𝑖 · ∑︁ 𝑢∈AE𝑆,𝑖 (𝑣) ℎ𝑙−1 𝑆 (𝑢)�� � , (70) where 𝛽𝑆,𝑖 are learnable weights representing the importance of each set AE𝑆,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that the focal node 𝑣 also belongs to an Ego-AE set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The final embedding of node 𝑣 at layer 𝑙 is then achieved through combining embeddings from a set of different subgraph patterns: ℎ𝑙 (𝑣) = � 𝑆 ∈Ω 𝛼𝑙 𝑆 · ℎ𝑙 𝑆 (𝑣), where Ω denotes the set of subgraph patterns, and 𝛼𝑙 𝑆 is learnable weight on a given subgraph pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This way GRAPE is able to differentiate neighbouring nodes according to their structural roles captured by Ego-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Certainly, the approach involves an extra step of choosing subgraph patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Layer-wise aggregation scope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Aggregator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Batch size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Type of graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='full-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GraphSAGE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood with sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Flexible choice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Multiset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood with attention scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Weighted sum/mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='FastGCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='layer-wise sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GraphSNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='structural coefficients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='DGCNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='full-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Directed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='MixHop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='full-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-hop GNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Outside-to-insides scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Adapt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='DGP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='full-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Directed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='DGCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='full-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Directed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='SGC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-hop neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Patchy-SAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='fixed number of nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='CNN-like (weighted sum) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Graph-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='PinSage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='random-walk neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Weighted sum/mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Bipartite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GraLSP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='random-walk neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum/Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-GNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-tuple neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Sum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Graph-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='GRAPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='subgraph neighbourhood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Weighted sum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Node & Graph level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='mini-batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='General ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Summary of approaches in the first category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 34 Message Content (What message is gathered and passed on) Count of subgraphs + X GSN ℱ-MPNN SMP Distance encoding + X Other information + X rGIN ID-GNN P-GNN DE-GNN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taxonomy from the message content perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Message Content The superior performance of the GCN lies in its ingenious combination of node attributes and a graph structure with node attributes used as initial representations and then subsequently propagated on the graph through certain convolutional operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In contrast, some learning approaches only exploit structural information, such as the matrix decomposition based methods [36, 158] and the random walk based methods [73, 161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In situations where no node attributes are provided, a simple structural metric like the node degree is often used as initial representations in the GCNs [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another group of approaches further propose to improve the GCN’s distinguishability through injecting more complicated structural features into the node representations, such as the count of graphlets, distance-based information, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The taxonomy and representative approaches are given in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Count of subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The number of certain subgraphs or substructures is often used as a node feature in traditional network studies [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Some approaches thus propose to include this type of structural information as part of a node representation in the GCN’s message passing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graph Substructure Network (GSN) proposes to capture structural features by counting the appearance of particular graphlet orbits and include them as part of node features in the convolutional operation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, node 𝑣’s representation at layer 𝑙 is defined as: h𝑙+1(𝑣) = MLP1 �� � ℎ𝑙 (𝑣), ∑︁ 𝑢∈N(𝑣) 𝑀𝐿𝑃2 �h𝑡 (𝑣), h𝑡 (𝑢), x𝑉 (𝑣), x𝑉 (𝑢), e(𝑢, 𝑣)��� � , (71) where x𝑉 (𝑣) and x𝑉 (𝑢) are structure features of nodes 𝑣 and 𝑢, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' e(𝑢, 𝑣) is an edge feature if provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The structural feature is a vector containing the counts of node orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, if subgraphs 2-path and 3-clique are con- sidered (𝐺1 and𝐺2 in Figure 4), the counts of three node orbits will be included in the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GSN further introduces a ver- sion based on edge orbits, which is formulated as: h𝑙+1(𝑣) = MLP1 � ℎ𝑙 (𝑣), � 𝑢∈N(𝑣) 𝑀𝐿𝑃2 �h𝑡 (𝑣), h𝑡 (𝑢), x𝐸 (𝑢, 𝑣), e(𝑢, 𝑣)�� , where e(𝑢, 𝑣) denotes the edge structural feature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a vector containing the count of edge orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The GSN has been proven to be strictly more powerful than the 1-WL test when the chosen subgraphs are not star graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Certainly, the choice of subgraphs is the core of this approach, and a larger subgraph will lead to higher computational complexity in a preprocessing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' F -MPNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A local graph parameter enabled GNN (F -MPNN) also proposes to include a subgraph count into the GCN [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' F = {𝑃𝑟 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', 𝑃𝑟 𝑘} is a set of pre-selected subgraph patterns with 𝑟 referring to a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The “homomorphism count” of each pattern 𝑃𝑟 𝑖 for node 𝑣 in the original graph 𝐺 is denoted as ℎ𝑜𝑚(𝑃𝑟 𝑖 ,𝐺𝑣), which is actually equivalent to the count of a given node orbit (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then a structural feature vector (ℎ𝑜𝑚(𝑃𝑟 1,𝐺𝑣), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=',ℎ𝑜𝑚(𝑃𝑟 𝑘,𝐺𝑣)) is added to 35 the one-hot encoding of node 𝑣’s label, serving as 𝑣’s initial feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, the framework is formulated as: h(0) 𝑣 := � 𝑥𝑣, hom �𝑃𝑟 1,𝐺𝑣� , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' , hom � 𝑃𝑘 ℓ ,𝐺𝑣�� , h(𝑙) 𝑣 := MERGE � x(𝑙−1) 𝑣 , AGGREGATE ��� x(𝑙−1) 𝑢 | 𝑢 ∈ 𝑁 (𝑣) ���� , (72) where 𝑥𝑣 is the one-hot encoding of node 𝑣’s label, MERGE and AGGREGATE are two MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Since this structural feature is only applied to enhance the initial feature of nodes, it can be used as an add-on to any GCN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Similar to the GSN, the choice of subgraph patterns is the core of F -MPNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Cycles of length smaller than 10 and cliques of size smaller than 5 are used as subgraph patterns in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ID-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Identity-aware GNN (ID-GNN) proposes to improve the expressivity of the GCN through distinguishing the root node of the extracted computation graphs from other nodes in its message passing scheme [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It contains two major steps: the first step, named inductive identity colouring, is to uniquely colour the root node in its k-hop ego network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' then in the second step, a heterogeneous message passing is applied to all the extracted ego networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the representation of any node 𝑣 in an extracted computation graph 𝐺𝑟 (rooted at node 𝑟) is formulated as: m(𝑙) 𝑢 = MSG(𝑙) 1[𝑢=𝑟 ] � h(𝑙−1) 𝑢 � , h(𝑙) 𝑣 = AGG(𝑙) �� m(𝑙) 𝑢 ,𝑢 ∈ N (𝑣) � , h(𝑙−1) 𝑣 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (73) MSG(𝑙) 1[𝑢=𝑟 ] (·) means that MSG(𝑙) 1 (·) is applied to the root node while MSG(𝑙) 0 (·) is applied to other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In this way, the representation of the root node is different from that of other nodes and will help distinguish other nodes when propagated to later layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The approach is inductive since the colouring is based on the extracted computation graphs instead of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Further, in order to avoid the overhead of extracting ego-networks, ID-GNN-Fast proposes to use the count of cycles as an augmented node feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore the input node feature is built from concatenating the original node feature and the augmented node feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Distance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Distance measures such as shortest paths between nodes are widely used in traditional network studies [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Naturally, some approaches propose to enhance the performance of the GCN through including distance information in their message passing scheme or as an additional initial node feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' P-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Position-aware graph neural network (P-GNN) proposes to let each node aggregate information from several randomly chosen subsets of nodes, instead of its own 1-hop neighbours [201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As every node shares the same neighbourhood in P-GNN, distance information is included to indicate the relative position of each node to those subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, given 𝑘 randomly sampled subsets, 𝑆𝑖 denoting the 𝑖th subset, the representation of node 𝑣 at layer 𝑙 is formulated as: h𝑙 𝑣 = AGG(𝑙) � M𝑙−1 𝑖 , ∀𝑖 ∈ [1,𝑘] � , M𝑙−1 𝑖 = {𝐹 (𝑑𝑢𝑣,ℎ𝑙−1 𝑢 ,ℎ𝑙−1 𝑣 ), ∀𝑢 ∈ 𝑆𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (74) 𝐹 is a message computation function accounting for both distance information and feature information of a pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The output at the last layer is constructed with M𝑖 being the 𝑖th embedding dimension, thus making the final representation “position aware”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that the subsets are resampled at each convolution layer, so that each node can aggregate information from different sets of nodes at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DE-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Distance-Encoding GNN (DE-GNN) also proposes to improve the GCN’s expressivity through adding distance information [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Intuitively, for any given node set 𝑆 whose representation is to be learnt, other nodes are encoded with their distances to each node of 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Formally, DE of node 𝑢 with regard to the target node set 𝑆 is defined as: 𝜁 (𝑢 | 𝑆) = AGG({𝜁 (𝑢 | 𝑣) | 𝑣 ∈ 𝑆}), 𝜁 (𝑢 | 𝑣) = 𝑓 �� (𝑀)𝑢𝑣, � 𝑀2� 𝑢𝑣 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' , � 𝑀𝑘� 𝑢𝑣 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' �� , (75) 36 where 𝑀 = 𝐴𝐷−1 is a matrix of landing probabilities through random walks,𝑓 can be a heuristic function or a learnable neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different distance measures can be captured by the above equation such as the shortest path distance or the generalised PageRank score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DE is denoted as DE-|𝑆| according to the size of set 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, DE-2 when |𝑆| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' One way of improving the GCN through distance encoding is to use it as an extra node feature:ℎ(0) 𝑣 = 𝐶𝑂𝑁𝐶𝐴𝑇 (𝑥𝑣,𝜁 (𝑣 | 𝑆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another approach is to use DE-1 in the layer-wise aggregation: ℎ(𝑙+1) 𝑣 = 𝑓1 � ℎ(𝑙) 𝑣 , AGG �� (𝑓2 � ℎ(𝑙) 𝑢 � ,𝜁 (𝑢 | 𝑣)) � 𝑢∈𝑉 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although DE-GNN adopts minibatch training, the distance information needs to be computed for every node set and for all nodes in its extracted L-hop ego-network, leading to a higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Also note that DE-GNN is flexible for tasks on different levels: DE-1 for node-level tasks, DE-2 for link prediction tasks and DE-3 for triangle prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To highlight that the DE-GNN is suitable for both node and link-level tasks, we use two colours in its block in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Apart from the count of certain subgraphs and the distance information, some other infor- mation such as the “local context matrix” or even random features are also used to enhance the performance of the GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' SMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to improve the GCN’s performances on structure-related tasks, Structural Message Passing (SMP) proposes to maintain a “local context matrix” at each node, instead of a feature vector as in the vanilla GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, each node is initialised as a one-hot encoding 𝑴 (0) 𝑖 = 1𝑖 ∈ R𝑛×1, and the additional node feature 𝑥𝑖 of 𝑣𝑖 is appended at the 𝑖th row: 𝑀 (0) 𝑖 [𝑖, :] = [1,𝑥𝑖] ∈ R1+𝑐𝑋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then the local context matrix of node 𝑣𝑖 at layer 𝑙 is formulated as: 𝑀 (𝑙) 𝑖 = 𝑀𝐿𝑃 (𝑙−1) 1 � 𝑀 (𝑙−1) 𝑖 ,𝐴𝐺𝐺 �� 𝑀𝐿𝑃 (𝑙−1) 2 � 𝑀 (𝑙−1) 𝑖 , 𝑀 (𝑙−1) 𝑗 ,𝑒𝑖𝑗 �� 𝑣𝑗 ∈𝑁𝑖 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (76) 𝐴𝐺𝐺 is an aggregation function which is by default a normalised sum aggregator: � 𝑣𝑗 ∈𝑁𝑖 𝑀𝐿𝑃 (𝑙−1) 2 � 𝑴 (𝑙) 𝑖 , 𝑴 (𝑙) 𝑗 , 𝒆𝑖𝑗 � /𝑑avg � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In this way, the 𝑗th row in 𝑀𝑖 is the representation node 𝑣𝑖 has of node 𝑣𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the vector form representation of node 𝑣𝑖 is obtained through applying an equivariant neural network for sets on the rows of its context matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although node ordering is needed when constructing the local context matrix, the learned representation is proven to be order-invariant when 𝑀𝐿𝑃1, 𝑀𝐿𝑃2 and 𝐴𝐺𝐺 are permutation equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' SMP is shown to excel in various tasks, such as the detection of structural properties including distance, eccentricity connectivity, diameter, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' rGIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Apart from various types of explicit structural features that are being added to the GCN, another work (termed rGIN) proves that the expressive power of the GCN can be enhanced by just adding random features to each node [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, rGIN first assigns a random value 𝑟𝑣 to each node and concatenates it with the original node feature 𝑥𝑣, then performs GIN’s convolutional operations: ℎ(0) 𝑣 = MLP(0) (𝐶𝑂𝑁𝐶𝐴𝑇 (𝑥𝑣,𝑟𝑣)) , ℎ(𝑙) 𝑣 = MLP(𝑙) �� � � 1 + 𝜀 (𝑙)� ℎ(𝑙−1) 𝑣 + ∑︁ 𝑢∈N(𝑣) ℎ(𝑙−1) 𝑢 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (77) With this simple modification on the initial node feature, rGIN is proven to be able to distinguish any local structure with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The idea of injecting random features into nodes is that the GCN fails to distinguish graphs with identical node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, a GCN with the node degree as an input feature cannot distinguish a node in a 3-cycle graph from a node in a 6-cycle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' rGIN is shown to perform well on structure-related tasks such as learning the existence of triangles, learning the local clustering coefficient, and learning the algorithm for the MDS (Minimum Dominating Set) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 37 Learning Scope (Input graph) Cluster-GCN G-Meta Subgraphs GraphSAINT SEAL Local subgraphs LGCN Other graphs DiffPool Shadow-GNN NGNN AM-GCN/kNN-GCN GNN-AK Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taxonomy from the learning scope perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Learning scope The third structural perspective on GCNs is regarding the learning scope or the input graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Whether it is full-batch or mini-batch training, most GCNs still have the whole graph as an input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', in an L-layer GCN, each node has the scope of its L-hop neighbourhood in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A higher number of layers leads to a neighbourhood explosion and thus higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To address this issue, many approaches limit the scope to subgraphs or localised subgraphs while some other methods propose to run the GCN on particular types of generated graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The taxonomy and related approaches are given in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Again, the block’s colour indicates the task the approach is proposed for: grey represents a node classification, blue represents a link prediction, and orange represents a network classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An intuitive idea is to limit the training scope to several selected subgraphs instead of the original whole graph, so the neighbourhood is restricted within the sphere of subgraphs no matter how many layers are stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GraphSAINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to enhance the scalability of the GCN, GraphSAINT proposes to train a GCN model iteratively on several sampled subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Each sampled subgraph 𝐺𝑠 ∈ G is a mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The representation of node 𝑣 in a sampled subgraph 𝐺𝑠 is formulated as: ℎ(𝑙+1) 𝑣 = ∑︁ 𝑢∈𝑁𝑣 |𝐺𝑠 ˜𝐴𝑣,𝑢 𝛼𝑢,𝑣 𝑊 (𝑙)ℎ(𝑙) 𝑢 , (78) where 𝑢 is 𝑣’s neighbour in 𝐺𝑠, and 𝛼𝑢,𝑣 is a coefficient to offset the biases from the sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, 𝛼𝑢,𝑣 is defined as the probability of edge (𝑢, 𝑣) being sampled, divided by the probability of node 𝑣 being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Given a set of pre-sampled subgraphs G, 𝛼𝑢,𝑣 = 𝐶𝑢,𝑣 𝐶𝑣 , where 𝐶𝑢,𝑣 and 𝐶𝑣 are the number of times edge (𝑢, 𝑣) and node 𝑣 appear in G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the batch loss is calculated as: 𝐿batch = 1 |G| � 𝐺𝑠 ∈G � 𝑣 𝐿𝑣 𝜆𝑣 , where 𝐿𝑣 is the loss on node 𝑣 in the GCN’s output layer, and 𝜆𝑣 is a loss normalisation term computed by the number of node 𝑣 appearing in G divided by the total number of nodes in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different samplers are integrated within the framework, such as random node sampler, random edge sampler and random walk based sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' According to the experiment, the random walk based sampler tends to have the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Cluster-GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Also to address the issue of neighbourhood explosion in large graphs, a Cluster-GCN proposes to first partition the whole graph into several clusters according to certain clustering algorithms, then run the GCN on those clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Given 𝑐 clusters, the original adjacency matrix 𝐴 is approximated as a list of submatrices 𝐴11, 𝐴11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', 𝐴𝑐𝑐 at diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The representation of nodes at layer 𝑙 in the 𝑡th cluster is thus formulated as: 𝐻 (𝑙) 𝑡 = ˆ𝐴𝑡𝑡𝐻 (𝑙−1) 𝑡 𝑊 (𝑙−1), (79) where ˆ𝐴𝑡𝑡 is the normalised version of 𝐴𝑡𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The loss is then calculated as: 𝐿𝑡 = 1 |𝑉𝑡 | � 𝑖 ∈𝑉𝑡 loss � 𝑦𝑖,ℎ(𝐿) 𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' At each iteration, the model weights are updated based on the loss of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This way, no matter how many convolutional layers are 38 involved, the neighbourhood scope is restricted to one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to offset the bias of clustering algorithms, a better version of the Cluster-GCN proposes to randomly form a subgraph with several randomly chosen clusters, then at each iteration, run GCN on one subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Experiments on very large datasets show that the Cluster-GCN is able to train a deeper GCN without time and space overhead and achieves advanced performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' LGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A learnable graph convolutional network (LGCN) proposes to transform graph data into a grid-like data structure and apply the traditional convolutional operation on it [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As traditional CNN requires a fixed number of ordered units in the receptive fields, the LGCN proposes to sort features at each dimension and select the k-largest ones to form a grid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The transformed data is then fed into a one-dimensional CNN to generate the final representation of the focal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the nodes’ representation at layer 𝑙 is formulated as: 𝐻 (𝑙+1) = 𝑐(𝑔(𝐻 (𝑙),𝐴,𝑘)), (80) where 𝑘 is a hyper parameter, 𝑔(·) is the function that performs k-largest selection to transfer the original graph data into grid data, and 𝑐(·) is a one-dimensional CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Furthermore, as Cluster-GCN and GraphSAINT, the LGCN also proposes to train the neural network on subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Each subgraph is built from randomly selecting a few initial nodes and then expanding adjacent nodes into it using a breadth-first-search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' At each training iteration, multiple subgraphs can be included in a mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The subgraph training strategy is shown to be more time and space efficient, with only negligible loss in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Local subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another popular idea to address the computational overhead is training the GCN on local subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that local subgraphs are different from subgraphs in that they are extracted around each node or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In contrast, subgraphs, as we have discussed earlier, have in general a wider range, without focusing on a node or a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' SEAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' SEAL is a GCN based framework specially designed for a link prediction task [209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Motivated by the fact that many successful link prediction heuristics, such as the common neighbour, Adamic-Adar and resource allocation, only involve the 1-hop or 2-hop neighbours around a node pair, SEAL proposes to train a GCN on the local subgraphs extracted around each target link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the local subgraph is the induced graph from each target node pair and their k-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' After having constructed the training data, it further introduces a node labelling procedure to give the target node pair special weights as well as to distinguish the neighbouring nodes in a given local subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, it labels each node in the target pair as “1”, and assigns larger labels to other nodes according to their distances to the target pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The assigned labels are then concatenated with other features to construct the feature matrix of the local subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the final step, a GCN is trained on the local subgraphs and their label-enhanced feature matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the experimental implementation, SEAL chooses to use DGCNN, a GCN model designed for graph classification (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1), as the default GCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Essentially, the link existence problem in the original graph is modelled as a graph classification problem on the extracted local subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' G-Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Motivated by the idea that local subgraphs may contain transferable knowledge that can be adapted to unseen tasks, G-Meta proposes to leverage local subgraph information in few-shot graph meta-learning [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For the node classification task, local subgraphs are constructed as induced graphs from each node and its k-hop neighbours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' and when it comes to link prediction, local subgraphs are built as in SEAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then a typical GCN is used on these local subgraphs to generate graph embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' At last, a prototypical loss and Model-Agnostic Meta-Learning (MAML) algorithm are used to update the GCN’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the prototype 𝑡𝑙 of label 𝑙 is calculated through averaging over subgraph embeddings in the support set: 𝑡𝑙 = 1 𝑁𝑙 � 𝑦𝑗=𝑙 h𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then for each local subgraph 𝑆𝑢 in both support and query set, a class distribution vector p is calculated as: p𝑙 = exp(−∥h𝑆𝑢 −t𝑙 ∥) � ˆ𝑙 exp(−∥h𝑆𝑢 −tˆ𝑙 ∥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Finally, the cross-entropy 39 loss is formulated as: L(p, y) = � 𝑗 y𝑗 log p𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Experiments on synthetic and real networks show that local subgraphs are vital for few-shot graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is worth mentioning that the best performance is yielded when 2-hop neighbours are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Shadow-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From the perspective of decoupling the scope (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a receptive field) and the depth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a number of layers) of the GCN, a Shadow-GNN also proposes to adopt local subgraph as an input [204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Typically on a full graph, the scope of the GCN increases with the number of layers — an L-layer GCN means an L-hop neighbourhood scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As the GCN model is also viewed as a form of Laplacian smoothing that mixes the feature of a node and its neighbours, when the scope becomes too large, node features may be oversmoothed [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To address this problem, the Shadow-GNN proposes to train the GCN on local subgraphs, so that the scope is bounded by the range of the local subgraphs, regardless of the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In this setting, the depth can be larger than the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It means that nodes in the subgraphs may exchange information multiple times, which could lead to better expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different subgraph extractors can be selected, such as an L-hop neighbourhood extractor or a random-walk-based extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In actual implementation, the scope is set as a 2- or 3-hop neighbourhood while the depth is deeper (3 or 5 layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' NGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A nested graph neural network (NGNN) proposes to apply the local subgraph training strategy on a graph classification task [211].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The extracted local subgraphs, termed rooted subgraphs, are also induced subgraphs from each node and its k-hop neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' First, a base GCN is applied on all rooted subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking root node 𝑣 for example, at layer 𝑙, any node 𝑢 in its k-hop rooted subgraph 𝐺𝑘𝑣 is formulated as: ℎ(𝑙) 𝑢,𝐺𝑘𝑣 = 𝑈𝑃𝐷𝐴𝑇𝐸(𝑙−1) ��� � ℎ(𝑙−1) 𝑢,𝐺𝑘𝑣 , ∑︁ 𝑤∈𝑁 (𝑢 |𝐺𝑘𝑣 ) 𝑀𝑆𝐺 (𝑙−1) � ℎ(𝑙−1) 𝑢,𝐺𝑘𝑣 ,ℎ(𝑙−1) 𝑤,𝐺𝑘𝑣 ,𝑒𝑢𝑤 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (81) Then, the final representation of root node 𝑣 at layer 𝐿 is set to be equal to its rooted subgraph representation obtained from applying a subgraph pooling on all nodes in the subgraph: ℎ𝑣 = ℎ𝐺𝑘𝑣 = 𝑃𝑂𝑂𝐿1 �� ℎ(𝐿) 𝑢,𝐺𝑘𝑣 | 𝑢 ∈ 𝐺𝑘𝑣 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' With the same base GCN applied on all nodes’ rooted subgraphs, the representation of each node can be obtained, and the graph representation can be generated from applying another GCN, termed outer GCN, on those updated node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To make it simple, the outer GCN can be just a graph pooling layer: ℎ𝐺 = 𝑃𝑂𝑂𝐿2(ℎ𝑣 | 𝑣 ∈ 𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The work theoretically proves that a proper NGNN can discriminate almost all 𝑟-regular graphs where the vanilla GCN cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GNN-AK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GNN-As Kernel (GNN-AK) is another local subgraph based approach for a graph classification problem [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Different from the NGNN which directly uses the rooted subgraph embedding to represent each node, the GNN-AK proposes to construct node representation from concatenating three types of embedding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', subgraph embedding, centroid embedding, and context embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Centroid embedding is simply the root node representation in its own subgraph, while context embedding is built from the representation of this node in other nodes’ rooted subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is argued that these two additional embeddings contain information which is not captured in the subgraph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Formally, the representation of node 𝑣 at layer 𝑙 is: ℎ(𝑙) 𝑣 = 𝐶𝑂𝑁𝐶𝐴𝑇 (ℎ(𝑙) 𝑣,centroid,ℎ(𝑙) 𝑣,subgraph,ℎ(𝑙) 𝑣,context), (82) with ℎ(𝑙) 𝑣,centroid = ℎ𝑣|𝐺𝑘𝑣 , ℎ(𝑙) 𝑣,subgraph = 𝑃𝑂𝑂𝐿1({ℎ𝑖 |𝐺𝑘𝑣 | 𝑖 ∈ N𝑘 (𝑣)}), and ℎ(𝑙) 𝑣,context = 𝑃𝑂𝑂𝐿2({ℎ𝑣|𝐺𝑘 𝑗 ∀𝑗 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑣 ∈ N𝑘 (𝑗)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ℎ𝑣|𝐺𝑘𝑢 denotes the representation of node 𝑣 in node 𝑢’s rooted subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, the final graph representation is obtained from another pooling at the output layer: ℎ𝐺 = 𝑃𝑂𝑂𝐿3({ℎ𝐿𝑣 | 𝑣 ∈ 𝑉 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is worth mentioning that a subgraph drop strategy is further introduced to improve the scalability of GNN-AK so that the number of local subgraphs can be much smaller than the number of nodes in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Other types of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Subgraphs or local subgraphs are still part of the original graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the third subcategory, we see approaches that use differently constructed graphs, such as the coarsened graph and the feature graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DiffPool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Analogous to the idea of spatial pooling in a traditional CNN, a DiffPool proposes to learn a graph representation in a hierarchical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Nodes at layer 𝑙 will be collapsed into higher-level cluster nodes at layer 𝑙 + 1 via a learned assignment matrix, and after stacking several hierarchical layers, the singular node’s embedding at the final layer is viewed as the representation for the whole graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Concretely, node embedding matrices 𝑍 (𝑙) are learned from a GCN (called an embedding GNN), and an assignment matrix 𝑆 (𝑙) is learned from another GCN, called a pooling GNN: 𝑍 (𝑙) = 𝐺𝑁𝑁 (𝑙) embed(𝐴(𝑙),𝑋 (𝑙)), 𝑆 (𝑙) = softmax(𝐺𝑁𝑁 (𝑙) pool(𝐴(𝑙),𝑋 (𝑙))), (83) where 𝐴(𝑙) and 𝑋 (𝑙) are the coarsened adjacency matrix and the cluster nodes feature matrix at layer 𝑙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The dimension of assignment matrix 𝑆 (𝑙) is 𝑛𝑙 × 𝑛𝑙+1, so that each role is one of the 𝑛𝑙 nodes at layer 𝑙 and each column is one of the cluster nodes at layer 𝑙 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, 𝐴(𝑙+1) and 𝑋 (𝑙+1) which are used as the next layer’s inputs are generated as: 𝑋 (𝑙+1) = 𝑆 (𝑙)𝑇𝑍 (𝑙), 𝐴(𝑙+1) = 𝑆 (𝑙)𝑇𝐴(𝑙)𝑆 (𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' (84) The assignment matrix 𝑆 (𝐿−1) at the penultimate layer is set to be a vector of 1’s, so that all nodes will collapse into a single cluster node at the final layer, and the corresponding node embedding is viewed as the representation for the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that the number of clusters is a predefined hyperparameter, which is usually set as a percentage of the number of nodes at the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' AM-GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An Adaptive Multi-channel Graph Conventional Network (AM-GCN) proposes to not only run the GCN on the original (topological) graph, but also on a feature graph constructed from a feature similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Specifically, the similarity matrix is computed using cosine similarity or heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, edges will be added between each node and 𝑘 other nodes of top similarity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The generated feature graph 𝐺𝑓 = (A𝑓 , X) is also called the k-nearest neighbour (kNN) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the embeddings on the feature graph are formulated as follows: H(𝑙) 𝑓 = ReLU � ˆA𝑓 H(𝑙−1) 𝑓 W(𝑙) 𝑓 � , (85) where ˆA𝑓 is the normalised feature graph adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another GCN is used to generate node embeddings on the original graph 𝐺 = (A, X): H(𝑙) 𝑡 = ReLU � ˆAH(𝑙−1) 𝑡 W(𝑙) 𝑡 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Further, in order to capture the correlation between a topological space and a feature space, a common convolutional module is introduced as: H(𝑙) 𝑐𝑡 = ReLU � ˆAH(𝑙−1) 𝑐𝑡 W(𝑙) 𝑐 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' H(𝑙) 𝑐𝑓 = ReLU � ˆA𝑓 H(𝑙−1) 𝑐𝑓 W(𝑙) 𝑐 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that the same weight matrix W(𝑙) 𝑐 is shared in H(𝑙) 𝑐𝑡 and H(𝑙) 𝑐𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Under this setting, node features are propagated not only in a topological space but also in a feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The final representation is then obtained through combining the above four embeddings with an attention scheme: Z = 𝛼𝑡 · H(𝐿) 𝑡 + 𝛼𝑓 · H(𝐿) 𝑓 + 𝛼𝑐 · ( H(𝐿) 𝑐𝑡 +H(𝐿) 𝑐𝑓 2 ), where 𝛼𝑡, 𝛼𝑓 and 𝛼𝑐 are attention vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4 Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Differences between layer-wise scope and overall learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Here we emphasize the differences between layer-wise scope and overall learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Layer-wise message aggregation scope, or a receptive field, is where a node receives the message from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It can be a 1-hop neighbourhood, k-hop neighbourhood, random-walk neighbourhood, or subgraph neighbourhood according to our taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although the receptive field is usually small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' distant nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Neighbourhood definition (representative approach) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Time complexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Space complexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood (GCN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �𝐿 �|𝑉 |2𝐶 + |𝑉 |𝐶2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1-hop neighbourhood (GraphSAGE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂(𝐿|𝑉 |(𝑠𝐶 + 𝐶2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='h-hop neighbourhood (MixHop) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �𝐿 �|𝑉 |2𝐶ℎ + |𝑉 |𝐶2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='h-hop neighbourhood (k-hop GNN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂(𝐿|𝑉 |(𝑘ℎ𝑚𝑎𝑥𝐶 + 𝐶2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Random-walk neighbourhood (PinSage) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂(𝐿|𝑉 |(𝑤𝑙 + 𝑣𝑙𝑜𝑔𝑣 + 𝑠𝐶 + 𝐶2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='k-node subgraph neighbourhood (k-GNN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂(𝐿�|𝑉 | ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='�(𝑘|𝑉 |𝐶 + 𝐶2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑂(𝐿�|𝑉 | ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='�𝐶 + 𝐶2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Time and space complexity from the perspective of a layer-wise message scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' |𝑉 | is the number of nodes in the graph, 𝐶 are node feature channels (assuming the number of features is fixed for all layers), 𝐿 is the number of convolutional layers, 𝑠 is the number of sampled nodes, ℎ is the number of hops away from a focal node, 𝑘𝑚𝑎𝑥 is maximum node degree, 𝑤 is the number of random walks, 𝑙 is the length of a random walk, 𝑣 is the number of visited nodes, and 𝑘 is the number of nodes in a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' can exchange messages after stacking multiple GCN layers, causing the well-known neighbourhood explosion issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Obviously, with a large enough number of layers, a node can exchange information with any other node in the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Overall learning scope, in contrast, is determined by the input graph, which can be the entire original graph, extracted subgraphs or local subgraphs, or coarsened graphs according to our taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking an extracted local subgraph for example, no matter how large the receptive field is or how many layers are stacked, a node can only exchange messages with other nodes in the same subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This naturally solves the neighbourhood explosion issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A large number of layers on a relatively small subgraph also means that nodes may exchange information multiple times, which is argued to help the GCN “better absorb and embed information” [204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Time and space complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the discussion about complexity, we focus on how the different definitions of the neighbourhood in a convolutional layer influence the cost of computation (corresponding to the layer-wise message scope taxonomy in Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The time and space complexities of each category are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' First, according to the propagation rule of the vanilla GCN (Equation 49), which is essentially the multiplication of three matrices 𝐴 ∈ R|𝑉 |×|𝑉 |, 𝐻 ∈ R|𝑉 |×𝐶, and 𝑊 ∈ R𝐶×𝐶, the time complexity at each layer is 𝑂 �|𝑉 |2𝐶 + |𝑉 |𝐶2�, and thus the overall complexity is 𝑂 �𝐿 �|𝑉 |2𝐶 + |𝑉 |𝐶2��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Certainly, when |𝑉 | >> 𝐶, and when the sparsity of adjacency matrix is exploited (for instance through the compressed sparse row format), its time complexity is sometimes expressed as 𝑂(𝐿|𝐸|𝐶) [44, 178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As the GCN’s space complexity is concerned, we need to store the embeddings of all nodes plus the weight matrix at each layer, which is 𝑂(𝐿|𝑉 |𝐶 +𝐿𝐶2), or 𝑂(𝐿|𝑉 |𝐶) when |𝑉 | >> 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GraphSAGE illustrates the same propagation procedure from a microscopic view, with a fixed number, denoted 𝑠, of sampled neighbours involved in the convolutional operation (Equation 51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The overall time complexity of GraphSAGE is, therefore: 𝑂(𝐿|𝑉 |(𝑠𝐶 + 𝐶2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notice that when 𝑠 equals |𝑉 |, the time complexity of GraphSAGE is the same as the vanilla GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Then, when each node aggregates messages from its higher-order neighbours, denoted h-hop neighbours here, the propagation rule can be put as: 𝐻 (𝑙) = 𝜎 � ˆ𝐴ℎ𝐻 (𝑙−1)𝑊 (𝑙)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A typical representative is MixHop [3] (refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thus, the time complexity at each layer is: 𝑂(|𝑉 |2𝐶ℎ + |𝑉 |𝐶2) or 𝑂(|𝑉 |2𝐶ℎ) when |𝑉 |ℎ >> 𝐶, and the space complexity stays unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' From a microscopic view, represented by the approach k-hop GNN [155], the time complexity of involving h-hop neighbours in convolutional operation would be 𝑂(𝐿|𝑉 |(𝑘ℎ𝑚𝑎𝑥𝐶 + 𝐶2)), or 𝑂(𝐿|𝑉 |𝑘ℎ𝑚𝑎𝑥𝐶) when 𝑘ℎ𝑚𝑎𝑥 >> 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Clearly, the time complexities of both macroscopic and microscopic algorithms grow with ℎ, and when ℎ equals one, they degrade to the versions of 1-hop neighbourhood algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the vanilla GCN and GraphSAGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 42 Message (representative approach) Time complexity Space complexity Node feature X (GCN) 𝑂 �𝐿 �|𝑉 |2𝐶 + |𝑉 |𝐶2�� 𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� Count of graphlets + X (GSN) 𝑂(|𝑉 |𝑘 |𝑆 |−1 𝑚𝑎𝑥 + 𝐿(|𝑉 |2(𝐶 + 𝑜) + |𝑉 |(𝐶 + 𝑜)2) 𝑂 �L|𝑉 |(𝐶 + 𝑜) + 𝐿(𝐶 + 𝑜)2� Distance information + X (P-GNN) 𝑂(|𝑉 |3 + 𝐿|𝑉 |(𝑇𝑛 + 𝐶2)) 𝑂 �L|𝑉 |𝐶 + 𝐿𝐶2� Random feature + X (rGIN) 𝑂(𝐿(|𝑉 |2(𝐶 + 𝑟) + |𝑉 |(𝐶 + 𝑟)2) 𝑂 �L|𝑉 |(𝐶 + 𝑟) + 𝐿(𝐶 + 𝑟)2� Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Time and space complexity from the perspective of a message content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' |𝑉 | is the number of nodes in the graph, 𝐶 is node feature channels (assuming the number of features is fixed for all layers), 𝐿 is the number of convolutional layers, |𝑆 | is the maximum size of a set of graphlets, 𝑜 is the number of orbits in graphlets, 𝑘𝑚𝑎𝑥 is maximum node degree, 𝑇 is the number of anchor sets, 𝑛 is the maximum number of nodes in an anchor set, and 𝑟 is the length of the random feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Thirdly, approaches with neighbourhood defined on random walks typically include the following steps (represented by PinSage [198]): performing 𝑤 times random walks of length 𝑙, ranking the visited 𝑣 nodes based on the visited times, aggregating messages from the top 𝑠 nodes, and finally applying weight matrix on node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the overall time complexity is termed as: 𝑂(𝐿|𝑉 |(𝑤𝑙 + 𝑣𝑙𝑜𝑔𝑣 + 𝑠𝐶 + 𝐶2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Normally, there is no need to record all the random walks, so the space complexity is still 𝑂(𝐿|𝑉 |𝐶 + 𝐿𝐶2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Comparing the time complexity of PinSage with that of GraphSAGE, we see that with the extra step of performing random walks and ranking visited nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the term 𝑤𝑙 and the term 𝑣𝑙𝑜𝑔𝑣, PinSage is more expensive in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In the fourth subcategory, we take k-GNN [146] as an example to analyse the complexity of having k-node subgraphs as neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The approach aims to learn embeddings for k-node tuples, and the neighbours of each k-tuple are defined as other k-tuples containing one node that is not in the focal k-tuple(refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Each k-tuple aggregates messages from all its k-tuple neighbours, with a time complexity of 𝑂(𝑘|𝑉 |𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, on all �|𝑉 | 𝑘 � k-tuples and 𝐿 layers, the overall time complexity is: 𝑂(𝐿�|𝑉 | 𝑘 �(𝑘|𝑉 |𝐶 + 𝐶2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' To store the embeddings of �|𝑉 | 𝑘 � node tuples and the weight matrices at all layers, it requires 𝑂(𝐿�|𝑉 | 𝑘 �𝐶 +𝐶2)) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This approach is essentially different from the previous ones, in that it is to generate embeddings for k-tuples instead of for each node, resulting in the term �|𝑉 | 𝑘 � appearing in both its time and space complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Clearly, its complexity grows combinatorially with 𝑘, and easily surpasses the complexities of all other algorithms when 𝑘 is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In practice, however, the value of 𝑘 generally does not exceed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In addition, Table 5 lists the time and space complexities of approaches that include extra node features in the GNNs (corresponding to the message content taxonomy in Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' First, when the count of graphlets, or more specifically, the count of node orbits is added to the node features (represented by the approach GSN [27]), it requires a preprocessing step to count the number of each node orbit, then performing the general convolutional operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The cost of counting orbits depends on the size of graphlet |𝑆| and the maximum degree of nodes 𝑘𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another difference from the vanilla GCN is that the node feature dimension will increase by the number of orbits, denoted 𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, its time complexity is 𝑂(|𝑉 |𝑘 |𝑆 |−1 𝑚𝑎𝑥 + 𝐿(|𝑉 |2(𝐶 + 𝑜) + |𝑉 |(𝐶 + 𝑜)2), and its space complexity is 𝑂 �L|𝑉 |(𝐶 + 𝑜) + 𝐿(𝐶 + 𝑜)2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Second, when distance information is included, as in the approach P-GNN [201], it requires first calculating the shortest path distances between all nodes (𝑂(|𝑉 |3) in the typical Floyd-Warshall algorithm), then aggregating message from a number of anchor sets (𝑇 anchor sets and each containing at most 𝑛 nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore the time complexity would be 𝑂(|𝑉 |3 + 𝐿|𝑉 |(𝑇𝑛 + 𝐶2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another less expensive version is to calculate a limited-hop, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', h-hop, shortest path distance in the preprocessing step, whose time complexity is 𝑂(|𝑉 |𝑘ℎ𝑚𝑎𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Third and lastly, when random features are 43 included, represented by the approach rGIN [169], the impact on time complexity is mainly due to the increase in feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This is because the cost of generating random features is generally negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We finally discuss the complexity of approaches that have different learning scopes (corresponding to the taxonomy in Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For GCNs running on subgraphs, represented by the GraphSAINT [205], the cost includes two steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', the subgraph sampling and the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Given the cost of sampling𝑇𝑠 and a set sampled subgraphs G (maximum number of nodes in sampled subgraphs denoted |𝑉𝑠 |), its complexity is: 𝑂(𝑇𝑠 + |G|𝐿(|𝑉𝑠 |2𝐶 + |𝑉𝑠 |𝐶2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The cost 𝑇𝑠, depending on the choice of the sampler, is normally less expensive than the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The key term is, therefore, |G|𝐿|𝑉𝑠 |2𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When |𝑉𝑠 | << |𝑉 |, subgraph-based approaches significantly reduce the training cost of the GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Similarly, for GCNs running on local subgraphs, exemplified by the Shadow-GNN [204], the two steps are extracting local subgraphs (extraction cost is denoted as 𝑇𝑒, the maximum number of nodes in extracted local subgraphs is denoted as |𝑉𝑙 |), and training the GCN on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Therefore, the time complexity is 𝑂(𝑇𝑒 + |𝑉 |𝐿(|𝑉𝑙 |2𝐶 + |𝑉𝑙 |𝐶2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Note that local subgraphs are usually extracted at each node, so the number of extracted subgraphs equals the number of nodes |𝑉 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Given that |𝑉𝑙 | << |𝑉 | (we should also have |𝑉𝑙 | << |𝑉𝑠 |), local subgraph based GCNs are generally much faster in training than full graph or subgraph based GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5 DISCUSSION AND OUTLOOK After reviewing the traditional structural measures and the graph convolutional networks, we are set to answer the following research question: How are these two classes of methods related, especially how traditional structural measures of Network Science can inform GCN methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In this section, we first briefly discuss the performance of GCNs in major learning tasks, then move on to drawing connections between GCNs and traditional structure based approaches, and finally introduce three future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 GCN’s performance in learning tasks Convolutional Neural Networks have been shown to be state-of-the-art in various tasks in the area of image processing, including image classification, object detection, and semantic segmentation [74, 120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' GCNs have also achieved promising performances in various graph-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As an extension of CNNs in graph data, GCNs, since their appearance, have received a lot of attention and are viewed as state-of-the-art by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, there are works showing that simple heuristics from traditional network science achieve a comparative performance of GCNs [203, 209], or even beat them in link prediction and network reconstruction tasks [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A recent paper shows that simply feeding heuristics derived from nodes similarity scores in a logistic regression model can achieve the best performance in link prediction among many deep learning approaches, including GCNs [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In addition, Katz index is the top performer in the network reconstruction task, followed by VGAE which uses GCN as the graph encoder [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Although the majority of GCN approaches focus on node classification and graph classification tasks, they rarely include structural heuristic-based methods as baselines in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This overlook could hinder a comprehensive evaluation of the performance of graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Additionally, comparing GCN approaches with traditional heuristic-based methods could help to better understand the strengths and limitations of GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We believe a closer integration of graph deep learning approaches and traditional network science approaches would immensely benefit both communities, and revealing the connections between the two classes of methods lays the foundation of this integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Connections between traditional network science approaches and GCNs Based on the current literature, the connections between GCNs and traditional structure based approaches are observed via the following four aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The first aspect covers the foundations of GCNs in traditional Network Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' the second aspect focuses on their similarities in dealing with directed networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' the third and final aspect cover two typical applications of traditional structural information in GCNs: (i) number of graphlets and (ii) distance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1 Message passing based approaches and GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' As we have seen in message passing based approaches (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4), a node’s influential score or centrality is calculated through iteratively aggregating the scores of its neighbours until it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking the eigenvector centrality, for example, the centrality of node 𝑖, denoted 𝑥(𝑖), is formulated as: 𝑥(𝑖) = 𝑐 ∑︁ 𝑗 ∈𝑁 (𝑖) 𝑥(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 𝑥, a vector of all nodes’ centralities, is found to converge to the dominant eigenvector of the adjacency matrix 𝐴, and 𝑐 converges to the reciprocal of the dominant eigenvalue of 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Interestingly, graph convolutional networks adopt the same idea of neighbourhood aggregation, and the iteration process is implemented through the usage of multiple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking the vanilla GCN for example, we have the following convolutional operation: ℎ(𝑙) 𝑣 = 𝜎 �� � ∑︁ 𝑢∈N(𝑣) 1 𝑐𝑣𝑢 ℎ(𝑙−1) 𝑢 𝑊 (𝑙)�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Comparing the above two expressions, one major difference is obviously the appearance of weight matrices: in eigenvector centrality, the influential score is directly calculated from forward propagation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', a power iteration), while in the GCN, weight matrices are updated in the backward propagation with the help of labelled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another subtle yet significant difference is that GCNs allow rich node features (n-dimensional vector for each node), while traditional message passing approaches, such as the eigenvector centrality, alpha centrality or PageRank, only support using a numeric value that represents the node’s importance or influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' These two points are also the main reasons why GCNs have quickly gained popularity — the learnable setting makes GCNs suitable for various types of tasks, and the support of rich node features makes them appropriate for different types of real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Despite the advancements and popularity of graph convolutional networks, traditional network science approaches remain important in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' They have a strong theoretical foundation, which can provide insights into the underlying mechanisms of networked systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Furthermore, traditional approaches are often more computationally efficient than deep learning approaches, making them more practical for certain types of tasks or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Overall, the continued use and development of traditional network science approaches alongside newer methods, such as GCNs, can help to deepen our understanding of complex networked systems and advance the field as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2 Dealing with link direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When directions of links are considered, we observe interesting connections between the traditional message passing approach HITS [109] and the recent graph convolutional approach DGP [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' HITS proposes to distinguish two roles in webpages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', authorities and hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Authorities, being reliable information sources, are pointed by hubs (based on incoming edges to the node), while hubs, acting as a home page or library, point to authorities (based on outgoing edges from the node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An authority score and a hub score are defined in a mutually dependent way: 𝑎(𝑖) = ∑︁ 𝑗 ∈𝑁 𝑖𝑛 𝑖 ℎ(𝑗), ℎ(𝑖) = ∑︁ 𝑗 ∈𝑁 𝑜𝑢𝑡 𝑖 𝑎(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 45 Interestingly, DGP, as a graph convolutional approach, proposes to distinguish link direction through a two-phase propagation scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=', one phase capturing outgoing connections and the other capturing incoming connections (find more in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2): 𝐻 = 𝜎 � 𝐾 ∑︁ 𝑘=0 𝛼𝑎 𝑘 ˆ𝐴𝑎 𝑘𝜎 � 𝐾 ∑︁ 𝑘=0 𝛼𝑑 𝑘 ˆ𝐴𝑑 𝑘𝑋𝑊𝑑 � 𝑊𝑎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Clearly, the major difference here is that in DGP one type of connection is stacked on top of another, and therefore only one representation is learnt, instead of two scores as in HITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Besides, k-hop outgoing/incoming connections are included at once in one convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another GCN approach that applies exactly the same idea of distinguishing outgoing edges and incoming edges is Asymmetric GNN, or AGNN [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It proposes a one-way message passing that only operates on the outgoing or incoming edges of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Two embeddings are then generated for each node to model their roles of sending and receiving information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It is also possible to design a one-way GCN at particular layers, while still considering both types of edges in other layers, which could allow the model to focus on different aspects of the graph structure at different stages of processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Number of graphlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The number of graphlets, or more specifically, node orbits or edge orbits are important topological features around individual nodes or edges (find more in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In a traditional non-learning setting, a vector composed of the counts of a chosen set of node orbits is used to distinguish the roles of nodes [97, 140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Weights of the orbits, when introduced, are calculated from hand-coded function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In graph convolutional networks, the count of graphlets is added as additional features in the message passing scheme, as we have seen in GSN [27], F -MPNN [14], and ID-GNN [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taken GSN for example, node orbits 𝑥𝑉 (𝑢), 𝑥𝑉 (𝑣) or edge orbits 𝑥𝐸 (𝑢, 𝑣) are introduced as follow: h𝑙+1(𝑣) = MLP1 �� � ℎ𝑙 (𝑣), ∑︁ 𝑢∈N(𝑣) 𝑀𝐿𝑃2 �h𝑡 (𝑣), h𝑡 (𝑢), x𝑉 (𝑣), x𝑉 (𝑢), e(𝑢, 𝑣)��� � , h𝑙+1(𝑣) = MLP1 �� � ℎ𝑙 (𝑣), ∑︁ 𝑢∈N(𝑣) 𝑀𝐿𝑃2 �h𝑡 (𝑣), h𝑡 (𝑢), x𝐸 (𝑢, 𝑣), e(𝑢, 𝑣)��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Obviously, in a learning setting, the weights on all types of features, including the count of graphlets, are learned in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Another interesting difference between non-learning approaches and GCN approaches is that the former chooses to include all node or edge orbits within a given size, while the latter tends to focus on specific substructures like cycles or cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' One open problem in using graphlets or orbits in GCNs is determining which ones to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Existing approaches have focused on using cliques and/or cycles within a specific range [14, 27, 200], without providing much rationale for this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' While these types of graphlets and orbits are crucial in some contexts, it is likely that other types could also be exploited to improve the performance of GCN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There is still much to be explored in terms of the utility of different graphlets and orbits in GCN models, and further research in this area could lead to advances in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='4 Distance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The path related information is largely used in traditional structural measures, such as in closeness centrality, betweenness centrality, 𝜅-path centrality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Taking the closeness centrality, for example, it is defined as the reciprocal of the average shortest path from the focal node 𝑖 to all other nodes: Θ𝐶 (𝑖) = |𝑉 | − 1 � 𝑗 ∈𝑉,𝑗≠𝑖 𝑑(𝑖, 𝑗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 46 The value of a node’s closeness centrality is directly used to describe the node’s capacity of spreading information on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Unsurprisingly, the distance information is also made of use in graph convolutional networks, as we have seen in P-GNN [201] and DE-GNN [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In P-GNN, for example, the distance between a node and several anchor sets is included in the convolutional operation: h𝑙 𝑣 = AGG(𝑙) � M𝑙−1 𝑖 , ∀𝑖 ∈ [1,𝑘] � , M𝑙−1 𝑖 = {𝐹 (𝑑𝑢𝑣,ℎ𝑙−1 𝑢 ,ℎ𝑙−1 𝑣 ), ∀𝑢 ∈ 𝑆𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In DE-GNN, the distance information between node 𝑣 and a target node set 𝑆 is used as an extra initial node feature: ℎ(0) 𝑣 = 𝐶𝑂𝑁𝐶𝐴𝑇 (𝑥𝑣,𝜁 (𝑣 | 𝑆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Recall that this idea of including extra structural features as additional initial node features is also found in F -MPNN [14], ID-GNN [200], and rGIN [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content='3 Future directions Although recent years have witnessed the great success of graph convolutional networks in various domains, there are still many open problems to be solved and a lot of room for further exploration [32, 216, 219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Except for the frequently mentioned directions, such as proposing GCNs for more complicated types of networks or to further increase the expressivity or scalability of GCNs, we would like to point out three potential directions which combine the traditional graph analysis approaches and GCN approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Exploring the applicability of more structural measures in GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We have seen appearances of various structural measures in GCNs, from the simplest node degree [78] to the much more complicated distance information [201] and graphlet orbits [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, there are many other traditional structural measures that have yet to be fully explored in the context of GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, subgraph formation based measures, such as the clustering coefficient [184] and the closure coefficient [94, 196], could be incorporated as node-level features or used to weight the edges of the graph [177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Global path based measures, such as the closeness or betweenness centrality measures, can be used to guide the sampling of nodes, edges or subgraphs when constructing the training set for a GCN [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, we could use closeness centrality to select the nodes that are most influential in the graph and build subgraphs based on these nodes as the input to the GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' It would be interesting to see how these and other structural measures could be utilised in GCNs to improve performance on certain tasks or in particular types of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Improving the explainability of GCNs / Guiding the choice of GCNs via traditional structural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When it comes to the explainability of GCN models, existing methods, represented by perturbation-based methods, mostly focus on generating explanations for a trained GCN [199, 202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There are, however, still many questions to be answered, such as how different GCNs perform differently on different types of networks, and what are the reasons for these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An analysis from a structural information perspective can provide more insights into how different GCN models extract and utilise graph structural information, and how the information may differ across different GCN models and graph types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This can help to better understand the strengths and limitations of different GCN models and how to effectively apply them in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Moreover, in view of the large collection of GCN models and their composition modules, it is difficult to decide which one to choose and how to set it up for the targeted dataset and task [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Traditional structural measures could be used as indices for selecting the appropriate GCN model and the related modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, for graphs that are rich in triangles, a particular GCN would be a better choice, while for graphs where quadrangles are overrepresented, another GCN model should be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Integrating edge features in GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' While the vanilla GCN primarily focuses on aggregating and passing information from neighbouring nodes, it is important to consider the role of edge attributes in many real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' For example, in consumer review networks, the ratings of products are often labelled on the edges, and in social networks, 47 the type and frequency of interactions are labelled on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Integrating edge features into GCNs could not only enhance the applicability of the model but also increase the accuracy and relevance of its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' There are works that naively include edge features in GCN or propose a tailor-made model to encompass them [27, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' However, there is still much to be learned about the utility of traditional edge-level structural measures in GCN models, such as the edge orbits [82], the edge clustering coefficient [181], the local path index [128], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Further research in this area is likely to yield valuable insights and improvements to the performance of GCN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 6 CONCLUSION The complexity of graph data mainly comes from its intricate topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Mining and exploiting graph structural information have always been one of the focal points in the study of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A large amount of work in traditional network science proposes various types of structural measures, especially local structural measures, to characterise and study complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When more nodes or edges are involved, such approaches, however, become infeasibly complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Graph convolutional networks, on the other hand, are proposed to automatically extract relevant features from nodes’ neighbourhoods, and in this manner, avoid choosing and manually calculating structural metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In order to reveal the connections between the two classes of methods, especially how traditional structural measures can inform GCNs, in this paper, we first reviewed the traditional structure-based approaches in Network Science and proposed a new taxonomy encompassing many seemingly unrelated concepts from a structural perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' With this prerequisite knowledge, we then extend the scope to the prominent and powerful graph convolutional networks, and provide a Network Science perspective on them — review and classify GCNs from three structural angles, which are the layer-wise message aggregation scope, the message content, and the overall learning scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Furthermore, we extensively discuss the connections between the traditional structural approaches and the graph convolutional networks and suggest three future research directions in the joint research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' We believe that the well-established foundations of traditional structure-based approaches in Network Science not only form the basis for GCNs but also could, and probably should, serve as a driving force for their future advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Yu-Xuan Qiu, Joakim Skarding and Xiaohan Zhang for their helpful comments and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' This work was supported by the Australian Research Council, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' DP190101087: “Dynamics and Control of Complex Social Networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' REFERENCES [1] Sergi Abadal, Akshay Jain, Robert Guirado, Jorge López-Alonso, and Eduard Alarcón.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Convolutional neural networks on graphs with fast localized spectral filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Advances in neural information processing systems 29 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [53] Hongbo Deng, Michael R Lyu, and Irwin King.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A generalized 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Heatmap centrality: A new measure to identify super-spreader nodes in scale-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Plos one 15, 7 (2020), e0235690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [58] Leo Egghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Theory and practise of the g-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Scientometrics 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Identifying influential spreaders by gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Scientific reports 9, 1 (2019), 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [122] Hao Liao, Manuel Sebastian Mariani, Matúš Medo, Yi-Cheng Zhang, and Ming-Yang Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Ranking in evolving complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physics Reports 689 (2017), 1–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [123] David Liben-Nowell and Jon Kleinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The link-prediction problem for social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Journal of the American society for information science and technology (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [124] Pedro G Lind, Marta C Gonzalez, and Hans J Herrmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Cycles and clustering in bipartite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physical review E (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [125] Qiang Liu, Yu-Xiao Zhu, Yan Jia, Lu Deng, Bin Zhou, Jun-Xing Zhu, and Peng Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Leveraging local h-index to identify and rank influential spreaders in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physica A: Statistical Mechanics and its Applications 512 (2018), 379–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [126] Zhiyuan Liu and Jie Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Introduction to graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Synthesis Lectures on Artificial Intelligence and Machine Learning 14, 2 (2020), 1–127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [127] Linyuan Lü, Duanbing Chen, Xiao-Long Ren, Qian-Ming Zhang, Yi-Cheng Zhang, and Tao Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Vital nodes identification in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physics Reports 650 (2016), 1–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [128] Linyuan Lü, Ci-Hang Jin, and Tao Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Similarity index based on local paths for link prediction of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physical Review E 80, 4 (2009), 046122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [129] Linyuan Lü, Yi-Cheng Zhang, Chi Ho Yeung, and Tao Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Leaders in social networks, the delicious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PloS one 6, 6 (2011), e21202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [130] Linyuan Lü, Tao Zhou, Qian-Ming Zhang, and H Eugene Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The H-index of a network node and its relation to degree and coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Nature communications 7, 1 (2016), 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [131] Ling-ling Ma, Chuang Ma, Hai-Feng Zhang, and Bing-Hong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Identifying influential spreaders in complex networks based on gravity formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physica A: Statistical Mechanics and its Applications 451 (2016), 205–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [132] Qian Ma and Jun Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Identifying and ranking influential spreaders in complex networks with consideration of spreading probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physica A: Statistical Mechanics and its Applications 465 (2017), 312–330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [133] Xiaoke Ma and Lin Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Biological network analysis: insights into structure and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Briefings in functional genomics 11, 6 (2012), 434–442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [134] Xiaojian Ma and Yinghong Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' The local triangle structure centrality method to rank nodes in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Complexity 2019 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [135] Yao Ma and Jiliang Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Deep learning on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [136] Alexandru Cristian Mara, Jefrey Lijffijt, and Tijl De Bie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' An Empirical Evaluation of Network Representation Learning Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Big Data (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [137] Travis Martin, Xiao Zhang, and Mark EJ Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Localization and centrality in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Physical review E 90, 5 (2014), 052808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 52 [138] Víctor Martínez, Fernando Berzal, and Juan-Carlos Cubero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A survey of link prediction in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' ACM computing surveys (CSUR) 49, 4 (2016), 1–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [139] Silvio Micali and Zeyuan Allen Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Reconstructing markov processes from independent and anonymous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Discrete Applied Mathematics 200 (2016), 108–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [140] Tijana Milenković and Nataša Pržulj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Uncovering biological network function via graphlet degree signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Cancer informatics 6 (2008), CIN–S680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [141] Ron Milo, Shalev Itzkovitz, Nadav Kashtan, Reuven Levitt, Shai Shen-Orr, Inbal Ayzenshtat, Michal Sheffer, and Uri Alon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Superfamilies of evolved and designed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Science 303, 5663 (2004), 1538–1542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [142] Ron Milo, Shai Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri Chklovskii, and Uri Alon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Network motifs: simple building blocks of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Science (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [143] Anders Møller and Michael I Schwartzbach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Static program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Feb (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [144] Enys Mones, Lilla Vicsek, and Tamás Vicsek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Hierarchy measure for complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PloS one 7, 3 (2012), e33799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [145] Flaviano Morone and Hernán A Makse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Influence maximization in complex networks through optimal percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Nature 524, 7563 (2015), 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 4602–4609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [147] Balcilar Muhammet, Renton Guillaume, Héroux Pierre, Gaüzère Benoit, Adam Sébastien, and Paul Honeine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' When spectral domain meets spatial domain in graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In Thirty-seventh International Conference on Machine Learning (ICML 2020)-Workshop on Graph Representation Learning and Beyond (GRL+ 2020).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Oxford university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [152] Mark EJ Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' A measure of betweenness centrality based on random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Social networks 27, 1 (2005), 39–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [153] Vincenzo Nicosia, John Tang, Cecilia Mascolo, Mirco Musolesi, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Learning convolutional neural networks for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' PMLR, 2014–2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [155] Giannis Nikolentzos, George Dasoulas, and Michalis Vazirgiannis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Asymmetric transitivity preserving graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 1105–1114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [159] Ashwin Paranjape, Austin R Benson, and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Using graph theory to analyze biological networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' BioData mining 4, 1 (2011), 1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' [161] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' Deepwalk: Online learning of social representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E3T4oBgHgl3EQf-QtO/content/2301.04824v1.pdf'} +page_content=' 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0000000000000000000000000000000000000000..33a0fe78483fca7cd101d5e20381681832c4e7d0 --- /dev/null +++ b/uNAzT4oBgHgl3EQfsP0n/content/tmp_files/2301.01655v1.pdf.txt @@ -0,0 +1,1906 @@ +FAST ABSOLUTE 3D CGO-BASED ELECTRICAL IMPEDANCE +TOMOGRAPHY ON EXPERIMENTAL TANK DATA +S. J. HAMILTON, P. A. MULLER, D. ISAACSON, V. KOLEHMAINEN, J. NEWELL, O. +RAJABI SHISHVAN, G. SAULNIER, AND J. TOIVANEN +Abstract. Objective: To present the first 3D CGO-based absolute EIT reconstructions from ex- +perimental tank data. Approach: CGO-based methods for absolute EIT imaging are compared to +traditional TV regularized non-linear least squares reconstruction methods. Additional robustness +testing is performed by considering incorrect modeling of domain shape. Main Results: The CGO- +based methods are fast, and show strong robustness to incorrect domain modeling comparable to +classic difference EIT imaging and fewer boundary artefacts than the TV regularized non-linear +least squares reference reconstructions. Significance: This work is the first to demonstrate fully 3D +CGO-based absolute EIT reconstruction on experimental data and also compares to TV-regularized +absolute reconstruction. The speed (1-5 seconds) and quality of the reconstructions is encouraging +for future work in absolute EIT. +1. Introduction +The main objective of this paper is to demonstrate the feasibility, speed, and robustness of +producing 3D absolute (static) images of the electrical conductivity inside a tank, from experimental +Electrical Impedance Tomography (EIT) data measured on an array of electrodes on the tank’s +surface by the ACT5 [65] adaptive current tomography system, using CGO-based reconstruction +algorithms. The primary contributions of this work are that it presents the first use of Complex +Geometrical Optics (CGO) based methods to produce absolute (static) images of the conductivity +from experimentally measured EIT data in 3D, and studies their robustness under modeling errors. +“Absolute”, or “static,” conductivity images are images of the conductivity inside a body made +from one set of experimental measurements made on the surface of the body at one time [24]. +Alternatively, “dynamic,” or “time-difference,” imaging uses two sets of data measured at two +different times to make an image of the change in the conductivity that took place between the +two times the measurements were made. We present both types of images made from experimental +data by CGO-based algorithms in 3D in this paper. +Applications of EIT began with geophysical exploration over a century ago [4,44] and have since +expanded into several fields including the transport of fluids and gases [26,40,80], and biomedical +imaging [2, 7, 20, 29, 35, 36, 42, 70]. +Systems for monitoring lung function in real-time are now +commercially available and clinical trials are in progress to determine the extent to which some of +these systems might be used to guide mechanical ventilation [12, 73, 75]. These systems typically +Key words and phrases. electrical impedance tomography, absolute imaging, conductivity. +S. J. Hamilton is with the Department of Mathematical and Statistical Sciences; Marquette University, Milwaukee, +WI 53233 USA, email: sarah.hamilton@marquette.edu. +D. Isaacson is with the Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, USA. +V. Kolehmainen and J. Toivanen are with the Department of Applied Physics, University of Eastern Finland, +FI-70210 Kuopio, Finland. +P. A. Muller is with the Department of Mathematics & Statistics; Villanova University, Villanova, PA 19085 USA. +J. Newell is with the Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, +USA. +O. Rajabi Shishvan and G. Saulnier are with the Department of Electrical and Computer Engineering, University +at Albany - SUNY, Albany, NY 12222, USA. +J. Toivanen is with the Department of Applied Physics, University of Eastern Finland, FI-70210 Kuopio, Finland. +1 +arXiv:2301.01655v1 [math.NA] 4 Jan 2023 + +2 +HAMILTON ET AL. +make and display two dimensional images of the differences in conductivity between two states, e.g., +lungs filled with air and lungs depleted of air, in order to take advantage of dynamic (difference) +imaging methods and algorithms [2,29]. The desire to improve the diagnosis of cancer and stroke +has motivated the development of systems and methods capable of imaging the absolute or static +internal conductivity and permittivity in 3D [17, 18, 33, 43, 45]. The interested reader is referred +to [15] for further review of applications of EIT. +Most absolute (static) EIT reconstruction focuses on solving a simplified linearized problem or +iteratively solving an optimization-based method which requires repeated solutions to the forward +problem which can become costly for highly dense meshes. CGO-based methods are direct meth- +ods in that they do not require iteration. +They have the capability to solve the full nonlinear +mathematical inverse problem, and do not require repeated solutions to the forward problem, e.g. +via the finite element method (FEM). The D-bar inversion algorithm, which is a specific type of +CGO-based inversion algorithm in 2D, has been used to make both absolute and difference images +in 2D in real-time from experimental data measured on tanks, and for difference imaging on human +subjects in laboratory and clinical settings. See [55] for a recent review of the 2D D-bar method +and its applications, and [59] for the theoretical foundation. +In 3D, existence and uniqueness of solutions [58,63] can be shown for a 3D D-bar type equation +but the constructive proof, upon which the reconstruction algorithms are built, bypasses the 3D D- +bar equation instead using a high (non-physical) frequency limit connecting the nonlinear scattering +data and the linear Fourier data. Advances in the numerical implementation of 3D CGO-based +methods are more recent. +The first numerical implementation of 3D CGO-based methods on +simulated electrode data using current injection on the surface of the sphere was presented in [34]. +A 3D CGO-based inversion algorithm, regularization scheme, rigorous proof of stability under +certain hypotheses, and examples reconstructing the conductivity inside a sphere from numerically +simulated Dirichlet data on the entire surface of the sphere (without electrodes) were given in [47]. +Until now it has been an open question if 3D CGO-based algorithms could be used with experimental +data. +This paper answers this question affirmatively by showing that the conductivity can be +recovered rapidly and robustly from experimental data measured on 32 electrodes on the surface +of a rectangular prism building on the work of [34]. +It was demonstrated in [10,32,36,42] that it is possible to make electrical impedance tomography +(EIT) Systems that produce both 3D static and dynamic images of the interior of the chest showing +heart and lung regions, as well as changes in those regions due to ventilation and perfusion. These +systems used linearized and iterative methods. The former are fast but less accurate and the latter +are slower. Recent 3D CGO inversion algorithms and their analysis, when applied to synthetic or +simulated data, suggested that they have the potential to be fast and more accurate [23, 34, 47]. +Here we test their capabilities on experimental EIT data. +In this paper we compare the following methods for making static 3D images from experimental +tank data, collected by the ACT5 imaging system: +• A CGO linearization method based on Calder´on’s original proposal introducing CGO ideas +into the subject [13]. We call this algorithm Calder´on’s method. +• Two CGO-based methods for solving the full non-linear inversion problem based on Sylvester +and Uhlmann’s original uniqueness proof and Nachman’s constructive uniqueness proof +in [58,71]. The first is the texp algorithm, and the second is called t0, [22,34] both of which +are simplifications of the original constructive proof. +• A more traditional iterative inversion method based on optimization using a Total Variation +(TV) regularization. +The EIT data were measured on 32 electrodes attached to the six surfaces of a rectangular tank +using the ACT5 system. The system can adaptively determine patterns of currents to apply to +all 32 electrodes that result in voltages on those electrodes that are proportional to the applied +currents. These “eigen currents” form a discrete orthogonal set and provide improved voltage data + +3D CGO-BASED EXPERIMENTAL EIT +3 +for reconstructing the conductivity inside the tank from a limited amount of current or power +that can be applied to the tank. The theory of adaptive current tomography systems is given +in [14,30,31,38,52,53,60,68]. +The remainder of the paper is organized as follows. Section 2 provides a brief review of the +mathematics of EIT and encompasses the methods used in the work: the CGO and reference +reconstruction algorithms, experimental setup for the ACT5 data collection, robustness tests that +will be explored, and metrics used to evaluate the results. The results section, Section 3, presents +slice, 3D, and isosurface renderings of the recovered conductivities for correct and incorrect domain +modeling. Section 4 contains a discussion of the results and conclusions are drawn in Section 5. +2. Methods +In this work we compare three CGO-based methods (Calder´on’s Method, the texp Method, and +the t0 Method) to a more common iterative Total Variation (TV) regularized non-linear least +squares method for absolute EIT image reconstruction. +For comparison, we also include time +difference EIT images for the CGO-based methods and compare them to a linearized difference +imaging method. We begin with a brief review of the mathematical problem. +2.1. Mathematical Background. The mathematical problem of reconstructing the internal con- +ductivity, when measurements can be made with infinite precision everywhere on the boundary of +a body, is currently called “Calder´on’s Problem” by much of the mathematical community since +A. Calder´on formulated this inverse problem as follows [13]: In two or more dimensions, can one +find the conductivity, σ(x), inside a body, Ω, from all possible electrical measurements made on the +surface, ∂Ω, of the body? Here the voltage or potential, u(x), inside the body, due to an applied +surface current density, j(x), is assumed to satisfy the following low frequency approximation to +Maxwell’s Equations, which we will refer to as the conductivity equation, with a Neumann boundary +condition: +∇ · σ(x)∇u(x) += +0, +x ∈ Ω ⊂ R3 +(2.1) +σ(x)∂u(x) +∂ν(x) += +j(x), +x ∈ ∂Ω . +Here ν = ν(x) denotes the outward pointing unit normal to the surface at x ∈ ∂Ω, and, ∂u(x) +∂ν(x) = +ν(x) · ∇u(x). We denote the mapping from applied current density to resulting voltage on the +surface, called the “Neumann-to-Dirchlet” (ND) map, by Rσ, where Rσj ≡ u(x) for x ∈ ∂Ω, and +u(x) solves the conductivity equation (2.1) with Neumann data j(x). +The Calder´on problem can also be formulated as the mathematical problem. If one is given the +ND map or operator, Rσ, or equivalently its inverse, the Dirichlet-to-Neumann (DN) operator, +Λσ := R−1 +σ , can one find σ(x)? Calder´on showed that if σ(x) does not differ too much from a +constant then one can recover an approximation to it from the ND map. His short paper showed +this could be done using Fourier Transforms in a very clever way, by introducing special solutions, +eζ·x, to the conductivity equation with constant conductivity, i.e. the Laplace equation, where ζ +is a complex valued vector with ζ · ζ = 0. This is possible in two or more dimensions when the +conductivity is close to a constant and gave birth to Calder´on’s method as described in this paper, +as well as more powerful methods for solving the full non-linear problem by generalizing Calder´on’s +solutions to what we now call Complex Geometrical Optics (CGO) solutions, introduced by [71,72] +in their landmark paper proving that the Calder´on problem in three or more dimensions has a unique +solution, where the inversion problem is reduced to a Fourier transform in the limit |ζ| → ∞. The +texp and t0 methods described below in § 2.3 follow this strategy, as opposed to the 2D D-bar +methods described in [55]. This strategy is not possible in 2D where the CGO solutions are found +by solving a first order linear PDE involving D-bar operators in the complex vector ζ and taking + +4 +HAMILTON ET AL. +it to 0. Constructive proofs using these CGO solutions and D-bar ideas from inverse scattering +theory, along with applications to scattering and acoustics were given in [58,63]. +CGO methods were used to prove uniqueness in the more difficult case of 2D where constructive +methods for reconstructing the conductivity were given in detail in [59] for σ ∈ C2(Ω) and later +in [5] for σ ∈ L∞(Ω). Other pioneering work proving uniqueness under a variety of hypotheses on +the conductivity include [25,48,50,63,64,74]. Extensive references to more recent progress in the +analysis and numerical analysis of the Calder´on problem can be found in [11,23,28,56,77]; [47,55]. +In what follows we will be interested in the problem of reconstructing an approximation to the +internal conductivity from finitely many experimental measurements made with finite precision. +Unfortunately, this is an ill-posed problem and, unlike the purely mathematical problem, it does not +have a unique solution. Nevertheless, it is sometimes possible to reconstruct useful approximations +to the internal conductivity with a finite number of degrees of freedom, or voxels, which we will +illustrate by making images from experimental data and comparing them to the actual interior +conductivity within the tanks. +The EIT problem for a body with L electrodes, eℓ, ℓ = 1, 2, . . . , L, on its surface, is to find an +approximation to the internal conductivity from all the possible electrical measurements made on +these L electrodes. In particular we will assume that we apply L − 1 linearly independent patterns +of currents, ⃗I(k), k = 1, 2, . . . , L − 1, to the L electrodes, and measure the resulting L − 1 voltage +patterns, ⃗V (k), where ⃗I(k) +ℓ +and ⃗V (k) +ℓ +denote the applied current, and measured voltage from the kth +pattern, on the ℓth electrode, for ℓ = 1, 2, . . . , L. From conservation of charge, and our choice of +ground, we assume +L +� +ℓ=1 +⃗I(k) +ℓ += +L +� +ℓ=1 +⃗V (k) +ℓ += 0. +The kth voltage or potential, u(k)(x), resulting inside the body is determined by the conductivity +equation, ∇ · σ∇u(k) = 0, and the “Complete Electrode Model”, [16,69] where: +� +eℓ +σ(x)∂u(k)(x) +∂ν(x) dS(x) = I(k) +ℓ +, +x ∈ eℓ +σ(x)∂u(k)(x) +∂ν(x) += 0, +x /∈ +L +� +ℓ=1 +eℓ +(2.2) +u(k)(x) + zℓσ(x)∂u(k)(x) +∂ν(x) += V (k) +ℓ +, +x ∈ eℓ. +Here zℓ is the effective contact, or surface, impedance on the ℓth electrode. The current patterns +used will be eigenvectors of the Current to Voltage map, which is a matrix approximation to the ND +map, for the homogeneous saline tank. They are found numerically by simulating a homogeneous +tank for static/absolute imaging, and experimentally by adaptive methods for difference/dynamic +imaging. The matrix approximations to the ND maps from the conductivity distribution σ are +denoted by the L × L matrices, Rσ, where Rσ⃗I(k) = ⃗V (k), and we define Rσ⃗1 = ⃗0. Here the vectors +⃗1, ⃗0, denote vectors all of whose components are 1, or 0, respectively. The discrete analog of the +DN map used in the CGO-based methods is given by Lσ := R−1 +σ . +2.2. Calder´on’s Method. Following Calder´on’s original paper [13], Calder´on’s method approx- +imates the conductivity, σ(x), from its Fourier transform. Here we present a brief description of +the method and refer the reader to [9,13,34,57] for further details. Calder´on’s method assumes the +conductivity is a small perturbation, δσ(x), from a constant background, σb, i.e. σ(x) = σb+δσ(x). +In this paper, we assume that the background conductivity σb = 1. If the background constant is +not one, then the problem can be scaled and unscaled as in [34,39]. +The three steps of Calder´on’s method in 3-D, as described in [34], are: + +3D CGO-BASED EXPERIMENTAL EIT +5 +Step 1: Use the DN maps Λσ and Λ1 to approximate the Fourier transform of the small +perturbation in conductivity, � +δσ(z), by +� +δσ(z) ≈ ˆF(z) := − +1 +2π2|z|2 +� +∂Ω +eπi(z·x)+π(a·x) (Λσ − Λ1) eπi(z·x)−π(a·x)dS(x), +(2.3) +where z and a satisfy +z, a ∈ R3, |z| = |a|, and z · a = 0, +(2.4) +and Λ1 is the DN map for a constant conductivity of 1. +Step 2: Take the inverse Fourier transform of ˆF(z): +δσCAL(x) ≈ F−1{ˆF(z)}(x) = +� +R3 +ˆF(z)e−2πi(x·z)dz. +(2.5) +Step 3: Add the background to the perturbation to recover the approximate conductivity, +σCAL(x): +σCAL(x) = σb + δσCAL(x). +(2.6) +The definition of the Fourier transform in Calder´on’s method is different from that used in the +texp and t0 methods described below. However, each method is consistent with its definition and +is consistent with the respective literature on that method. +In this paper, we compute ˆF(z) in spherical Fourier coordinates. As such, we choose +z = |z|(cos ˜φ sin ˜θ, sin ˜φ sin ˜θ, cos ˜θ) and a = |z|(cos ˜φ cos ˜θ, sin ˜φ cos ˜θ, − sin ˜θ), +for |z| ≥ 0, 0 ≤ ˜φ ≤ 2π and 0 ≤ ˜θ ≤ π so that z and a satisfy (2.4). Then, the inverse Fourier +transform in (2.5) becomes +δσCAL(x) = +� ∞ +0 +� 2π +0 +� π +0 +|z|2 sin ˜θˆF(|z|, ˜φ, ˜θ)e−2πi(x·z)d˜θd˜φd|z|. +(2.7) +Additionally, we implement the use of a mollifier, ˆη +� +z +y +� +, as introduced in [13] for some parameter +y ∈ R to reduce Gibbs phenomenon caused by jump discontinuities in σ(x) while recovering δσCAL +δσCAL(x) = +� ∞ +0 +� 2π +0 +� π +0 +|z|2 sin ˜θˆF(|z|, ˜φ, ˜θ)ˆη +�z +y +� +e−2πi(x·z)d˜θd˜φd|z|. +(2.8) +We implement the same mollifier as used in [9,34], +ˆη +�z +y +� += e−πt|z|2, +(2.9) +where y = 1/ +√ +t and t acts as a smoothing parameter with larger t values producing smoother +reconstructions with smaller jumps at points of discontinuity in σ(x). +Since noise causes (2.3) to blow up at large |z|, we use a non-uniform truncation regularization +strategy. A similar regularization strategy was proved stable for the 2-D D-bar method in [46], +which also noted non-uniform truncation also produces reliable reconstructions. In our case, we +will first compute ˆF(z) for |z| within an outer radius of Tz2. We keep values of ˆF(z) whose real and +imaginary amplitudes are below a threshold determined by the amplitudes of ˆF within a smaller +radius |z| ≤ Tz1; ˆF is set to 0 everywhere else. Both radii are chosen empirically. The inner radius +Tz1 is chosen as a region in z-space where noise does not cause ˆF to blow up and Tz2 is chosen to +keep as much reasonable information from ˆF without introducing holes in the non-zero region of ˆF. +As such, our truncated ˆF is computed by + +6 +HAMILTON ET AL. +ˆFR(z) = +� ˆF(z), +if |z| ≤ Tz2, and |Re(ˆF(z))| ≤ max +|z|≤Tz1 +|Re(ˆF(z))|, |Im(ˆF(z))| ≤ max +|z|≤Tz1 +|Im(ˆF(z))| +0, +else. +(2.10) +With our truncated ˆF, equation (2.8) is truncated in the radial variable, leading to the approxi- +mation +δσCAL +R +(x) = +� Tz2 +0 +� 2π +0 +� π +0 +|z|2 sin ˜θ ˆFR(|z|, ˜φ, ˜θ)e−πt|z|2e−2πi(x·z)d˜θd˜φd|z|. +(2.11) +For difference images shown in this paper, we only perform Steps 1 and 2 of the method and +replace Λ1 in (2.3) with a reference DN map, Λσref before computing (2.11). Thus, the flow for +absolute images is +(Λσ, Λ1) +1 +−→ ˆF(z) +2 +−→ δσCAL +3 +−→ σCAL +and the flow for difference images in this paper is +� +Λσ, Λσref +� +1 +−→ ˆF(z) +2 +−→ δσCAL. +2.2.1. Numerical Implementation. In this section, we review the implementation details of Calder´on’s +method introduced in [34]. +For Step 1, we compute (2.3) for |z| ≤ Tz2 by discretizing the boundary integral as follows, +ˆF(z) += +− +1 +2π2|z|2 +� +∂Ω +eπi(z·x)+π(a·x) (Λσ − Λ1) eπi(z·x)−π(a·x)dS(x) +≈ +− +1 +2π2|z|2 +�|∂Ω| +L +� +(eπi(z·x)+π(a·x))TQ (Lσ − L1) QT � +eπi(z·x)−π(a·x)� +, +(2.12) +where |∂Ω| is the surface area of the domain; L is the number of electrodes; x ∈ R(L × 3) denotes +the vector of the Cartesian centers of the L electrodes; T denotes the traditional, non-conjugate, +matrix transpose; Lσ and L1 denote the discrete matrix approximations to the DN maps Λσ and Λ1 +respectively; and Q ∈ RL × Nli denotes an orthonormal basis created using Nli linearly independent +applied currents over L electrodes as was done in [34]. The matrix L1 is based on the FEM solution +of the CEM (2.2). Problem-specific mesh details are given below in section 2.6. +We then compute ˆFR according to equations (2.10) and (2.12). For Step 2, on an equally-spaced +16 × 16 × 16 rectangular grid in x, the conductivity difference δσCAL +R +(x) is computed via (2.11) +using a 3D Simpson’s rule using N|z| = 10, N˜θ = 10, and N˜φ = 30 uniformly-spaced nodes on +the |z|, ˜θ, and ˜φ grids, respectively. As the number of nodes in the Fourier domain increases, so +does computation time, but some artefact reduction can be achieved. Empirically, the artefact +reduction did not seem significant enough to warrant an increased computational time beyond +these parameter choices. +Difference images are computed using (2.11) and (2.12), replacing L1 with a discrete reference +DN map, Lσref in (2.12). For the phantom tank experiments in this paper, this reference map +is from data collected with a tank filled only with saline matching the experiment and no other +inclusions. +The absolute reconstructions of σCAL +R +(x) in this paper are produced using (2.6) replacing σb with +σbest +σCAL +R +(x) = σbest + δσCAL +R +(x). +The solution is then interpolated to a 64 × 64 × 64 rectangular grid. +Following [39], σbest is given by +σbest = +�K +k=1 +�L +ℓ=1 U k +ℓ U k +ℓ +�K +k=1 +�L +ℓ=1 U k +ℓ V k +ℓ +, +(2.13) + +3D CGO-BASED EXPERIMENTAL EIT +7 +where U k +ℓ is the kth simulated voltage pattern measured on electrode ℓ with a homogeneous conduc- +tivity of 1 S/m and V k +ℓ is the kth voltage pattern measured on electrode ℓ for the inhomogeneous +conductivity σ. The simulated voltages are the same voltages used to compute L1, as described in +§2.6.1. +2.3. The texp and t0 methods. Both the texp and t0 methods are derived from the construc- +tive proofs of [58, 62, 63] and involve special solutions called Complex Geometrical Optics (CGO) +solutions [71]. A brief summary is included here for the reader’s convenience. For further details +see [22,23,34,58]. +Assuming that the conductivity is a constant σc = 1 in a neighborhood of the boundary ∂Ω, the +real-valued conductivity equation (2.1) can be transformed to the Schr¨odinger equation +(−∆ + q(x)) �u(x) = 0, +x ∈ R3, +(2.14) +via the change of variables q(x) = +∆√ +σ(x) +√ +σ(x) +and �u(x) = u(x) +� +σ(x), by extending σ(x) ≡ 1 for all +x ∈ R3 \¯Ω. For ζ(ξ) ∈ Vξ, unique CGO solutions exist to the transformed problem +(−∆ + q(x)) ψ(x, ζ) = 0, +x ∈ R3, +where ψ(x, ζ) ∼ eix·ζ for large |x| or |ζ|, and +Vξ = +� +ζ ∈ C3���ζ2 = 0, (ξ + ζ)2 = 0 +� +, +for each ξ ∈ R3, +(2.15) +where ζ2 = ζ ·ζ and ζ is a purely auxiliary parameter. The conductivity σ(x) can then be recovered +from the DN map Λσ as follows. +For each x ∈ ∂Ω and ζ ∈ Vξ, solve the Fredholm integral equation of the Second Kind, +ψ(x, ζ) = eix·ζ − +� +∂Ω +Gζ(x − y) (Λσ − Λ1) ψ(y, ζ) dS(y), +(2.16) +where +Gζ(x) = eix·ζ +(2π)3 +� +R3 +eix·k +|k|2 + 2k · ζ dk, +x ∈ R3 \{0}, +denotes the Faddeev Green’s function [27]. Then, evaluate the scattering data +t(ξ, ζ) = +� +∂Ω +e−ix·(ξ+ζ) (Λσ − Λ1) ψ(x, ζ) dS(x). +(2.17) +For |ζ| large, the Schr¨odinger potential q(x) can be recovered via the inverse Fourier transform +q(x) ≈ F−1 {t(ξ, ζ)} (x) = +1 +(2π)3 +� +R3 eix·ξt(ξ, ζ) dξ, +x ∈ R3. +(2.18) +The conductivity is then recovered by solving the boundary value problem +� +(−∆ + q(x))˜u(x) += +0 +x ∈ Ω ⊂ R3 +˜u(x) += +1 +x ∈ ∂Ω, +(2.19) +for �u(x) and evaluating σ(x) = (˜u(x))2. This is the full nonlinear reconstruction method. +Replacing the CGO solutions ψ(x, ζ) by their asymptotic behavior eix·ζ in the scattering data +(2.17) via +texp(ξ, ζ) = +� +∂Ω +e−ix·(ξ+ζ) (Λσ − Λ1) eix·ζ dS(x). +(2.20) +yields a ‘Born approximation’ typically called the texp approximation for EIT. Using this approx- +imate scattering data in place of the fully nonlinear t(ξ, ζ), one proceeds with the recovery of an +approximate potential qexp(x) via (2.18) and conductivity σexp(x) via (2.19). The flow is: +(Λσ, Λ1) +1 +−→ texp(ξ, ζ) +2 +−→ qexp(x) +3 +−→ σexp(x). + +8 +HAMILTON ET AL. +An intermediate approximation can be computed by replacing the Faddeev Green’s function +Gζ(x) in the single layer potential, in (2.16), for the traces of the CGOs, with the standard Green’s +function G0(x) = +1 +4π|x| for the Laplacian operator. Thus, one solves +ψ0(x, ζ) = eix·ζ − +� +∂Ω +G0(x − y) (Λσ − Λ1) ψ0(y, ζ) dS(y), +(2.21) +for the CGOs ψ0(x, ζ), avoiding the exponentially growing Faddeev Green’s function. A correspond- +ing approximation to the scattering data is then computed by using ψ0(x, ζ) in place of ψ(x, ζ) in +(2.17) and then continuing to recover q0(x) and σ0(x). The flow is then +(Λσ, Λ1) +1 +−→ ψ0(x, ζ) +2 +−→ t0(ξ, ζ) +3 +−→ q0(x) +4 +−→ σ0(x). +We point out that the methods, as outlined above, assumed that the conductivity was a constant +σc = 1 near the boundary of the domain. As mentioned in section 2.2, the problem can be scaled +and unscaled as in [34, 39] when the constant σc is not one. In practice, we estimate the best-fit +constant conductivity fit to the data σbest as given by (2.13). Explicitly, as in [34], we scale the DN +map by using +1 +σbest Λσ in place of Λσ, and re-scale at the end using σexp(x) = σbest (˜uexp(x)) and +σ0(x) = σbest +� +˜u0(x) +� +. +In this work we consider both the texp and t0 methods. We note that this is the first time that +t0 has been implemented on non-continuum DN data and the first time that either texp or t0 have +been demonstrated on experimental 3D EIT data, for absolute or time-difference EIT imaging. +2.3.1. Numerical Implementation. Here we provide the numerical details pertinent to the imple- +mentation of the texp and t0 algorithms outlined above. As with the Calder´on method above, the +main idea is to expand functions in the same orthonormal basis Q as described in section 2.2. +Following [21], for each electrode center xℓ, expand ψ(x, ζ) and eix·ζ as +ψ(xℓ, ζ) ≈ +Nli +� +j=1 +bj(ζ)Qj +ℓ, +eixℓ·ζ ≈ +Nli +� +j=1 +cj(ζ)Qj +ℓ, +ℓ = 1, . . . L, +(2.22) +where Qj +ℓ denotes the (ℓ, j) entry of Q. +Then, the boundary integral equation (2.21) can be +approximated as follows +ψ0(xℓ, ζ) += += eixℓ·ζ − +� +∂Ω +G0(xℓ − y) (Λσ − Λ1) ψ0(y, ζ) dS(y) +≈ +eixℓ·ζ − +L +� +ℓ′=1 +� +Eℓ′ +G0(xℓ − y) (Λσ − Λ1) ψ0(y, ζ) dS(y) +≈ +eixℓ·ζ − +� L +� +ℓ′=1 +� +Eℓ′ +G0(xℓ − y)dS(y) +� +� +(Lσ − L1) ψ0(yℓ′, ζ) +� +, +(2.23) +where Eℓ′ denotes the ℓ′th extended electrode where �L +ℓ′=1 Eℓ′ = ∂Ω and the Eℓ′ are mutually +disjoint [37]. Note the true electrodes need not cover the surface ∂Ω, only the extended (math- +ematical) electrodes that we will use to discretize the integral. Then, using the expansions from +(2.22) in (2.23), we have +Nli +� +j=1 +bj(ζ)Qj +ℓ +≈ +Nli +� +j=1 +cj(ζ)Qj +ℓ − +� L +� +ℓ′=1 +� +Eℓ′ +G0(xℓ − y)dS(y) +� � +�(Lσ − L1) +Nli +� +j=1 +bj(ζ)Qj +ℓ′ +� +� += +Nli +� +j=1 +cj(ζ)Qj +ℓ − +� L +� +ℓ′=1 +� +Eℓ′ +G0(xℓ − y)dS(y) +� � +� +Nli +� +j=1 +bj(ζ)fj (yℓ′) +� +� , +(2.24) + +3D CGO-BASED EXPERIMENTAL EIT +9 +where fj (yℓ′) represents the action of (Lσ − L1) on Qj evaluated at yℓ′, which can be computed as +the (ℓ′, j) entry of Q (Lσ − L1). Define +�G0(ℓ, ℓ′) = +� +1 +4π|xℓ−yℓ′| +ℓ ̸= ℓ′ +0 +ℓ = ℓ′, +(2.25) +where we have removed the singularities at xℓ = yℓ. Assuming |Eℓ| = | ∂Ω | +L +for each ℓ = 1, . . . , L, +and using (2.25) in (2.24) we find +Nli +� +j=1 +bj(ζ)Qj +ℓ +≈ +Nli +� +j=1 +cj(ζ)Qj +ℓ − | ∂Ω | +L +Nli +� +j=1 +bj(ζ) +L +� +ℓ′=1 +�G0(ℓ, ℓ′)fj (yℓ′) +≈ +Nli +� +j=1 +cj(ζ)Qj +ℓ − | ∂Ω | +L +Nli +� +j=1 +bj(ζ) +� +�G0Q (Lσ − L1) +� +(ℓ, j) +or in matrix form, +Q⃗b = Q⃗c − | ∂Ω | +L +�G0Q (Lσ − L1)⃗b. +The solution to this equation can be found by solving the following system for the unknowns ⃗b, +(I + A)⃗b = ⃗c, +(2.26) +where A = | ∂Ω | +L QT �G0Q (Lσ − L1), and I is the identity matrix. If the extended electrodes Eℓ +are not uniform in size, then one could compute as weighted sum replacing the uniform weight +|Eℓ| = | ∂Ω | +L +as appropriate. +Next, following [34], the scattering data t0(ξ, ζ(ξ)) can be computed for all |ξ| less than a chosen +truncation radius Tξ via +t0(ξ, ζ) ≈ +� | ∂Ω | +L +� +e−ix·(ξ+ζ)�T Q (Lσ − L1)⃗b +|ξ| < Tξ +0 +else, +(2.27) +since ⃗b = QTψ0(x, ζ), where x ∈ R(L × 3) is the same as in Section 2.2.1, i.e. a vector storing the +centers of the electrodes. +Next, the approximate potential q0 is recovered by computing the inverse Fourier transform of +the truncated scattering data t0 using a Simpson’s rule in 3D, +q0(x) = +1 +(2π)3 +� +[−Tξ,Tξ]3 eix·ξ t0(ξ, ζ(ξ)) dξ. +Alternatively, one could use an IFFT to achieve additional speedup, taking care with quadrature +points and the particular form of the kernel. Following [34], the conductivity σ0(x) was recovered +by first solving the boundary value problem (2.19), using the PDE toolbox in Matlab using +a mesh with approximately 21, 000 3D elements, then computing σ0(x) = σbest (˜u(x))2. The 3D +visualizations were obtained by interpolation to a 64 × 64 × 64 rectangular grid using Matlab’s +scatteredInterpolant function. The ζ values were computed following the minimal-zeta approach +outlined in [23]. We remark that while the non-existence of exceptional points for the solution of +the boundary integral equation (2.16) is proven for large |ζ| [58,71], as well as small |ζ| [19]; it is +still an open question for the intermediate values required here to perform the computation on a +computer. + +10 +HAMILTON ET AL. +The texp reconstructions were obtained in an analogous fashion to those of t0, this time bypassing +the boundary integral equation (2.17) and directly computing +texp(ξ, ζ) ≈ +� | ∂Ω | +L +� +e−ix·(ξ+ζ)�T Q (Lσ − L1) QT � +eix·ζ� +|ξ| < Tξ +0 +else, +where we have replaced the vector of coefficients ⃗b with the expansion of the asymptotic behavior +of eix·ζ given by QT � +eix·ζ� +. +Difference imaging can be performed with the t0 and texp methods by replacing the matrix L1 +with Lσref and computing σdiff(x) = σ(x) − σbest in the final step. +2.4. Reference methods. Total Variation regularized non-linear least squares reconstructions +will serve as the reference reconstructions for the CGO-based absolute imaging cases considered +here. A classic linear difference imaging scheme, also reviewed below, will serve as the reference for +the CGO-based difference images. +2.4.1. Absolute imaging with TV regularization. A widely used numerical approach for absolute +EIT is the total variation (TV) regularized non-linear least squares minimization +ˆσ = arg min +σ>0{∥Le (V − U(σ, z∗)) ∥2 + αTVβ(σ)}, +(2.28) +where U(σ, z) is the finite dimensional forward map, z∗ ∈ RL are the fixed electrode contact +impedances obtained from an initialization step of the minimization, Le is the Cholesky factor of +the noise precision matrix of e so that LT +e Le = Γ−1 +e , scalar valued α is the regularization parameter +and TVβ(σ) is the (smooth) TV regularization functional [66] +TVβ(σ) = +� +Ω +� +∥∇σ∥2 + β dr, +(2.29) +where β is the (fixed) smoothing parameter. +The forward model U(σ, z) in (2.28) is based on +the finite element (FEM) discretization of the complete electrode model [69]. For details of the +FEM model, see [41, 78, 79]. In the FEM model, the electric conductivity is approximated as a +linear combination of the piecewise linear nodal basis functions in a uniform tetrahedral mesh +of N nodes, leading to vector of unknowns σ ∈ RN, and the electric potential is approximated +similarly in a significantly more dense tetrahedral mesh with refinements near the electrodes. The +non-linear optimization in (2.28) is solved by a lagged Gauss-Newton method equipped with a +line search algorithm. The line search is implemented using bounded minimization such that the +non-negativity σ > 0 is enforced. For more details of the method, see [76]. +The fixed contact impedances z∗ and initial (constant) conductivity estimate for (2.28) are ob- +tained from the solution of the non-linear least squares problem +(σ0, z∗) = arg min +σc,z>0{∥Le (V − U(σc, z)) ∥2}, +(2.30) +where the scalar σc ∈ R is the coefficient of a spatially constant conductivity image σc1 and z ∈ RL. +The non-linear least squares problem (2.30) is solved by a Gauss Newton optimization. +2.4.2. Linear difference imaging. In linear difference imaging, see e.g. [6, 8], the objective is to +reconstruct the change in conductivity between two measurements (V1; V2). The reference linear +difference imaging approach of this paper uses linearized approximations of the observation models +V1 ≈ U(σ0, z∗) + Jσ(σ1 − σ0) + e1 +(2.31) +V2 ≈ U(σ0, z∗) + Jσ(σ2 − σ0) + e2, +(2.32) + +3D CGO-BASED EXPERIMENTAL EIT +11 +where the Jacobian matrix Jσ of U(σ, z∗) is evaluated at σ = σ0. With these linearized models, the +difference in the measurements becomes +δV += V2 − V1 += (U(σ0) + Jσ(σ2 − σ0) + e2) − (U(σ0) + Jσ(σ1 − σ0) + e1) += Jσδσ + δe, +(2.33) +where δσ = σ2 − σ1 and δe = e2 − e1. Now, the inverse problem is to reconstruct δσ based on the +difference data δV and the model (2.33). A widely used formulation for the linear problem is to use +the (generalized) Tikhonov regularization with a smoothness promoting regularization functional +ˆδσ = arg min +δσ +� +||Lδe(δV − Jσδσ)||2 + ||Lpδσ||2� +, +(2.34) +where Lδe is the Cholesky factor of the noise precision matrix of δe so that LT +δeLδe = Γ−1 +δe += +(Γe1 + Γe2)−1. In this paper, the regularization matrix Lp is constructed by utilization of a distance +based covariance function. More specifically, we set LT +p Lp = Γ−1 +p , where the (prior covariance) +matrix Γp is constructed using the distance based correlation function [54] +Γp(i, j) = std(σ)2 exp +� +−∥xi − xj∥2 +2a2 +� +, +i, j = 1, . . . , N, +(2.35) +where the parameter a controls the correlation length and can be solved by setting the distance +∥xi − xj∥ to a selected value d (e.g. half the radius of the target) and setting Γp(i, j) to the desired +covariance for that distance (e.g. 1% of variance). +2.5. Experimental Setup. ACT5 [65] is a 32 electrode parallel EIT instrument that uses 32 +current sources to apply patterns of current to the target and 32 voltmeters to measure the resulting +voltages. The current sources in ACT5 adjust the delivered current to compensate for current lost +through shunt impedance, including that introduced by the capacitance of cables that connect +the instrument to the electrodes, enabling the desired current patterns to be applied with high +precision [67]. ACT5 can apply sinusoidal currents at frequencies in the range of 11 kHz to 1 MHz +with a peak amplitude of 0.25 mA. The voltmeters measure voltages up to 0.5 V peak with a +maximum signal-to-noise ratio (SNR) of 96 dB. Because the current source compensates for shunt +capacitance of the cables, the system operates with grounded-shield cables [1]. +A test tank was built from 3/8-inch-thick Plexiglas with interior dimensions of 17.0cm x 25.5cm +x 17.0cm. +Thirty-two electrodes were fabricated of 16-gauge 316 stainless steel, each 80.0 mm +square. The inside surfaces of the tank were milled out in the shape and depth of the electrodes +so that the interior surfaces were flat. The nominal gaps between electrodes were 5.0 mm. The +32 electrodes were placed four on each end and six on each side. +Bolts through the center of +each electrode passed through the Plexiglas, allowing connections to the cables from the ACT5 +electronics. Five sides of the tank were glued together, and provision was made to secure the top +with threaded rods at all four corners. Access holes of 0.6 mm at various sites in the top to allow +threads to suspend targets, and 25-Ga hypodermic needles to complete the filling of the tank after +the lid was in place. In use, a mixture of saline at room temperature at the desired background +conductivity is made and measured using a Oakton CON 6+ handheld conductivity meter. +Targets were made the day before experiments by adding NaCl and a few drops of food dye to +distilled water until the desired conductivity was obtained, as measured by an immersible conduc- +tivity meter. The solution was then heated slowly with stirring as 4% agar (Fisher Scientific) was +added until the temperature reached 85◦C−90◦C. The mixture was poured into 5.27 cm inside- +diameter spherical molds and allowed to harden overnight. Test cells were also filled at this time +so the final conductivities could be verified using the ACT5 instrument. +Data were first collected of the tank filled with saline of the desired conductivity (24 mS/m) +with no targets present. Then, targets were added to the tank supported on fine toothpicks from + +12 +HAMILTON ET AL. +Figure 1. Experimental setups. +Left: Top view of the one target experiment. +Middle: Top view of two target experiment. Right: side view of height of targets +above floor of tank. Note that the targets were measured at 290 mS/m in test cells. +below. Photos were captured of the tank, with each target in place, to verify its/their position. +The conductivity values of the spheres used were approximately 290 mS/m. Averaged data, over +100 frames, were used in the subsequent conductivity reconstructions. The experimental data will +be made publicly available on github.1 +Note that even though we are testing on a rectangular prism tank, the mathematical reconstruc- +tion algorithms are not limited to this geometry. +2.6. Robustness Tests. In addition to testing the reconstruction methods with the correct domain +modeling, we consider a moderate, as well as strong, mis-modeling of the domain. The incorrect +domain modeling comes into play in the following places for the CGO-based methods. First in +the simulation of the L1 DN data that requires solving the forward EIT problem with a known +conductivity σ ≡ 1 S/m and the applied current patterns. Next, the incorrect domain modeling +presents as incorrect information about the centers of the electrodes x and surface area for the +domain | ∂Ω |; the subsequent reconstructions of σCAL, σexp, and σ0 are recovered on the incorrect +domain. For the TV regularized method, the incorrect domain modeling is present in the FEM +based forward map U(σ, z) used in the minimization problem (2.28) and computational mesh. For +the linear difference imaging method, the incorrect domain modeling is present in the Jacobian +matrix of the forward map Jδσ in the minimization problem (2.34) and computational mesh. +Accurate domain modeling can be challenging in practice. To address this we explored a domain +with a moderately incorrect domain shape, 18cm x 27cm x 19cm, as well as a domain with a +significantly incorrect domain shape, 20cm x 35cm x 25cm2. Electrodes were uniformly distributed +as in the original box design, but clearly had different locations and spacing than the truth due +to the new box dimensions. +Figure 2 (center and right) depict the larger box domains. +Each +reconstruction method was then tested assuming that the actual experimental voltage and current +measurements were coming from the boxes shown in Figure 2. +Time-difference reconstructions were performed using basal measurements with only 24 mS/m +saline in the experimental tank. This data was then used in the reconstruction methods discussed +in §2.3.1 and §2.4.2. +2.6.1. Forward modeling and Regularization Parameters. For the CGO-based methods, the matrix +approximation L1 to the DN map Λ1 was formed using simulated voltage data, using EIDORS [3,78]. +The EIDORS data is produced by solving the forward conductivity problem with the finite element +approximation of the Complete Electrode Model (2.2) with σ ≡ 1 S/m. In each domain modeling +case, see Figure 2, the forward problem was solved on a box of the corresponding dimensions, +with 8cm by 8cm electrodes, using approximately 250, 000 nodes and 1.2 million elements with +high refinement used near the electrodes, and the default contact impedance in EIDORS. For the +1See https://github.com/sarahjhamilton/open3D_EIT_data +2Due to the smaller ratio of longest side to shortest side, the 16 × 16 × 16 X-grid used for Calder´on’s method as +mentioned in §2.2.1 does not capture the whole domain, so a 32 × 32 × 32 x-grid is used in the significantly incorrect +domain case + +C +DD12 +28 +19 +273D CGO-BASED EXPERIMENTAL EIT +13 +Correct Model (17x25.5x17) +Box 18x27x19 +Box 20x35x25 +Figure 2. 3D renderings of simulated boxes used in robustness testing. +reference methods, the FEM based forward problem was solved using the same meshes for the +electric potential as with the CGO-based methods and the conductivities were approximated in +sparser, uniform meshes with 8, 918; 9, 410; 14, 678 nodes and 43, 690; 45, 106; 72, 438 elements, +respectively, for the different domain modeling cases. +The regularization parameters for each of the reconstruction methods were chosen as follows. +The regularization parameters Tz1 and Tz2 for Calder´on’s method were empirically chosen to be 1.4 +and 1.7, respectively, and the mollifying parameter was chosen to be t = 0.1 for all reconstructions. +These values were chosen empirically to provide the best visual reconstructions. The truncation +radius Tξ for the nonlinear scattering data used in the texp and t0 methods was chosen empirically +in the range [10.5, 12] for each case shown here. The localization of the target(s) did not appear to +change significantly with Tξ, however the contrast does appear sensitive to this parameter choice. +Further discussion is given below in Section 4. For the total variation regularized method, the +regularization parameter α and the smoothing parameter β were selected by computing a series of +reconstructions with varying α and β values, in the range [1e-6, 100] for α and [1e-4, 0.1] for β, +and by choosing the parameters that gave the best visual quality of reconstructions in the correct +domain model case. The chosen values α = 0.01 and β = 0.001 were used in all the test cases with +different domain models. For the linear difference imaging method, the visually best reconstructions +were obtained using a standard deviation of conductivity std(σ) = 16σ0 and parameter a calculated +using a covariance of 1 % of the variance at a correlation distance d = lbox/8, where lbox is the +length of the longest side of the box that is used as the domain model. +2.7. Evaluation Metrics. As the main focus of this work is experimental data reconstruction, +comparing to a known ‘truth’ with zero error is not possible. The conductivity values for the targets +were measured at approximately 290 mS/m and the targets were placed to be roughly centered in +a sub-cube of the overall box. These data allow us to approximate the localization error (LE) of +our reconstructions and maximum conductivity in each reconstructed target. +LE, as in [34], measures the distance between the centroids of the reconstructed targets and +those of the true targets by +LE = +� +(xrecon +1 +− xtruth +1 +)2 + (xrecon +2 +− xtruth +2 +)2 + (xrecon +3 +− xtruth +3 +)2. +(2.36) +A localization error of 0 is ideal, and signifies the targets are reconstructed in the correct loca- +tion. In our experiments, the centers of the spherical targets were placed at the centers of the +nearest electrode in each direction and in (2.36), (xtruth +1 +, xtruth +2 +, xtruth +3 +) is our best estimate of the +true location of the target center, acknowledging possible errors within millimeters of the truth. +Following [34], to obtain the centers of the reconstructed targets, we use a thresholded segmenta- +tion of the reconstructions to identify targets with MATLAB’s regionprops3. The thresholds for +segmentation were used to empirically identify regions with more than a certain percentage of the + +22 +21 +13 +8 +5 +6 +28 +20 +2 +12 +0 +-2 +-4 ~ +27 +9- +3 +19 +-8 +11 +26 +10 +-5 +5 +0 +0 +5 +10 +-1022 +14 +21 +13 +8 +6 +28 +20 +4 - +12 +2 +0 ++3 +-2 +-4 +27 +-6 +19 +-8 - +11 +26 +10 +-5 +9 +18 +0 +5 +10 +-10 ++122 +21 +14 +10 +28 +20 ++3 +70 +X +-5 ~ +-10~ +15 +10 +26 +-10 +0 +-5 +18-5 +90 +-10 +5 +1010 +-15 +×2 +X114 +HAMILTON ET AL. +maximum reconstructed conductivity. We note that the volume of the segmented targets changes +with the choice of threshold, but as we increase the percentage (i.e. threshold) the locations of the +centroids stabilize (up to millimeters), from which we computed the LE. +As a major focus of this work is to study how the algorithms tolerate modeling errors, we report +a scaled version of the LE, +Scaled LE = +� +(xrecon +1 +− xtruth +1 +)2 + (xrecon +2 +− xtruth +2 +)2 + (xrecon +3 +− xtruth +3 +)2 +25.5 +, +(2.37) +where we have divided the LE from (2.36) by 25.5cm, the length of the longest side of the true +tank. This scaling value for the computing the LE metric is kept fixed across incorrect modeling +scenarios. +3. Results +We first present absolute EIT reconstructions with correct domain modeling, for the one and +two targets, and for the four reconstruction methods (Figure 3). Cross-sections in the center of +the bottom row of electrodes of the box are shown on the left; this slice cuts through the cen- +ter(s) of the physical target(s). Isosurfaces, shown in the middle column, are extracted from the +3D reconstructions to indicate the locations, size, and number of targets. The isosurfaces were +extracted using MATLAB’s regionprops3 with threshold segmentation in the range of 60% to +85% of the maximum recovered conductivity in the tank. A uniform threshold across all examples +and reconstruction methods was not used as the blurring varied across methods. Note that since +the isosurfaces are dependent on this threshold, they can omit information from the full recon- +struction. Therefore, the third column shows a full 3D rendering of conductivity in the domain. +This 3D rendering is produced in MATLAB by displaying 100 equally-spaced x1-x3 slices set to +0.1 transparency of each slice. +Table 1. One target absolute imaging evaluation metrics across all domain mod- +eling scenarios. +Calder´on +texp +t0 +TV +Scaled LE +correct domain +0.042 +0.094 +0.098 +0.043 +box 18x27x19 +0.075 +0.131 +0.119 +0.021 +box 20x35x25 +0.241 +0.250 +0.231 +n/a +Max. Target +Conductivity (mS/m) +correct domain +63.77 +300.00 +308.67 +44.26 +box 18x27x19 +76.94 +303.32 +284.67 +70.14 +box 20x35x25 +85.51 +305.63 +260.88 +n/a +Figure 4 shows the absolute EIT reconstructions for the moderately mismodelled domain whereas +Figure 5 shows the absolute EIT reconstructions for the significantly mismodelled domain. Dif- +ference images are shown in Figure 6. Metrics of maximum conductivity per target and scaled +localization error are shown in Tables 1 and 2 for the absolute images and Table 3 for the difference +images. Bolded entries correspond to the best metric value in each row. +4. Discussion +Beginning with the correct domain modeling scenario, each method, Calder´on, texp, t0, and +TV produced reconstructions that clearly show the target(s) with good localization. +The TV +images are sharpest, as expected. The texp and t0 methods achieved the best contrast and best +approximated the estimated experimental conductivity. The Calder´on and TV methods achieved +lower contrast but better scaled localization error. Notably, as shown in Figure 6, in terms of +artefacts, the absolute CGO reconstructions are as clean as their corresponding difference images. +The CGO methods required simulated voltages data for the same experimental setup but with a + +3D CGO-BASED EXPERIMENTAL EIT +15 +Table 2. Two target absolute imaging evaluation metrics across all domain mod- +eling scenarios. Note: Target 1 is the same target as in the one-target case. +Calder´on +texp +t0 +TV +Scaled LE +Target 1 +correct domain +0.056 +0.098 +0.105 +0.034 +box 18x27x19 +0.103 +0.135 +0.133 +0.021 +box 20x35x25 +0.268 +0.267 +0.257 +n/a +Target 2 +correct domain +0.062 +0.195 +0.180 +0.031 +box 18x27x19 +0.120 +0.191 +0.214 +0.057 +box 20x35x25 +0.277 +0.311 +0.295 +n/a +Max. Target +Conductivity (mS/m) +Target 1 +correct domain +55.61 +310.95 +315.50 +55.57 +box 18x27x19 +73.53 +216.29 +322.39 +68.29 +box 20x35x25 +80.00 +334.58 +264.72 +n/a +Target 2 +correct domain +60.47 +234.85 +226.35 +53.94 +box 18x27x19 +67.94 +153.82 +203.84 +64.22 +box 20x35x25 +77.81 +285.06 +244.80 +n/a +Table 3. Two target difference imaging evaluation metrics across all domain mod- +eling scenarios. Note: Target 1 is the same target as in the one-target case. +Calder´on +texp +t0 +Linear +Scaled LE +Target 1 +correct domain +0.034 +0.095 +0.103 +0.049 +box 18x27x19 +0.022 +0.116 +0.124 +0.065 +box 20x35x25 +0.113 +0.242 +0.243 +n/a +Target 2 +correct domain +0.053 +0.100 +0.094 +0.050 +box 18x27x19 +0.043 +0.116 +0.115 +0.029 +box 20x35x25 +0.112 +0.230 +0.233 +n/a +Max. Target +Conductivity (mS/m) +Target 1 +correct domain +57.18 +277.83 +295.63 +484.83 +box 18x27x19 +61.72 +272.80 +282.05 +227.95 +box 20x35x25 +56.30 +253.46 +243.88 +227.95 +Target 2 +correct domain +53.84 +266.91 +284.25 +490.88 +box 18x27x19 +59.91 +294.17 +306.30 +208.05 +box 20x35x25 +55.07 +293.61 +282.38 +n/a +conductivity σ ≡ 1 S/m. Even though this data was not tuned to the EIT machine (noise and +contact impedances) the methods produced high quality absolute reconstructions. +Moving into incorrect domain modeling, we see all methods handled the moderate domain mis- +modeling quite well and the CGO methods handled the severe domain mismodeling as well. No +significant boundary artefacts are seen in the CGO reconstructions for the 18cm x 27cm x 19cm +and only moderate boundary artefacts in the TV reconstructions. In the more severe mis-modeling +case, assuming the data is coming from a much larger box (20cm x 35cm x 25cm) instead of the +true domain (17cm x 25.5cm x 17cm), the texp and t0 methods still produce good reconstructions +showing the correct number of targets and correct region of the tank. The localization is worst in +the x3 direction showing the targets slightly elongated. While artefacts in the Calder´on 20x35x25 +reconstructions are present, the method still does detect the true objects as the most conductive. +The TV algorithm failed to identify any of the targets for both cases with the 20x35x25 box. The +artefacts in the TV reconstructions in these cases with incorrect domain model could be expected +as the regularized non-linear least squares minimization-based absolute imaging EIT algorithms are +known to be highly sensitive to modeling errors such as errors in modelling of the domain shape, +see e.g. [49,51,61]. +We can see the effect of the underlying assumption for Calder´on’s method that the conductivity +is a small perturbation from a constant. In the physical experiments, the conductor(s) had roughly +twelve times the conductivity of the background saline. This is seen in the underestimation of + +16 +HAMILTON ET AL. +0 +20 +40 +60 +0 +20 +40 +60 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +300 +0 +10 +20 +30 +40 +50 +10 +20 +30 +40 +Slice +Isosurface +3D +Slice +Isosurface +3D +One Target +Two Targets +Truth +Cald +texp +t0 +TV +Figure 3. +Absolute image reconstructions comparing the CGO methods to +the regularized method with correct domain modeling. Slices, isosurfaces, and 3D +renderings of the conductivity are shown. Note the truth targets had a measured +conductivity of approx 290 mS/m. +maximum conductivity from Calder´on’s method. +However, unsurprisingly, this method does a +good job of localizing the targets, in all cases except for the absolute image from the significant +domain modeling error. The artefacts seen in the Calder´on reconstructions are consistent with +what has been observed for absolute imaging with Calder´on in 2D, such as Gibbs phenomenon +for all absolute reconstructions and higher reconstructed conductivity at electrode locations when +the domains were incorrectly modelled. Across all modeling scenarios, the regularized TV method +struggled with the high contrast targets, giving maximum target conductivities well below the +measured values. +The same effect was seen in simulated scenarios where, however, maximum +conductivities of lower contrast targets were reliably recovered. This is at least partially explained +by saturation of contrast distinquishability of the measurements, i.e. the effect of increasing the +conductivity of the target inclusion on the measurements gradually diminishes and until certain +level the TV regularized method can no longer discern between high and even higher conductivities. + +C +DD3D CGO-BASED EXPERIMENTAL EIT +17 +-20 +0 +20 +40 +60 +0 +20 +40 +60 +50 +100 +150 +200 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +0 +20 +40 +60 +0 +20 +40 +60 +Slice +Isosurface +3D +Slice +Isosurface +3D +One Target +Two Targets +Truth +Cald +texp +t0 +TV +Figure 4. +Absolute image reconstructions comparing the CGO methods to the +regularized method with moderately incorrect domain modeling, using a box of size +18cm x 27cm x 19cm. Note the truth targets had a measured conductivity of approx +290 mS/m. +We note that the electrodes used in this experiment were very large and the structure of the +domain, a box with corners, may exacerbate some of the modeling and/or hardware challenges. +Nevertheless, the study provides informative results on the feasibility of absolute EIT reconstruction +in 3D. +The difference images from Calder´on are able to handle the stronger mismodeling of the domain, +as are the texp and t0 methods. The strong mismodeling proved too severe for the linear difference +imaging reference method, which did not manage to identify the targets, see Figure 6. +While the texp and t0 CGO methods did reliably recover the contrast and approximate location +of the targets across examples studied here, they do appear more sensitive than their 2D D-bar +based counterparts in regards to the regularization parameter, Tξ, used in the truncation of the +nonlinear scattering data. +Figure 7 displays the effect that Tξ plays on the scaled localization +error and maximum recovered conductivity value for the single target, correct domain modeling + +C +DD18 +HAMILTON ET AL. +-50 +0 +50 +-50 +0 +50 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +0 +50 +100 +0 +50 +100 +Slice +Isosurface +3D +Slice +Isosurface +3D +One Target +Two Targets +Truth +Cald +texp +t0 +TV +Figure 5. +Absolute image reconstructions comparing the CGO methods to the +regularized method with largely incorrect domain modeling, using a box of size 20cm +x 35cm x 25cm. Note the truth targets had a measured conductivity of approx 290 +mS/m. +case. Recall that a secondary nonuniform truncation is also enforced where scattering data with +magnitudes exceeding 20 for the real or imaginary parts are set to zero. Adjusting that value will +also have an effect on the reconstruction. The value of 20 was chosen in this work for its overall +reliability across examples. The contrast appears more sensitive than the localization error. As +in [23], the minimum ζ parameterization of ξ was used, as the scattering data is a function of +both ζ ∈ C3 and ξ ∈ R3 in 3D instead of just k ∈ C as in 2D D-bar based methods. Alternative +parameterizations and a more detailed study of the effect of the regularization parameters, while +interesting, are outside the scope of this work. +In terms of speed, when running reconstructions on a MacBook Pro with a 2.3 GHz Dual-Core +Intel® Core i5 processor, Calder´on reconstructions on a 16 × 16 × 16 x-grid which are interpolated +to a 64 × 64 × 64 x-grid take 1 to 2 seconds without optimizing for parallelization. This increases +to 6-8 seconds per reconstruction when the initial x-grid is 32 × 32 × 32. When running on a + +C +DD3D CGO-BASED EXPERIMENTAL EIT +19 +20 +30 +40 +50 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +300 +50 +100 +150 +200 +250 +-50 +0 +50 +100 +150 +-100 +0 +100 +200 +300 +-100 +0 +100 +200 +300 +400 +Correct modeling +18x27x19 modeling +20x35x25 modeling +Truth +Cald +texp +t0 +Linear +Figure 6. +Difference image reconstructions comparing the CGO methods to +a typical linear method. Slices and 3D renderings of the conductivity are shown +for the correct domain modeling, and increasing levels of error in domain modeling. +Note the truth targets had a measured conductivity difference from the background +of approx 266 mS/m. +PC with a AMD EPYC 7702P 64-Core Processor 2.00 GHz, the reconstruction times are 0.6-0.7 +seconds and 4 seconds, respectively, again without optimizing for parallelization. +On an 2015 +iMac with a 4 GHz Quad-Core Intel® Core i7 processor, the texp and t0 methods require 3-4 +seconds/recon using a 21 × 21 × 21 x-grid for the potential q(x) or 6-8 seconds/recon when using a +41 × 41 × 41 x-grid for q(x). The timings are non-optimized with the highest computational cost +coming from computing the inverse Fourier transform and solving the boundary value problem using +FEM. The regularized TV, and linear difference, reconstructions averaged 2-3 hrs, and 3 minutes, +respectively, when computed on a server with 256GB of RAM and two 10 core Intel® Xeon® CPU +E5-2630 v4 @2.20GHz processors. We remark that the rather long computation times of the TV +regularized non-linear least squares approach are caused by the 3D problem with a computationally + +D20 +HAMILTON ET AL. +6 +8 +10 +12 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +texp +t0 +6 +8 +10 +12 +0 +200 +400 +600 +800 +texp +t0 +Scaled Localization Error +Max Target Cond. (mS/m) +Tξ +Tξ +Figure 7. Comparison of the effect of the truncation value Tξ of the scattering +radius in the texp and t0 CGO methods for Scaled Localization Error (left) and +Maximum value of the recovered target. Max conductivity values (right) for Tξ = +11, 11.5, off the plot, spiked into the 1500-5000 mS/m range. +rather challenging geometry as the sufficient accuracy of the CEM forward model (2.2) necessitates +significant mesh refinement near the electrodes, leading to the large number of degrees of freedom +(approx 250,000 nodes) in the FEM based forward model. The FEM model needs to be solved +multiple times in the line search at each iteration of the Gauss-Newton method and with the mesh +used each forward solution takes approximately 80s computation time. +5. Conclusions +In this work, we presented the first 3D absolute EIT reconstructions from CGO-based methods +on experimental 3D tank data, and compared them to the current standard, a total variation reg- +ularized non-linear least squares approach. We demonstrated that, with correct domain modeling, +quality 3D absolute reconstructions can be obtained by all of the methods, comparable to the qual- +ity seen in linear difference imaging. All methods, Calder´on, texp, t0, and TV reasonably handled +the moderate domain modeling error within little noticeable change in localization error and target +contrast. For the large modeling error case, the texp and t0 methods correctly identified the targets +with high contrast, additional artefacts were introduced into the Calder´on reconstruction, and the +error proved too significant for the TV method. The computational cost of the CGO reconstruc- +tion is trivial compared to TV (non-optimized, less than 1 sec/recon for Calder´on, approximately +5 sec/recon for texp and t0, compared to 2-3 hours per reconstruction for TV). +Acknowledgments +Research reported in this paper was supported by the National Institute of Biomedical Imaging +and Bioengineering of the National Institutes of Health under award numbers R21EB028064 (SH +and JN) 1R01EB026710-01A1 (GS, DI, JN, ORS, and the development of the ACT5 device). The +content is solely the responsibility of the authors and does not necessarily represent the official +views of the National Institutes of Health. JT and VK were supported by the Academy of Finland +(Project 336791, Finnish Centre of Excellence in Inverse modeling and Imaging), the Jane and +Aatos Erkko Foundation and Neurocenter Finland. +References +[1] Ahmed Abdelwahab, Omid Rajabi Shishvan, and Gary J. Saulnier. 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Elsevier, 2015. + diff --git a/uNAzT4oBgHgl3EQfsP0n/content/tmp_files/load_file.txt b/uNAzT4oBgHgl3EQfsP0n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e969ed393866b023caa404639b250202b2788e94 --- /dev/null +++ b/uNAzT4oBgHgl3EQfsP0n/content/tmp_files/load_file.txt @@ -0,0 +1,1136 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf,len=1135 +page_content='FAST ABSOLUTE 3D CGO-BASED ELECTRICAL IMPEDANCE TOMOGRAPHY ON EXPERIMENTAL TANK DATA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' HAMILTON, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' MULLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' ISAACSON, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' KOLEHMAINEN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' NEWELL, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' RAJABI SHISHVAN, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' SAULNIER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' TOIVANEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Objective: To present the first 3D CGO-based absolute EIT reconstructions from ex- perimental tank data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Approach: CGO-based methods for absolute EIT imaging are compared to traditional TV regularized non-linear least squares reconstruction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Additional robustness testing is performed by considering incorrect modeling of domain shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Main Results: The CGO- based methods are fast, and show strong robustness to incorrect domain modeling comparable to classic difference EIT imaging and fewer boundary artefacts than the TV regularized non-linear least squares reference reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Significance: This work is the first to demonstrate fully 3D CGO-based absolute EIT reconstruction on experimental data and also compares to TV-regularized absolute reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The speed (1-5 seconds) and quality of the reconstructions is encouraging for future work in absolute EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Introduction The main objective of this paper is to demonstrate the feasibility, speed, and robustness of producing 3D absolute (static) images of the electrical conductivity inside a tank, from experimental Electrical Impedance Tomography (EIT) data measured on an array of electrodes on the tank’s surface by the ACT5 [65] adaptive current tomography system, using CGO-based reconstruction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The primary contributions of this work are that it presents the first use of Complex Geometrical Optics (CGO) based methods to produce absolute (static) images of the conductivity from experimentally measured EIT data in 3D, and studies their robustness under modeling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' “Absolute”, or “static,” conductivity images are images of the conductivity inside a body made from one set of experimental measurements made on the surface of the body at one time [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Alternatively, “dynamic,” or “time-difference,” imaging uses two sets of data measured at two different times to make an image of the change in the conductivity that took place between the two times the measurements were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We present both types of images made from experimental data by CGO-based algorithms in 3D in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Applications of EIT began with geophysical exploration over a century ago [4,44] and have since expanded into several fields including the transport of fluids and gases [26,40,80], and biomedical imaging [2, 7, 20, 29, 35, 36, 42, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Systems for monitoring lung function in real-time are now commercially available and clinical trials are in progress to determine the extent to which some of these systems might be used to guide mechanical ventilation [12, 73, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' These systems typically Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' electrical impedance tomography, absolute imaging, conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Hamilton is with the Department of Mathematical and Statistical Sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Marquette University, Milwaukee, WI 53233 USA, email: sarah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='hamilton@marquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Isaacson is with the Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY 12180, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Kolehmainen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Toivanen are with the Department of Applied Physics, University of Eastern Finland, FI-70210 Kuopio, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Muller is with the Department of Mathematics & Statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Villanova University, Villanova, PA 19085 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Newell is with the Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Rajabi Shishvan and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Saulnier are with the Department of Electrical and Computer Engineering, University at Albany - SUNY, Albany, NY 12222, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Toivanen is with the Department of Applied Physics, University of Eastern Finland, FI-70210 Kuopio, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='01655v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='NA] 4 Jan 2023 2 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' make and display two dimensional images of the differences in conductivity between two states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=', lungs filled with air and lungs depleted of air, in order to take advantage of dynamic (difference) imaging methods and algorithms [2,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The desire to improve the diagnosis of cancer and stroke has motivated the development of systems and methods capable of imaging the absolute or static internal conductivity and permittivity in 3D [17, 18, 33, 43, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The interested reader is referred to [15] for further review of applications of EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Most absolute (static) EIT reconstruction focuses on solving a simplified linearized problem or iteratively solving an optimization-based method which requires repeated solutions to the forward problem which can become costly for highly dense meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' CGO-based methods are direct meth- ods in that they do not require iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' They have the capability to solve the full nonlinear mathematical inverse problem, and do not require repeated solutions to the forward problem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' via the finite element method (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The D-bar inversion algorithm, which is a specific type of CGO-based inversion algorithm in 2D, has been used to make both absolute and difference images in 2D in real-time from experimental data measured on tanks, and for difference imaging on human subjects in laboratory and clinical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' See [55] for a recent review of the 2D D-bar method and its applications, and [59] for the theoretical foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In 3D, existence and uniqueness of solutions [58,63] can be shown for a 3D D-bar type equation but the constructive proof, upon which the reconstruction algorithms are built, bypasses the 3D D- bar equation instead using a high (non-physical) frequency limit connecting the nonlinear scattering data and the linear Fourier data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Advances in the numerical implementation of 3D CGO-based methods are more recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The first numerical implementation of 3D CGO-based methods on simulated electrode data using current injection on the surface of the sphere was presented in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A 3D CGO-based inversion algorithm, regularization scheme, rigorous proof of stability under certain hypotheses, and examples reconstructing the conductivity inside a sphere from numerically simulated Dirichlet data on the entire surface of the sphere (without electrodes) were given in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Until now it has been an open question if 3D CGO-based algorithms could be used with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This paper answers this question affirmatively by showing that the conductivity can be recovered rapidly and robustly from experimental data measured on 32 electrodes on the surface of a rectangular prism building on the work of [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' It was demonstrated in [10,32,36,42] that it is possible to make electrical impedance tomography (EIT) Systems that produce both 3D static and dynamic images of the interior of the chest showing heart and lung regions, as well as changes in those regions due to ventilation and perfusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' These systems used linearized and iterative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The former are fast but less accurate and the latter are slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Recent 3D CGO inversion algorithms and their analysis, when applied to synthetic or simulated data, suggested that they have the potential to be fast and more accurate [23, 34, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here we test their capabilities on experimental EIT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In this paper we compare the following methods for making static 3D images from experimental tank data, collected by the ACT5 imaging system: A CGO linearization method based on Calder´on’s original proposal introducing CGO ideas into the subject [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We call this algorithm Calder´on’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Two CGO-based methods for solving the full non-linear inversion problem based on Sylvester and Uhlmann’s original uniqueness proof and Nachman’s constructive uniqueness proof in [58,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The first is the texp algorithm, and the second is called t0, [22,34] both of which are simplifications of the original constructive proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A more traditional iterative inversion method based on optimization using a Total Variation (TV) regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The EIT data were measured on 32 electrodes attached to the six surfaces of a rectangular tank using the ACT5 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The system can adaptively determine patterns of currents to apply to all 32 electrodes that result in voltages on those electrodes that are proportional to the applied currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' These “eigen currents” form a discrete orthogonal set and provide improved voltage data 3D CGO-BASED EXPERIMENTAL EIT 3 for reconstructing the conductivity inside the tank from a limited amount of current or power that can be applied to the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The theory of adaptive current tomography systems is given in [14,30,31,38,52,53,60,68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Section 2 provides a brief review of the mathematics of EIT and encompasses the methods used in the work: the CGO and reference reconstruction algorithms, experimental setup for the ACT5 data collection, robustness tests that will be explored, and metrics used to evaluate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The results section, Section 3, presents slice, 3D, and isosurface renderings of the recovered conductivities for correct and incorrect domain modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Section 4 contains a discussion of the results and conclusions are drawn in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Methods In this work we compare three CGO-based methods (Calder´on’s Method, the texp Method, and the t0 Method) to a more common iterative Total Variation (TV) regularized non-linear least squares method for absolute EIT image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For comparison, we also include time difference EIT images for the CGO-based methods and compare them to a linearized difference imaging method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We begin with a brief review of the mathematical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Mathematical Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The mathematical problem of reconstructing the internal con- ductivity, when measurements can be made with infinite precision everywhere on the boundary of a body, is currently called “Calder´on’s Problem” by much of the mathematical community since A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on formulated this inverse problem as follows [13]: In two or more dimensions, can one find the conductivity, σ(x), inside a body, Ω, from all possible electrical measurements made on the surface, ∂Ω, of the body?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here the voltage or potential, u(x), inside the body, due to an applied surface current density, j(x), is assumed to satisfy the following low frequency approximation to Maxwell’s Equations, which we will refer to as the conductivity equation, with a Neumann boundary condition: ∇ · σ(x)∇u(x) = 0, x ∈ Ω ⊂ R3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1) σ(x)∂u(x) ∂ν(x) = j(x), x ∈ ∂Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here ν = ν(x) denotes the outward pointing unit normal to the surface at x ∈ ∂Ω, and, ∂u(x) ∂ν(x) = ν(x) · ∇u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We denote the mapping from applied current density to resulting voltage on the surface, called the “Neumann-to-Dirchlet” (ND) map, by Rσ, where Rσj ≡ u(x) for x ∈ ∂Ω, and u(x) solves the conductivity equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1) with Neumann data j(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The Calder´on problem can also be formulated as the mathematical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' If one is given the ND map or operator, Rσ, or equivalently its inverse, the Dirichlet-to-Neumann (DN) operator, Λσ := R−1 σ , can one find σ(x)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on showed that if σ(x) does not differ too much from a constant then one can recover an approximation to it from the ND map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' His short paper showed this could be done using Fourier Transforms in a very clever way, by introducing special solutions, eζ·x, to the conductivity equation with constant conductivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' the Laplace equation, where ζ is a complex valued vector with ζ · ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This is possible in two or more dimensions when the conductivity is close to a constant and gave birth to Calder´on’s method as described in this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' as well as more powerful methods for solving the full non-linear problem by generalizing Calder´on’s solutions to what we now call Complex Geometrical Optics (CGO) solutions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' introduced by [71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='72] in their landmark paper proving that the Calder´on problem in three or more dimensions has a unique solution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' where the inversion problem is reduced to a Fourier transform in the limit |ζ| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The texp and t0 methods described below in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3 follow this strategy, as opposed to the 2D D-bar methods described in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This strategy is not possible in 2D where the CGO solutions are found by solving a first order linear PDE involving D-bar operators in the complex vector ζ and taking 4 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Constructive proofs using these CGO solutions and D-bar ideas from inverse scattering theory, along with applications to scattering and acoustics were given in [58,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' CGO methods were used to prove uniqueness in the more difficult case of 2D where constructive methods for reconstructing the conductivity were given in detail in [59] for σ ∈ C2(Ω) and later in [5] for σ ∈ L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Other pioneering work proving uniqueness under a variety of hypotheses on the conductivity include [25,48,50,63,64,74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Extensive references to more recent progress in the analysis and numerical analysis of the Calder´on problem can be found in [11,23,28,56,77];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' [47,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In what follows we will be interested in the problem of reconstructing an approximation to the internal conductivity from finitely many experimental measurements made with finite precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Unfortunately, this is an ill-posed problem and, unlike the purely mathematical problem, it does not have a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Nevertheless, it is sometimes possible to reconstruct useful approximations to the internal conductivity with a finite number of degrees of freedom, or voxels, which we will illustrate by making images from experimental data and comparing them to the actual interior conductivity within the tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The EIT problem for a body with L electrodes, eℓ, ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' , L, on its surface, is to find an approximation to the internal conductivity from all the possible electrical measurements made on these L electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In particular we will assume that we apply L − 1 linearly independent patterns of currents, ⃗I(k), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' , L − 1, to the L electrodes, and measure the resulting L − 1 voltage patterns, ⃗V (k), where ⃗I(k) ℓ and ⃗V (k) ℓ denote the applied current, and measured voltage from the kth pattern, on the ℓth electrode, for ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' From conservation of charge, and our choice of ground, we assume L � ℓ=1 ⃗I(k) ℓ = L � ℓ=1 ⃗V (k) ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The kth voltage or potential, u(k)(x), resulting inside the body is determined by the conductivity equation, ∇ · σ∇u(k) = 0, and the “Complete Electrode Model”, [16,69] where: � eℓ σ(x)∂u(k)(x) ∂ν(x) dS(x) = I(k) ℓ , x ∈ eℓ σ(x)∂u(k)(x) ∂ν(x) = 0, x /∈ L � ℓ=1 eℓ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2) u(k)(x) + zℓσ(x)∂u(k)(x) ∂ν(x) = V (k) ℓ , x ∈ eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here zℓ is the effective contact, or surface, impedance on the ℓth electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The current patterns used will be eigenvectors of the Current to Voltage map, which is a matrix approximation to the ND map, for the homogeneous saline tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' They are found numerically by simulating a homogeneous tank for static/absolute imaging, and experimentally by adaptive methods for difference/dynamic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The matrix approximations to the ND maps from the conductivity distribution σ are denoted by the L × L matrices, Rσ, where Rσ⃗I(k) = ⃗V (k), and we define Rσ⃗1 = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here the vectors ⃗1, ⃗0, denote vectors all of whose components are 1, or 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The discrete analog of the DN map used in the CGO-based methods is given by Lσ := R−1 σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on’s Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Following Calder´on’s original paper [13], Calder´on’s method approx- imates the conductivity, σ(x), from its Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here we present a brief description of the method and refer the reader to [9,13,34,57] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on’s method assumes the conductivity is a small perturbation, δσ(x), from a constant background, σb, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' σ(x) = σb+δσ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In this paper, we assume that the background conductivity σb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' If the background constant is not one, then the problem can be scaled and unscaled as in [34,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The three steps of Calder´on’s method in 3-D, as described in [34], are: 3D CGO-BASED EXPERIMENTAL EIT 5 Step 1: Use the DN maps Λσ and Λ1 to approximate the Fourier transform of the small perturbation in conductivity, � δσ(z), by � δσ(z) ≈ ˆF(z) := − 1 2π2|z|2 � ∂Ω eπi(z·x)+π(a·x) (Λσ − Λ1) eπi(z·x)−π(a·x)dS(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3) where z and a satisfy z, a ∈ R3, |z| = |a|, and z · a = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4) and Λ1 is the DN map for a constant conductivity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Step 2: Take the inverse Fourier transform of ˆF(z): δσCAL(x) ≈ F−1{ˆF(z)}(x) = � R3 ˆF(z)e−2πi(x·z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5) Step 3: Add the background to the perturbation to recover the approximate conductivity, σCAL(x): σCAL(x) = σb + δσCAL(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6) The definition of the Fourier transform in Calder´on’s method is different from that used in the texp and t0 methods described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' However, each method is consistent with its definition and is consistent with the respective literature on that method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In this paper, we compute ˆF(z) in spherical Fourier coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As such, we choose z = |z|(cos ˜φ sin ˜θ, sin ˜φ sin ˜θ, cos ˜θ) and a = |z|(cos ˜φ cos ˜θ, sin ˜φ cos ˜θ, − sin ˜θ), for |z| ≥ 0, 0 ≤ ˜φ ≤ 2π and 0 ≤ ˜θ ≤ π so that z and a satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Then, the inverse Fourier transform in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5) becomes δσCAL(x) = � ∞ 0 � 2π 0 � π 0 |z|2 sin ˜θˆF(|z|, ˜φ, ˜θ)e−2πi(x·z)d˜θd˜φd|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='7) Additionally, we implement the use of a mollifier, ˆη � z y � , as introduced in [13] for some parameter y ∈ R to reduce Gibbs phenomenon caused by jump discontinuities in σ(x) while recovering δσCAL δσCAL(x) = � ∞ 0 � 2π 0 � π 0 |z|2 sin ˜θˆF(|z|, ˜φ, ˜θ)ˆη �z y � e−2πi(x·z)d˜θd˜φd|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='8) We implement the same mollifier as used in [9,34], ˆη �z y � = e−πt|z|2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='9) where y = 1/ √ t and t acts as a smoothing parameter with larger t values producing smoother reconstructions with smaller jumps at points of discontinuity in σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Since noise causes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3) to blow up at large |z|, we use a non-uniform truncation regularization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A similar regularization strategy was proved stable for the 2-D D-bar method in [46], which also noted non-uniform truncation also produces reliable reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In our case, we will first compute ˆF(z) for |z| within an outer radius of Tz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We keep values of ˆF(z) whose real and imaginary amplitudes are below a threshold determined by the amplitudes of ˆF within a smaller radius |z| ≤ Tz1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' ˆF is set to 0 everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Both radii are chosen empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The inner radius Tz1 is chosen as a region in z-space where noise does not cause ˆF to blow up and Tz2 is chosen to keep as much reasonable information from ˆF without introducing holes in the non-zero region of ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As such, our truncated ˆF is computed by 6 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' ˆFR(z) = � ˆF(z), if |z| ≤ Tz2, and |Re(ˆF(z))| ≤ max |z|≤Tz1 |Re(ˆF(z))|, |Im(ˆF(z))| ≤ max |z|≤Tz1 |Im(ˆF(z))| 0, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='10) With our truncated ˆF, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='8) is truncated in the radial variable, leading to the approxi- mation δσCAL R (x) = � Tz2 0 � 2π 0 � π 0 |z|2 sin ˜θ ˆFR(|z|, ˜φ, ˜θ)e−πt|z|2e−2πi(x·z)d˜θd˜φd|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='11) For difference images shown in this paper, we only perform Steps 1 and 2 of the method and replace Λ1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3) with a reference DN map, Λσref before computing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Thus, the flow for absolute images is (Λσ, Λ1) 1 −→ ˆF(z) 2 −→ δσCAL 3 −→ σCAL and the flow for difference images in this paper is � Λσ, Λσref � 1 −→ ˆF(z) 2 −→ δσCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Numerical Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In this section, we review the implementation details of Calder´on’s method introduced in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For Step 1, we compute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3) for |z| ≤ Tz2 by discretizing the boundary integral as follows, ˆF(z) = − 1 2π2|z|2 � ∂Ω eπi(z·x)+π(a·x) (Λσ − Λ1) eπi(z·x)−π(a·x)dS(x) ≈ − 1 2π2|z|2 �|∂Ω| L � (eπi(z·x)+π(a·x))TQ (Lσ − L1) QT � eπi(z·x)−π(a·x)� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='12) where |∂Ω| is the surface area of the domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' L is the number of electrodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' x ∈ R(L × 3) denotes the vector of the Cartesian centers of the L electrodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' T denotes the traditional, non-conjugate, matrix transpose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Lσ and L1 denote the discrete matrix approximations to the DN maps Λσ and Λ1 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' and Q ∈ RL × Nli denotes an orthonormal basis created using Nli linearly independent applied currents over L electrodes as was done in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The matrix L1 is based on the FEM solution of the CEM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Problem-specific mesh details are given below in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We then compute ˆFR according to equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For Step 2, on an equally-spaced 16 × 16 × 16 rectangular grid in x, the conductivity difference δσCAL R (x) is computed via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='11) using a 3D Simpson’s rule using N|z| = 10, N˜θ = 10, and N˜φ = 30 uniformly-spaced nodes on the |z|, ˜θ, and ˜φ grids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As the number of nodes in the Fourier domain increases, so does computation time, but some artefact reduction can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Empirically, the artefact reduction did not seem significant enough to warrant an increased computational time beyond these parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Difference images are computed using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='12), replacing L1 with a discrete reference DN map, Lσref in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the phantom tank experiments in this paper, this reference map is from data collected with a tank filled only with saline matching the experiment and no other inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The absolute reconstructions of σCAL R (x) in this paper are produced using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6) replacing σb with σbest σCAL R (x) = σbest + δσCAL R (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The solution is then interpolated to a 64 × 64 × 64 rectangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Following [39], σbest is given by σbest = �K k=1 �L ℓ=1 U k ℓ U k ℓ �K k=1 �L ℓ=1 U k ℓ V k ℓ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='13) 3D CGO-BASED EXPERIMENTAL EIT 7 where U k ℓ is the kth simulated voltage pattern measured on electrode ℓ with a homogeneous conduc- tivity of 1 S/m and V k ℓ is the kth voltage pattern measured on electrode ℓ for the inhomogeneous conductivity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The simulated voltages are the same voltages used to compute L1, as described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The texp and t0 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Both the texp and t0 methods are derived from the construc- tive proofs of [58, 62, 63] and involve special solutions called Complex Geometrical Optics (CGO) solutions [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A brief summary is included here for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For further details see [22,23,34,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Assuming that the conductivity is a constant σc = 1 in a neighborhood of the boundary ∂Ω, the real-valued conductivity equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1) can be transformed to the Schr¨odinger equation (−∆ + q(x)) �u(x) = 0, x ∈ R3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='14) via the change of variables q(x) = ∆√ σ(x) √ σ(x) and �u(x) = u(x) � σ(x), by extending σ(x) ≡ 1 for all x ∈ R3 \\¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For ζ(ξ) ∈ Vξ, unique CGO solutions exist to the transformed problem (−∆ + q(x)) ψ(x, ζ) = 0, x ∈ R3, where ψ(x, ζ) ∼ eix·ζ for large |x| or |ζ|, and Vξ = � ζ ∈ C3���ζ2 = 0, (ξ + ζ)2 = 0 � , for each ξ ∈ R3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='15) where ζ2 = ζ ·ζ and ζ is a purely auxiliary parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The conductivity σ(x) can then be recovered from the DN map Λσ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For each x ∈ ∂Ω and ζ ∈ Vξ, solve the Fredholm integral equation of the Second Kind, ψ(x, ζ) = eix·ζ − � ∂Ω Gζ(x − y) (Λσ − Λ1) ψ(y, ζ) dS(y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='16) where Gζ(x) = eix·ζ (2π)3 � R3 eix·k |k|2 + 2k · ζ dk, x ∈ R3 \\{0}, denotes the Faddeev Green’s function [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Then, evaluate the scattering data t(ξ, ζ) = � ∂Ω e−ix·(ξ+ζ) (Λσ − Λ1) ψ(x, ζ) dS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='17) For |ζ| large, the Schr¨odinger potential q(x) can be recovered via the inverse Fourier transform q(x) ≈ F−1 {t(ξ, ζ)} (x) = 1 (2π)3 � R3 eix·ξt(ξ, ζ) dξ, x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='18) The conductivity is then recovered by solving the boundary value problem � (−∆ + q(x))˜u(x) = 0 x ∈ Ω ⊂ R3 ˜u(x) = 1 x ∈ ∂Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='19) for �u(x) and evaluating σ(x) = (˜u(x))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This is the full nonlinear reconstruction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Replacing the CGO solutions ψ(x, ζ) by their asymptotic behavior eix·ζ in the scattering data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='17) via texp(ξ, ζ) = � ∂Ω e−ix·(ξ+ζ) (Λσ − Λ1) eix·ζ dS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='20) yields a ‘Born approximation’ typically called the texp approximation for EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Using this approx- imate scattering data in place of the fully nonlinear t(ξ, ζ), one proceeds with the recovery of an approximate potential qexp(x) via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='18) and conductivity σexp(x) via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The flow is: (Λσ, Λ1) 1 −→ texp(ξ, ζ) 2 −→ qexp(x) 3 −→ σexp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 8 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' An intermediate approximation can be computed by replacing the Faddeev Green’s function Gζ(x) in the single layer potential, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='16), for the traces of the CGOs, with the standard Green’s function G0(x) = 1 4π|x| for the Laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Thus, one solves ψ0(x, ζ) = eix·ζ − � ∂Ω G0(x − y) (Λσ − Λ1) ψ0(y, ζ) dS(y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='21) for the CGOs ψ0(x, ζ), avoiding the exponentially growing Faddeev Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A correspond- ing approximation to the scattering data is then computed by using ψ0(x, ζ) in place of ψ(x, ζ) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='17) and then continuing to recover q0(x) and σ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The flow is then (Λσ, Λ1) 1 −→ ψ0(x, ζ) 2 −→ t0(ξ, ζ) 3 −→ q0(x) 4 −→ σ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We point out that the methods, as outlined above, assumed that the conductivity was a constant σc = 1 near the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As mentioned in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2, the problem can be scaled and unscaled as in [34, 39] when the constant σc is not one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In practice, we estimate the best-fit constant conductivity fit to the data σbest as given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Explicitly, as in [34], we scale the DN map by using 1 σbest Λσ in place of Λσ, and re-scale at the end using σexp(x) = σbest (˜uexp(x)) and σ0(x) = σbest � ˜u0(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In this work we consider both the texp and t0 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We note that this is the first time that t0 has been implemented on non-continuum DN data and the first time that either texp or t0 have been demonstrated on experimental 3D EIT data, for absolute or time-difference EIT imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Numerical Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Here we provide the numerical details pertinent to the imple- mentation of the texp and t0 algorithms outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As with the Calder´on method above, the main idea is to expand functions in the same orthonormal basis Q as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Following [21], for each electrode center xℓ, expand ψ(x, ζ) and eix·ζ as ψ(xℓ, ζ) ≈ Nli � j=1 bj(ζ)Qj ℓ, eixℓ·ζ ≈ Nli � j=1 cj(ζ)Qj ℓ, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' L, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='22) where Qj ℓ denotes the (ℓ, j) entry of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Then, the boundary integral equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='21) can be approximated as follows ψ0(xℓ, ζ) = = eixℓ·ζ − � ∂Ω G0(xℓ − y) (Λσ − Λ1) ψ0(y, ζ) dS(y) ≈ eixℓ·ζ − L � ℓ′=1 � Eℓ′ G0(xℓ − y) (Λσ − Λ1) ψ0(y, ζ) dS(y) ≈ eixℓ·ζ − � L � ℓ′=1 � Eℓ′ G0(xℓ − y)dS(y) � � (Lσ − L1) ψ0(yℓ′, ζ) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='23) where Eℓ′ denotes the ℓ′th extended electrode where �L ℓ′=1 Eℓ′ = ∂Ω and the Eℓ′ are mutually disjoint [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note the true electrodes need not cover the surface ∂Ω, only the extended (math- ematical) electrodes that we will use to discretize the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Then, using the expansions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='22) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='23), we have Nli � j=1 bj(ζ)Qj ℓ ≈ Nli � j=1 cj(ζ)Qj ℓ − � L � ℓ′=1 � Eℓ′ G0(xℓ − y)dS(y) � � �(Lσ − L1) Nli � j=1 bj(ζ)Qj ℓ′ � � = Nli � j=1 cj(ζ)Qj ℓ − � L � ℓ′=1 � Eℓ′ G0(xℓ − y)dS(y) � � � Nli � j=1 bj(ζ)fj (yℓ′) � � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='24) 3D CGO-BASED EXPERIMENTAL EIT 9 where fj (yℓ′) represents the action of (Lσ − L1) on Qj evaluated at yℓ′, which can be computed as the (ℓ′, j) entry of Q (Lσ − L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Define �G0(ℓ, ℓ′) = � 1 4π|xℓ−yℓ′| ℓ ̸= ℓ′ 0 ℓ = ℓ′, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='25) where we have removed the singularities at xℓ = yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Assuming |Eℓ| = | ∂Ω | L for each ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' , L, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='25) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='24) we find Nli � j=1 bj(ζ)Qj ℓ ≈ Nli � j=1 cj(ζ)Qj ℓ − | ∂Ω | L Nli � j=1 bj(ζ) L � ℓ′=1 �G0(ℓ, ℓ′)fj (yℓ′) ≈ Nli � j=1 cj(ζ)Qj ℓ − | ∂Ω | L Nli � j=1 bj(ζ) � �G0Q (Lσ − L1) � (ℓ, j) or in matrix form, Q⃗b = Q⃗c − | ∂Ω | L �G0Q (Lσ − L1)⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The solution to this equation can be found by solving the following system for the unknowns ⃗b, (I + A)⃗b = ⃗c, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='26) where A = | ∂Ω | L QT �G0Q (Lσ − L1), and I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' If the extended electrodes Eℓ are not uniform in size, then one could compute as weighted sum replacing the uniform weight |Eℓ| = | ∂Ω | L as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Next, following [34], the scattering data t0(ξ, ζ(ξ)) can be computed for all |ξ| less than a chosen truncation radius Tξ via t0(ξ, ζ) ≈ � | ∂Ω | L � e−ix·(ξ+ζ)�T Q (Lσ − L1)⃗b |ξ| < Tξ 0 else, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='27) since ⃗b = QTψ0(x, ζ), where x ∈ R(L × 3) is the same as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' a vector storing the centers of the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Next, the approximate potential q0 is recovered by computing the inverse Fourier transform of the truncated scattering data t0 using a Simpson’s rule in 3D, q0(x) = 1 (2π)3 � [−Tξ,Tξ]3 eix·ξ t0(ξ, ζ(ξ)) dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Alternatively, one could use an IFFT to achieve additional speedup, taking care with quadrature points and the particular form of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Following [34], the conductivity σ0(x) was recovered by first solving the boundary value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='19), using the PDE toolbox in Matlab using a mesh with approximately 21, 000 3D elements, then computing σ0(x) = σbest (˜u(x))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The 3D visualizations were obtained by interpolation to a 64 × 64 × 64 rectangular grid using Matlab’s scatteredInterpolant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The ζ values were computed following the minimal-zeta approach outlined in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We remark that while the non-existence of exceptional points for the solution of the boundary integral equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='16) is proven for large |ζ| [58,71], as well as small |ζ| [19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' it is still an open question for the intermediate values required here to perform the computation on a computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 10 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The texp reconstructions were obtained in an analogous fashion to those of t0, this time bypassing the boundary integral equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='17) and directly computing texp(ξ, ζ) ≈ � | ∂Ω | L � e−ix·(ξ+ζ)�T Q (Lσ − L1) QT � eix·ζ� |ξ| < Tξ 0 else, where we have replaced the vector of coefficients ⃗b with the expansion of the asymptotic behavior of eix·ζ given by QT � eix·ζ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Difference imaging can be performed with the t0 and texp methods by replacing the matrix L1 with Lσref and computing σdiff(x) = σ(x) − σbest in the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Reference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Total Variation regularized non-linear least squares reconstructions will serve as the reference reconstructions for the CGO-based absolute imaging cases considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A classic linear difference imaging scheme, also reviewed below, will serve as the reference for the CGO-based difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Absolute imaging with TV regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A widely used numerical approach for absolute EIT is the total variation (TV) regularized non-linear least squares minimization ˆσ = arg min σ>0{∥Le (V − U(σ, z∗)) ∥2 + αTVβ(σ)}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='28) where U(σ, z) is the finite dimensional forward map, z∗ ∈ RL are the fixed electrode contact impedances obtained from an initialization step of the minimization, Le is the Cholesky factor of the noise precision matrix of e so that LT e Le = Γ−1 e , scalar valued α is the regularization parameter and TVβ(σ) is the (smooth) TV regularization functional [66] TVβ(σ) = � Ω � ∥∇σ∥2 + β dr, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='29) where β is the (fixed) smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The forward model U(σ, z) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='28) is based on the finite element (FEM) discretization of the complete electrode model [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For details of the FEM model, see [41, 78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In the FEM model, the electric conductivity is approximated as a linear combination of the piecewise linear nodal basis functions in a uniform tetrahedral mesh of N nodes, leading to vector of unknowns σ ∈ RN, and the electric potential is approximated similarly in a significantly more dense tetrahedral mesh with refinements near the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The non-linear optimization in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='28) is solved by a lagged Gauss-Newton method equipped with a line search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The line search is implemented using bounded minimization such that the non-negativity σ > 0 is enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For more details of the method, see [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The fixed contact impedances z∗ and initial (constant) conductivity estimate for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='28) are ob- tained from the solution of the non-linear least squares problem (σ0, z∗) = arg min σc,z>0{∥Le (V − U(σc, z)) ∥2}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='30) where the scalar σc ∈ R is the coefficient of a spatially constant conductivity image σc1 and z ∈ RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The non-linear least squares problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='30) is solved by a Gauss Newton optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Linear difference imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In linear difference imaging, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' [6, 8], the objective is to reconstruct the change in conductivity between two measurements (V1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The reference linear difference imaging approach of this paper uses linearized approximations of the observation models V1 ≈ U(σ0, z∗) + Jσ(σ1 − σ0) + e1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='31) V2 ≈ U(σ0, z∗) + Jσ(σ2 − σ0) + e2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='32) 3D CGO-BASED EXPERIMENTAL EIT 11 where the Jacobian matrix Jσ of U(σ, z∗) is evaluated at σ = σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' With these linearized models, the difference in the measurements becomes δV = V2 − V1 = (U(σ0) + Jσ(σ2 − σ0) + e2) − (U(σ0) + Jσ(σ1 − σ0) + e1) = Jσδσ + δe, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='33) where δσ = σ2 − σ1 and δe = e2 − e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Now, the inverse problem is to reconstruct δσ based on the difference data δV and the model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A widely used formulation for the linear problem is to use the (generalized) Tikhonov regularization with a smoothness promoting regularization functional ˆδσ = arg min δσ � ||Lδe(δV − Jσδσ)||2 + ||Lpδσ||2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='34) where Lδe is the Cholesky factor of the noise precision matrix of δe so that LT δeLδe = Γ−1 δe = (Γe1 + Γe2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In this paper, the regularization matrix Lp is constructed by utilization of a distance based covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' More specifically, we set LT p Lp = Γ−1 p , where the (prior covariance) matrix Γp is constructed using the distance based correlation function [54] Γp(i, j) = std(σ)2 exp � −∥xi − xj∥2 2a2 � , i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' , N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='35) where the parameter a controls the correlation length and can be solved by setting the distance ∥xi − xj∥ to a selected value d (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' half the radius of the target) and setting Γp(i, j) to the desired covariance for that distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 1% of variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Experimental Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' ACT5 [65] is a 32 electrode parallel EIT instrument that uses 32 current sources to apply patterns of current to the target and 32 voltmeters to measure the resulting voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The current sources in ACT5 adjust the delivered current to compensate for current lost through shunt impedance, including that introduced by the capacitance of cables that connect the instrument to the electrodes, enabling the desired current patterns to be applied with high precision [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' ACT5 can apply sinusoidal currents at frequencies in the range of 11 kHz to 1 MHz with a peak amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='25 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The voltmeters measure voltages up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5 V peak with a maximum signal-to-noise ratio (SNR) of 96 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Because the current source compensates for shunt capacitance of the cables, the system operates with grounded-shield cables [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A test tank was built from 3/8-inch-thick Plexiglas with interior dimensions of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='0cm x 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5cm x 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='0cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Thirty-two electrodes were fabricated of 16-gauge 316 stainless steel, each 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='0 mm square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The inside surfaces of the tank were milled out in the shape and depth of the electrodes so that the interior surfaces were flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The nominal gaps between electrodes were 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='0 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The 32 electrodes were placed four on each end and six on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Bolts through the center of each electrode passed through the Plexiglas, allowing connections to the cables from the ACT5 electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Five sides of the tank were glued together, and provision was made to secure the top with threaded rods at all four corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Access holes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6 mm at various sites in the top to allow threads to suspend targets, and 25-Ga hypodermic needles to complete the filling of the tank after the lid was in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In use, a mixture of saline at room temperature at the desired background conductivity is made and measured using a Oakton CON 6+ handheld conductivity meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Targets were made the day before experiments by adding NaCl and a few drops of food dye to distilled water until the desired conductivity was obtained, as measured by an immersible conduc- tivity meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The solution was then heated slowly with stirring as 4% agar (Fisher Scientific) was added until the temperature reached 85◦C−90◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The mixture was poured into 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='27 cm inside- diameter spherical molds and allowed to harden overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Test cells were also filled at this time so the final conductivities could be verified using the ACT5 instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Data were first collected of the tank filled with saline of the desired conductivity (24 mS/m) with no targets present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Then, targets were added to the tank supported on fine toothpicks from 12 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Left: Top view of the one target experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Middle: Top view of two target experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Right: side view of height of targets above floor of tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note that the targets were measured at 290 mS/m in test cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Photos were captured of the tank, with each target in place, to verify its/their position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The conductivity values of the spheres used were approximately 290 mS/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Averaged data, over 100 frames, were used in the subsequent conductivity reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The experimental data will be made publicly available on github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1 Note that even though we are testing on a rectangular prism tank, the mathematical reconstruc- tion algorithms are not limited to this geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Robustness Tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In addition to testing the reconstruction methods with the correct domain modeling, we consider a moderate, as well as strong, mis-modeling of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The incorrect domain modeling comes into play in the following places for the CGO-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' First in the simulation of the L1 DN data that requires solving the forward EIT problem with a known conductivity σ ≡ 1 S/m and the applied current patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Next, the incorrect domain modeling presents as incorrect information about the centers of the electrodes x and surface area for the domain | ∂Ω |;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' the subsequent reconstructions of σCAL, σexp, and σ0 are recovered on the incorrect domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the TV regularized method, the incorrect domain modeling is present in the FEM based forward map U(σ, z) used in the minimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='28) and computational mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the linear difference imaging method, the incorrect domain modeling is present in the Jacobian matrix of the forward map Jδσ in the minimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='34) and computational mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Accurate domain modeling can be challenging in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' To address this we explored a domain with a moderately incorrect domain shape, 18cm x 27cm x 19cm, as well as a domain with a significantly incorrect domain shape, 20cm x 35cm x 25cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Electrodes were uniformly distributed as in the original box design, but clearly had different locations and spacing than the truth due to the new box dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Figure 2 (center and right) depict the larger box domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Each reconstruction method was then tested assuming that the actual experimental voltage and current measurements were coming from the boxes shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Time-difference reconstructions were performed using basal measurements with only 24 mS/m saline in the experimental tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This data was then used in the reconstruction methods discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1 and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Forward modeling and Regularization Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the CGO-based methods, the matrix approximation L1 to the DN map Λ1 was formed using simulated voltage data, using EIDORS [3,78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The EIDORS data is produced by solving the forward conductivity problem with the finite element approximation of the Complete Electrode Model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2) with σ ≡ 1 S/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In each domain modeling case, see Figure 2, the forward problem was solved on a box of the corresponding dimensions, with 8cm by 8cm electrodes, using approximately 250, 000 nodes and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2 million elements with high refinement used near the electrodes, and the default contact impedance in EIDORS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the 1See https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='com/sarahjhamilton/open3D_EIT_data 2Due to the smaller ratio of longest side to shortest side, the 16 × 16 × 16 X-grid used for Calder´on’s method as mentioned in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1 does not capture the whole domain, so a 32 × 32 × 32 x-grid is used in the significantly incorrect domain case C DD12 28 19 273D CGO-BASED EXPERIMENTAL EIT 13 Correct Model (17x25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5x17) Box 18x27x19 Box 20x35x25 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 3D renderings of simulated boxes used in robustness testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' reference methods, the FEM based forward problem was solved using the same meshes for the electric potential as with the CGO-based methods and the conductivities were approximated in sparser, uniform meshes with 8, 918;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 9, 410;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 14, 678 nodes and 43, 690;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 45, 106;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 72, 438 elements, respectively, for the different domain modeling cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The regularization parameters for each of the reconstruction methods were chosen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The regularization parameters Tz1 and Tz2 for Calder´on’s method were empirically chosen to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='7, respectively, and the mollifying parameter was chosen to be t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1 for all reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' These values were chosen empirically to provide the best visual reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The truncation radius Tξ for the nonlinear scattering data used in the texp and t0 methods was chosen empirically in the range [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5, 12] for each case shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The localization of the target(s) did not appear to change significantly with Tξ, however the contrast does appear sensitive to this parameter choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Further discussion is given below in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the total variation regularized method, the regularization parameter α and the smoothing parameter β were selected by computing a series of reconstructions with varying α and β values, in the range [1e-6, 100] for α and [1e-4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1] for β, and by choosing the parameters that gave the best visual quality of reconstructions in the correct domain model case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The chosen values α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='01 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='001 were used in all the test cases with different domain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the linear difference imaging method, the visually best reconstructions were obtained using a standard deviation of conductivity std(σ) = 16σ0 and parameter a calculated using a covariance of 1 % of the variance at a correlation distance d = lbox/8, where lbox is the length of the longest side of the box that is used as the domain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As the main focus of this work is experimental data reconstruction, comparing to a known ‘truth’ with zero error is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The conductivity values for the targets were measured at approximately 290 mS/m and the targets were placed to be roughly centered in a sub-cube of the overall box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' These data allow us to approximate the localization error (LE) of our reconstructions and maximum conductivity in each reconstructed target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' LE, as in [34], measures the distance between the centroids of the reconstructed targets and those of the true targets by LE = � (xrecon 1 − xtruth 1 )2 + (xrecon 2 − xtruth 2 )2 + (xrecon 3 − xtruth 3 )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='36) A localization error of 0 is ideal, and signifies the targets are reconstructed in the correct loca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In our experiments, the centers of the spherical targets were placed at the centers of the nearest electrode in each direction and in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='36), (xtruth 1 , xtruth 2 , xtruth 3 ) is our best estimate of the true location of the target center, acknowledging possible errors within millimeters of the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Following [34], to obtain the centers of the reconstructed targets, we use a thresholded segmenta- tion of the reconstructions to identify targets with MATLAB’s regionprops3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The thresholds for segmentation were used to empirically identify regions with more than a certain percentage of the 22 21 13 8 5 6 28 20 2 12 0 2 4 ~ 27 9- 3 19 8 11 26 10 5 5 0 0 5 10 1022 14 21 13 8 6 28 20 4 - 12 2 0 +3 2 4 27 6 19 8 - 11 26 10 5 9 18 0 5 10 10 +122 21 14 10 28 20 +3 70 X 5 ~ 10~ 15 10 26 10 0 5 18-5 90 10 5 1010 15 ×2 X114 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' maximum reconstructed conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We note that the volume of the segmented targets changes with the choice of threshold, but as we increase the percentage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' threshold) the locations of the centroids stabilize (up to millimeters), from which we computed the LE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As a major focus of this work is to study how the algorithms tolerate modeling errors, we report a scaled version of the LE, Scaled LE = � (xrecon 1 − xtruth 1 )2 + (xrecon 2 − xtruth 2 )2 + (xrecon 3 − xtruth 3 )2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='37) where we have divided the LE from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='36) by 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5cm, the length of the longest side of the true tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This scaling value for the computing the LE metric is kept fixed across incorrect modeling scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Results We first present absolute EIT reconstructions with correct domain modeling, for the one and two targets, and for the four reconstruction methods (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Cross-sections in the center of the bottom row of electrodes of the box are shown on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' this slice cuts through the cen- ter(s) of the physical target(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Isosurfaces, shown in the middle column, are extracted from the 3D reconstructions to indicate the locations, size, and number of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The isosurfaces were extracted using MATLAB’s regionprops3 with threshold segmentation in the range of 60% to 85% of the maximum recovered conductivity in the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' A uniform threshold across all examples and reconstruction methods was not used as the blurring varied across methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note that since the isosurfaces are dependent on this threshold, they can omit information from the full recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Therefore, the third column shows a full 3D rendering of conductivity in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This 3D rendering is produced in MATLAB by displaying 100 equally-spaced x1-x3 slices set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1 transparency of each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' One target absolute imaging evaluation metrics across all domain mod- eling scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on texp t0 TV Scaled LE correct domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='043 box 18x27x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='021 box 20x35x25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='231 n/a Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Target Conductivity (mS/m) correct domain 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='77 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='00 308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='67 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='26 box 18x27x19 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='94 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='32 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='67 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='14 box 20x35x25 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='51 305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='63 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='88 n/a Figure 4 shows the absolute EIT reconstructions for the moderately mismodelled domain whereas Figure 5 shows the absolute EIT reconstructions for the significantly mismodelled domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Dif- ference images are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Metrics of maximum conductivity per target and scaled localization error are shown in Tables 1 and 2 for the absolute images and Table 3 for the difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Bolded entries correspond to the best metric value in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Discussion Beginning with the correct domain modeling scenario, each method, Calder´on, texp, t0, and TV produced reconstructions that clearly show the target(s) with good localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The TV images are sharpest, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The texp and t0 methods achieved the best contrast and best approximated the estimated experimental conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The Calder´on and TV methods achieved lower contrast but better scaled localization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Notably, as shown in Figure 6, in terms of artefacts, the absolute CGO reconstructions are as clean as their corresponding difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The CGO methods required simulated voltages data for the same experimental setup but with a 3D CGO-BASED EXPERIMENTAL EIT 15 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Two target absolute imaging evaluation metrics across all domain mod- eling scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note: Target 1 is the same target as in the one-target case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on texp t0 TV Scaled LE Target 1 correct domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='034 box 18x27x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='021 box 20x35x25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='257 n/a Target 2 correct domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='031 box 18x27x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='057 box 20x35x25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='295 n/a Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Target Conductivity (mS/m) Target 1 correct domain 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='61 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='95 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='50 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='57 box 18x27x19 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='53 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='29 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='39 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='29 box 20x35x25 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='00 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='58 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='72 n/a Target 2 correct domain 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='47 234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='85 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='35 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='94 box 18x27x19 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='94 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='82 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='84 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='22 box 20x35x25 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='81 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='06 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='80 n/a Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Two target difference imaging evaluation metrics across all domain mod- eling scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note: Target 1 is the same target as in the one-target case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Calder´on texp t0 Linear Scaled LE Target 1 correct domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='049 box 18x27x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='065 box 20x35x25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='243 n/a Target 2 correct domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='050 box 18x27x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='029 box 20x35x25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='233 n/a Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Target Conductivity (mS/m) Target 1 correct domain 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='18 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='83 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='63 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='83 box 18x27x19 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='72 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='80 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='05 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='95 box 20x35x25 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='30 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='46 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='88 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='95 Target 2 correct domain 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='84 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='91 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='25 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='88 box 18x27x19 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='91 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='17 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='30 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='05 box 20x35x25 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='07 293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='61 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='38 n/a conductivity σ ≡ 1 S/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Even though this data was not tuned to the EIT machine (noise and contact impedances) the methods produced high quality absolute reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Moving into incorrect domain modeling, we see all methods handled the moderate domain mis- modeling quite well and the CGO methods handled the severe domain mismodeling as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' No significant boundary artefacts are seen in the CGO reconstructions for the 18cm x 27cm x 19cm and only moderate boundary artefacts in the TV reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In the more severe mis-modeling case, assuming the data is coming from a much larger box (20cm x 35cm x 25cm) instead of the true domain (17cm x 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5cm x 17cm), the texp and t0 methods still produce good reconstructions showing the correct number of targets and correct region of the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The localization is worst in the x3 direction showing the targets slightly elongated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' While artefacts in the Calder´on 20x35x25 reconstructions are present, the method still does detect the true objects as the most conductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The TV algorithm failed to identify any of the targets for both cases with the 20x35x25 box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The artefacts in the TV reconstructions in these cases with incorrect domain model could be expected as the regularized non-linear least squares minimization-based absolute imaging EIT algorithms are known to be highly sensitive to modeling errors such as errors in modelling of the domain shape, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' [49,51,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We can see the effect of the underlying assumption for Calder´on’s method that the conductivity is a small perturbation from a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In the physical experiments, the conductor(s) had roughly twelve times the conductivity of the background saline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This is seen in the underestimation of 16 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 0 20 40 60 0 20 40 60 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 0 10 20 30 40 50 10 20 30 40 Slice Isosurface 3D Slice Isosurface 3D One Target Two Targets Truth Cald texp t0 TV Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Absolute image reconstructions comparing the CGO methods to the regularized method with correct domain modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Slices, isosurfaces, and 3D renderings of the conductivity are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note the truth targets had a measured conductivity of approx 290 mS/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' maximum conductivity from Calder´on’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' However, unsurprisingly, this method does a good job of localizing the targets, in all cases except for the absolute image from the significant domain modeling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The artefacts seen in the Calder´on reconstructions are consistent with what has been observed for absolute imaging with Calder´on in 2D, such as Gibbs phenomenon for all absolute reconstructions and higher reconstructed conductivity at electrode locations when the domains were incorrectly modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Across all modeling scenarios, the regularized TV method struggled with the high contrast targets, giving maximum target conductivities well below the measured values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The same effect was seen in simulated scenarios where, however, maximum conductivities of lower contrast targets were reliably recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This is at least partially explained by saturation of contrast distinquishability of the measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' the effect of increasing the conductivity of the target inclusion on the measurements gradually diminishes and until certain level the TV regularized method can no longer discern between high and even higher conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' C DD3D CGO-BASED EXPERIMENTAL EIT 17 20 0 20 40 60 0 20 40 60 50 100 150 200 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 0 20 40 60 0 20 40 60 Slice Isosurface 3D Slice Isosurface 3D One Target Two Targets Truth Cald texp t0 TV Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Absolute image reconstructions comparing the CGO methods to the regularized method with moderately incorrect domain modeling, using a box of size 18cm x 27cm x 19cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note the truth targets had a measured conductivity of approx 290 mS/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We note that the electrodes used in this experiment were very large and the structure of the domain, a box with corners, may exacerbate some of the modeling and/or hardware challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Nevertheless, the study provides informative results on the feasibility of absolute EIT reconstruction in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The difference images from Calder´on are able to handle the stronger mismodeling of the domain, as are the texp and t0 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The strong mismodeling proved too severe for the linear difference imaging reference method, which did not manage to identify the targets, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' While the texp and t0 CGO methods did reliably recover the contrast and approximate location of the targets across examples studied here, they do appear more sensitive than their 2D D-bar based counterparts in regards to the regularization parameter, Tξ, used in the truncation of the nonlinear scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Figure 7 displays the effect that Tξ plays on the scaled localization error and maximum recovered conductivity value for the single target, correct domain modeling C DD18 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 50 0 50 50 0 50 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 50 100 150 200 250 0 50 100 0 50 100 Slice Isosurface 3D Slice Isosurface 3D One Target Two Targets Truth Cald texp t0 TV Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Absolute image reconstructions comparing the CGO methods to the regularized method with largely incorrect domain modeling, using a box of size 20cm x 35cm x 25cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note the truth targets had a measured conductivity of approx 290 mS/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Recall that a secondary nonuniform truncation is also enforced where scattering data with magnitudes exceeding 20 for the real or imaginary parts are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Adjusting that value will also have an effect on the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The value of 20 was chosen in this work for its overall reliability across examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The contrast appears more sensitive than the localization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' As in [23], the minimum ζ parameterization of ξ was used, as the scattering data is a function of both ζ ∈ C3 and ξ ∈ R3 in 3D instead of just k ∈ C as in 2D D-bar based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Alternative parameterizations and a more detailed study of the effect of the regularization parameters, while interesting, are outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' In terms of speed, when running reconstructions on a MacBook Pro with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='3 GHz Dual-Core Intel® Core i5 processor, Calder´on reconstructions on a 16 × 16 × 16 x-grid which are interpolated to a 64 × 64 × 64 x-grid take 1 to 2 seconds without optimizing for parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' This increases to 6-8 seconds per reconstruction when the initial x-grid is 32 × 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' When running on a C DD3D CGO-BASED EXPERIMENTAL EIT 19 20 30 40 50 10 20 30 40 50 60 10 20 30 40 50 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 300 50 100 150 200 250 50 0 50 100 150 100 0 100 200 300 100 0 100 200 300 400 Correct modeling 18x27x19 modeling 20x35x25 modeling Truth Cald texp t0 Linear Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Difference image reconstructions comparing the CGO methods to a typical linear method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Slices and 3D renderings of the conductivity are shown for the correct domain modeling, and increasing levels of error in domain modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Note the truth targets had a measured conductivity difference from the background of approx 266 mS/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' PC with a AMD EPYC 7702P 64-Core Processor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='00 GHz, the reconstruction times are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='7 seconds and 4 seconds, respectively, again without optimizing for parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' On an 2015 iMac with a 4 GHz Quad-Core Intel® Core i7 processor, the texp and t0 methods require 3-4 seconds/recon using a 21 × 21 × 21 x-grid for the potential q(x) or 6-8 seconds/recon when using a 41 × 41 × 41 x-grid for q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The timings are non-optimized with the highest computational cost coming from computing the inverse Fourier transform and solving the boundary value problem using FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The regularized TV, and linear difference, reconstructions averaged 2-3 hrs, and 3 minutes, respectively, when computed on a server with 256GB of RAM and two 10 core Intel® Xeon® CPU E5-2630 v4 @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='20GHz processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We remark that the rather long computation times of the TV regularized non-linear least squares approach are caused by the 3D problem with a computationally D20 HAMILTON ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='25 texp t0 6 8 10 12 0 200 400 600 800 texp t0 Scaled Localization Error Max Target Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' (mS/m) Tξ Tξ Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Comparison of the effect of the truncation value Tξ of the scattering radius in the texp and t0 CGO methods for Scaled Localization Error (left) and Maximum value of the recovered target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Max conductivity values (right) for Tξ = 11, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='5, off the plot, spiked into the 1500-5000 mS/m range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' rather challenging geometry as the sufficient accuracy of the CEM forward model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content='2) necessitates significant mesh refinement near the electrodes, leading to the large number of degrees of freedom (approx 250,000 nodes) in the FEM based forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The FEM model needs to be solved multiple times in the line search at each iteration of the Gauss-Newton method and with the mesh used each forward solution takes approximately 80s computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Conclusions In this work, we presented the first 3D absolute EIT reconstructions from CGO-based methods on experimental 3D tank data, and compared them to the current standard, a total variation reg- ularized non-linear least squares approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' We demonstrated that, with correct domain modeling, quality 3D absolute reconstructions can be obtained by all of the methods, comparable to the qual- ity seen in linear difference imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' All methods, Calder´on, texp, t0, and TV reasonably handled the moderate domain modeling error within little noticeable change in localization error and target contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' For the large modeling error case, the texp and t0 methods correctly identified the targets with high contrast, additional artefacts were introduced into the Calder´on reconstruction, and the error proved too significant for the TV method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The computational cost of the CGO reconstruc- tion is trivial compared to TV (non-optimized, less than 1 sec/recon for Calder´on, approximately 5 sec/recon for texp and t0, compared to 2-3 hours per reconstruction for TV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Acknowledgments Research reported in this paper was supported by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under award numbers R21EB028064 (SH and JN) 1R01EB026710-01A1 (GS, DI, JN, ORS, and the development of the ACT5 device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' JT and VK were supported by the Academy of Finland (Project 336791, Finnish Centre of Excellence in Inverse modeling and Imaging), the Jane and Aatos Erkko Foundation and Neurocenter Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' References [1] Ahmed Abdelwahab, Omid Rajabi Shishvan, and Gary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Saulnier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} +page_content=' Performance of an adaptive current source for EIT driving loads through a shielded coaxial cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfsP0n/content/2301.01655v1.pdf'} 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Glyph Design for Showing +Large-Magnitude-Range Quantum Spins +Henan Zhao, Garnett W. Bryant, Wesley Griffin, Judith E. Terrill, Jian Chen +Abstract—We present experimental results to explore a form of bivariate glyphs for representing large-magnitude-range vectors. The +glyphs meet two conditions: (1) two visual dimensions are separable; and (2) one of the two visual dimensions uses a categorical +representation (e.g., a categorical colormap). We evaluate how much these two conditions determine the bivariate glyphs’ +effectiveness. The first experiment asks participants to perform three local tasks requiring reading no more than two glyphs. The +second experiment scales up the search space in global tasks when participants must look at the entire scene of hundreds of vector +glyphs to get an answer. Our results support that the first condition is necessary for local tasks when a few items are compared. But it +is not enough for understanding a large amount of data. The second condition is necessary for perceiving global structures of +examining very complex datasets. Participants’ comments reveal that the categorical features in the bivariate glyphs trigger emergent +optimal viewers’ behaviors. This work contributes to perceptually accurate glyph representations for revealing patterns from large +scientific results. We release source code, quantum physics data, training documents, participants’ answers, and statistical analyses +for reproducible science at https : //osf.io/4xcf5/?viewonly = 94123139df9c4ac984a1e0df811cd580. +Index Terms—Separable and integral dimension pairs, bivariate glyph, 3D glyph, quantitative visualization, large-magnitude-range. +! +1 +INTRODUCTION +B +IVARATE glyph visualization is a common form of +visual design in which a dataset is depicted by two +visual variables, often chosen from a set of perceptually +independent graphical dimensions of shape, color, texture, +size, orientation, curvature, and so on [1], [2]. A bivariate +glyph design [3] has been broadly applied to reveal atom +spin behaviors for quantum physicists at National Institute +of Standards and Technology (NIST) to examine experi- +mental results; thanks to their team’s Nobel-prize-winning +simulations [4]. Quantum physicists world-wide can now +manipulate many individual quantum systems to study +complex atom and sub-atom interactions. Because atoms +can be in multiple states simultaneously and because these +spin magnitudes are large in range and often vary greatly +in local regions, computational solutions still do not exist to +characterize the spin behaviors. Today’s quantum physicists +rely on visualization to interpret simulation results. +On the visualization side, the initial design and eval- +uation of large-magnitude-range spin vector visualizations +use scientific notation to depict digit and exponent as two +concentric cylinders [3]: inside and outside tube-lengths +(lengthylengthy or LyLy or splitVectors) are mapped to digit +and power of spin magnitude accordingly (Figure 1e). A +three-dimensional (3D) bivariate glyph scene of this splitVec- +tors design (Figure 2e) achieves up to ten times greater +accuracy than the traditional direct approach (Linear, Fig- +ure 2f) for reading a vector magnitude of a single spin or +deriving ratios between two spin magnitudes. However, this +• +Henan Zhao is with University of Maryland, Baltimore County. E-mail: +henan1@umbc.edu. +• +Garnett W. Bryant and Judith E. Terrill are with the National +Institute of Standards and Technology. E-mail: {garnett.bryant, ju- +dith.terrill}@nist.gov. +• +Wesley Griffin is with Stellar Science. E-mail: griffin5@umbc.edu. +• +Jian Chen is with The Ohio State University. E-mail: chen.8028@osu.edu. +design also increases task completion time for an apparently +simple comparison task between two magnitudes in three +dimensions (3D): the traditional direct approach of Linear +(Figure 2f) is significantly faster than splitVectors (Figure 2e). +One may frame this large-magnitude-range issue as a +visual design problem: how can we depict a scalar value using +bivariate visual features to help quantum physicists examine com- +plex spatial data? Intuitively, since all tasks in previous study +involve a single or at most two spin locations, human visual +system would integrate the two component parts (digit and +exponent terms) of a quantum spin into one gestalt before +comparing the results [5]. Since relating the digit and expo- +nent to the two size features demands a focused attention +mode of visual processing, a viewer would take longer to +process two component parts in splitVectors compared to a +single linear mapping. We term this explanation the object- +level hypothesis where a viewer responds to combine two +component parts of a value represented in a glyph to its +original scalar value (here the magnitude). +However, the object-level processing may be neither effi- +cient nor necessary. For example, Borgo et al. [6] state that +“... effective glyph design should encompass a non-conflicting set +of separable retinal variables”. Now, for our examples, if we +increase the bivariate feature separability by replacing the +exponent-to-length mapping in Figures 1e and 2e to the +exponent-to-color mapping in Figures 1c and 2c for compar- +ison tasks, it would be counterproductive for our attention +first to visit each glyph to compute the magnitude. Instead, +the global categorical color (hue) can guide our attention to +first compare the exponent, prior to compare vector lengths +(digit). In these cases, no object-level attentive processing +of bivariate features is needed as long as the two color hues +can be easily recognizable. +Further considering the quantum physicists’ task rele- +vant to multiple objects (e.g., find maximum among hun- +arXiv:2301.00002v1 [cs.HC] 25 Dec 2022 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +Fig. 1: Illustration of five bivariate configurations of vector magnitudes ∈ (0, 9, 999]. Three examples show vector +magnitudes 440 (4.4 × 102), 9, 999 (9.9 × 103), and 1 (1 × 100). Take 440 as an example, lengthylengthx (a) maps 4.4 +(digit) and 2 (exponent) to lengths along the y and x axes accordingly ( lengthy lengthx); (b)-(e) have the same digit- +to-lengthy representation as (a). The exponent representations are manipulated to be (1) more integral or separable from +lengthy and (2) more or less categorical. (b) lengthycolor/lengthx uses color to double-code exponent compared to (a). The +exponents in (c), (d), and (e) use color, texture, or outer cylinder length accordingly. Our experimental results support that +more separable dimensions lead to more perceptually accurate glyphs. The higher the separability, the higher the accuracy. +Also, using a more categorical feature (e.g., color in (c)) of one of the variables benefited efficiency and accuracy. +dreds of vectors) (Figure 2), viewers are likely to check the +color legend and then use color to first divide the scene +into subregions, prior to use length for detailed comparisons +within the yellow region (Figures 2b and 2c). The colorful +scene context benefits the reduction of search to a much +smaller scale via global statistics of the scene. Coinciden- +tally, this first impression of the data to drive structural and +statistical information is also called scene-level processing [7]; +Wolfe called features guiding this top-down task-driven +attention behaviors as scene features. Scene features are also +preattentive and can guide attention in visual search toward +a target [8], perhaps due to fast ensemble processing [9]. +Taken together, an effective design of bivariate glyphs +is likely to be influenced by two conditions: separable +dimensions, with one of them being a pre-attentive scene +feature. These two factors are not necessarily independent. +For example, For the first factor, we can follow Borgo et +al. [6] and Ware [10] for “a non-conflicting set of separable +retinal variables”. To meet the both conditions to choose the +scene feature, we can give preferences of the separable pair +when one of the variables is categorical. This is because +categorical features are likely to be better at facilitating the +perception of a group of objects in the scene [7], [11], [12]. +We in this work compared several separable-integral pairs, +length-color (Figures 1b, 2b, 1c, 2c), length-texture (Figures 1d, +2d), and length-length (Figures 1a, +2a). Among the three +features of color, texture, and size, color is categorical and +thus “most recognizable”. Color ensembles are preattentive +and permit visual selection at a glance [13]. We purposefully +select texture patterns by varying the amount of dark on +white, thus introducing luminance variations when many +vectors are examined together (Figure 2d). Compared to the +continuous random noise in Urness et al. [14], ours is for +discrete quantities and thus uses regular scale variations. +When coupled with separable features, we hypothesize that +highly distinguishable separable dimension pairs, with one being +categorical might encourage preattentive global processing to +reduce task completion time and be more accurate. +We +tested +this +hypothesis +in +two +experiments +with +six +tasks +using +four +pairs +to +compare +against +the +lengthylengthy +(separable) +in +Zhao +et +al. +[3]: +lengthylengthx +(integral), +lengthycolor +(separable), +lengthytexture +(separable), +and +lengthycolor/lengthx +(redundant +and +separable). +Since +we +predicted +that +separable +dimensions +with +more +preattentive +features +would reduce the task completion time, lengthycolor +and lengthycolor/lengthx might achieve more efficiency +without hampering accuracy than other bivariate pairs. +This work makes the following contributions: +• Empirically validates that bivariate-glyphs encoded by + +Y +V +VV +VV +Digit +Length +Length +Length +Length +Length +5个 +X +3 +0 +0123 +Exponent +1 +Length +Color and length +Color +Texture +Length +five-unitlongandthefeathers alwaysface theuser. +Example: +440 +4.4 +4.4 +4.4 +4.4 +Maximum: +9999 +(Note: these +glyphs are +shownata +different scale.) +Minimum: 1 +(Note: these +glyphs are +shown at a +differentscale.) +(b) Lengthy +(a) Lengthy lengthx +(c) Lengthy color +(d) Length, texture (e) Length, lengthy (splitVectors ) +color/lengthxJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +(a) Lengthylengthx (LyLx) (integral) +(b) Lengthycolor/lengthx (LCL) (redun- +dant encoding) +(c) Lengthycolor (LC) (separable) +(d) Lengthytexture (LT) (separable) +(e) Lengthylengthy (splitVectors, LyLy) [3] +(f) Linear +Fig. 2: Real-world large-magnitude-range quantum physics simulation results shown using (a)-(e) five bivariate feature- +pairs and (f) a traditional linear representation. LC, LCL, and LT can reveal scene spatial structures. We anticipate that +two conditions determine the glyph efficiency: (1) the bivariate glyph uses two separable dimensions; and (2) one of the +two dimensions uses a categorical representation thus can reveal global structures in data. The first condition is necessary +for local tasks when a few items are compared. The second condition is needed for inspecting the entire scene. +highly separable dimensions would improve compari- +son task completion time (Exp I). +• Is the first to evaluate categorical features in bivirate- +glyphs to leverage the benefits of the global scene +features (Exp II). +• Offers a rank order of separable variables for 3D glyph +design and shows that the separable pairs lengthycolor +and lengthytexture are among the most effective and +efficient feature pairs. +• Reveals a novel visual design method for scalable +search in big-data. +2 +THEORETICAL FOUNDATIONS +IN PERCEPTION +AND VISION SCIENCES +At least four perceptual and vision science theories have +inspired our work: integral and separable dimensions [15], +preattentive scene features [7], [8], [16], [17], feature ranking, +and monotonicity [2]. +Integral +and +Separable +Dimensions. +Garner +and +Felfoldy’s seminal work on integral and separable dimen- +sions [15] has inspired many visualization design guide- +lines. Ware [10] suggests a continuum from more inte- +gral to more separable pairs: (red-green)-(yellow-blue), sizex- +sizey, color-shape/size/orientation, motion-shape/size/orientation, +motion-color, and group position-color. His subsequent award- +winning bivariate study [2] using hue-size, hue-luminance, +and hue-texton (texture) supports the idea that more sep- +arable dimensions of hue-texton lead to higher accuracy. +Our work follows the same ideas of quantifying integral +and separable dimensions but differs from Ware’s texton +selection in two important aspects. First, the Ware study +focuses on finding relationships between two independent +data variables. In contrast, ours demands participants to +examine a complex scene for item discrimination when the +two variables are component parts of a vector magnitude. +Second, our texture uses the amount of black and white to +show luminance variations, in contrast to the discrete shape +variation in textons. We anticipate that ours will be more +suitable to continuous quantitative values so it is easier +to compare large and small to divide the regions [18]. No +existing work we know of has studied whether or not one of +the separable features being categorical can facilitate global +comparisons and can be scaled to large and more complex +3D vector magnitude analysis. +Scene-Guidance and Feature Distance. In order to rec- +ognize items, viewers do not “see” features and instead +“bind” these features to objects. This binding studies how +our visual systems separate object features such as shape, +color, motion trajectories, sizes, and distances into the whole + +0 +2 +30 +2 +30 +2 +3JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +object [5]. What we “see” also depends on our goals and +expectations. Wolfe et al. propose the theory of “guided +search” [8], a first attempt to incorporate users’ goals into +viewing. For example, if the viewer’s goal is to search largest +values, s/he can just check the yellow ones in Figure 2. +Wolfe et al. [8] further suggest that color, texture, size, and +spatial frequency are among the most effective features in +attracting the user’s attention. +When we combine features together, Duncan and +Humphreys articulate some of the most basic principles. +In general, guidance to a target will be stronger when the +feature differences between the target (T) and distractor (D) +are larger (TD differences), and when the feature differ- +ences amongst distractors are smaller (DD similarity) [19]. +For example, Ts are 2.3 (digit) and 2 (exponent) for 230 +(2.3 × 102). Ds include all numbers but 2.3 and 2. Using the +TD differences between features may explain why splitVec- +tors was time consuming. For example, to compare 230 +(2.3 × 102) to 2,300 (2.3 × 103), viewers have to differentiate +the two lengths of 2 (T) and 3 (T) from other distractors +(Ds other than 2 or 3). The heterogeneity of Ds or small DD +distances from 3D lengths may make the use of splitVectors +challenging, thus introducing temporal cost. +Preattentive and Attentive Feature Ranking. Human +visual processing can be faster when it is preattentive. Wolfe +called a feature preattentive when it guides attention in +search and cannot be decomposed into simpler features [7]. +The idea of preattentive pop-out has historically highlighted +that a single object has been considered compelling because +it captures the user’s attention against a background of +other objects (e.g., in showing spatial highlights [20]). Visual +features such as orientation and color (hue, saturation, light- +ness) can generate pop-out effects [21]. This type of pop- +out was also used visualizations. For example, Ropinski, +Oeltze, and Preim [22] summarized two groups of glyph +design: “parameter mapping” from shape and appearance +(color, transparency, and texture) and “placement” driven by +features or data. Our study concerns appearance. +Recent vision science development also suggests that +the preattentive feature is not limited to single items +but expanded to high-level structures. Global statistical and +structural features can be also preattentive [7]. Unlike the +now outdated Treisman’s 1988 preattentive processing [23], +where preattentive features were considered to be perceived +before it is given focused attention [23], these preattentive +features are persistent during viewers’ data exploration thus +can continue to provide guidance [7], [8]. Viewers can use +peripheral vision to compare in parallel to confidently tell +apart regions relevant or irrelevant to tasks [24]. +Visual features also can be responsible for different at- +tention speeds, and color (hue) and size (length and spatial +frequency) are among those that guide attention [9], [18]. +Healey and Enns [25] in their comprehensive review further +remark that these visual features are not popped-out at the +same speed: hue has higher priority than shape and tex- +ture [26]. Also, when data size increased, some preattentive +features diminished [27] [28]. +For visualizing quantitative data, MacKinlay [29] and +Cleveland and McGill [30] leverage the ranking of visual +features and suggest that position and size are quantitative +and can be compared in 2D. For example, MacKinlay’s +A Presentation Tool (APT) [29] automatically recommends +visualizations using effectiveness and expressive criteria and +outputs a ranked set of encoding to enumerate candidate +visualizations based on data types. +Casner [31] expands +MacKinlay’s APT by incorporating user tasks to guide +visualization generation. McColeman et al. [32] revise the +ranking of visual features based on the number of items. +All these studies almost exclusively consider only single +item mappings. Demiralp et al. [33] evaluate a crowdsourc- +ing method to study subjective perceptual distances of 2D +bivariate pairs of shape-color, shape-size, and size-color. +When adopted in 3D glyph design, the authors further +suggest that the most important data attributes should be +displayed with the most salient visual features, to avoid sit- +uations in which secondary data values mask the informa- +tion the viewer wants to see. Ours also emphasizes the use +of global scene features to optimize viewing experiences. +Monotonicity. Quantitative data encoding must nor- +mally be monotonic, and various researchers have recom- +mended a coloring sequence that increases monotonically in +luminance [34]. In addition, the visual system mostly uses +luminance variation to determine shape information [35]. +There has been much debate about the proper design of +a color sequence for displaying quantitative data, mostly +in 2D [36] and in 3D shape volume variations [37]. Our +primary requirement is to use categorical colormaps that +users be able to read large or small exponents at a glance. We +used four color steps in the first study and up to seven steps +in the second study from ColorBrewer +[36] for showing +areas of large and small exponents that are mapped to a hue- +varying sequence. We claim not that these color sequences +are optimal, only that they are reasonable solutions to the +design problem. +3 +EXPERIMENT I: EFFECT OF SEPARABLE PAIRS +ON LOCAL DISCRIMINATION AND COMPARISON +The goal in this first experiment is to quantify the benefits of +separable pairs with preattentive features for visual process- +ing of a few items. This section discusses the experiment, the +design knowledge we can gain from it, and the factors that +influence our design. +3.1 +Methods +3.1.1 +Bivariate Feature-Pairs +We chose five bivariate feature-pairs to examine the com- +parison task efficiency of separable-integral pairs. +Lengthylengthx (integral) (Figure 1a). Lengths encoded +digits and exponents shown as the height and radius of +cylinder glyphs. +Lengthycolor/lengthx (redundant and separable) (Fig- +ure 1b). This pair compared to lengthylengthx added a +redundant color (luminance and hue variations) dimension +to the exponent and the four sequential colors were chosen +from Colorbrewer [36] (Appendix A shows the sequences.) +Lengthycolor (separable) (Figure 1c). This pair mapped +exponents to color. Pilot testing showed that the least incor- +rect exponent levels were selected among these five feature- +pairs. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +Lengthytexture (separable) (Figure 1d). Texture repre- +sented exponents. The percentage of black color (Bertin [38]) +was used to represent the exponential terms 0 (0%), 1 (30%), +2 (60%) and 3 (90%), wrapped around the cylinders in five +segments to make them visible from any viewpoint. +Lengthylengthy (splitVectors [3], separable) (Figure 1e). +This glyph used splitVectors [3] as the baseline and mapped +both digit and exponent to lengths. The glyphs were semi- +transparent so that the inner cylinders showing the digit +terms were legible. +Feather-like fishbone legends were added at each location +when the visual variable length was used. The tick-mark band +was depicted as subtle light-gray lines around each cylinder. +Distances between neighboring lines show a unit length +legible at certain distance (Figure 1, rows 1 and 2). +3.1.2 +Hypotheses +Given the analysis below and recommendations in the liter- +ature, we arrived at the following working hypotheses: +• Exp I. H1. (Overall). The lengthycolor feature-pair can lead +to the most accurate answers. +• Exp I. H2. (Integral-separable). Among the three separable +dimensions, lengthycolor may lead to the greatest speed and +accuracy and lengthytexture will be more effective than +lengthylengthy (splitVectors). +• Exp I. H3. (Redundancy on time). The redundant pair +lengthycolor/lengthx will reduce task completion time +compared to splitVectors. +Several reasons led to H1 and H2. They are related to +the two conditions of glyph design we evaluate. Color and +length were separable dimensions, so comparing length to +color is simple (condition 1). And color was preattentive +and could be detected quickly (condition 2). Compared to +the redundant lengthycolor/lengthx, lengthycolor reduced +crowding since the feature-pairs were generally smaller +than those in lengthycolor/lengthx. Also, distinguishing +two lengths in splitVectors might be less efficient than +lengthytexture. H3 could be supported because redun- +dancy increased information processing capacity [10]. Re- +dundancy contributes to efficiency by increasing the feature +distances between exponents. We did not expect accuracy +gain from redundancy because splitVectors achieved the +same level of accuracy as reading texts in Zhao et al. [3]. +It may not be useful to decode quantitative data in this +experiment at least for showing a few items. +3.1.3 +Tasks +Participants performed the following three task types as in +Zhao et al. [3] so that results were comparable. They had +unlimited time to perform these three tasks. +Exp I. Task 1 (MAG): magnitude reading (Figure 3a). +What is the magnitude at point A? One vector was marked by +a red triangle labeled “A”, and participants should report +the magnitude of that vector. This task required precise +numerical input. +Exp I. Task 2 (RATIO): ratio estimation (Figure 3b). +What is the ratio of magnitudes of points A and B? Two vectors +are marked with two red triangles labeled “A” and “B”, and +participants should estimate the ratio of magnitudes of these +two vectors. The ratio judgment is the most challenging +(a) MAG task: What is the magnitude of the vector at point A? +(answer: 636.30) +(b) RATIO task: What is the ratio of the magnitude between the +vectors at points A and B? (answer: 3.60) +(c) COMP task: Which magnitude is larger, point A or point B? +(answer: A on the right.) +Fig. 3: Experiment I: Local discrimination and comparison +tasks. These two red equilateral triangles are rendered on +the screen coordinate and are thus always visible. +quantitative task [29]. Participants could either compare +the glyph shapes or decipher each vector magnitude and +compute the ratio mentally. +Exp I. Task 3 (COMP): comparison (Figure 3c). Which +magnitude is larger, point A or B? Two vectors are marked +with red triangles and labeled “A” and “B”. Participants +select their answer by directly clicking the “A” or “B” +answer buttons. This task was a simple comparison between +two values and offered a binary choice of large or small. +3.1.4 +Data Selection +Because we were also interested in comparing our results +to those in Zhao et al. [3], we replicated their data selection +method by randomly sampling some quantum physics sim- +ulation results and produce samples within 3D boxes of size + +done3/80 +pause +done +Task 1. What is the magnitude at point A?done +sk 2.What is the ratio betweenJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +TABLE 1: Experiment I design: 20 participants are as- +signed to one of the five blocks and use all five bivari- +ate pairs. Here, LyLy: lengthylengthy (splitVectors), LyLx: +lengthylengthx, LC: lengthycolor, LT: lengthytexture, +and LCL: lengthycolor/lengthx. +Block +Participant +Feature-pair +1 +P1, P6, P11, P16 +splitVectors, LyLx, LC, LT , LCL +2 +P2, P7, P12, P17 +LyLx, LC, LT , LCL, splitVectors +3 +P3, P8, P13, P18 +LC, LT , LCL, splitVectors, LyLx +4 +P4, P9, P14, P19 +LT , LCL, splitVectors, LyLx, LC +5 +P5, P10, P15, P20 +LCL, splitVectors, LyLx, LC, LT +5 × 3 × 3. There were 445 to 455 sampling locations in each +selected data region. +We selected the data satisfying the same following con- +ditions: (1) the answers must be at locations where some +context information was available, i.e., not too close to +the boundary of the testing data; (2) no data sample was +repeated to the same participant; (3) since data must include +a broad measurement, we selected the task-relevant data +from each exponential term of 0 to 3. +3.1.5 +Empirical Study Design +Design and Order of Trials. We used a within-subject de- +sign with one independent variable of bivariate quantitative +feature-pair (five types). Dependent variables were error +and task completion time. We also collected participants’ +confidence levels. Table 1 showed that participants were +assigned into five blocks in a Latin-square order, and within +one block the order of the five feature-pair types is the +same. Participants performed tasks with randomly selected +datasets. Each participant performed 60 trials (3 tasks × 4 +random data × 5 feature-pairs). These four random data +were from four exponent ranges. +Participants. We diversified the participant pool as much +as possible, since all tasks could be carried out by those +with only some science background. Twenty participants +(15 male and 5 female, mean age = 23.3, and standard +deviation = 4.02) participated in the study, with ten in com- +puter science, three in engineering, two in chemistry, one in +physics, one in linguistics, one in business administration, +one double-major in computer science and math, and one +double-major in biology and psychology. The five females +were placed in each of the five blocks (Table 1). On average, +participants spent about 40 minutes on the tasks. +Procedure. Participants were greeted and completed +an Institutional Review Board (IRB) consent form (which +described the procedure, risks and benefits of the study) +and the demographic survey. All participants had nor- +mal or corrected-to-normal vision and passed the Ishihara +color-blindness test. We showed feature-pair examples and +trained the participants with one trial for every feature-pair +per task. They were told to be as accurate and as quickly as +possible, and that accuracy was more important than time. +They could ask questions during the training but were told +they could not do so during the formal study. Participants +practiced until they fully understood the feature-pairs and +tasks. After the formal study, participants filled in a post- +questionnaire asking how these feature pairs supported +their tasks and were interviewed for their comments. Pilot +studies were conducted to examine the procedures. +Environment. Participants sat at a 27 ′′ BenQ GTG XL +2720Z, gamma-corrected display with resolution 1920 × +1080 to ensure the colors were displayed properly. The +distance between the participants and the display was about +50cm. The minimum visual angle of task-associated glyphs +was 0.2◦ in the default view where all data points were +visible and the scene filled the screen. +Interaction. Participants could rotate the data and zoom +in and out. Lighting placement and intensity were chosen to +produce visualization with contrast and lighting properties +appropriate for human assumptions and the spatial data. +The screen background color was neutral stimulus-free gray +background to minimize the discriminability and appear- +ance of colors [10]. Using black or white background colors +makes the black and white texture stimuli disappear thus +bias the results (See Appendix B for examples). +3.2 +Experiment I: Results and Discussion +3.2.1 +Analysis Approaches +We collected 400 data points for each task. In preparing +the accuracy and task completion time for analysis, we +differentiated two error metrics related to the perceptual +accuracy of the bivariate pairs: +• Correspondence error (C-Error): A trial is considered to +have an answer of C-Error if response’s exponent value +does not match the correct one. Having a C-Error would +mean that participants have trouble differentiating the +exponent levels within a glyph. +• Relative error (R-Error): This R-Error follows Zhao et +al. [3] to study how sensitive a method is to error +uncertainty based on fractional uncertainty, calculated +as R-Error = | correct answer - participant answer | / (correct +answer). This measure was used for MAG and RATIO +tasks. The benefit of this metric was that it took into +account the value of the quantity being compared and +thus provided an accurate view of the overall errors. +In subsequent analysis, we separated these two error +measurements since Combining these two errors in the +analysis would also be problematic. The C-Errors are at +least one order of magnitude larger or smaller than the +ground truth. We also did not remove participants’ data +with C-Errors, since the source of errors was caused by +glyph design methods independent of trials. +A post-hoc analysis using Tukey’s Studentized Range +test (HSD) was performed when we observed a significant +main effect on R-Errors. When the dependent variable was +binary (i.e., answer correct or wrong), we used a logistic +regression and reported the p value from the Wald χ2 test. +When the p value was less than 0.05, variable levels with +95% confidence interval of odds ratios not overlapping were +considered significantly different. All error bars represent +95% confidence intervals. We also evaluated effect sizes +using eta-square, labeled “small” (0.01 − 0.06), “medium” +[0.06 − 0.14), and “large” (≥ 0.14) effects following Co- +hen [39]. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +(a) Task 1 (MAG) +(b) Task 2 (RATIO) +(c) Task 3 (COMP) +Fig. 4: Experiment I task completion time and relative error or accuracy by tasks. The horizontal axis represents the mean +task completion time while the vertical axis showing the accuracy or relative error. Same letters represent the same post-hoc +analysis group. Colors label the feature-pair types. All error bars represent 95% confidence interval. +TABLE 2: Summary statistics by tasks. The significant main +effects and the high effect size (ES) are in bold (none in these +observations) and the medium effect size is in italic. Effect +size is eta-square labeled “small” (0.01 − 0.06), “medium” +[0.06 − 0.14), and “large” (≥ 0.14) effects following Co- +hen [39]. Post-hoc Tukey grouping results are reported for +significant main effects, where > means statistically signif- +icantly better and enclosing parentheses mean they belong +to the same Tukey group. +Task +Variables +Significance +ES +MAG +time +F(4, 384) = 6.8, p < 0.0001 +0.07 +(LC, LT , LCL, splitVectors) > LyLx +relative error +F(4, 384) = 0.9, p = 0.46 +0.01 +RATIO +time +F(4, 395) = 6.2, p < 0.0001 +0.06 +Three groups: A: LC, splitVectors, LT +B: splitVectors, LT , LCL +C: LT , LCL, LyLx +relative error +F(4, 395) = 0.8, p = 0.50 +0.01 +COMP +time +F(4, 395) = 10.4, p < 0.0001 +0.09 +Three groups: A: LCL, LC, LT +B: LC, splitVectors +C: splitVectors, LyLx +accuracy +χ2 = 0.4, p = 0.98 +0.03 +3.2.2 +Overview of Study Results +Figure 5 show all C-Error occurrences. Table 2 and Fig- +ure 4 show the F and p values computed with SAS one- +way measures of variance for task completion time and +relative error. Our results clearly demonstrated the benefits +in terms of task completion time of separable dimensions +for comparison. We observed a significant main effect of +feature-pair type on task completion time for all three tasks +MAG, RATIO, and COMP, and the effect sizes were in +the medium range. Lengthycolor was the most efficient +approach. For COMP, lengthycolor, lengthytexture and +lengthycolor/lengthx were most efficient for simple two- +point comparisons (Figure 4c). +3.2.3 +Separable Dimension Coupled with Categorical Fea- +tures had the Least Correspondence Errors. +We only observed C-Errors in MAG, but not in the RATIO +and COMP tasks. The total count was relatively small (11 +instances of 400 data points). They came from 9 partic- +ipants (error mean = 1.22 and 95% confidence intervals +Fig. 5: Experiment I (Task MAG): All instances of cor- +respondence errors by participant. The most separable +lengthycolor glyph had no instances of correspondence er- +ror whilst the lengthylengthx had the most. The redundant +color dimensions helped removed some correspondence +errors (Two instances of lengthycolor/lengthx vs. five in- +stances of lengthylengthx). +(CI)=[0.96, 1.48]). Figure 5 shows all instances of these er- +rors by participant and by encoding methods. It appeared +that the degree of separability of integral-separable dimen- +sions influenced the errors: the most integral dimension +lengthylengthx had the highest number (5 instances) of C- +Errors and the most separable lengthycolor had none. +3.2.4 +Separable Dimensions Are Better Than Integral Di- +mensions for Local Comparisons. But Categorical Feature +was not a Statistically Significant Effect. +Our first two hypotheses H1 and H2 are supported. In the +MAG task, the integral lengthylengthx was the least ef- +ficient and all other separable-pairs were in a separate +group, the most efficient one (Figure 4a). In RATIO, +lengthycolor, lengthytexture, and splitVectors were the +most efficient group (Figure 4b); in COMP, the redundant +lengthycolor/lengthx, lengthycolor, and lengthytexture +were in the most efficient group (Figure 4c). SplitVectors was +not as bad as we originally thought in perceiving correct +exponents. SplitVectors belonged to the same efficient post- + +0.12 +SplitVectors +0.1 +Lengthy lengthx +Lengthy color +Error +0.08 +Length, texture +Length, color/length, +Relative +0.06 +0.04 +0.02 +A +A AA +B +0 +0 +10 +20 +30 +40 +Task Completion Time (s)A +AA +1 +BB +B +SplitVectors +0.9 +C +C +Lengthy length, +0.8 +Lengthy color +0.7 +Error +Length, texture +0.6 +Length, color/length. +Relative +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +10 +20 +30 +40 +50 +60 +70 +Task Completion Time (s)1 +0.9 +A +A +A +B +B +Accuracy +0.8 +c +SplitVectors +0.7 +Lengthy lengthx +Lengthy color +0.6 +Length, texture +Lengthy color/length, +0.5 +0 +5 +10 +15 +20 +25 +Task Completion Time (s)TaskMAG:NumberofCorrespondenceErrors +byParticipant(X-axis)and GlyphType (Y-axis) +The numbers overlaid on bars show the (ground truth, participant's answer) exponent pair. +SplitVectors +(3.0) +(3,2) +(2,3) +LengthyLengthx +(1.0) +(2.3) +LengthColor +LengthTexture +(2.4) +Lengthy/Color Lengthx +(3.0) +(1.2) +0 +12 +4 +6 +7 +910 11 12 13 14 15 16 17 18 19 20 +ParticipantIDJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +hoc group as lengthycolor and lengthytexture for RATIO +and these three were also most efficient for MAG. +3.2.5 +Separable +Pairs +of +Lengthycolor +And +Lengthycolor/lengthx Achieved Comparable Efficiency To +Direct Linear Glyph +One aspect for motivating this experiment was to quantify +the benefits of separable pairs [6], [10]: whether the sepa- +rable pairs supported COMP and how the separable pairs +compared in efficiency to the direct mapping (Figure 2(f)). +Since our study had the same numbers of sample data +as Zhao et al. [3], we then performed a one-way t-test +to compare against the direct linear encoding in Zhao et +al. [3]. Our results indicated that results for COMP (judging +large or small) from separable variables was no more time- +consuming than direct linear glyphs, and our post-hoc anal- +ysis showed that lengthycolor, lengthycolor/lengthx, and +linear were in the same post-hoc group. We also observed +that splitVectors dropped to the least efficient post-hoc group +(Figure 4c). This result replicated the former study results +in Zhao et al. [3] by showing that splitVectors impaired +comparison efficiency. +3.2.6 +Redundant Feature-Pairs Were Efficient +We also confirmed hypothesis H3. We were surprised by +the large performance gain with the redundant encoding +lengthycolor/lengthx of mapping color and length to the +exponents in splitVectors. With the redundant encoding, +the task completion time was significantly shorter than +lengthylengthx for MAG and COMP tasks. While Ware [10] +confirmed that the efficiency might not be improved by +using separable dimensions, in our case, where color and +size (separable) represent the same quantitative value, we +suggested that the redundancy worked because participants +could use either length or color in different task conditions. +We could also consider that lengthycolor/lengthx is a re- +dundant encoding of lengthycolor, and those two feature- +pairs had similar efficiency and accuracy for all local tasks. +3.3 +Summary +The separable-pair condition is necessary for effective glyph +design because all separable pairs were more efficient +than the integral ones. The pre-attentive condition enabled +by categorical encoding among the separable pairs may +be not since not all conditions were statistically different +performance-wise. All tasks (MAG, RATIO, and COMP) +lacked of significant main effect on relative errors (in MAG +or RATIO) or accuracy (in COMP). Note that none of these +three tasks required initial visual search, and target answers +were labeled. Wolfe called this type of task-driven with +known target guided tasks [8]. Lengthycolor was the most +accurate in all tasks. +We also did not see the needs for the second condition +for perceptually accurate glyphs in this experiment. We did +not observe differences among categorical dimensions color, +texture, and length. We suspect that the reason for this lack +of significance could well be their similarities in mentally +computing load. The load was relatively small when com- +paring two values. We suspected that when search-space +set-size increases, and when tasks are more complex in- +volving all items, participants will need preattentive global +scene features to guide their search. We subsequently ran +the second experiment to increase the set size in tasks to the +entire scene to study the benefits of categorical features to +show quantitative exponent values to benefit global search. +4 +EXPERIMENT +II: +SCALABILITY +OF +GLOBAL +SCENE FEATURES +The goal in this second experiment is to quantify the benefits +of separable feature-pairs when they introduce categorical +features of scene guidance for global tasks in search spaces, +as large as the entire dataset of several hundreds items. In +other word, we measure scene feature scalability of global +tasks. +4.1 +Overview +We had three design considerations for us to carefully +choose the categorical features in setting up this experiment, +concerning the use of glyphs for showing complex simula- +tion results. Intriguingly, all of these considerations support +our second glyph design consideration of using a categorical +variable in one of the separable pairs. +The first reason is that the initial at-a-glance global +statistical summary of the scene depends on categorical +information [7]. +One of the most important advances in +vision science is to find that viewers can summarize the +scene without attending to the specific items [40]. Visual +dimensions facilitating this summary process become global +scene features and these features are pre-attentive [8]. While +visualization is mainly about mapping data values to visual +variables, the new theory concerns how features form the +structural and content of the scene that can affect efficiency. +If the quantum spins contain one object at a time, then +the first condition of glyph design considering integral and +separable dimensions is sufficient to explain the experience +as we have shown in Experiment I. For complex tasks, in +general, our visual system has a limited capacity. To cope +with this limit, humans first visually summarize the scene +to find specific regions of interests [6], [8]. If categorical fea- +tures stimulate population responses from multiple items, +we should observe fewer errors and better efficiency. For ex- +ample, we have exemplified in the Introduction section for +search of “largest” values by looking up “yellow” regions, +without attending to every single items of “yellow”. +The second concerns scalability to feature distances. Here +feature distance is meant to represent target-distractor simi- +larity. It is not the absolute features (e.g., yellow) that direct +our attention towards the answer; rather, what determines +performance is the result of a comparison between target +(yellow) and other data features (such as pink and orange) +in the scene (e.g., yellow is different from other colors and +the yellow regions stand out) [8]. In other words, one must +also look at feature distractors [14], [41], [42], whether or +not they are heterogeneous, and that the efficiency of a +scene guidance will decline as a function of the degree of +distractor variation [19], [24], [43]. While generally, subjec- +tive reports from Experiment I indicate that lengthycolor +and lengthytexture show the similar perceptual speed. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +Fig. 6: Visual mapping using color and texture in Experi- +ment II. From the top to bottom, colors and texture segments +are mapped to exponent values from the largest to the +smallest. The three numbers next to the 7-level colormap are +the RGB values. The numbers next to the texture columns +are the proportion of black-on-white for the last 7-level +texture configuration. +Performance of texture may decline faster than color as +the exponent range increases because our vision is not as +sensitive to luminance-variation as to hues. For example, at +the exponent-range of 7 in Figure 6, the differences between +yellow and pink could be more differentiable than the two +top-level textures of different amount of black. +In this +study, we expanded the data range from the single level in +Experiment I to five ranges ∈ [3, 7] to understand feature- +pair scalability to feature distances. The efficiency of color in +Experiment I could well arise because the range (of 4) was +not large enough. +The third concerns the density effects on color choices. +Figure 7 shows two densities and two colormaps (a cate- +gorical colormap from Colorbrewer [36] and a segmented +continuous colormap by the number of exponents generated +from the extended blackbody colormap). For a feature to +actually guide attention, we can see from Figure 7, the +boundary detection with these colormaps is associated with +data density. Unless the data density was reasonably high, +detecting the boundaries using continuous colormaps (Fig- +ures 7a, +7b) is harder than the ColorBrewer colormaps +(Figures 7c, 7d). +4.2 +Method +4.2.1 +Feature-Pairs +We +used +lengthycolor, +lengthytexture, +and +baseline +splitVectors in Experiment II. These three visualizations +were chosen because lengthycolor and lengthytexture are +among the best feature-pairs from Experiment I and because +color and texture are among the most separable features ac- +cording to Ware [10]. To introduce a “distractor” experience +to measure scalability to feature distances, we vary the data +range from the 4 levels in Experiment I to 3 − 7 levels in +Experiment II (See mapping in Figure 13, Appendix C.) +4.2.2 +Hypotheses +We had the following hypotheses: +• Exp II.H1 (Accuracy). More categorical feature in the sep- +arable pairs will be more effective. We thus anticipate a +rank order of effectiveness from high to low: lengthycolor, +lengthytexture, and splitVectors. +• Exp II.H2 (Correspondence Errors). More categorical feature +of color in the separable pairs will reduce C-Errors, when +participants will choose the correct exponent level. +• Exp II.H3 (User behavior). More categorical dimension in the +separable feature-pairs will lead to optimal users’ behaviors: +i.e., participants can quickly locate task-related regions for +tasks that demand looking among many vectors due to global +scene features. +4.2.3 +Tasks +Participants performed three tasks in which they had to +compare all vectors to obtain an answer. +Exp II. Task 1 (SEARCH): visual search. A vector search +within 20 seconds (Figure 8a). Find the vector with magnitude +X within 20 seconds. The target vector was shown at the +bottom-right corner of the screen. Participants were asked +to find this vector. +Exp II. Task 2 (MAX): find maximum. An extreme value +search within 20 seconds (Figure 8b). Within 20 seconds, lo- +cate the point of maximum magnitude when the exponent is X. X +in the study was a number from 0 to the maximum exponent +(∈ [2, 6]). This was a global task requiring participants to +find the extremum among many vectors. +Exp II. Task 3 (NUMEROSITY): estimate the total +number of unique vector exponents (Figure 8c). Estimate the +total number of unique vector exponents in the entire vector field +within 2 seconds. Data are randomly chosen and modified to +produce the 3 to 7 range. +4.2.4 +Task Choices +Tasks are use-inspired by real-world quantum physics data +analyses. Experiment I drilled down to a single or at most +two spins. But global tasks are also of quantum physicists’ +interests, such as those involving understanding the dis- +tributions of quantum spin magnitudes. Practically, a spin +represents charge density or the measure of the probability +of an electron being present at an infinitesimal element of +space surrounding any given point. This probability varies +due to electron traveling from one grid point to another and +is often interpreted together with its neighbors. Quantum +physicists are thus interested in searches for regions, where +local regions are defined by spin magnitude and different +regions would correspond to changes in exponent. Often +the most interesting regions are also those with specific +charge densities (Task 1) or largest magnitudes (Task 2) . The +regional task is related to learning the number of interesting +regions or magnitude exponent clusters (Task 3). +Performing tasks was limited to 20 seconds as a pilot +study showed that it took participants about ∈ [15, 25] +seconds or on average about 20 seconds to finish search +tasks 1 and 2. Also, preattentive processing when used for +scene guidance involving a group of similar objects are +often fast for viewers to see and increasing the number of +items should not significantly impair the search time. From +the practical side for the last experiment, participants who +would want a perfect score could just spend time counting. +Constraining the time allowed us to measure the accuracy +when they may have to use the scene feature. + +(255,255,179 +100% +252,205,229) +83% +(253, 280, 98) +67% +(190, 186, 218) +50% +(141,211,199) +33% +(128,177,211) +17% +(251,128,114) +0% +3 +4 +6 +3 +4 +5 +6 +7 +Exponent-range +Exponent-rangeJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +(a) Continuous colormap and high-density data +(b) Continuous colormap and low-density data +(c) Categorical colormap and high-density data +(d) Categorical colormap and low-density data +Fig. 7: Density effects on color choices to justify the use of dense sampling and categorical colormap (c) in Experiment II. +This example dataset shows two colormaps: ( segmented-continuous (a and b) and categorical (c and d) colormaps), at two +different data densities. (a) and (c) show data with the raw density from the simulation results; (b) and (d) were produced by removing +around 70% vector glyphs. The boundaries between the data categories are more recognizable when the data are dense in +(a) and (c) (comparing the 1st column and the 2nd column). At the same density (comparing the 1st and 2nd row), the +boundaries between levels are easier to recognize when spin vectors are rendered using a categorical colormap of (c) and +(d). We thus use the raw dense and categorical colormaps (c) in Experiment II. +4.2.5 +Data Choices +Data were first sampled using the same approach as Exper- +iment I, and no data is used repeatedly in this experiment. +We then modified the exponent range from 3 to 7 for the +three tasks by normalizing the data to the desired new data +range. +Prior literature used both synthetic data and real-world +data to construct the data visualization as test scenarios, en- +abling tight control over the stimulus parameters (e.g., [44]). +Most of the synthetic data in literature were to replicate +real-world data characteristics; and others were explained +in fictitious use scenarios. The goal was primarily to prevent +preconceived user knowledge about the domain-specific +attributes. As a result, the synthetic data strike the right bal- +ance between real-world uses and the data characteristics. +In our cases, replicating characteristics in quantum +physics data was challenging and indeed impossible, since +atom behaviors in high-dimensional space were largely +unknown and thus were not easily simulated. Our approach +was therefore to randomly sample quantum physics simu- +lation results to capture domain-specific attributes and then +modify the data to suit evaluation purposes. We showed +our data to our physicist collaborators to ensure their va- +lidity. We confirmed that these modifications preserved the +domain-specific schema of a scene in terms of the domain- +specific structures and complexity from real simulations. +These modifications represented less than 4% of overall data +points in each scene. Finally, It improves the reuse of our +study results. +4.2.6 +Empirical Study Design +Dependent and Independent Variables. We used a within- +subject design with two independent variables of feature- +pair (three levels: baseline splitVectors, lengthycolor, and +lengthytexture) and exponent range (five levels: 3 − 7). The +dependent variable was relative error. We did not measure +time since all tasks were time-constrained. +Participants performed 3 (feature-pairs) × 5 (magnitude- +ranges) × 3 (repetitions) = 45 trials for the first two tasks. +Three repetitions were used to give participants enough +time to develop strategies. For NUMEROSITY tasks, the +design runs 4 repetitions, resulting in 3 (feature-pairs) × +5 (exponent-ranges) × 4 (repetitions) = 60 trials. Each par- +ticipant thus executed 45+45+60 = 150 trials. Completing +all tasks took about 32 minutes. +Self-Reporting Strategies. Several human-computer inter- +action (HCI) approaches can help observe users’ behaviors. +Answering questions can assist us to determine not just +which technique is better but also the strategies humans +adopt. For example, cognitive walkthrough (CTW) mea- +sures whether or not the users’ actions match the designers’ +pre-designed steps. Here we predicted that participants + +456 +-7 +-8 +-9 +-10 +-11.4 +-5 +9- +-7 +-8 +-9 +-10 +-11-4 +-5 +-6 +-7 +-8 +-9 +-10 +-11-4 +-5 +9- +-7 +-8 +-9 +-10 +-11JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +(a) SEARCH: Find the vector with magnitude X. (X: 731, +answer: the point marked by two yellow triangles.) +(b) MAX: Which point has the maximum magnitude when +the exponent is X? (X: 1, answer: the point marked by two +yellow triangles.) +(c) NUMEROSITY (NUM): Estimate the total number of +unique vector exponents of the entire vector field within 2 +seconds. (answer: 7) +Fig. 8: Experiment II three task types. The callouts show the +task-relevant feature-pair(s). +would use the global scene-features as guidance to accom- +plish tasks. We interviewed participants and asked them to +verbalize their visual observations in accomplishing tasks. +4.2.7 +Participants +Eighteen new participants (12 male and 6 female, mean +age = 23.8, and standard deviation = 4.94) of diverse +backgrounds participated in the study (seven in computer +science, four in computer engineering, two in information +systems, three in engineering, one in business school, and +one in physics). +Procedure, interaction, and environment were the same +as those in the Experiment I. +4.3 +Experiment II: Results and Discussion +We collected 810 data points per task for the first two +tasks of SEARCH and MAX and 1080 points for the third +NUMEROSITY task. +4.3.1 +Analysis Approaches +For SEARCH and MAX tasks, we measured relative error +(which was the percentage the reported value was away +from the ground truth and the same as that of Experiment +I) with SAS repeated measure. +The last NUMEROSITY +task used error rate which was the percentage of incorrect +answers of all trials for each participant. We also used +the same outlier removal methods to remove instances of +correspondence errors for SEARCH and MAX. +4.3.2 +Overview of Study Results +Table 3 and Figure 10 show the summary statistics; And +all error bars again represent 95% confidence intervals. We +observed a significant main effect of feature-pair type on all +three tasks. For the first two tasks, the post-hoc analysis +revealed that lengthycolor and lengthytexture were in +the same group, the most efficient one and that relative +errors were statistically significantly lower than those of +the splitVectors. Lengthycolor remained the most accurate +pair for the NUMEROSITY tasks. Exponent-range was only +a significant main effect for NUMEROSITY, with power +ranges 3 and 4 were significantly better than 5, which was +better than 6 and 7. +4.3.3 +More Categorical Features of Separable Dimensions +Improved Accuracy +We were interested to see if we could observe significant +main effects of categorical features in the separable pairs +in this experiment. Here we did observe the significant +main effect and confirmed our first hypothesis (H1) for +both SEARCH and MAX: in the general trend, more separa- +ble lengthycolor was more effective than lengthytexture +which was better than splitVectors, and lengthycolor and +lengthytexture were in the same Tukey group, when view- +ers were in the correct data sub-categories. +Lengthycolor led to the most accurate answers, and +splitVectors was better than lengthytexture for NUMEROS- +ITY task. This result can be explained by participants’ be- +haviors - more than half the participants suggested they +simply look for the longest cylinder from splitVectors since +they know the numerical values in the test were continu- +ous. This behavior deviated from our original purpose of +testing the global estimate but did show two perspectives in +favor of this work: (1) participants developed task-specific +strategies during the experiment for efficiency; (2) 3D length +still supported judging large and small and it was not as +effective as color perhaps due to ensemble perception from +categorical features. +4.3.4 +Color Categories of Separable Pairs Reduced Corre- +spondence Errors by a Large Margin +Our second hypothesis H2 was also supported. We first +tested the number of correspondence errors in SEARCH +and MAX in the same way as in Experiment I. These results +when combined with those in Experiment I confirmed again + +ind +7.31×102 +DoneFind the vector with max magnitudle +yhan power is 1 +DoneTask 3: 1/30 +Number of different powers +DoneJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +12 +Fig. 9: Experiment II (Tasks SEARCH and MAX): All in- +stances of correspondence errors by participant. Again, the +lengthycolor has the least instances of correspondence error +whilst the lengthytexture had the most. +that the lengthycolor reduced correspondence errors. For +SEARCH, There were only a single instance of correspon- +dence error. 36 instances of correspondence errors came +from 14 participants (mean= 2.57, 95% CIs=[2.1, 3.04]) +(Figure 9 top). Another 59 instances for MAX came from +16 of 18 participants, mean= 3.68, 95% CIs= [2.85, 4.51]) +(Figure 9 bottom). +4.3.5 +Compensating The Cost of Search in Complex Data +through Preattentive Scene Feature +The visualizations in our study contained hundreds of +items from realistic uses. Subjective behaviors through self- +report suggested that they adopted a sequential task-driven +viewing strategy to first obtain gross regional distribution +of task-relevant exponents. After this, a visual comparison +within the same exponent region were achieved. With these +two steps, judging large or small or perceiving quantities +TABLE 3: Exp II: Summary statistics by tasks. The significant +main effects and the high effect size are in bold and the +medium effect size is in italic. Effect size is Cohen’s d +for tasks SEARCH and MAX, and Cramer’s V for task +NUMEROSITY (NUM). Post-hoc Tukey grouping results +are reported for significant main effects, where > means +statistically significantly better and enclosing parentheses +mean they belong to the same Tukey group. Here, LC: +lengthycolor and LT: lengthytexture. +Task +Variables +Significance +ES +SEARCH +feature-pair +F(2, 261) = 18.4, p < 0.0001 +0.46 +(LC, LT) > splitVectors +power-range +F(4, 261) = 3.0, p = 0.20 +0.86 +MAX +feature-pair +F(2, 261) = 15.4, p < 0.0001 +0.47 +(LC, LT) > splitVectors +power-range +F(4, 261) = 0.3, p = 0.87 +0.11 +NUM +feature-pair +χ2 = 63.2, p < 0.0001 +0.25 +LC > splitVectors > LT +power-range +χ2 = 47.4, p < 0.0001 +0.35 +(3, 4) > 5 > (6, 7) +Fig. 10: Relative error for Tasks SEARCH and MAX was the +percentage the reported value was away from the ground +truth. Error rate for NUMEROSITY was the percentage of +wrong answers of all trials for each participant. The vertical +axis shows the relative error or error rate. Same letters +represent the same post-hoc analysis group. All error bars +represent 95% confidence intervals. + +TaskSEARCH:NumberofCorrespondenceErrors +byParticipant (X-axis)and GlyphType (Y-axis) +The numbers overlaid on bars show the (ground truth, participant's answer) exponent pair. +3 +1:2) +(4.6) +2 +（5.6) +(0.3) +(3,2) +(5.4) +SplitVectors +(2.1) +.5 +(5,4) +2.3 +(0.2) +(2,0) +(5.6) +（13） +LengthColor +(0.2) +(7,2) +3 +LengthTexture +（4.3) +2 +(1.2) +(2.1) +(3.4) +(4,5) +(2.4) +(1,2) +(3,1) +(3.4) +1 +2 +3 +5 +6 +7 +8 +9101112 131415161718 +Participant IDTask MAX: Number of Correspondence Errors +by Participant (X-axis) and Glyph Type (Y-axis) +The numbers overlaid on bars show the (ground truth, participant's answer) exponent pair. +4 +SplitVectors +(3,4) +3 +4.3) +2 +(4.3) +(4,0) +1 +5.6 +(5,6) +(3,4) +(5,4) +3 +(0,4) +(3:2) +LengthColor +2 +（4,2) +1 +(0,3) +(3,5) +(2.0) +8 +LengthTexture +7 +6 +5 +4 +(0,2) +(1,4) +3 +(6.0) +2 +(3,4) +(2,3) +(4,2) +(1,0) +(2.3 +(5,4) +1 +(3,2) +3.2 +(6,5) +(2,3) +(4,3） +(3,2) +(3,4) +(4,3) +0,2) +1 +2 +3 +5 +6 +8910 11 12 13 14 15 16 17 18 +Participant ID0.15 +0.15 +1.0 +SplitVectors +SplitVectors +SplitVectors +Length,Color +LengthyColor +Length,Color +Length,Texture +Length,Texture +LengthyTexture +0.75 +lative Error +Relative Error +0.10 +0.10 +Rate +B +0.5 +B +rror +e +E +0.05 +0.05 +B +R +0.25 +A +A +A +A +A +0 +0 +0 +SEARCH +MAX +NUMEROSITY0.15 +0.15 +1.0 +0.75 +Relative Error + Error +0.10 +0.10 +Rate +ative +0.5 +Error +e +0.05 +0.05 +B +0.25 +A +A +0 +0 +0 +3 +7 +Exponent-range +Exponent-range +Exponent-range +SEARCH +MAX +NUMEROSITYJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +13 +accurately from separable variables would not use object- +level information process. +Many participants commented on how the number +of powers in the data affected their effectiveness. For +lengthytexture, 10 participants remarked that it was dif- +ficult to differentiate adjacent powers when the total power +level is around 4-5 for lengthytexture. The white and black +textures were very easy to perceive. All but two participants +agreed that lengthycolor could perhaps support up to 6. +Chung et al. [42] studied ordering effects and it would be +challenging to compare ours to their results because their +visual features were not shown as a scene but an isolated +feature. More than half of the participants felt that effec- +tiveness of lengthylengthy was not affected by changing +the number of powers, since they looked for the longest +outer cylinder to help find the answer. These results may +suggest that subregion selection with lengthytexture can +perhaps be better designed with interfaces when the users +can interactively select a texture level. +5 +GENERAL DISCUSSION +We discuss the results from both experiments and suggest +future directions. +5.1 +Separable Dimensions with Preattentive Guidance +for Large-Magnitude-Range Quantum Physics Spins +Our first principle in glyph design is to follow the conven- +tion to use separable variable pairs [6], [10]. The results +of Experiment I showed that separable dimensions could +achieve the same efficiency as direct linear visualizations for +COMP and was always more efficient than integral pairs. +For these local-tasks, we didn’t observe significant error +reduction. +Our second principle in glyph design is to include cate- +gorical features in separable pairs. The results from Exper- +iment II studied the rank order of the separable pairs and +found that they indeed improved accuracy for global tasks. +Lengthytexture and splitVectors in both experiments led to +more correspondence errors than lengthycolor. Achieving +integrated numerical readings by combining two separable +visual features at object level seems not necessary. +The separable-dimension pairs of lengthycolor and +lengthytexture worked because they were preattentive +scene features. Our experiments show that viewers adopted +a sequential task-driven viewing strategy based on a view +hierarchy: viewers first obtain global distributions of the +scene. Then, a visual scrutiny is possible within a subregion. +Although splitVectors are separable, visual search for length +among length would be unguided because both targets and +distractors contained the same visual variable. The more +separable, the easier it would be to guide the attention. +Using coloring to provide some initial regional division may +be always better than not. Texture (luminance) could achieve +similar accuracy and efficiency as long as the task-relevant +regions could be detected. +5.2 +Feature Guidance vs. Scene Guidance +Taking into account both study results, we think an impor- +tant part of the answer to visualization design is guidance +of attention. It is guided to some objects or locations over +others by two broad methods: feature guidance (seeing objects) +and scene guidance (seeing global structures). +Feature guidance refers to guidance by properties of the +task-target as well as the distractors (leading to correspon- +dence errors). These features are limited to a relatively small +subset of visual dimensions: color, size, texture, orientation, +shape, blur or shininess and so on. These features have been +broadly studied in 3D glyph design (see reviews by Healey +and Enns [25], Borgo et al. [6], Lie et al. [46], Ropinski et +al. [22], and McNabb and Laramee [28]). Take one more +example from quantum physics simulation results, but with +a different task of searching for the structural distributions +in the power of 3 in Figure 11 will guide attention to either +the fat cylinders (Figure 11a) or the bright yellow color +(Figure 11d, +11b) or the very dark texture (Figure 11c), +depending on the feature-pair types. +Working with quantum physicists, we have noticed that +the structure and content of the scene strongly constrain the +possible location of meaningful structures, guided “scene +guidance” constraints [8], [47]. Scientific data are not ran- +dom and are typically structured. Contextual and global +structural influences can arise from different sources of +visual information. If we return to the MAX search task in +Figure 11 again, we will note that the chunk of darker or +lighter texture patterns and colors on these regular contour +structures strongly influence our quick detection. This is +a structural and physical constraint that can be utilized +effectively by viewers. This observation coupled with the +empirical study results may suggest an interesting future +work and hypothesis: adding scene structure guidance +would speed up quantitative discrimination, improve the +accuracy of comparison tasks, and reduce the perceived +data complexity. +Another structure acting as guidance is the size itself. +It was used by participants seeking to resolve the NU- +MEROSTIY tasks to look for the longest outside cylinders. +We have showed several examples like Figure 11, our +collaborator suggested that the cylinder-bases of the same +size with the redundant encoding (Figure 11b) also helped +locate and group glyphs belonging to the same magnitude. +This observation agrees with the most recent literature that +guidance-by-size in 3D must take advantage of knowledge +of the layout of the scene [45]. +Though feature guidance can be preattentive and fea- +tures are detected within a fraction of a second, scene +guidance is probably just about as fast (though precise +experiments have not been done and our Experiment II only +merely shows this effect). Scene ‘gist’ can be extracted from +complex images after very brief exposures [47] [48]. This +doesn’t mean that a viewer instantly knows, say, where the +answer is located. However, with a fraction of a second’s +exposure, a viewer will know enough about the spatial lay- +out of the scene to guide his or her attention towards vector +groups in the regions of interest. For example, categorical +color becomes scene features since these colorful glyphs +were perceived as a whole +A future direction, and also an approach to understand- +ing the efficiency and the effectiveness of scene guidance, +is to conduct an eye-tracking study to give viewers a flash- +view of our spatial structures and then let the viewer see the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +14 +(a) Lengthylengthx feature-pair +(b) Lengthycolor/lengthx feature-pair +(c) Lengthytexture feature-pair +(d) Lengthycolor feature-pair +Fig. 11: Contours of simulation data. Size from this viewpoint can guide visual grouping and size in 3D must take advantage +of knowledge of the layout of the scene [45]. +display only in a narrow range around the point of fixation: +does this brief preview guide attention and the gaze effectively? +Work in vision and visualization [49], [50], [51], [52] domain +has measured and correlated performance on the glance or +global structure formation. Vision science discovered long +ago that seeing global scene structures in medical imaging +decision making guides experts’ attention (experts always +know where to look) [53] [54]. +5.3 +Redundancy and Ensemble Graphical Perception +Our results showed that adding categorical colors, in which +the correspondence parts could be quickly discriminated, is +scalable to a large number of items. Our result agrees with +that of Northelfer and Gleicher [55]. They observed that +redundant encoding using color and shape could strengthen +grouping when searching for targets from multiple objects. +Their explanation was a race model [55]: for separable +dimensions, the performance of a glyph with the redundant +encoding might be dominated by the feature with greater + +0 +2. +30 +1 +2 +30 +2 +3JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +15 +efficiency. We did not find efficiency improvement - this +suggested that the grouping is generally fast. So it might +not be the redundancy itself that contributed to scene un- +derstanding. +Another possible theory is perhaps ensemble perception, +i.e., “the visual system’s ability to extract summary sta- +tistical information from groups of similar objects - often +in a brief glance” [40]. Also ensemble features are best +represented using the categorical features. To model parallel +processing, the target contrast signal theory by Buetti et +al. [24] may suit our scenario better. It describes more +specific time estimate it takes to evaluate items in parallel. +In visualization, we just began to understand the ensemble +averages (e.g., Chen [11] and Alberts et al. [56]) but have +limited understanding of ensemble visual encoding choices +to guide attention to optimize behaviors. We leave this to +future work. +5.4 +Use Our Results in Visualization Tools and Limita- +tions of Our Work +Visualization is used when the goal is to augment human +capabilities in situations where the problems might not be +sufficiently defined for algorithms to communicate certain +information. One of our showcase areas is quantum physics. +We believe that the design principle of prompting the ad- +dition of categorical features in bivariate glyphs would be +broadly applicable to glyph design. Also, application do- +mains carrying similar data attributes could reuse of work. +Our current study concerns bivariate data visualization in +which the bivariate variables are component parts of scalar +variables. +Our design could have been improved by following +advanced tensor glyph design methods. Both generic [57] +and +domain-specific +requirements +for +glyph +designs +[37] [58] [59] have led to the summary of glyph properties +(e.g., invariant, uniqueness, continuity) to guide design and +to render 2D and 3D tensors. A logic step is to truly un- +derstand the quantum physics principles to combine data +attributes and human perception to improve our domain- +specific solutions. +One limitation of this work is that we measure only +a subset of tasks crucial to showing structures and omit- +ted all tasks relevant to orientation. However, one may +argue that the vectors naturally encode orientation. When +orientation is considered, we could address the multiple- +channel mappings in two ways. The first solution is to use +the lengthytexture to encode the quantitative glyphs and +color to encode the orientation clusters. The second solution +is to treat magnitude and orientation as two data facets +and use multiple views to display them separately, with +one view showing magnitude and the other for orientation +(using Munzner’s multiform design recommendations [60]). +The second limitation here was that our experiments were +limited to a relatively small subset of visual dimensions: +color, texture, and size. A future direction would be to try +shapes and glyphs to produce novel and useful design. +6 +CONCLUSION +Our findings in general suggest that, as we hypothe- +sized, distinguishable separable dimensions with preatten- +tive categorical features perform better. The separable pair +lengthycolor was the most efficient and effective for both +local and global tasks. The categorical features enable effec- +tive complex scene inspections. Our empirical study results +provide the following recommendations for designing 3D +bivariate glyphs. . +• Highly separable pairs can be used for quantitative +comparisons as long as these glyphs could guide at- +tention (i.e., category forming). We recommend using +lengthycolor. +• Texture-based glyphs (lengthytexture) that introduces +luminance variation will only be recommended when +task-relevant structures can be isolated. +• Integral and separable bivariate feature-pairs have the +similar accuracy when the tasks are local. +When the +search tasks are more complex, introducing categorical +features in the separable feature-pairs will lead to per- +ceptually accurate glyphs. +• 3D glyph scene would shorten task completion time +by combing two glyph design factors: separability and +visual guidance from categorical features. +• The +redundant +encoding +(lengthycolor/lengthx) +greatly improved on task completion time of integral +dimensions (lengthylengthx) by adding separable and +preattentive color features. +ACKNOWLEDGMENTS +The work is supported in part by NSF IIS-1302755, NSF +CNS-1531491, and NIST-70NANB13H181. The user study +was funded by NSF grants with the OSU IRB approval +number 2018B0080. Non-User Study design work was sup- +ported by grant from NIST-70NANB13H181. The authors +would like to thank Katrina Avery for her excellent editorial +support and all participants for their time and contributions. +Any opinions, findings, and conclusions or recommen- +dations expressed in this material are those of the author(s) +and do not necessarily reflect the views of the National +Science Foundation. Certain commercial products are iden- +tified in this paper in order to specify the experimental +procedure adequately. 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[Online]. +Available: https://doi.org/10.1109/tvcg.2006.134 +[60] T. Munzner, Visualization Analysis and Design. +A K Peters +Visualization +Series. +CRC +Press, +2014. +[Online]. +Available: +https://doi.org/10.1201/b17511 +Henan Zhao was a PhD student in Department +of Computer Science and Electrical Engineer- +ing at University of Maryland, Baltimore County. +She received B.E. degree in Computer Science +and Information Security from Nankai University, +China. Her research interests include design and +evaluation of perceptually accurate visualization +techniques. This work was conducted while she +was visiting The Ohio State University. +Garnett Bryant received his PhD at Indiana Uni- +versity in theoretical condensed matter physics. +After research positions at Washington State +University, the National Bureau of Standards, +McDonnell Research Labs and the Army Re- +search Laboratory, he has worked at the Na- +tional Institute of Standards and Technology +(NIST) since 1994. He is directing the Quan- +tum Processes and Metrology Group at NIST +with experimental and theoretical programs on +nanoscale, condensed matter systems for quan- +tum information science and metrology. He is a Fellow of the Joint Quan- +tum Institute of NIST/University of Maryland, a Fellow of the American +Physical Society and a member of the IEEE. His theoretical research +program focuses on nanosystems, nanooptics and quantum science. +Wesley Griffin received his PhD degree in Com- +puter Science from the University of Maryland, +Baltimore County. He is a developer at Stellar +Science. His research interests include real-time +graphics and graphics hardware. He is a mem- +ber of ACM SIGGRAPH, the IEEE and the IEEE +Computer Society. +Judith E. Terrill is a Computer Scientist and the +Leader of the High Performance Computing and +Visualization Group at the National Institute of +Standards and Technology. She is a member of +the IEEE Computer Society, the Association for +Computing Machinery, and the Association for +the Advancement of Artificial Intelligence. +Jian Chen is an Associate Professor in Com- +puter Science and Engineering at The Ohio +State University. She received her PhD degree +in Computer Science from Virginia Tech, and +her MS degree in Mechanical Engineering | Pre- +cision Instrument from Tianjin University | Ts- +inghua University, China. She was a postdoc- +toral fellow at Brown University and a visiting +researcher at Harvard University. Her current +research interests include visual design, 3D in- +teraction, and human-AI teaming. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +18 +Evaluating Glyph Design for Showing Large-Magnitude-Range +Quantum Spins +Additional Material +Empirical study training documents, source code, study data, and results are online at https : //osf.io/4xcf5/?viewonly = +94123139df9c4ac984a1e0df811cd580. +A. BACKGROUND COLOR +Fig. 12 shows an example represented by lengthytexture with gray, white, and black background colors. Gray background +color was selected for the experiments. We could observe that both white and black cylinders with lengthytexture encoding +could be displayed more clearly in the gray background (Fig. 12, left). +B. VISUAL MAPPING FOR COLOR AND TEXTURE IN THE Lengthycolor AND Lengthytexture PAIRS +Fig. 6 shows the visual mapping using color and texture in Experiment II. The horizontal axis represents the exponent +range ∈ [3, 7]. We selected those categorical colors from ColorBrewer [36]. For texture, the percentage of black is mapped +to the exponent-range. Examples with three different exponent-ranges of 3, 5, and 7 are shown in Fig. 13, in which color +and texture are used for the visual mapping of study data. +C. VISUAL FEATURES AND EXPONENT-RANGE +Fig. 13 shows examples for visual features and three exponent-ranges of 3, 5, and 7. The figures with the same exponent- +range were generated using the same data and different visual features. The dataset used in this figure is for illustration +purpose only and does not necessarily reflect all image features used in the vector magnitude experiments. +Fig. 12: Examples using different background colors: gray, white, and black. Figures on the top row are magnified views +of region 1, marked by orange-box on the left image, and the bottom row shows region 2. With white background, the +white cylinders would be washed out (top right image). With black background, the black cylinders would be washed out +(bottom right image). In this study, the neutral stimulus-free gray background was chosen. + +Gray background +White background +Gray background +Black backgroundJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +19 +(a) Lengthylengthy (splitVectors) +(b) Lengthycolor +(c) Lengthytexture +Fig. 13: Experiment II: examples of selected exponent ranges of 3, 5, and 7 (from the second left to right). We could see that +the pattern of magnitude distribution is more revealing by categorical colors than by texture glyphs. Coloring may show +more steps with large exponent ranges and also give us a better understanding of data distribution. For example, we could +quickly focus on the orange region. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +20 +D. SPATIAL PROXIMITY +Figures 14 and +15 show spatial distributions of the identified targets (participants’ answers) to the correct targets in +the search and max tasks in Experiment II. Here locations of the correct targets are translated to the origin (0, 0, 0). +Participants’ answers are depicted in green and each dot represents a trial. Dots may overlap. Dots in orange illustrate +some of the nearest spins whose exponent values differ from the target (located at the origin). Comparing the distribution +of participants’ answers and the orange dot locations illustrates one of the key quantum physics data attributes: quantum +physics data are discrete; and spatial proximity is not correlated with the spin magnitude proximity. For complex data like +this, using the structural features (e.g., from color) in search will help them be more efficient and reduce errors. +(a) lengthycolor +(b) lengthytexture +(c) lengthylengthy +Fig. 14: Experiment II: Search task. The spatial proximity of the locations of the identified targets, to the ground truth, +for all trials in the study. Here the ground truth locations are translated to the origin (0, 0, 0). This task was time- +constrained. among the 810 trials (or 270 trials for each bivariate glyph type), participants completed 262 lengthycolor, 261 +lengthytexture, and 251 lengthylengthy trials. +(a) lengthycolor +(b) lengthytexture +(c) lengthylengthy +Fig. 15: Experiment II: Max task. The spatial proximity of the locations of the identified targets, to the ground truth (centered +at the origin (0, 0, 0), for all trials in this task. The yellow dots show the closest points from other-than-target-exponent +regions. Here the ground truth locations are translated to the origin (0, 0, 0). Among the 810 trials, participants gave an +answer to 270 trials for each bivariate glyph type. Among each of these 270, participants completed 269 lengthycolor, 269 +lengthytexture, and 259 lengthylengthy trials in total. + +3 +3 +2 +2 +1 +1 +No +0 +-1 +-1 +-2 +-2 +3.3 +-35 +2 +1 +0 +43-2-1012 +34 +X +3 +2 +Z +> +0 +-1 +-2 +3 +-4-3 +4 +X +X3 +m +2 +2 +1 +1 +No +-1 +-1 +-2 +-2 +33 +35 +2 +-1 +0 +-4-3 -2-1 +0 +X +3 +2 +1 +0 +-1 +-2 +2 +0 +-4-3-2-1 +0 +12345 +X +x +33 +3 +2 +N +1 +No +No +-1 +-2 +-2 +-4.3 +5 +3 +X +3 +2 +1 +Y +0 +-1 +2 +3 +5-43-2-1 +1 +X +X3 +3 +2 +2 +1 +1 +No +No +-1 +-1 +-2 +-2 +-3 +-5-4-3-2-101 +2345 +-3 +.3 +-2 +0 +2 +3 +X +Y +3 +2 +1 +Z +Y +0 +-1 +2 +2 +3 +5-4-3-2-1 +0 +X +2 +X3 +3 +2 +2 +1 +1 +N +0 +No +-1 +-1 +-2 +-2 +-3 +-5 -4 -3 -2 -1 0 +-3 +.3 +-2 +.1 +0 +2 +3 +X +3 +2 +3210 +1 +N +0 +-1 +-2 +2 +-3 +0 +5 -4 -3-2-1 +0 +X +X3 +3 +2 +2 +1 +1 +No +No +-1 +-1 +-2 +-2 +-3 +-5-4-3-2-1012345 +-3 +-3 +-2 +-1 +0 +2 +3 +x +Y +m +2 +321 +1 +N +0 +-1 +-2 +0 +1 +5-4-3-2-10 +12345 +X +X \ No newline at end of file diff --git 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Bryant, Wesley Griffin, Judith E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Terrill, Jian Chen Abstract—We present experimental results to explore a form of bivariate glyphs for representing large-magnitude-range vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The glyphs meet two conditions: (1) two visual dimensions are separable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' and (2) one of the two visual dimensions uses a categorical representation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', a categorical colormap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We evaluate how much these two conditions determine the bivariate glyphs’ effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The first experiment asks participants to perform three local tasks requiring reading no more than two glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The second experiment scales up the search space in global tasks when participants must look at the entire scene of hundreds of vector glyphs to get an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our results support that the first condition is necessary for local tasks when a few items are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' But it is not enough for understanding a large amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The second condition is necessary for perceiving global structures of examining very complex datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants’ comments reveal that the categorical features in the bivariate glyphs trigger emergent optimal viewers’ behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This work contributes to perceptually accurate glyph representations for revealing patterns from large scientific results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We release source code, quantum physics data, training documents, participants’ answers, and statistical analyses for reproducible science at https : //osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='io/4xcf5/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='viewonly = 94123139df9c4ac984a1e0df811cd580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Index Terms—Separable and integral dimension pairs, bivariate glyph, 3D glyph, quantitative visualization, large-magnitude-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 1 INTRODUCTION B IVARATE glyph visualization is a common form of visual design in which a dataset is depicted by two visual variables, often chosen from a set of perceptually independent graphical dimensions of shape, color, texture, size, orientation, curvature, and so on [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A bivariate glyph design [3] has been broadly applied to reveal atom spin behaviors for quantum physicists at National Institute of Standards and Technology (NIST) to examine experi- mental results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' thanks to their team’s Nobel-prize-winning simulations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Quantum physicists world-wide can now manipulate many individual quantum systems to study complex atom and sub-atom interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Because atoms can be in multiple states simultaneously and because these spin magnitudes are large in range and often vary greatly in local regions, computational solutions still do not exist to characterize the spin behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Today’s quantum physicists rely on visualization to interpret simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' On the visualization side, the initial design and eval- uation of large-magnitude-range spin vector visualizations use scientific notation to depict digit and exponent as two concentric cylinders [3]: inside and outside tube-lengths (lengthylengthy or LyLy or splitVectors) are mapped to digit and power of spin magnitude accordingly (Figure 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A three-dimensional (3D) bivariate glyph scene of this splitVec- tors design (Figure 2e) achieves up to ten times greater accuracy than the traditional direct approach (Linear, Fig- ure 2f) for reading a vector magnitude of a single spin or deriving ratios between two spin magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' However, this Henan Zhao is with University of Maryland, Baltimore County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' E-mail: henan1@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Garnett W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Bryant and Judith E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Terrill are with the National Institute of Standards and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' E-mail: {garnett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='bryant, ju- dith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='terrill}@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wesley Griffin is with Stellar Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' E-mail: griffin5@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Jian Chen is with The Ohio State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' E-mail: chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='8028@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' design also increases task completion time for an apparently simple comparison task between two magnitudes in three dimensions (3D): the traditional direct approach of Linear (Figure 2f) is significantly faster than splitVectors (Figure 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' One may frame this large-magnitude-range issue as a visual design problem: how can we depict a scalar value using bivariate visual features to help quantum physicists examine com- plex spatial data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Intuitively, since all tasks in previous study involve a single or at most two spin locations, human visual system would integrate the two component parts (digit and exponent terms) of a quantum spin into one gestalt before comparing the results [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Since relating the digit and expo- nent to the two size features demands a focused attention mode of visual processing, a viewer would take longer to process two component parts in splitVectors compared to a single linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We term this explanation the object- level hypothesis where a viewer responds to combine two component parts of a value represented in a glyph to its original scalar value (here the magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' However, the object-level processing may be neither effi- cient nor necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, Borgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [6] state that “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' effective glyph design should encompass a non-conflicting set of separable retinal variables”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Now, for our examples, if we increase the bivariate feature separability by replacing the exponent-to-length mapping in Figures 1e and 2e to the exponent-to-color mapping in Figures 1c and 2c for compar- ison tasks, it would be counterproductive for our attention first to visit each glyph to compute the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Instead, the global categorical color (hue) can guide our attention to first compare the exponent, prior to compare vector lengths (digit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In these cases, no object-level attentive processing of bivariate features is needed as long as the two color hues can be easily recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Further considering the quantum physicists’ task rele- vant to multiple objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', find maximum among hun- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='00002v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='HC] 25 Dec 2022 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 1: Illustration of five bivariate configurations of vector magnitudes ∈ (0, 9, 999].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Three examples show vector magnitudes 440 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 × 102), 9, 999 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='9 × 103), and 1 (1 × 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Take 440 as an example, lengthylengthx (a) maps 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 (digit) and 2 (exponent) to lengths along the y and x axes accordingly ( lengthy lengthx);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (b)-(e) have the same digit- to-lengthy representation as (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The exponent representations are manipulated to be (1) more integral or separable from lengthy and (2) more or less categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (b) lengthycolor/lengthx uses color to double-code exponent compared to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The exponents in (c), (d), and (e) use color, texture, or outer cylinder length accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our experimental results support that more separable dimensions lead to more perceptually accurate glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The higher the separability, the higher the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Also, using a more categorical feature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', color in (c)) of one of the variables benefited efficiency and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' dreds of vectors) (Figure 2), viewers are likely to check the color legend and then use color to first divide the scene into subregions, prior to use length for detailed comparisons within the yellow region (Figures 2b and 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The colorful scene context benefits the reduction of search to a much smaller scale via global statistics of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Coinciden- tally, this first impression of the data to drive structural and statistical information is also called scene-level processing [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wolfe called features guiding this top-down task-driven attention behaviors as scene features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Scene features are also preattentive and can guide attention in visual search toward a target [8], perhaps due to fast ensemble processing [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Taken together, an effective design of bivariate glyphs is likely to be influenced by two conditions: separable dimensions, with one of them being a pre-attentive scene feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These two factors are not necessarily independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, For the first factor, we can follow Borgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [6] and Ware [10] for “a non-conflicting set of separable retinal variables”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' To meet the both conditions to choose the scene feature, we can give preferences of the separable pair when one of the variables is categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This is because categorical features are likely to be better at facilitating the perception of a group of objects in the scene [7], [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We in this work compared several separable-integral pairs, length-color (Figures 1b, 2b, 1c, 2c), length-texture (Figures 1d, 2d), and length-length (Figures 1a, 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Among the three features of color, texture, and size, color is categorical and thus “most recognizable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Color ensembles are preattentive and permit visual selection at a glance [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We purposefully select texture patterns by varying the amount of dark on white, thus introducing luminance variations when many vectors are examined together (Figure 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Compared to the continuous random noise in Urness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [14], ours is for discrete quantities and thus uses regular scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When coupled with separable features, we hypothesize that highly distinguishable separable dimension pairs, with one being categorical might encourage preattentive global processing to reduce task completion time and be more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We tested this hypothesis in two experiments with six tasks using four pairs to compare against the lengthylengthy (separable) in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3]: lengthylengthx (integral), lengthycolor (separable), lengthytexture (separable), and lengthycolor/lengthx (redundant and separable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Since we predicted that separable dimensions with more preattentive features would reduce the task completion time, lengthycolor and lengthycolor/lengthx might achieve more efficiency without hampering accuracy than other bivariate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This work makes the following contributions: Empirically validates that bivariate-glyphs encoded by Y V VV VV Digit Length Length Length Length Length 5个 X 3 0 0123 Exponent 1 Length Color and length Color Texture Length five-unitlongandthefeathers alwaysface theuser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Example: 440 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 Maximum: 9999 (Note: these glyphs are shownata different scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') Minimum: 1 (Note: these glyphs are shown at a differentscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') (b) Lengthy (a) Lengthy lengthx (c) Lengthy color (d) Length, texture (e) Length, lengthy (splitVectors ) color/lengthxJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 3 (a) Lengthylengthx (LyLx) (integral) (b) Lengthycolor/lengthx (LCL) (redun- dant encoding) (c) Lengthycolor (LC) (separable) (d) Lengthytexture (LT) (separable) (e) Lengthylengthy (splitVectors, LyLy) [3] (f) Linear Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 2: Real-world large-magnitude-range quantum physics simulation results shown using (a)-(e) five bivariate feature- pairs and (f) a traditional linear representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' LC, LCL, and LT can reveal scene spatial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We anticipate that two conditions determine the glyph efficiency: (1) the bivariate glyph uses two separable dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' and (2) one of the two dimensions uses a categorical representation thus can reveal global structures in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The first condition is necessary for local tasks when a few items are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The second condition is needed for inspecting the entire scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' highly separable dimensions would improve compari- son task completion time (Exp I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Is the first to evaluate categorical features in bivirate- glyphs to leverage the benefits of the global scene features (Exp II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Offers a rank order of separable variables for 3D glyph design and shows that the separable pairs lengthycolor and lengthytexture are among the most effective and efficient feature pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Reveals a novel visual design method for scalable search in big-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 2 THEORETICAL FOUNDATIONS IN PERCEPTION AND VISION SCIENCES At least four perceptual and vision science theories have inspired our work: integral and separable dimensions [15], preattentive scene features [7], [8], [16], [17], feature ranking, and monotonicity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Integral and Separable Dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Garner and Felfoldy’s seminal work on integral and separable dimen- sions [15] has inspired many visualization design guide- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Ware [10] suggests a continuum from more inte- gral to more separable pairs: (red-green)-(yellow-blue), sizex- sizey, color-shape/size/orientation, motion-shape/size/orientation, motion-color, and group position-color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' His subsequent award- winning bivariate study [2] using hue-size, hue-luminance, and hue-texton (texture) supports the idea that more sep- arable dimensions of hue-texton lead to higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our work follows the same ideas of quantifying integral and separable dimensions but differs from Ware’s texton selection in two important aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' First, the Ware study focuses on finding relationships between two independent data variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In contrast, ours demands participants to examine a complex scene for item discrimination when the two variables are component parts of a vector magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Second, our texture uses the amount of black and white to show luminance variations, in contrast to the discrete shape variation in textons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We anticipate that ours will be more suitable to continuous quantitative values so it is easier to compare large and small to divide the regions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' No existing work we know of has studied whether or not one of the separable features being categorical can facilitate global comparisons and can be scaled to large and more complex 3D vector magnitude analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Scene-Guidance and Feature Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In order to rec- ognize items, viewers do not “see” features and instead “bind” these features to objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This binding studies how our visual systems separate object features such as shape, color, motion trajectories, sizes, and distances into the whole 0 2 30 2 30 2 3JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 4 object [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' What we “see” also depends on our goals and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wolfe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' propose the theory of “guided search” [8], a first attempt to incorporate users’ goals into viewing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, if the viewer’s goal is to search largest values, s/he can just check the yellow ones in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wolfe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [8] further suggest that color, texture, size, and spatial frequency are among the most effective features in attracting the user’s attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When we combine features together, Duncan and Humphreys articulate some of the most basic principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In general, guidance to a target will be stronger when the feature differences between the target (T) and distractor (D) are larger (TD differences), and when the feature differ- ences amongst distractors are smaller (DD similarity) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, Ts are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 (digit) and 2 (exponent) for 230 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 × 102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Ds include all numbers but 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Using the TD differences between features may explain why splitVec- tors was time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, to compare 230 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 × 102) to 2,300 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 × 103), viewers have to differentiate the two lengths of 2 (T) and 3 (T) from other distractors (Ds other than 2 or 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The heterogeneity of Ds or small DD distances from 3D lengths may make the use of splitVectors challenging, thus introducing temporal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Preattentive and Attentive Feature Ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Human visual processing can be faster when it is preattentive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wolfe called a feature preattentive when it guides attention in search and cannot be decomposed into simpler features [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The idea of preattentive pop-out has historically highlighted that a single object has been considered compelling because it captures the user’s attention against a background of other objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', in showing spatial highlights [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Visual features such as orientation and color (hue, saturation, light- ness) can generate pop-out effects [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This type of pop- out was also used visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, Ropinski, Oeltze, and Preim [22] summarized two groups of glyph design: “parameter mapping” from shape and appearance (color, transparency, and texture) and “placement” driven by features or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our study concerns appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Recent vision science development also suggests that the preattentive feature is not limited to single items but expanded to high-level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Global statistical and structural features can be also preattentive [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Unlike the now outdated Treisman’s 1988 preattentive processing [23], where preattentive features were considered to be perceived before it is given focused attention [23], these preattentive features are persistent during viewers’ data exploration thus can continue to provide guidance [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Viewers can use peripheral vision to compare in parallel to confidently tell apart regions relevant or irrelevant to tasks [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Visual features also can be responsible for different at- tention speeds, and color (hue) and size (length and spatial frequency) are among those that guide attention [9], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Healey and Enns [25] in their comprehensive review further remark that these visual features are not popped-out at the same speed: hue has higher priority than shape and tex- ture [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Also, when data size increased, some preattentive features diminished [27] [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For visualizing quantitative data, MacKinlay [29] and Cleveland and McGill [30] leverage the ranking of visual features and suggest that position and size are quantitative and can be compared in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, MacKinlay’s A Presentation Tool (APT) [29] automatically recommends visualizations using effectiveness and expressive criteria and outputs a ranked set of encoding to enumerate candidate visualizations based on data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Casner [31] expands MacKinlay’s APT by incorporating user tasks to guide visualization generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' McColeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [32] revise the ranking of visual features based on the number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All these studies almost exclusively consider only single item mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Demiralp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [33] evaluate a crowdsourc- ing method to study subjective perceptual distances of 2D bivariate pairs of shape-color, shape-size, and size-color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When adopted in 3D glyph design, the authors further suggest that the most important data attributes should be displayed with the most salient visual features, to avoid sit- uations in which secondary data values mask the informa- tion the viewer wants to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Ours also emphasizes the use of global scene features to optimize viewing experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Quantitative data encoding must nor- mally be monotonic, and various researchers have recom- mended a coloring sequence that increases monotonically in luminance [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In addition, the visual system mostly uses luminance variation to determine shape information [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' There has been much debate about the proper design of a color sequence for displaying quantitative data, mostly in 2D [36] and in 3D shape volume variations [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our primary requirement is to use categorical colormaps that users be able to read large or small exponents at a glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We used four color steps in the first study and up to seven steps in the second study from ColorBrewer [36] for showing areas of large and small exponents that are mapped to a hue- varying sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We claim not that these color sequences are optimal, only that they are reasonable solutions to the design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3 EXPERIMENT I: EFFECT OF SEPARABLE PAIRS ON LOCAL DISCRIMINATION AND COMPARISON The goal in this first experiment is to quantify the benefits of separable pairs with preattentive features for visual process- ing of a few items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This section discusses the experiment, the design knowledge we can gain from it, and the factors that influence our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Bivariate Feature-Pairs We chose five bivariate feature-pairs to examine the com- parison task efficiency of separable-integral pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthylengthx (integral) (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengths encoded digits and exponents shown as the height and radius of cylinder glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthycolor/lengthx (redundant and separable) (Fig- ure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This pair compared to lengthylengthx added a redundant color (luminance and hue variations) dimension to the exponent and the four sequential colors were chosen from Colorbrewer [36] (Appendix A shows the sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') Lengthycolor (separable) (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This pair mapped exponents to color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Pilot testing showed that the least incor- rect exponent levels were selected among these five feature- pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 5 Lengthytexture (separable) (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Texture repre- sented exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The percentage of black color (Bertin [38]) was used to represent the exponential terms 0 (0%), 1 (30%), 2 (60%) and 3 (90%), wrapped around the cylinders in five segments to make them visible from any viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthylengthy (splitVectors [3], separable) (Figure 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This glyph used splitVectors [3] as the baseline and mapped both digit and exponent to lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The glyphs were semi- transparent so that the inner cylinders showing the digit terms were legible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Feather-like fishbone legends were added at each location when the visual variable length was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The tick-mark band was depicted as subtle light-gray lines around each cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Distances between neighboring lines show a unit length legible at certain distance (Figure 1, rows 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Hypotheses Given the analysis below and recommendations in the liter- ature, we arrived at the following working hypotheses: Exp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (Overall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The lengthycolor feature-pair can lead to the most accurate answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (Integral-separable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Among the three separable dimensions, lengthycolor may lead to the greatest speed and accuracy and lengthytexture will be more effective than lengthylengthy (splitVectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (Redundancy on time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The redundant pair lengthycolor/lengthx will reduce task completion time compared to splitVectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Several reasons led to H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' They are related to the two conditions of glyph design we evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Color and length were separable dimensions, so comparing length to color is simple (condition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' And color was preattentive and could be detected quickly (condition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Compared to the redundant lengthycolor/lengthx, lengthycolor reduced crowding since the feature-pairs were generally smaller than those in lengthycolor/lengthx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Also, distinguishing two lengths in splitVectors might be less efficient than lengthytexture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' H3 could be supported because redun- dancy increased information processing capacity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Re- dundancy contributes to efficiency by increasing the feature distances between exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We did not expect accuracy gain from redundancy because splitVectors achieved the same level of accuracy as reading texts in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' It may not be useful to decode quantitative data in this experiment at least for showing a few items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 Tasks Participants performed the following three task types as in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3] so that results were comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' They had unlimited time to perform these three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task 1 (MAG): magnitude reading (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' What is the magnitude at point A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' One vector was marked by a red triangle labeled “A”, and participants should report the magnitude of that vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This task required precise numerical input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task 2 (RATIO): ratio estimation (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' What is the ratio of magnitudes of points A and B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Two vectors are marked with two red triangles labeled “A” and “B”, and participants should estimate the ratio of magnitudes of these two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The ratio judgment is the most challenging (a) MAG task: What is the magnitude of the vector at point A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (answer: 636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='30) (b) RATIO task: What is the ratio of the magnitude between the vectors at points A and B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (answer: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='60) (c) COMP task: Which magnitude is larger, point A or point B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (answer: A on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3: Experiment I: Local discrimination and comparison tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These two red equilateral triangles are rendered on the screen coordinate and are thus always visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' quantitative task [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants could either compare the glyph shapes or decipher each vector magnitude and compute the ratio mentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task 3 (COMP): comparison (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Which magnitude is larger, point A or B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Two vectors are marked with red triangles and labeled “A” and “B”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants select their answer by directly clicking the “A” or “B” answer buttons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This task was a simple comparison between two values and offered a binary choice of large or small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 Data Selection Because we were also interested in comparing our results to those in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3], we replicated their data selection method by randomly sampling some quantum physics sim- ulation results and produce samples within 3D boxes of size done3/80 pause done Task 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' What is the magnitude at point A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='done sk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='What is the ratio betweenJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 6 TABLE 1: Experiment I design: 20 participants are as- signed to one of the five blocks and use all five bivari- ate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here, LyLy: lengthylengthy (splitVectors), LyLx: lengthylengthx, LC: lengthycolor, LT: lengthytexture, and LCL: lengthycolor/lengthx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Block Participant Feature-pair 1 P1, P6, P11, P16 splitVectors, LyLx, LC, LT , LCL 2 P2, P7, P12, P17 LyLx, LC, LT , LCL, splitVectors 3 P3, P8, P13, P18 LC, LT , LCL, splitVectors, LyLx 4 P4, P9, P14, P19 LT , LCL, splitVectors, LyLx, LC 5 P5, P10, P15, P20 LCL, splitVectors, LyLx, LC, LT 5 × 3 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' There were 445 to 455 sampling locations in each selected data region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We selected the data satisfying the same following con- ditions: (1) the answers must be at locations where some context information was available, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', not too close to the boundary of the testing data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (2) no data sample was repeated to the same participant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (3) since data must include a broad measurement, we selected the task-relevant data from each exponential term of 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 Empirical Study Design Design and Order of Trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We used a within-subject de- sign with one independent variable of bivariate quantitative feature-pair (five types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Dependent variables were error and task completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We also collected participants’ confidence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Table 1 showed that participants were assigned into five blocks in a Latin-square order, and within one block the order of the five feature-pair types is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants performed tasks with randomly selected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Each participant performed 60 trials (3 tasks × 4 random data × 5 feature-pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These four random data were from four exponent ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We diversified the participant pool as much as possible, since all tasks could be carried out by those with only some science background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Twenty participants (15 male and 5 female, mean age = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3, and standard deviation = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='02) participated in the study, with ten in com- puter science, three in engineering, two in chemistry, one in physics, one in linguistics, one in business administration, one double-major in computer science and math, and one double-major in biology and psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The five females were placed in each of the five blocks (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' On average, participants spent about 40 minutes on the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants were greeted and completed an Institutional Review Board (IRB) consent form (which described the procedure, risks and benefits of the study) and the demographic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All participants had nor- mal or corrected-to-normal vision and passed the Ishihara color-blindness test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We showed feature-pair examples and trained the participants with one trial for every feature-pair per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' They were told to be as accurate and as quickly as possible, and that accuracy was more important than time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' They could ask questions during the training but were told they could not do so during the formal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants practiced until they fully understood the feature-pairs and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' After the formal study, participants filled in a post- questionnaire asking how these feature pairs supported their tasks and were interviewed for their comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Pilot studies were conducted to examine the procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants sat at a 27 ′′ BenQ GTG XL 2720Z, gamma-corrected display with resolution 1920 × 1080 to ensure the colors were displayed properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The distance between the participants and the display was about 50cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The minimum visual angle of task-associated glyphs was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2◦ in the default view where all data points were visible and the scene filled the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants could rotate the data and zoom in and out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lighting placement and intensity were chosen to produce visualization with contrast and lighting properties appropriate for human assumptions and the spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The screen background color was neutral stimulus-free gray background to minimize the discriminability and appear- ance of colors [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Using black or white background colors makes the black and white texture stimuli disappear thus bias the results (See Appendix B for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Experiment I: Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Analysis Approaches We collected 400 data points for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In preparing the accuracy and task completion time for analysis, we differentiated two error metrics related to the perceptual accuracy of the bivariate pairs: Correspondence error (C-Error): A trial is considered to have an answer of C-Error if response’s exponent value does not match the correct one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Having a C-Error would mean that participants have trouble differentiating the exponent levels within a glyph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Relative error (R-Error): This R-Error follows Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3] to study how sensitive a method is to error uncertainty based on fractional uncertainty, calculated as R-Error = | correct answer - participant answer | / (correct answer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This measure was used for MAG and RATIO tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The benefit of this metric was that it took into account the value of the quantity being compared and thus provided an accurate view of the overall errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In subsequent analysis, we separated these two error measurements since Combining these two errors in the analysis would also be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The C-Errors are at least one order of magnitude larger or smaller than the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We also did not remove participants’ data with C-Errors, since the source of errors was caused by glyph design methods independent of trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A post-hoc analysis using Tukey’s Studentized Range test (HSD) was performed when we observed a significant main effect on R-Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When the dependent variable was binary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', answer correct or wrong), we used a logistic regression and reported the p value from the Wald χ2 test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When the p value was less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='05, variable levels with 95% confidence interval of odds ratios not overlapping were considered significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All error bars represent 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We also evaluated effect sizes using eta-square, labeled “small” (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='06), “medium” [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='06 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='14), and “large” (≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='14) effects following Co- hen [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 7 (a) Task 1 (MAG) (b) Task 2 (RATIO) (c) Task 3 (COMP) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4: Experiment I task completion time and relative error or accuracy by tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The horizontal axis represents the mean task completion time while the vertical axis showing the accuracy or relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Same letters represent the same post-hoc analysis group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Colors label the feature-pair types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All error bars represent 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' TABLE 2: Summary statistics by tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The significant main effects and the high effect size (ES) are in bold (none in these observations) and the medium effect size is in italic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Effect size is eta-square labeled “small” (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='06), “medium” [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='06 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='14), and “large” (≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='14) effects following Co- hen [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Post-hoc Tukey grouping results are reported for significant main effects, where > means statistically signif- icantly better and enclosing parentheses mean they belong to the same Tukey group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task Variables Significance ES MAG time F(4, 384) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='8, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='07 (LC, LT , LCL, splitVectors) > LyLx relative error F(4, 384) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='9, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='01 RATIO time F(4, 395) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='06 Three groups: A: LC, splitVectors, LT B: splitVectors, LT , LCL C: LT , LCL, LyLx relative error F(4, 395) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='8, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='01 COMP time F(4, 395) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='09 Three groups: A: LCL, LC, LT B: LC, splitVectors C: splitVectors, LyLx accuracy χ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Overview of Study Results Figure 5 show all C-Error occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Table 2 and Fig- ure 4 show the F and p values computed with SAS one- way measures of variance for task completion time and relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our results clearly demonstrated the benefits in terms of task completion time of separable dimensions for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We observed a significant main effect of feature-pair type on task completion time for all three tasks MAG, RATIO, and COMP, and the effect sizes were in the medium range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthycolor was the most efficient approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For COMP, lengthycolor, lengthytexture and lengthycolor/lengthx were most efficient for simple two- point comparisons (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 Separable Dimension Coupled with Categorical Fea- tures had the Least Correspondence Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We only observed C-Errors in MAG, but not in the RATIO and COMP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The total count was relatively small (11 instances of 400 data points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' They came from 9 partic- ipants (error mean = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='22 and 95% confidence intervals Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5: Experiment I (Task MAG): All instances of cor- respondence errors by participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The most separable lengthycolor glyph had no instances of correspondence er- ror whilst the lengthylengthx had the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The redundant color dimensions helped removed some correspondence errors (Two instances of lengthycolor/lengthx vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' five in- stances of lengthylengthx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (CI)=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='96, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Figure 5 shows all instances of these er- rors by participant and by encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' It appeared that the degree of separability of integral-separable dimen- sions influenced the errors: the most integral dimension lengthylengthx had the highest number (5 instances) of C- Errors and the most separable lengthycolor had none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 Separable Dimensions Are Better Than Integral Di- mensions for Local Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' But Categorical Feature was not a Statistically Significant Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our first two hypotheses H1 and H2 are supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In the MAG task, the integral lengthylengthx was the least ef- ficient and all other separable-pairs were in a separate group, the most efficient one (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In RATIO, lengthycolor, lengthytexture, and splitVectors were the most efficient group (Figure 4b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' in COMP, the redundant lengthycolor/lengthx, lengthycolor, and lengthytexture were in the most efficient group (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' SplitVectors was not as bad as we originally thought in perceiving correct exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' SplitVectors belonged to the same efficient post- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='12 SplitVectors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Lengthy lengthx Lengthy color Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='08 Length, texture Length, color/length, Relative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='02 A A AA B 0 0 10 20 30 40 Task Completion Time (s)A AA 1 BB B SplitVectors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='9 C C Lengthy length, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='8 Lengthy color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='7 Error Length, texture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6 Length, color/length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Relative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 0 0 10 20 30 40 50 60 70 Task Completion Time (s)1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='9 A A A B B Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='8 c SplitVectors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='7 Lengthy lengthx Lengthy color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6 Length, texture Lengthy color/length, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content="5 0 5 10 15 20 25 Task Completion Time (s)TaskMAG:NumberofCorrespondenceErrors byParticipant(X-axis)and GlyphType (Y-axis) The numbers overlaid on bars show the (ground truth, participant's answer) exponent pair." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' SplitVectors (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0) (3,2) (2,3) LengthyLengthx (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3) LengthColor LengthTexture (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4) Lengthy/Color Lengthx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2) 0 12 4 6 7 910 11 12 13 14 15 16 17 18 19 20 ParticipantIDJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 8 hoc group as lengthycolor and lengthytexture for RATIO and these three were also most efficient for MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 Separable Pairs of Lengthycolor And Lengthycolor/lengthx Achieved Comparable Efficiency To Direct Linear Glyph One aspect for motivating this experiment was to quantify the benefits of separable pairs [6], [10]: whether the sepa- rable pairs supported COMP and how the separable pairs compared in efficiency to the direct mapping (Figure 2(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Since our study had the same numbers of sample data as Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3], we then performed a one-way t-test to compare against the direct linear encoding in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our results indicated that results for COMP (judging large or small) from separable variables was no more time- consuming than direct linear glyphs, and our post-hoc anal- ysis showed that lengthycolor, lengthycolor/lengthx, and linear were in the same post-hoc group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We also observed that splitVectors dropped to the least efficient post-hoc group (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This result replicated the former study results in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [3] by showing that splitVectors impaired comparison efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6 Redundant Feature-Pairs Were Efficient We also confirmed hypothesis H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We were surprised by the large performance gain with the redundant encoding lengthycolor/lengthx of mapping color and length to the exponents in splitVectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' With the redundant encoding, the task completion time was significantly shorter than lengthylengthx for MAG and COMP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' While Ware [10] confirmed that the efficiency might not be improved by using separable dimensions, in our case, where color and size (separable) represent the same quantitative value, we suggested that the redundancy worked because participants could use either length or color in different task conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We could also consider that lengthycolor/lengthx is a re- dundant encoding of lengthycolor, and those two feature- pairs had similar efficiency and accuracy for all local tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 Summary The separable-pair condition is necessary for effective glyph design because all separable pairs were more efficient than the integral ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The pre-attentive condition enabled by categorical encoding among the separable pairs may be not since not all conditions were statistically different performance-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All tasks (MAG, RATIO, and COMP) lacked of significant main effect on relative errors (in MAG or RATIO) or accuracy (in COMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Note that none of these three tasks required initial visual search, and target answers were labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wolfe called this type of task-driven with known target guided tasks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthycolor was the most accurate in all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We also did not see the needs for the second condition for perceptually accurate glyphs in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We did not observe differences among categorical dimensions color, texture, and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We suspect that the reason for this lack of significance could well be their similarities in mentally computing load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The load was relatively small when com- paring two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We suspected that when search-space set-size increases, and when tasks are more complex in- volving all items, participants will need preattentive global scene features to guide their search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We subsequently ran the second experiment to increase the set size in tasks to the entire scene to study the benefits of categorical features to show quantitative exponent values to benefit global search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4 EXPERIMENT II: SCALABILITY OF GLOBAL SCENE FEATURES The goal in this second experiment is to quantify the benefits of separable feature-pairs when they introduce categorical features of scene guidance for global tasks in search spaces, as large as the entire dataset of several hundreds items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In other word, we measure scene feature scalability of global tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Overview We had three design considerations for us to carefully choose the categorical features in setting up this experiment, concerning the use of glyphs for showing complex simula- tion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Intriguingly, all of these considerations support our second glyph design consideration of using a categorical variable in one of the separable pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The first reason is that the initial at-a-glance global statistical summary of the scene depends on categorical information [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' One of the most important advances in vision science is to find that viewers can summarize the scene without attending to the specific items [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Visual dimensions facilitating this summary process become global scene features and these features are pre-attentive [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' While visualization is mainly about mapping data values to visual variables, the new theory concerns how features form the structural and content of the scene that can affect efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' If the quantum spins contain one object at a time, then the first condition of glyph design considering integral and separable dimensions is sufficient to explain the experience as we have shown in Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For complex tasks, in general, our visual system has a limited capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' To cope with this limit, humans first visually summarize the scene to find specific regions of interests [6], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' If categorical fea- tures stimulate population responses from multiple items, we should observe fewer errors and better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For ex- ample, we have exemplified in the Introduction section for search of “largest” values by looking up “yellow” regions, without attending to every single items of “yellow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The second concerns scalability to feature distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here feature distance is meant to represent target-distractor simi- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' It is not the absolute features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', yellow) that direct our attention towards the answer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' rather, what determines performance is the result of a comparison between target (yellow) and other data features (such as pink and orange) in the scene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', yellow is different from other colors and the yellow regions stand out) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In other words, one must also look at feature distractors [14], [41], [42], whether or not they are heterogeneous, and that the efficiency of a scene guidance will decline as a function of the degree of distractor variation [19], [24], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' While generally, subjec- tive reports from Experiment I indicate that lengthycolor and lengthytexture show the similar perceptual speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 6: Visual mapping using color and texture in Experi- ment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' From the top to bottom, colors and texture segments are mapped to exponent values from the largest to the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The three numbers next to the 7-level colormap are the RGB values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The numbers next to the texture columns are the proportion of black-on-white for the last 7-level texture configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Performance of texture may decline faster than color as the exponent range increases because our vision is not as sensitive to luminance-variation as to hues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, at the exponent-range of 7 in Figure 6, the differences between yellow and pink could be more differentiable than the two top-level textures of different amount of black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In this study, we expanded the data range from the single level in Experiment I to five ranges ∈ [3, 7] to understand feature- pair scalability to feature distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The efficiency of color in Experiment I could well arise because the range (of 4) was not large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The third concerns the density effects on color choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Figure 7 shows two densities and two colormaps (a cate- gorical colormap from Colorbrewer [36] and a segmented continuous colormap by the number of exponents generated from the extended blackbody colormap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For a feature to actually guide attention, we can see from Figure 7, the boundary detection with these colormaps is associated with data density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Unless the data density was reasonably high, detecting the boundaries using continuous colormaps (Fig- ures 7a, 7b) is harder than the ColorBrewer colormaps (Figures 7c, 7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Method 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Feature-Pairs We used lengthycolor, lengthytexture, and baseline splitVectors in Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These three visualizations were chosen because lengthycolor and lengthytexture are among the best feature-pairs from Experiment I and because color and texture are among the most separable features ac- cording to Ware [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' To introduce a “distractor” experience to measure scalability to feature distances, we vary the data range from the 4 levels in Experiment I to 3 − 7 levels in Experiment II (See mapping in Figure 13, Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Hypotheses We had the following hypotheses: Exp II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='H1 (Accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' More categorical feature in the sep- arable pairs will be more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We thus anticipate a rank order of effectiveness from high to low: lengthycolor, lengthytexture, and splitVectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='H2 (Correspondence Errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' More categorical feature of color in the separable pairs will reduce C-Errors, when participants will choose the correct exponent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='H3 (User behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' More categorical dimension in the separable feature-pairs will lead to optimal users’ behaviors: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', participants can quickly locate task-related regions for tasks that demand looking among many vectors due to global scene features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 Tasks Participants performed three tasks in which they had to compare all vectors to obtain an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task 1 (SEARCH): visual search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A vector search within 20 seconds (Figure 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Find the vector with magnitude X within 20 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The target vector was shown at the bottom-right corner of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants were asked to find this vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task 2 (MAX): find maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' An extreme value search within 20 seconds (Figure 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Within 20 seconds, lo- cate the point of maximum magnitude when the exponent is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' X in the study was a number from 0 to the maximum exponent (∈ [2, 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This was a global task requiring participants to find the extremum among many vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exp II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task 3 (NUMEROSITY): estimate the total number of unique vector exponents (Figure 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Estimate the total number of unique vector exponents in the entire vector field within 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Data are randomly chosen and modified to produce the 3 to 7 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 Task Choices Tasks are use-inspired by real-world quantum physics data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Experiment I drilled down to a single or at most two spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' But global tasks are also of quantum physicists’ interests, such as those involving understanding the dis- tributions of quantum spin magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Practically, a spin represents charge density or the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This probability varies due to electron traveling from one grid point to another and is often interpreted together with its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Quantum physicists are thus interested in searches for regions, where local regions are defined by spin magnitude and different regions would correspond to changes in exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Often the most interesting regions are also those with specific charge densities (Task 1) or largest magnitudes (Task 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The regional task is related to learning the number of interesting regions or magnitude exponent clusters (Task 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Performing tasks was limited to 20 seconds as a pilot study showed that it took participants about ∈ [15, 25] seconds or on average about 20 seconds to finish search tasks 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Also, preattentive processing when used for scene guidance involving a group of similar objects are often fast for viewers to see and increasing the number of items should not significantly impair the search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' From the practical side for the last experiment, participants who would want a perfect score could just spend time counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Constraining the time allowed us to measure the accuracy when they may have to use the scene feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (255,255,179 100% 252,205,229) 83% (253, 280, 98) 67% (190, 186, 218) 50% (141,211,199) 33% (128,177,211) 17% (251,128,114) 0% 3 4 6 3 4 5 6 7 Exponent-range Exponent-rangeJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 10 (a) Continuous colormap and high-density data (b) Continuous colormap and low-density data (c) Categorical colormap and high-density data (d) Categorical colormap and low-density data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 7: Density effects on color choices to justify the use of dense sampling and categorical colormap (c) in Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This example dataset shows two colormaps: ( segmented-continuous (a and b) and categorical (c and d) colormaps), at two different data densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (a) and (c) show data with the raw density from the simulation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (b) and (d) were produced by removing around 70% vector glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The boundaries between the data categories are more recognizable when the data are dense in (a) and (c) (comparing the 1st column and the 2nd column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' At the same density (comparing the 1st and 2nd row), the boundaries between levels are easier to recognize when spin vectors are rendered using a categorical colormap of (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We thus use the raw dense and categorical colormaps (c) in Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 Data Choices Data were first sampled using the same approach as Exper- iment I, and no data is used repeatedly in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We then modified the exponent range from 3 to 7 for the three tasks by normalizing the data to the desired new data range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Prior literature used both synthetic data and real-world data to construct the data visualization as test scenarios, en- abling tight control over the stimulus parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Most of the synthetic data in literature were to replicate real-world data characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' and others were explained in fictitious use scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The goal was primarily to prevent preconceived user knowledge about the domain-specific attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' As a result, the synthetic data strike the right bal- ance between real-world uses and the data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In our cases, replicating characteristics in quantum physics data was challenging and indeed impossible, since atom behaviors in high-dimensional space were largely unknown and thus were not easily simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our approach was therefore to randomly sample quantum physics simu- lation results to capture domain-specific attributes and then modify the data to suit evaluation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We showed our data to our physicist collaborators to ensure their va- lidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We confirmed that these modifications preserved the domain-specific schema of a scene in terms of the domain- specific structures and complexity from real simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These modifications represented less than 4% of overall data points in each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Finally, It improves the reuse of our study results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6 Empirical Study Design Dependent and Independent Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We used a within- subject design with two independent variables of feature- pair (three levels: baseline splitVectors, lengthycolor, and lengthytexture) and exponent range (five levels: 3 − 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The dependent variable was relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We did not measure time since all tasks were time-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants performed 3 (feature-pairs) × 5 (magnitude- ranges) × 3 (repetitions) = 45 trials for the first two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Three repetitions were used to give participants enough time to develop strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For NUMEROSITY tasks, the design runs 4 repetitions, resulting in 3 (feature-pairs) × 5 (exponent-ranges) × 4 (repetitions) = 60 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Each par- ticipant thus executed 45+45+60 = 150 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Completing all tasks took about 32 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Self-Reporting Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Several human-computer inter- action (HCI) approaches can help observe users’ behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Answering questions can assist us to determine not just which technique is better but also the strategies humans adopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, cognitive walkthrough (CTW) mea- sures whether or not the users’ actions match the designers’ pre-designed steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here we predicted that participants 456 7 8 9 10 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 5 9- 7 8 9 10 11-4 5 6 7 8 9 10 11-4 5 9- 7 8 9 10 11JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 11 (a) SEARCH: Find the vector with magnitude X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (X: 731, answer: the point marked by two yellow triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') (b) MAX: Which point has the maximum magnitude when the exponent is X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (X: 1, answer: the point marked by two yellow triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=') (c) NUMEROSITY (NUM): Estimate the total number of unique vector exponents of the entire vector field within 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (answer: 7) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8: Experiment II three task types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The callouts show the task-relevant feature-pair(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' would use the global scene-features as guidance to accom- plish tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We interviewed participants and asked them to verbalize their visual observations in accomplishing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='7 Participants Eighteen new participants (12 male and 6 female, mean age = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='8, and standard deviation = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='94) of diverse backgrounds participated in the study (seven in computer science, four in computer engineering, two in information systems, three in engineering, one in business school, and one in physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Procedure, interaction, and environment were the same as those in the Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 Experiment II: Results and Discussion We collected 810 data points per task for the first two tasks of SEARCH and MAX and 1080 points for the third NUMEROSITY task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Analysis Approaches For SEARCH and MAX tasks, we measured relative error (which was the percentage the reported value was away from the ground truth and the same as that of Experiment I) with SAS repeated measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The last NUMEROSITY task used error rate which was the percentage of incorrect answers of all trials for each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We also used the same outlier removal methods to remove instances of correspondence errors for SEARCH and MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Overview of Study Results Table 3 and Figure 10 show the summary statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' And all error bars again represent 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We observed a significant main effect of feature-pair type on all three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For the first two tasks, the post-hoc analysis revealed that lengthycolor and lengthytexture were in the same group, the most efficient one and that relative errors were statistically significantly lower than those of the splitVectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthycolor remained the most accurate pair for the NUMEROSITY tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Exponent-range was only a significant main effect for NUMEROSITY, with power ranges 3 and 4 were significantly better than 5, which was better than 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 More Categorical Features of Separable Dimensions Improved Accuracy We were interested to see if we could observe significant main effects of categorical features in the separable pairs in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here we did observe the significant main effect and confirmed our first hypothesis (H1) for both SEARCH and MAX: in the general trend, more separa- ble lengthycolor was more effective than lengthytexture which was better than splitVectors, and lengthycolor and lengthytexture were in the same Tukey group, when view- ers were in the correct data sub-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthycolor led to the most accurate answers, and splitVectors was better than lengthytexture for NUMEROS- ITY task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This result can be explained by participants’ be- haviors - more than half the participants suggested they simply look for the longest cylinder from splitVectors since they know the numerical values in the test were continu- ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This behavior deviated from our original purpose of testing the global estimate but did show two perspectives in favor of this work: (1) participants developed task-specific strategies during the experiment for efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (2) 3D length still supported judging large and small and it was not as effective as color perhaps due to ensemble perception from categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 Color Categories of Separable Pairs Reduced Corre- spondence Errors by a Large Margin Our second hypothesis H2 was also supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We first tested the number of correspondence errors in SEARCH and MAX in the same way as in Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These results when combined with those in Experiment I confirmed again ind 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='31×102 DoneFind the vector with max magnitudle yhan power is 1 DoneTask 3: 1/30 Number of different powers DoneJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 9: Experiment II (Tasks SEARCH and MAX): All in- stances of correspondence errors by participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Again, the lengthycolor has the least instances of correspondence error whilst the lengthytexture had the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' that the lengthycolor reduced correspondence errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For SEARCH, There were only a single instance of correspon- dence error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 36 instances of correspondence errors came from 14 participants (mean= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='57, 95% CIs=[2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='04]) (Figure 9 top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Another 59 instances for MAX came from 16 of 18 participants, mean= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='68, 95% CIs= [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='85, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='51]) (Figure 9 bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 Compensating The Cost of Search in Complex Data through Preattentive Scene Feature The visualizations in our study contained hundreds of items from realistic uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Subjective behaviors through self- report suggested that they adopted a sequential task-driven viewing strategy to first obtain gross regional distribution of task-relevant exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' After this, a visual comparison within the same exponent region were achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' With these two steps, judging large or small or perceiving quantities TABLE 3: Exp II: Summary statistics by tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The significant main effects and the high effect size are in bold and the medium effect size is in italic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Effect size is Cohen’s d for tasks SEARCH and MAX, and Cramer’s V for task NUMEROSITY (NUM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Post-hoc Tukey grouping results are reported for significant main effects, where > means statistically significantly better and enclosing parentheses mean they belong to the same Tukey group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here, LC: lengthycolor and LT: lengthytexture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Task Variables Significance ES SEARCH feature-pair F(2, 261) = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='46 (LC, LT) > splitVectors power-range F(4, 261) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='86 MAX feature-pair F(2, 261) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='47 (LC, LT) > splitVectors power-range F(4, 261) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='11 NUM feature-pair χ2 = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='25 LC > splitVectors > LT power-range χ2 = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='35 (3, 4) > 5 > (6, 7) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 10: Relative error for Tasks SEARCH and MAX was the percentage the reported value was away from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Error rate for NUMEROSITY was the percentage of wrong answers of all trials for each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The vertical axis shows the relative error or error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Same letters represent the same post-hoc analysis group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All error bars represent 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=" TaskSEARCH:NumberofCorrespondenceErrors byParticipant (X-axis)and GlyphType (Y-axis) The numbers overlaid on bars show the (ground truth, participant's answer) exponent pair." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3 1:2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6) 2 （5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3) (3,2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4) SplitVectors (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 (5,4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2) (2,0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6) （13） LengthColor (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2) (7,2) 3 LengthTexture （4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3) 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4) (4,5) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4) (1,2) (3,1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content="4) 1 2 3 5 6 7 8 9101112 131415161718 Participant IDTask MAX: Number of Correspondence Errors by Participant (X-axis) and Glyph Type (Y-axis) The numbers overlaid on bars show the (ground truth, participant's answer) exponent pair." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 4 SplitVectors (3,4) 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3) 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3) (4,0) 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='6 (5,6) (3,4) (5,4) 3 (0,4) (3:2) LengthColor 2 （4,2) 1 (0,3) (3,5) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0) 8 LengthTexture 7 6 5 4 (0,2) (1,4) 3 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0) 2 (3,4) (2,3) (4,2) (1,0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 (5,4) 1 (3,2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 (6,5) (2,3) (4,3） (3,2) (3,4) (4,3) 0,2) 1 2 3 5 6 8910 11 12 13 14 15 16 17 18 Participant ID0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0 SplitVectors SplitVectors SplitVectors Length,Color LengthyColor Length,Color Length,Texture Length,Texture LengthyTexture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='75 lative Error Relative Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='10 Rate B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 B rror e E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='05 B R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='25 A A A A A 0 0 0 SEARCH MAX NUMEROSITY0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='75 Relative Error Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='10 Rate ative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='5 Error e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='05 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='25 A A 0 0 0 3 7 Exponent-range Exponent-range Exponent-range SEARCH MAX NUMEROSITYJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 13 accurately from separable variables would not use object- level information process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Many participants commented on how the number of powers in the data affected their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For lengthytexture, 10 participants remarked that it was dif- ficult to differentiate adjacent powers when the total power level is around 4-5 for lengthytexture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The white and black textures were very easy to perceive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' All but two participants agreed that lengthycolor could perhaps support up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [42] studied ordering effects and it would be challenging to compare ours to their results because their visual features were not shown as a scene but an isolated feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' More than half of the participants felt that effec- tiveness of lengthylengthy was not affected by changing the number of powers, since they looked for the longest outer cylinder to help find the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These results may suggest that subregion selection with lengthytexture can perhaps be better designed with interfaces when the users can interactively select a texture level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5 GENERAL DISCUSSION We discuss the results from both experiments and suggest future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 Separable Dimensions with Preattentive Guidance for Large-Magnitude-Range Quantum Physics Spins Our first principle in glyph design is to follow the conven- tion to use separable variable pairs [6], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The results of Experiment I showed that separable dimensions could achieve the same efficiency as direct linear visualizations for COMP and was always more efficient than integral pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For these local-tasks, we didn’t observe significant error reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our second principle in glyph design is to include cate- gorical features in separable pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The results from Exper- iment II studied the rank order of the separable pairs and found that they indeed improved accuracy for global tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Lengthytexture and splitVectors in both experiments led to more correspondence errors than lengthycolor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Achieving integrated numerical readings by combining two separable visual features at object level seems not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The separable-dimension pairs of lengthycolor and lengthytexture worked because they were preattentive scene features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our experiments show that viewers adopted a sequential task-driven viewing strategy based on a view hierarchy: viewers first obtain global distributions of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Then, a visual scrutiny is possible within a subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Although splitVectors are separable, visual search for length among length would be unguided because both targets and distractors contained the same visual variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The more separable, the easier it would be to guide the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Using coloring to provide some initial regional division may be always better than not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Texture (luminance) could achieve similar accuracy and efficiency as long as the task-relevant regions could be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2 Feature Guidance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Scene Guidance Taking into account both study results, we think an impor- tant part of the answer to visualization design is guidance of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' It is guided to some objects or locations over others by two broad methods: feature guidance (seeing objects) and scene guidance (seeing global structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Feature guidance refers to guidance by properties of the task-target as well as the distractors (leading to correspon- dence errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These features are limited to a relatively small subset of visual dimensions: color, size, texture, orientation, shape, blur or shininess and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' These features have been broadly studied in 3D glyph design (see reviews by Healey and Enns [25], Borgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [6], Lie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [46], Ropinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [22], and McNabb and Laramee [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Take one more example from quantum physics simulation results, but with a different task of searching for the structural distributions in the power of 3 in Figure 11 will guide attention to either the fat cylinders (Figure 11a) or the bright yellow color (Figure 11d, 11b) or the very dark texture (Figure 11c), depending on the feature-pair types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Working with quantum physicists, we have noticed that the structure and content of the scene strongly constrain the possible location of meaningful structures, guided “scene guidance” constraints [8], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Scientific data are not ran- dom and are typically structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Contextual and global structural influences can arise from different sources of visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' If we return to the MAX search task in Figure 11 again, we will note that the chunk of darker or lighter texture patterns and colors on these regular contour structures strongly influence our quick detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This is a structural and physical constraint that can be utilized effectively by viewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This observation coupled with the empirical study results may suggest an interesting future work and hypothesis: adding scene structure guidance would speed up quantitative discrimination, improve the accuracy of comparison tasks, and reduce the perceived data complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Another structure acting as guidance is the size itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' It was used by participants seeking to resolve the NU- MEROSTIY tasks to look for the longest outside cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We have showed several examples like Figure 11, our collaborator suggested that the cylinder-bases of the same size with the redundant encoding (Figure 11b) also helped locate and group glyphs belonging to the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This observation agrees with the most recent literature that guidance-by-size in 3D must take advantage of knowledge of the layout of the scene [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Though feature guidance can be preattentive and fea- tures are detected within a fraction of a second, scene guidance is probably just about as fast (though precise experiments have not been done and our Experiment II only merely shows this effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Scene ‘gist’ can be extracted from complex images after very brief exposures [47] [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This doesn’t mean that a viewer instantly knows, say, where the answer is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' However, with a fraction of a second’s exposure, a viewer will know enough about the spatial lay- out of the scene to guide his or her attention towards vector groups in the regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, categorical color becomes scene features since these colorful glyphs were perceived as a whole A future direction, and also an approach to understand- ing the efficiency and the effectiveness of scene guidance, is to conduct an eye-tracking study to give viewers a flash- view of our spatial structures and then let the viewer see the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 14 (a) Lengthylengthx feature-pair (b) Lengthycolor/lengthx feature-pair (c) Lengthytexture feature-pair (d) Lengthycolor feature-pair Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 11: Contours of simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Size from this viewpoint can guide visual grouping and size in 3D must take advantage of knowledge of the layout of the scene [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' display only in a narrow range around the point of fixation: does this brief preview guide attention and the gaze effectively?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Work in vision and visualization [49], [50], [51], [52] domain has measured and correlated performance on the glance or global structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Vision science discovered long ago that seeing global scene structures in medical imaging decision making guides experts’ attention (experts always know where to look) [53] [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 Redundancy and Ensemble Graphical Perception Our results showed that adding categorical colors, in which the correspondence parts could be quickly discriminated, is scalable to a large number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our result agrees with that of Northelfer and Gleicher [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' They observed that redundant encoding using color and shape could strengthen grouping when searching for targets from multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Their explanation was a race model [55]: for separable dimensions, the performance of a glyph with the redundant encoding might be dominated by the feature with greater 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 30 1 2 30 2 3JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 15 efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We did not find efficiency improvement - this suggested that the grouping is generally fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' So it might not be the redundancy itself that contributed to scene un- derstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Another possible theory is perhaps ensemble perception, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', “the visual system’s ability to extract summary sta- tistical information from groups of similar objects - often in a brief glance” [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Also ensemble features are best represented using the categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' To model parallel processing, the target contrast signal theory by Buetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [24] may suit our scenario better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' It describes more specific time estimate it takes to evaluate items in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In visualization, we just began to understand the ensemble averages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', Chen [11] and Alberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [56]) but have limited understanding of ensemble visual encoding choices to guide attention to optimize behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We leave this to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='4 Use Our Results in Visualization Tools and Limita- tions of Our Work Visualization is used when the goal is to augment human capabilities in situations where the problems might not be sufficiently defined for algorithms to communicate certain information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' One of our showcase areas is quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We believe that the design principle of prompting the ad- dition of categorical features in bivariate glyphs would be broadly applicable to glyph design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Also, application do- mains carrying similar data attributes could reuse of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our current study concerns bivariate data visualization in which the bivariate variables are component parts of scalar variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our design could have been improved by following advanced tensor glyph design methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Both generic [57] and domain-specific requirements for glyph designs [37] [58] [59] have led to the summary of glyph properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', invariant, uniqueness, continuity) to guide design and to render 2D and 3D tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A logic step is to truly un- derstand the quantum physics principles to combine data attributes and human perception to improve our domain- specific solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' One limitation of this work is that we measure only a subset of tasks crucial to showing structures and omit- ted all tasks relevant to orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' However, one may argue that the vectors naturally encode orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When orientation is considered, we could address the multiple- channel mappings in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The first solution is to use the lengthytexture to encode the quantitative glyphs and color to encode the orientation clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The second solution is to treat magnitude and orientation as two data facets and use multiple views to display them separately, with one view showing magnitude and the other for orientation (using Munzner’s multiform design recommendations [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The second limitation here was that our experiments were limited to a relatively small subset of visual dimensions: color, texture, and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A future direction would be to try shapes and glyphs to produce novel and useful design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 6 CONCLUSION Our findings in general suggest that, as we hypothe- sized, distinguishable separable dimensions with preatten- tive categorical features perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The separable pair lengthycolor was the most efficient and effective for both local and global tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The categorical features enable effec- tive complex scene inspections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Our empirical study results provide the following recommendations for designing 3D bivariate glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Highly separable pairs can be used for quantitative comparisons as long as these glyphs could guide at- tention (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', category forming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We recommend using lengthycolor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Texture-based glyphs (lengthytexture) that introduces luminance variation will only be recommended when task-relevant structures can be isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Integral and separable bivariate feature-pairs have the similar accuracy when the tasks are local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' When the search tasks are more complex, introducing categorical features in the separable feature-pairs will lead to per- ceptually accurate glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3D glyph scene would shorten task completion time by combing two glyph design factors: separability and visual guidance from categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The redundant encoding (lengthycolor/lengthx) greatly improved on task completion time of integral dimensions (lengthylengthx) by adding separable and preattentive color features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work is supported in part by NSF IIS-1302755, NSF CNS-1531491, and NIST-70NANB13H181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The user study was funded by NSF grants with the OSU IRB approval number 2018B0080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Non-User Study design work was sup- ported by grant from NIST-70NANB13H181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The authors would like to thank Katrina Avery for her excellent editorial support and all participants for their time and contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Any opinions, findings, and conclusions or recommen- dations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Certain commercial products 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='120 [59] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Kindlmann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Westin, “Diffusion tensor visualization with glyph packing,” IEEE Transactions on Visualization and Computer Graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 1329–1336, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1109/tvcg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='134 [60] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Munzner, Visualization Analysis and Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A K Peters Visualization Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' CRC Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1201/b17511 Henan Zhao was a PhD student in Department of Computer Science and Electrical Engineer- ing at University of Maryland, Baltimore County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' She received B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' degree in Computer Science and Information Security from Nankai University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Her research interests include design and evaluation of perceptually accurate visualization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This work was conducted while she was visiting The Ohio State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Garnett Bryant received his PhD at Indiana Uni- versity in theoretical condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' After research positions at Washington State University, the National Bureau of Standards, McDonnell Research Labs and the Army Re- search Laboratory, he has worked at the Na- tional Institute of Standards and Technology (NIST) since 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' He is directing the Quan- tum Processes and Metrology Group at NIST with experimental and theoretical programs on nanoscale, condensed matter systems for quan- tum information science and metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' He is a Fellow of the Joint Quan- tum Institute of NIST/University of Maryland, a Fellow of the American Physical Society and a member of the IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' His theoretical research program focuses on nanosystems, nanooptics and quantum science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Wesley Griffin received his PhD degree in Com- puter Science from the University of Maryland, Baltimore County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' He is a developer at Stellar Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' His research interests include real-time graphics and graphics hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' He is a mem- ber of ACM SIGGRAPH, the IEEE and the IEEE Computer Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Judith E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Terrill is a Computer Scientist and the Leader of the High Performance Computing and Visualization Group at the National Institute of Standards and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' She is a member of the IEEE Computer Society, the Association for Computing Machinery, and the Association for the Advancement of Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Jian Chen is an Associate Professor in Com- puter Science and Engineering at The Ohio State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' She received her PhD degree in Computer Science from Virginia Tech, and her MS degree in Mechanical Engineering | Pre- cision Instrument from Tianjin University | Ts- inghua University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' She was a postdoc- toral fellow at Brown University and a visiting researcher at Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Her current research interests include visual design, 3D in- teraction, and human-AI teaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 18 Evaluating Glyph Design for Showing Large-Magnitude-Range Quantum Spins Additional Material Empirical study training documents, source code, study data, and results are online at https : //osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='io/4xcf5/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='viewonly = 94123139df9c4ac984a1e0df811cd580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' BACKGROUND COLOR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 12 shows an example represented by lengthytexture with gray, white, and black background colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Gray background color was selected for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We could observe that both white and black cylinders with lengthytexture encoding could be displayed more clearly in the gray background (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 12, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' VISUAL MAPPING FOR COLOR AND TEXTURE IN THE Lengthycolor AND Lengthytexture PAIRS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 6 shows the visual mapping using color and texture in Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The horizontal axis represents the exponent range ∈ [3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We selected those categorical colors from ColorBrewer [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For texture, the percentage of black is mapped to the exponent-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Examples with three different exponent-ranges of 3, 5, and 7 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 13, in which color and texture are used for the visual mapping of study data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' VISUAL FEATURES AND EXPONENT-RANGE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 13 shows examples for visual features and three exponent-ranges of 3, 5, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The figures with the same exponent- range were generated using the same data and different visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The dataset used in this figure is for illustration purpose only and does not necessarily reflect all image features used in the vector magnitude experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 12: Examples using different background colors: gray, white, and black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Figures on the top row are magnified views of region 1, marked by orange-box on the left image, and the bottom row shows region 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' With white background, the white cylinders would be washed out (top right image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' With black background, the black cylinders would be washed out (bottom right image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' In this study, the neutral stimulus-free gray background was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Gray background White background Gray background Black backgroundJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 19 (a) Lengthylengthy (splitVectors) (b) Lengthycolor (c) Lengthytexture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 13: Experiment II: examples of selected exponent ranges of 3, 5, and 7 (from the second left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' We could see that the pattern of magnitude distribution is more revealing by categorical colors than by texture glyphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Coloring may show more steps with large exponent ranges and also give us a better understanding of data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For example, we could quickly focus on the orange region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 8, AUGUST 2015 20 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' SPATIAL PROXIMITY Figures 14 and 15 show spatial distributions of the identified targets (participants’ answers) to the correct targets in the search and max tasks in Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here locations of the correct targets are translated to the origin (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Participants’ answers are depicted in green and each dot represents a trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Dots may overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Dots in orange illustrate some of the nearest spins whose exponent values differ from the target (located at the origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Comparing the distribution of participants’ answers and the orange dot locations illustrates one of the key quantum physics data attributes: quantum physics data are discrete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' and spatial proximity is not correlated with the spin magnitude proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' For complex data like this, using the structural features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=', from color) in search will help them be more efficient and reduce errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (a) lengthycolor (b) lengthytexture (c) lengthylengthy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 14: Experiment II: Search task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The spatial proximity of the locations of the identified targets, to the ground truth, for all trials in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here the ground truth locations are translated to the origin (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' This task was time- constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' among the 810 trials (or 270 trials for each bivariate glyph type), participants completed 262 lengthycolor, 261 lengthytexture, and 251 lengthylengthy trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' (a) lengthycolor (b) lengthytexture (c) lengthylengthy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 15: Experiment II: Max task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The spatial proximity of the locations of the identified targets, to the ground truth (centered at the origin (0, 0, 0), for all trials in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' The yellow dots show the closest points from other-than-target-exponent regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Here the ground truth locations are translated to the origin (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Among the 810 trials, participants gave an answer to 270 trials for each bivariate glyph type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' Among each of these 270, participants completed 269 lengthycolor, 269 lengthytexture, and 259 lengthylengthy trials in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content=' 3 3 2 2 1 1 No 0 1 1 2 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 35 2 1 0 43-2-1012 34 X 3 2 Z > 0 1 2 3 4-3 4 X X3 m 2 2 1 1 No 1 1 2 2 33 35 2 1 0 4-3 -2-1 0 X 3 2 1 0 1 2 2 0 4-3-2-1 0 12345 X x 33 3 2 N 1 No No 1 2 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 5 3 X 3 2 1 Y 0 1 2 3 5-43-2-1 1 X X3 3 2 2 1 1 No No 1 1 2 2 3 5-4-3-2-101 2345 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 2 0 2 3 X Y 3 2 1 Z Y 0 1 2 2 3 5-4-3-2-1 0 X 2 X3 3 2 2 1 1 N 0 No 1 1 2 2 3 5 -4 -3 -2 -1 0 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} +page_content='1 0 2 3 X 3 2 3210 1 N 0 1 2 2 3 0 5 -4 -3-2-1 0 X X3 3 2 2 1 1 No No 1 1 2 2 3 5-4-3-2-1012345 3 3 2 1 0 2 3 x Y m 2 321 1 N 0 1 2 0 1 5-4-3-2-10 12345 X X' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAyT4oBgHgl3EQfOPYv/content/2301.00002v1.pdf'} diff --git a/vtFKT4oBgHgl3EQf4S7l/content/tmp_files/load_file.txt b/vtFKT4oBgHgl3EQf4S7l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..294e261959b9424e4760878dc06950b57386ff7e --- /dev/null +++ b/vtFKT4oBgHgl3EQf4S7l/content/tmp_files/load_file.txt @@ -0,0 +1,1536 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf,len=1535 +page_content='In our mind’s eye: Visible and invisible in quantum theory, with Schrödinger’s cat experiment Arkady Plotnitsky* Literature, Theory, Cultural Studies Program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Philosophy and Literature Program, Purdue University, West Lafayette, IN 47907, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Email: plotnits@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This article aims to reconsider E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger’s famous thought experiment, the cat-paradox experiment, and its place in quantum foundations from a new perspective, grounded in the type of interpretation of quantum phenomena and quantum mechanics, which belongs to the class of interpretations designated here as “reality without realism” (RWR) interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such interpretations have not been previously brought to bear on the cat experiment, including by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr, whose interpretation in its ultimate form (as he changed his interpretation a few times) is an RWR interpretation, but who does not appear to have commented on the cat experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The interpretation adopted in this article follows Bohr’s interpretation, as based on two assumptions or postulates, the Heisenberg and Bohr postulates, but it adds a third postulate, the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The article also introduces, in conjunction with the concept of reality without realism, the concepts of visible and invisible to thought and considers their role in the cat-paradox experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Key words: the cat paradox experiment, the cut, classical objects, quantum objects, quantum phenomena, reality without realism, visible to thought, invisible to thought Hamlet: My father — methinks I see my father.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Horatio: Where, my lord?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hamlet: In my mind’s eye, Horatio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' --William Shakespeare, Hamlet, Act 1, Scene 2, ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 183-185 To die for the invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is metaphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' --Emmanuel Levinas, Totality and infinity: An essay on exteriority 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Introduction This article aims to reconsider E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger’s famous thought experiment, the cat-paradox experiment (hereafter “the cat experiment”), and its place in quantum foundations from a new perspective, which removes any paradox from it [Schrödinger 1935].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' So, admittedly, do some other views of the experiment, and Schrödinger himself did not call it a paradox, rather a “ridiculous situation” [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The present view of it, however, is grounded the type of interpretation of quantum phenomena and quantum mechanics (QM) that has not, to my knowledge, been previously brought to bear on the cat experiment, including by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr, whose interpretation, especially in its ultimate form, is the closest to the one adopted here, but who does not appear to have commented on the cat experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is worth keeping in mind that, while an interpretation of QM, commonly, including Bohr’s or the present interpretation, involves an interpretation of quantum phenomena, the latter have separate interpretable aspects (noted whenever necessary in this article) that do not depend on, and hence could be interpreted independently of, any theory predicting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum phenomena will be assumed here to be defined by the fact that in considering them (technically, the data found in them, pertinent to quantum experiment), the Planck constant, h, which is a classically measurable quantity, must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='1 1 I put aside qualifications of this definition, necessary in general but not germane for this article, because all quantum phenomena and measurements considered involve h (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 37-38], also [Khrennikov 2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I might only add that all quantum-mechanical equations used for actually predicting the data observed in 2 Bohr eventually came to see quantum phenomena as revealing “a novel feature of atomicity in the laws of nature,” “disclosed” by “Planck’s discovery of the quantum of action [h], supplementing in such unexpected manner the old [Democritean] doctrine of the limited divisibility of matter” [Bohr 1938, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Atomicity and, thus, discreteness or discontinuity initially emerged on this Democritean model, with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Planck’s discovery of the quantum nature of radiation in 1900, which led Planck to his concept of the quantum of action, h, physically defining this discontinuity, and then A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Einstein’s introduction of the concept of a photon, as a particle of light, in 1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The situation, however, gradually, especially with the discovery of QM in 1925 by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Heisenberg, revealed itself to be more complex, eventually leading Bohr to his concepts of phenomenon and atomicity (essentially equivalent to that of phenomenon, but highlighting some of the features of the latter concept, such as discreteness), and the interpretation of quantum phenomena and QM based in these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This interpretation, developed in the later 1930s, became the ultimate version of Bohr’s interpretation, following a decade of the development of, and some significant changes in, his views (with only a few minor refinements added later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This requires one to specify to which version of his interpretation one refers, which I shall do as necessary, while focusing on his ultimate interpretation, unavoidably, in the present interpretation of his interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Unless qualified, “Bohr’s interpretation” will refer to his ultimate interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (The designation “the Copenhagen interpretation” requires even more qualifications as concerns whose interpretation it is, say, that of Heisenberg, Dirac, or von Neumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Accordingly, I avoid this designation altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') The interpretation adopted in this article follows this interpretation, in particular as based on two assumptions or postulates, the Heisenberg and Bohr postulates, but it adds a third postulate, the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All three postulates are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I would like, however, to emphasize from the outset that these postulates are interpretive assumptions that could, in principle, be falsified, even though, as discussed later in this article, a falsification of an interpretation is not the same (and a more complex matter than) that of a theory by experimental evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By virtue of the first two postulates, especially the Heisenberg postulate, both interpretations belong to the class of interpretations of quantum phenomena and QM, or quantum field theory (QFT), designated here as “reality without realism” (RWR) interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This article is only concerned with QM and, marginally, QFT (in high-energy regimes) in their currently standard forms, and puts aside, except in passing, alternative quantum theories, such as Bohmian mechanics or spontaneous collapse theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='2 The Heisenberg postulate, most essentially defining RWR interpretations, was in effect introduced by Bohr’s 1913 atomic theory, in considering the transitions, “quantum jumps,” between stationary states of electrons, while retaining a realist view of stationary states by assuming them to be represented as orbits of electrons around nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The RWR understanding of quantum phenomena and QM emerged in its full form in Heisenberg’s approach to quantum theory that led him to his discovery of QM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which is why I use the designation “the Heisenberg postulate.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The Heisenberg postulate places the emergence of quantum phenomena beyond representation or knowledge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' or even conceptions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' beyond the reach of quantum phenomena contain h or ℏ (or something mathematically equivalent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' by suitable changing the values of the parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' such as time),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which fact is sometimes hidden in more abstract,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' such as Hilbert-space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' versions of the formalism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' unless one properly unfolds its relevant elements to make actual predictions possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2 The interpretation offered in this article was considered previously in [Plotnitsky 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2022a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The last article cited expressly adopts the three postulates in question, under the headings of Heisenberg, Bohr, and Dirac discontinuity, in considering the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' RWR interpretations without the Bohr postulate may be possible, but they will be put aside here, because both Bohr and the present interpretation adopt this postulate, along with the Heisenberg postulate, which defines all RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is possible and technically more rigorous to see a different interpretation of a given theory as forming a different theory, because each interpretation may involve concepts not be shared by other interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is the case, for example, in different versions of “the Copenhagen interpretation,” not all of which are RWR interpretations, which too may be different, as are Bohr’s and the present interpretation, because the present interpretation assumes the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' What is shared is the mathematical formalism used, at least in terms of the equivalence or mutual translatability of its different versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For simplicity, however, I shall continue to speak of different interpretations of a theory itself containing a given mathematical formalism, specifically, of different interpretations of QM or QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 3 thought, or in terms I shall adopt here, make this emergence invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Realism, by contrast, is defined by the assumption of the possibility of either representation or knowledge, or at least conception of how the phenomena considered are possible, thus making them visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I shall speak of weak RWR interpretations (or the weak form of the Heisenberg postulate) when this emergence is assumed to be beyond representation or knowledge, and of strong RWR interpretation (or of the strong form of Heisenberg postulate) when it is assumed to be beyond conception and thus made invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This article adopts a strong RWR interpretation, as did Bohr in the ultimate version of his interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Unless qualified, the term “RWR interpretation” will, hereafter, refer to strong RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concepts of classical physics, specifically classical mechanics, emerged as mathematized refinements of our daily concepts—concepts arising from our general phenomenal experience of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All modern physics (classical, relativistic, or quantum) only deals with suitably mathematized idealizations of physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This connection between physical and daily concepts has proven to be difficult to use in quantum theory, even in realist interpretations, which would assume that QM or QFT provides a mathematized representation of quantum objects and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such a representation is no longer a mathematical refinement of our general phenomenal experience of the world, although QM or QFT formally adopts some (but only some) mathematics used in classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' RWR interpretations preclude any representation, including any mathematical representation, or even conception of the ultimate reality responsible for quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I qualify because classical physics remains an essential part of quantum theory, including in RWR interpretations, if they assume, as both Bohr’s and the present interpretation do, the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By the Bohr postulate, quantum phenomena, defined by what is observed in measuring instruments, along with the observable parts of these instruments, are represented by classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='3 By classical physics I mean (as Bohr appears to have done, although he did not always specify the term) to classical mechanics and classical electromagnetic theory, with the addition of special relativity in high-energy (QFT) quantum regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By classical mechanics I refer to Newton’s mechanics defined by its three main laws, the law of inertia, the law of the changes a force can have on the motion of a body, F = ma, and the law of action and reaction between interacting bodies, as equal in magnitude by opposite in direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These three theories—classical physics, relativity, and quantum theory—are sufficient for the present purposes for representing the observable part of measuring instruments in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (Gravity, governed by Newton’s forth law, is a special case of Newton’s mechanics, which can be put aside for the present purposes, as can be the equivalence principle, not involved in any phenomena considered in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') Accordingly, by a quantum instrument I understand any technological device able of establishing quantum phenomena and registering quantum data, the data that is represented by classical physics but that cannot be predicted by classical physics (including as concerns the role of h in these data), and thus requires an alternative, quantum theory, such as QM or QFT, of possibly some alternative theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (This article, again, will put such alternatives, be they actual or hypothetical, aside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') At least, such is the case, as things stand now (a qualification assumed throughout this article), for it is in principle possible that one or another form of classical theory able to predict this data can be developed, and there are attempt in this direction, which will be put aside here, because this article is only concerned with RWR interpretations of QM or QFT and, a subject not considered previously, how the cat paradox appears in this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I make no other claims here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Measuring instruments, it follows, also have quantum strata through which they interact with quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Eventually, Bohr adopted the term “phenomenon” to refer strictly to what is observed in measuring instruments, as effects of their interaction with quantum objects [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The Bohr postulate, thus, also reflects the transition, via measuring instruments, from the ultimate, “quantum,” reality considered to the classical reality of observation, and conversely, in the initial stage (preparation) of an experiment, from the classical reality of observation to the ultimate, “quantum,” reality considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In strong RWR interpretations, when referring to this ultimate reality, “quantum,” or, in the first place 3 RWR interpretations on the Heisenberg postulate alone (defining all RWR interpretations), without assuming the Bohr postulate, are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Conversely, the Bohr postulate need not be limited to RWR interpretations and is found in realist interpretations of quantum phenomena and QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 4 “reality,” is a term to which no concept we can form can be associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any such association is only possible and necessary at the level of observation or phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At this level, that of visible to thought or even available to immediate human perception, thus defining quantum phenomena or quantum events, the term “quantum” has (classical) physical concepts, such as discreteness or individuality, associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By the Heisenberg postulate, how quantum phenomena come about cannot, in RWR interpretations, be represented by QM or QFT, but only predicted by it, in general probabilistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM or QFT, has, in these interpretations, no physical connections apart from making these predictions, to either the ultimate nature of reality responsible for quantum phenomena or, because they are described by classical physics, to these phenomena themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, in these interpretations, the capacity of the mathematics of QM or QFT to predict the outcomes of quantum experiments, even if only probabilistically (which is, however, in accord with the experimental evidence now in place), becomes in turn beyond knowledge or even conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' We know how this mathematics works (how to use it), but we do not know and perhaps cannot know or even conceive of why it works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Fortunately for us, however, it does work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, in classical physics or relativity (special or general), the mathematical formalism (ideally) represents, make visible to thought, the physical reality responsible for the phenomena considered and connects, by continuous processes, these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter can, moreover, be identified with the physical objects considered, because the interference of measuring instruments can be neglected for all practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This identification is no longer possible in considering quantum phenomena, in the constitution of which the role of measuring instruments is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nobody has ever seen a moving electron or photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is invisible to an observation and, in the RWR view, even invisible to thought, is entirely beyond the reach of thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is only possible to observe traces, which are visible even to our immediate sense perception and consciousness (such traces may also be “clicks” that we hear rather than see) of their interactions with measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These traces make it difficult and, in strong RWR interpretations, impossible to reconstitute the ultimate nature of the reality responsible for them, whether one sees this reality in terms of quantum objects or assumes, as I do here, that a quantum object is an idealization applicable only at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (This assumption is the content of the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') Either way, this situation entails an unavoidable discrimination between quantum objects and instruments, and hence phenomena, a discrimination that, according to Bohr, “may indeed be said to form a principal distinction between classical and quantum-mechanical description of physical phenomena” [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 701].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='4 While adopting this structure of observation in quantum physics, the present interpretation further stratifies it by the Dirac postulate, not found in Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s argumentation might be seen as, at certain points, suggesting the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr, however, never formulated this type of postulate or the corresponding view and appears to have always assumed that the concept of a quantum object is an entity that exist independently, while still being beyond representation or even conception, by the Heisenberg postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' According to the Dirac postulate, the concept of a quantum object is only applicable at the time of observation, but not to anything assumed to exist independently in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In Bohr’s interpretation in 4 The difference between phenomena and objects has its genealogy, in modern times (it had earlier precursors, even in ancient Greek philosophy), in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Kant’s distinction between objects as things-in-themselves in their independent existence and phenomena as representations created by our mind, which may not correspond to the objects which they are aiming to represent or to which they may be representationally unrelated at all [Kant 1997].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter is in fact the case in considering quantum phenomena vis-à-vis quantum objects because quantum phenomena represent classical physical objects observed in measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As the strong RWR view, Bohr’s or the present view is more radical than that of Kant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While Kant’s things-in-themselves are assumed to be beyond knowledge, they are not beyond conception, at least a hypothetical conception, even if such a conception cannot be guaranteed to be correct and is only practically justified in its applications [Kant 1997, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, in the strong RWR view what is practically justified is not a possible conception of the ultimate nature of reality responsible for quantum phenomena, but the impossibility of such a conception, thus in precluding this reality from being visible to thought, even hypothetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' No other justification than practical is possible by virtue of the impossibility of this conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concept of a quantum object can of course be considered from alternative, including realist, perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' See, for example, [Jaeger 2014], which offers a rigorous argument for such a concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 5 all its versions, the ultimate reality responsible for quantum phenomena was associated with quantum objects, eventually as independent RWR-type entities, different from quantum phenomena, defined by the irreducible role of measuring instruments, the observable parts of which are described by classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A quantum object is, in Bohr view, a physical object responsible for the existence of a quantum phenomenon, as an effect of the interaction between this object and a measuring instrument or some (classical) object existing in nature that function as an instrument for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nothing, either built by us or by nature, can be defined as an instrument apart from us in the present and, I would argue, in Bohr’s view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As RWR type entities quantum objects were assumed by Bohr to be beyond conception, and hence could not be assigned any properties, including h, even at the time of observation and measurement, as all physical properties were only assignable to observables parts of measuring instruments, described classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As will be seen, it is possible in a quantum experiment to consider as the object under investigation an object that also contains a classical part, such as the cat in the cat experiment, but this composite object must still contain a properly quantum object for the observed phenomenon to be a quantum phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM would predict such observed properties of measuring instruments and only them, rather than any properties of quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The Dirac postulate introduces a triple rather the double, stratification into this situation, following [Plotnitsky 2021a, 2022a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The ultimate RWR reality responsible for quantum phenomena is an idealization assumed to exist independently of our interactions with it, and thus independently of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, the concept of a quantum object, elementary, such as a photon or electron, or composite (possibly macroscopic) is an idealization that, while still of the RWR-type, only applies at the time of an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' An observation becomes a creation of a quantum phenomenon by the interaction between the ultimate RWR-type reality, and the instrument we use, and the capacity of our thought to observe the phenomena thus created, which also allows enables one to apply the concept of quantum object, by the Dirac postulate only after an observation has taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In all three cases—the (independent) ultimate RWR reality responsible for quantum phenomena, quantum objects, and quantum phenomena—one only deals with idealizations created by our thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The reason for using the designation “Dirac postulate” is that, while, unlike the Heisenberg postulate by Heisenberg and the Bohr postulate by Bohr (even if without using these designations as such), this postulate was not considered by Dirac himself, it may be seen as having emerged from Dirac’s famous equation for a relativistic electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While originally written for an electron, Dirac’s equation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='𝛽𝑚𝑐!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' + \' 𝛼"𝑝" # "$% 𝑐* 𝜓(𝑥, 𝑡) = 𝑖ℏ 𝜕𝜓(𝑥, 𝑡) 𝜕𝑡 (I4 is the identity matrix) revealed itself to an equation for both the (free) electron and the (free) positron, including their spins, which the equation contains automatically, in contrast to QM, where predicting the spin of an electron needs to be handled separately, via Pauli matrices combined with Schrödinger’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Dirac’s equation reflected and, as it happened, led to the discovery that a different particle (in the present view, defined in terms of effects observed in measuring instruments) can be registered in a single experiment: the initial observation can register an electron, while the next one a positron, or a photon, or an electron- positron pair, with the probabilities defined by the same equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Once one moves to still higher energies, the panoply of possible outcomes becomes even greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In QED, one only deals with electrons, positrons, and photons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' in QFT, depending how high the energy is, one can find any known elementary particle or combination, that is, the corresponding effects will be registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Accordingly, it is reasonable to apply the concept of a quantum object (still as an RWR-type entity) exclusively at the time of ai 2 = b2 = I4 aib + bai = 0 aia j + a jai = 0 6 observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are, however, reasons to adopt this view in low-energy (QM) quantum regimes, including in order more effectively to interpret quantum conundrums, such as that of the double-slit experiment, a paradigmatic and, arguably, the most famous quantum experiment [Plotnitsky 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='5 Do quantum phenomena or QM require the Dirac postulate, or the Heisenberg and Bohr postulates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It would be difficult to argue such a case, and it is not my aim to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' My only claim is the logical consistency of the interpretations, such as the one adopted here, grounded in these postulates, and their accord with the experimental evidence currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As I said, new experimental evidence can change the present situation of fundamental physics, just as then new evidence changed its situation around 1900, leading to Planck’s discovery of quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The next section outlines the (RWR-type) interpretation adopted in this article, cast in terms of the relationships between visible and invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Section 3 discusses, by way of the bridge to the cat experiment, the letter exchange between Schrödinger and Bohr (at the time Schrödinger’s work on his paper containing the cat paradox) concerning the use of classical concepts in quantum measurement, thus, essentially the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Section 4 considers Schrödinger’s cat experiment from the (RWR) perspective established by the preceding analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The conclusion offers philosophical reflections on the role of metaphysics in physics, via the relationships between visible and invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Reality without realism, and visible and invisible to thought in fundamental physics This section outlines the RWR view of quantum phenomena and quantum theory, a view cast in terms of the relationships between visible and invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For simplicity, I shall primarily discuss QM, only briefly referring to QFT, although my argument applies to and can be further supported by QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I speak, more generally, of “the RWR view” because it can lead to various interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These interpretations share the Heisenberg postulate, defining RWR interpretations, but beyond being either 5 One can translate the argument of this article into quantum-informational terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, information is human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nature has no information, only we do, possibly about nature, or what we assume to be nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All information obtainable in quantum experiments is contained in the data observed in measuring instruments, described classically, by the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, this information qua information is classical, Shannon information (measured in classical bits), and as such is visible to thought and communicable unambiguously given the mathematical nature of Shannon information (as opposed to other forms of information with a semantic content, which may allow for ambiguity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, this information and its organization, as manifested in quantum experiments cannot be predicted or processed by classical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The emergence of this information requires the assumption of quantum objects or in the present view, by the Dirac postulate, an RWR type reality ultimately responsible for quantum phenomena and the use measuring instruments capable on interacting with this reality, and a theory, such as QM, different from classical theories, that is capable of handling this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In this sense, while all actual information obtained in experiments is classical, one can speak, as is common, of “quantum information.” In the present view, the “quantum,” as the ultimate reality responsible for quantum phenomena, in only a particular way, defined by out interaction with nature, to create and communicate classical information, which cannot be predicted by classical physics (or relativity) and the structure of which cannot be generate by classical objects and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Dealing with quantum information requires quantum information science, a vast subject of its own, including as concerning its impact on quantum foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum information theory brings new features to the differences between classical and quantum information in relation to QM or QFT, because at this stage quantum information theory is primarily concerned with discrete variables, physically represented by spin, rather than continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While there are classical and quantum versions of continuous variables, spin is a quantum variable, corresponding to a strictly quantum aspect of nature, and has no classical analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By the same token, a spin may be seen as reflecting something in nature strictly beyond our thought’s capacity to form a conception of, a strictly RWR-type entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I am indebted to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' D’Ariano for exchanges concerning of the visible and communicable or (his preferred term) “sharable” nature of classical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I am not claiming that he subscribes to the argument of this article or all of this argument, which builds on [Plotnitsky 2021a,b, 2022a,b], as concerns the RWR view and the idea of the invisibility to thought, as extending to all thought, including mathematical and physical thought, rather than only our immediate (conscious) phenomenal intuition or visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In general, this article’s argument is independent of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7 weak or strong RWR interpretations, some of them may contain additional postulates, such as the Bohr postulate or the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, while Bohr’s interpretation and the present interpretation both assume the Bohr postulate, only the present interpretation assumes the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The philosophical position grounding the present interpretation implies that modern physics, as a mathematical-experimental science, contains two forms of thinking, which may be designated as “classical” and “quantum” in view of their respective origins in classical and quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Both assume the physical reality they consider to exist independently of our existence and thinking as humans, but each treats this reality differently as concerns the ultimate nature of this reality, assumed to be representable and, thus, visible to thought in classical thinking, and to be beyond not only representation but also conception and, thus, invisible to thought in quantum thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I speak of the ultimate nature of the reality considered or (for the sake of economy) just the ultimate reality considered, because quantum thinking assumes that classical thinking applies at some levels of the reality considered, specifically, by the Bohr postulate, that of the observable parts of measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum thinking also involves classical thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In adopting only classical thinking one assumes it to apply at all levels of physical reality, without allowing for quantum thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These two forms of thinking are as follows: (1) Classical thinking, which is essentially a realist thinking, deals with a form of reality that is visible to thought, as what can be perceived, imagined, visualized, represented, known, conceptualized, and so forth, and as such allows one to have statements or images of this reality that can, at least in principle, be communicable unambiguously, as is necessary for the practice of science, as constituted now;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (2) Quantum thinking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which is essentially RWR thinking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' contains classical thinking but also assumes the existence of a form of reality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as the ultimate reality considered,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' that is no longer available to classical thinking and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as such,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' is invisible to thought,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as what cannot be perceived,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' imagined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' visualized,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' represented,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' known,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' conceptualized,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and so forth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which also means that nothing about it can be communicated unambiguously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' if at all,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' apart from the claim that it is beyond the reach thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='6 Thus, in the case of this (RWR) form of reality, quantum thinking, divorces the term “reality” from any possible concept associated to it, making it akin to a mathematical symbol, like R or X, which could have been used instead of the word reality here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Reality without realism” or RWR functions in this way as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I emphasize that quantum thinking also assumes forms of reality, such as that observed, as phenomena, in quantum experiments, that handled by classical thinking by the Bohr postulate and as such is visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The very existence of any RWR-type reality is inferred from certain configurations (such those defining the data observed in quantum experiments) of classical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Classical and quantum thinking are not the same as physical theories or interpretations using either thinking, because such theories contain additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, while different theories, both classical physics and relativity, special or general, conform to classical thinking, which quantum theory does not at least in RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is true that special relativity severely limits the capacity of our immediate phenomenal intuition to represent or visualize the kinematic used by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These qualifications, however, do not prevent this kinematic from being visible to thought and allow for a realist treatment, because relativity, special or general, represents all reality considered in it in terms of suitably mathematized physical concepts, just as does classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In considering classical physical phenomena or (they can, again, be identified with each other in classical physics or relativity) objects the role of both h and c can be disregarded and is by classical mechanics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' in considering relativistic phenomena or (they can, again, be identified with each other) objects, only h can be disregarded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and in considering quantum phenomena or (they can no longer be identified with each other) objects, h be taken 6 By thus relating the concepts of “visible to thought” and “unambiguously communicable,” specifically, by means of language (although language is not the only form of unambiguous communication, which can be visual, for example), speaking of visible and invisible to thought suggests a connection to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bell’s title Speakable and Unspeakable in Quantum Mechanics [Bell 2004], a collection of his writings, primarily on QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The philosophical position adopted in this article is, however, opposite of that of Bell, which also leads Bell to his discontent with Bohr’s view, including the Bohr postulate, and with QM in the first place, as discussed in [Plotnitsky 2021b, 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 8 into account, and in high-energy relativistic (QFT) regimes, c must be taken into account as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='7 This formulation is different from saying that c is assumed to be infinite in classical physics and h equal to zero in classical physics and relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In particular, classical mechanics need not be, and in the present view is not, assumed to be the limit of QM by putting h equal to zero or express in this way Bohr’s correspondence principle (explained below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These are, in the present view, two different theories, dealing with two different types of objects, even though both are ultimately composed of quantum objects or (in the present view) the same ultimate reality: classical mechanics is not a special (limit) form of QM and classical objects are not a special type of quantum objects, although quantum objects can sometimes be treated classically, which claim, however, requires important qualifications explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present interpretation, moreover, quantum objects are only defined at the time of observation by the Dirac postulate, which does not apply to classical objects, defined independently of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum thinking emerged in quantum theory in RWR interpretations in view of the nature of quantum phenomena, assumed to be the effects on the interactions between the ultimate reality responsible for these phenomena and suitable measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These phenomena, or rather numerical data they contain, are predicted by quantum theory, QM or QFT, without, in RWR interpretations, representing the ultimate reality responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM or QFT does not represent these effects either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' They are, by the Bohr postulate, represented by classical physics, which enables them to be as unambiguously communicable, as is the mathematics of QM or QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Along with the predictive capacities of both theories, the possibility of this unambiguous communication and, in this sense (a qualification discussed below), objectivity make these theories conform to “the basic principles of science,” as stressed by Bohr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 700;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 67-68, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, classical theories cannot predict these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It follows that both types of theories are necessary in fundamental physics, including quantum theory, because, by the Bohr postulate, quantum phenomena are represented by classical physics, with adding special relativity in high-energy (QFT) regimes, while they can only be predicted by quantum theory, which in RWR interpretations represents neither these phenomena not, by the Heisenberg postulate how they come about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, classical physics is necessary for describing measuring instruments in relativity as well, specifically in all measurements within each local reference frame, even though the instrument are subjects to the relativistic laws of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While philosophically classical, that is, conforming to classical thinking in the sense defined here, general relativity is a separate part of fundamental physics as currently constituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It may sometime play a role in dealing with quantum phenomena, keeping in mind that the emergence of quantum phenomena considered thus far does not involve gravity as a such, as we do not have a quantum theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='8 Classical theories are grounded, in Bohr’s words, “the idea [and hence the assumption] that the phenomena concerned may be observed without disturbing them appreciably,” which enables one to identify these phenomena with the objects considered [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This assumption no longer appears possible in considering quantum phenomena, empirically and hence regardless of interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, it may not be rigorously possible even in classical physics and relativity, insofar as all phenomena considered are still created by our thought, which is a product of our bodies and brains, as experimental technologies created by nature [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' vii-xxiv].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, the assumption that “the phenomena concerned may be observed without disturbing them appreciably” is workable in these 7 See, however, note 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 8 It may in principle be contended that, even if necessary for the description of the observable phenomena in quantum or relativistic measurements, classical physics is not a separate theory, for example, by assuming or arguing that ultimately all physical objects considered are quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, however, classical physics is a necessary separate part of fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It may not, as such, deal with the ultimate constitution of matter, but I would argue, it is unavoidable in considering many, even most, macroscopic phenomena, and, again, even in dealing with fundamental, such as quantum, physics, where classical physics is unavoidable in dealing with observation and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Given that gravity plays no role quantum (or classical) phenomena considered in this article, I put it aside, although it may ultimately bear on the set of questions posed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 9 theories for all practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' in Bohr’s or the present interpretation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' quantum theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM or QFT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' are defined by dealing with the combination of fours features: (1) the ultimate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “quantum,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' reality responsible for quantum phenomena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' a reality invisible to thought by the Heisenberg postulate and commonly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' including by Bohr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' identified with quantum objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' by the Dirac postulate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' defined (still as RWR-type entities and thus invisible to thought) only at the time of observation in the present interpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (2) observational technology, commonly understood as comprised of measuring instruments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (3) observed phenomena, created by the interaction between quantum objects and measuring instruments, phenomena that are always visible to thought and even to our immediate phenomenal perception, while the numerical data observed and, in the first place, the observable parts of measuring instruments, are described by classical physics by the Bohr postulate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (4) the mathematical formalism of QM (cum Born’s rule), probabilistically or statistically predicting the outcomes of quantum experiments, as observed, via measuring instruments, in quantum phenomena, without representing the ultimate reality responsible for them by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Measuring instruments, it follows, contain both classical, observable, strata of reality and unobservable, ultimately invisible-to-thought, quantum strata of reality, which enables this interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The reasons for my emphasis on visible and invisible, extended to the idea of visible and invisible to thought (essentially thinkable and unthinkable) are both conceptual and historical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Conceptually, our capacity for visualizing the world, which is also related to the neurological functioning of our brain (about 60% of which is dealing with vision), is crucial to our thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This capacity has shaped classical physics, as a mathematical refinement of the world we observe, but it was defeated, first by special relativity, the kinematic of which is beyond it, and then, more radically, by quantum theory, which brought into physics that which is invisible to thought altogether, is beyond thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Historically, this emphasis follows Bohr’s appeal to the impossibility of visualization of the ultimate responsible for quantum phenomena, defined as effects of the interaction between this reality and our agencies of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s use of visualization and its avatars was in part shaped by the German term for intuition, Anschaulichkeit, which etymologically relates to what is phenomenally visualizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Even before (albeit only by a few months) Heisenberg’s discovery of QM, based on “abandoning the ordinary spacetime description,” and hence on an RWR type view, at least in its weak form [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 48], Bohr said: I am forcing myself these days with all my strength to familiarize myself with the mysticism of nature and am attempting to prepare myself for all eventualities, indeed even for the assumption of a coupling of quantum processes in separated atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, the cost of this assumption are so great that they cannot be estimated within the ordinary spacetime description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [A Letter to Heisenberg, April 18, 1925, Bohr 1972–1996, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 79–80, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 237] In a letter to Born, a few days later, he added: [Quantum experiments] preclude the possibility of a simple description of the physical occurrences [at the quantum level] by means of visualizable pictures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [S]uch pictures are of even more limited applicability than is ordinarily supposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is of course almost a purely negative assertion, but I feel that .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' one must have recourse to symbolic analogies to an even greater extent than hitherto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Just recently I have been racking my brain to dream up such analogies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Letter to Born, 1 May 1925, Bohr 1972–1996, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 311] The word “mysticism” will soon disappear from Bohr’s writings, replaced by an emphasis on QM as a rational theory, free from any “mysticism incompatible with the true spirit of science” [Bohr 1937, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 83, Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By referring to “the assumption of a coupling of quantum processes in separated atoms,” the statement also captures the core of the dilemma later posed by the Einstein-Podolsky-Rosen (EPR) experiment [Einstein et al 1935].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr links this dilemma to the impossibility of visualization, 10 ultimately making how what is observed there, or in any quantum experiment, come about invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' What makes Bohr’s statements remarkable is that they were made in 1925, 10 years before EPR’s article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is true that Einstein brought up related considerations in his exchanges with Bohr already in 1927, which was, however, still two years away in 1925, and followed the invention of QM [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 41-58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are numerous invocations of the limits and in effect the impossibility of visualization throughout Bohr’s writing, with an increasing emphasis, ultimately amounting to dealing with invisible to thought, even if without using this language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='9 To cite some key passages, proceeding chronologically: “In atomic problems we have apparently met with such a limitation of our usual means of visualization” (1925) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “On the whole, it would scarcely seem justifiable, in the case of the interaction problem, to demand a visualization by means of ordinary space-time pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, all our knowledge concerning the internal properties of atoms is derived from experiments on their radiation or collision reactions, such that the interpretation of experimental facts ultimately depends on the abstractions of radiation in free space, and free material particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, our whole space–time view of physical phenomena, as well as the definition of energy and momentum, depends ultimately upon these abstractions” (1927) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 77];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “The resignation with regard to the desires for visualization which gives our whole language its character, to which we are compelled by the situation [in QM]” (1929) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 98];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Indeed, only a conscious resignation of our usual demands of visualization and [classical] causality] was it possible to make Planck’s discovery fruitful in explaining the properties if the [chemical] elements of the basis of our knowledge of the building stones of atoms” (1929) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 108];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “We must only be prepared for the necessity for the necessity of ever extending abstraction from our customary demands for a directly visualizable description of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Above all, we might expect new surprises in the domain [of QED and QFT] where the quantum theory meets with the theory of relativity and where unsolved difficulties still stand as hindrance to a complete fusion of the extension of our knowledge and the expedients to account for natural phenomena which these theories have given us” (1929) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 108];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The fundamental indeterminacy which we meet here [in Heisenberg’s uncertainty relations] may, as the writer [Bohr] has shown, be considered as a direct expression of the absolute limitation of the applicability of visualizable conception in the description of [the ultimate reality of] atomic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' … The resignation as regards visualization and [classical] causality, to which we are thus forced in our description of atomic phenomena, might well be regarded as a frustration of the hopes which formed the [Democritean] starting-point of atomic conceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nevertheless, from the present standpoint of the atomic theory, we must consider this very renunciation as an essential advance in our understanding” (1929) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 114-115];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Only [the] limitation of our visualizable conception of motion, which is characteristic of quantum theory, enables us to understand how electrons can make their way between the metal atoms in the wire” (1929) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 118];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “We must be prepared for a more comprehensive generalization of the complementary mode of description [in QFT] which will demand a still more radical renunciation of the usual claims of so-called visualization” (1937) [Bohr 1937, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 88];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “The extent to which renunciation of the visualization of atomic phenomena [technically, how they come about] is imposed upon us is strikingly illustrated by the following example to which Einstein very early called attention and often has reverted [in effect that of the alternative, complementarity, behavior of photon, depending up the two alternative set-up of the experiment, in essence equivalently to the double slit experiment” (1949) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “… [T]he ingenious formalism of quantum mechanics … abandons pictorial representation and aims directly at the statistical account of quantum processes” (1951) [Bohr 1998, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 152];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Indeed, renouncing pictorial description of electronic constitution of the atomic system and only making use of empirical knowledge of threshold and binding energies of molecular processes, we can within a wide field of experiences treat the reaction of such systems by ordinary chemical kinetics, based on the well- established laws of thermodynamics” (1962) [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 9 I have considered this aspect of Bohr’s argumentation in [Plotnitsky 2012, 2016, 2021a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 11 “Certainly the issue [raised by the EPR experiment] is of a very subtle character and suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 59;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' emphasis added].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The statement closing my traversal is Bohr’s 1949 comment on the EPR experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As already Bohr’s 1925 statements cited above make clear, for Bohr the cost of “the assumption of a coupling of the processes in separated atoms,” at stake in the EPR experiment (which deals with two spatially separated quantum objects) was the impossibility of “the ordinary spacetime description” of how phenomena is observed there and, as such, described in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The EPR experiment and, in fact, all quantum experiments “preclude the possibility of a simple description of the physical occurrences [of phenomena considered] by means of visualizable pictures” [Letter to Born, 1 May 1925, Bohr 1972–1996, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr, thus, assumed that such may be the case well before EPR’s article, which might be one of the reasons why he thought that EPR’s experiment didn’t contain anything essentially new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' He might not have been entirely right on this point, given the role of entanglement and correlations brought about by the EPR experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' EPR, however, did not consider these concepts either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' That of entanglement was introduced by Schrödinger in response to EPR’s paper, including in the cat-paradox paper [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Correlations became prominent even later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Be it as it may on that score, Bohr argued that, although “the issue [raised by the EPR experiment] is of a very subtle character and suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization,” EPR’s argument does not demonstrate, as EPR claimed, either the incompleteness of QM or else its nonlocal nature (in the sense of allowing an instantaneous action at a distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' My main point at the moment is Bohr’s view that the ultimate reality responsible for quantum phenomena is invisible to thought, a reality defined here as a reality without realism (RWR), while quantum phenomena, as, by the Bohr postulate, observed classically, are visible to thought or in fact available to our immediate phenomenal perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It might be useful to briefly consider, as a simple representative example, which illustrate and will help to guide my discussion of RWR interpretations, how predicting the polarization of a photon appears in these interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are two possible outcomes of measurement (after the initial preparation): for example, the horizontal state x and the vertical state 𝑦, observed classically by the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In RWR interpretations, one could not say, as it is said sometimes, that before it is measured, the photon is (or is prepared) in a superposition of two physical states, and in the present view, moreover, the very concept of a photon, while it cannot be observed as such (only the corresponding effect in measuring instruments can) is only applicable at the time of observation by the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The wave function allowing one to predict either physical state x or y is written as |𝜓⟩ = 𝛼|𝑋⟩ + 𝛽|𝑌⟩ with probability amplitudes of |𝜓⟩ associated with state vector |𝑋⟩ given by 𝛼 and |𝑌⟩ given by 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In a random experiment, the probability of the photon, when its polarization will be measured, to be horizontally polarized is |𝛼|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and to be vertically polarized is |𝛽|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (by Born’s rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (Actual predictions will involve h, which does not appear in these abstract notations, but will once there are properly unfolded to make actual predictions possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') That, however, need not, and in the RWR view does not, mean that |𝜓⟩ = 𝛼|𝑋⟩ + 𝛽|𝑌⟩ represents the photon in a superposition of two physical states, x and y, as nothing can be said, by the Heisenberg postulate, concerning what happens between observations in the RWR view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Only the mathematical state vectors, designated |𝑋⟩ and |𝑌⟩ (in capital letters), in the Hilbert space used, are in a linear (mathematical) superposition, with given amplitudes, and not quantum objects, let alone the outcomes of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM, then, in Bohr’s or the present interpretation, does not represent either, by the Heisenberg postulate, the physical emergence of quantum phenomena or, by the Bohr postulate, the observed quantum phenomena, represented by classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The only relationship between quantum phenomena and QM in these interpretations is defined by the fact that QM predicts, in general probabilistically, the outcomes of quantum experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The probabilistic (or statistical) nature of these predictions is in accord with what is experimentally observed because no other predictions concerning such outcomes are in general possible, as concerns kinematic or dynamical variables, such as the position or the momentum, or the direction of spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such quantities as mass, charge, or spin are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (There 12 are certain specific situations, such those of the EPR-type experiments, where exact predictions are ideally possible, bur with important qualifications, explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') These predictions are, moreover, only possible by using rules added to the formalism rather than being part of it, such as Born’s rule, which relates (essentially, by using complex conjugation) complex quantities of the formalism to real numbers corresponding to the probabilities of quantum events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Arguments to the effect that such rules are inherent in the formalism have been offered, but they are not commonly accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The Heisenberg postulate remains the grounding postulate of all RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concept itself of reality-without-realism is based in more general concepts of reality and existence, assumed here to be primitive concepts and not given analytical definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By “reality” I refer to that which is assumed to exist, without making any claims concerning the character of this existence or reality, claims that, as explained below, define realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The absence of such claims allows one to place this character beyond representation or even conception, which defines the RWR view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I understand existence as a capacity to have effects on the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The assumption that something is real, including of the RWR-type, is made, by inference, on the basis of such effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The RWR view is grounded in the assumption that observable (either immediately or via a mediation of observational instruments) effects of physical reality allow for a representation these effects but not necessarily a representation (the weak RWR view) or even a conception (the strong RWR view) of how these effects are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter representation or conception may not be possible and is not in RWR interpretation in the case of the ultimate reality responsible for quantum phenomena, making this reality invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It follows this reality or nature or matter, in the first place, are assumed to exist independently in the first place, just as it would be in classical theories or relativity or other realist theories, where, however, all strata of nature or matter considered are assumed to be visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The assumption of the independent existence of nature or matter essentially amounts to the assumption that it has existed before we existed and will continue to exist when we will no longer exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Even this assumption, which still belongs to thought, has been challenged, even to the point of denying that the ultimate nature of reality is material rather than mental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Plato is the most famous ancient and Bishop Berkeley as the most famous modern case of this questioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such views are useful in suggesting that any conception of how anything exists, or even that it exists, including when assumed to be unavailable to human thought, still belongs to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It need not follow, however, that something which such concepts represent or to which they relate otherwise, possibly placing it beyond representation or even conception, does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum phenomena would not be possible without our interaction with nature by means of experimental technology and our specific (human) ways of observing phenomena and thinking about them, which makes them visible to thought or even to our immediate phenomenal perception or consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (Not all perceptions or forms of thought are conscious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') In RWR interpretations, nature has no quantum objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and when it comes to its ultimate workings, at least those responsible for quantum phenomena, nature is beyond knowledge or, in strong RWR interpretations, conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, the term “workings,” “nature,” or “existence” would not ultimately apply either, any more than any other terms or concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Modern physics gave us new, mathematical-experimental, means of dealing with the world by interacting with it by means of experimental technology and mathematics (as a form of thought).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, however, it gave us no more than such means, even in classical physics or relativity, where the assumption that the theory actually (ideally) represent nature is workable for all practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In any RWR interpretation, the concept of a quantum object is an idealization created in response to our interactions with nature by means experimental technology resulting in quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The present interpretation goes further by assuming the Dirac postulate, which makes the concept of quantum object an idealization (of the RWR type) only applicable at the time of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Importantly, however, the present (strong RWR) interpretation does not a uniform or otherwise unified character of the ultimate, RWR-type, reality considered in QM or QFT, a character only manifesting itself differently in quantum experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This assumption is in conflict with strong RWR interpretations, which preclude any conception of this reality and, hence, that of its unity or oneness, uniform or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While each time unknowable or even unthinkable, invisible to thought, an RWR-type reality is assumed to be each time different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is what makes each quantum phenomenon, as an effect 13 of this reality, individual and unrepeatable, unique, manifesting the unique, but still inconceivable, nature of the reality ultimately responsible for it each time one encounters this reality through its effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One can always repeat the setup of a given measurement, because this setup can be classically controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Not so, however, as concerns the outcome of this repeated measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such outcomes are ideally the same and are ideally predictable exactly in in classical or relativistic experiments dealing with individual or simple systems, with probability only entering when these systems have a great mechanical complexity, as in classical statistical physics or chaos theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, these outcomes will in general (apart from special cases) be different in identically prepared quantum experiments, no matter how elementary the quantum object considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As explained below, while possible, even preparing a given state, say, that of a “spin-up,” as manifested in the corresponding measurement, cannot in general be done in a single preparation, but only by post-selecting the required preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A brief outline of realist thinking may help to sharpen the nature of the RWR view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Realist thinking is manifested in the corresponding theories, commonly representational in character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such theories aim to represent the reality they consider, in modern, post-Galilean, physics primarily by mathematized models, suitably idealizing this reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is even possible to aim, including in quantum theory, for a strictly mathematical representation of this reality apart from physical concepts, at least as they are ordinarily understood, say, in classical physics or relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is also possible to assume an independent structure (defined by properties and relationships among them) of the reality considered, while admitting that it is either (A) not possible to represent this architecture or (B) even to form a rigorously specified concept of it, either at a given moment in history or even ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Under (A), a theory that is merely predictive could be accepted for lack of a realist alternative, usually with the hope that a future theory will do better by being a representational theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Einstein and, often following him, others held this type of view of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' What, then, grounds realism most fundamentally is the assumption that the ultimate constitution of reality possesses properties and the relationships between them, or, as in (ontic) structural realism, just a structure, the more elemental constituents of which are not defined in terms of properties [Ladyman 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such properties, relationships, or structures may either be ideally represented and hence known, or be unrepresentable or unknown or even unknowable, but are still conceivable, usually with a hope that they will be eventually so represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Most realist theories are representational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In considering physics, the concept of realism just outlined is often called “scientific realism.” However, this outline would apply to most forms of realism in science or philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It does not cover all forms of realism, which would be impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I shall also refer, as is common, to realist theories as ontological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='10 Thus, classical mechanics (used in dealing with individual objects and small systems, apart from chaotic ones), classical statistical mechanics (used in dealing, statistically, with large classical systems), chaos theory (used in dealing with classical systems that exhibit a highly nonlinear behavior), or relativity, special and general, are realist theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While classical statistical mechanics does not expressly represent the overall behavior of the systems considered because their mechanical complexity prevents such a representation, it assumes that the individual constituents of these systems are represented by classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In chaos theory, which, too, deals with systems consisting of large numbers of atoms, one assumes a mathematical representation of the behavior of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Relativity posed major, even insurmountable, difficulties for our immediate spatiotemporal phenomenal intuition, because the relativistic law of addition of velocities (defined by the Lorentz transformation) in special relativity, 𝑠 = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' "# $"(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content="#/')), for collinear motion (c is the speed of light in a vacuum), runs contrary to any possible intuitive conception." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Our phenomenal intuition cannot conceive of, visualize, this kind of motion, thus, making this concept of motion no longer a mathematical refinement of a daily sense of motion as the concept of motion is in classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Relativity was the first physical theory that 10 Although the terms “realist” and “ontological” sometimes designate more diverging concepts, these terms are commonly close in their meaning and will be used, as adjectives, interchangeably here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I shall adopt “realism,” as a noun, as a more general term and refer by an “ontology” to the representation or conception of the reality considered by a given theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Another, relatively common, term for realist theories, sometimes with additional specificities (not important for my argument here), is “ontic,” coming, as does ontological, from the ancient Greek on (Being).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 14 defeated our ability to form a phenomenal visualization of an elementary individual physical process, although the concept of (classical) field in classical electromagnetism already posed certain complexities in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr did not miss this point: “I am glad to have the opportunity of emphasizing the great significance of Einstein’s theory of relativity in recent development of physics with respect to our emancipation from the demands of visualization” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 115-116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Emancipation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is not a casual word choice, rarely, if ever, found in Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Special, as well as general, relativity, however, still offer mathematically idealized conceptual representations of the physical reality they consider, and in this respect allowed this reality still to be visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum physics brought this emancipation to a more radical level, that of the invisible to thought, mathematically reflected in the Hilbert space (or analogous) formalism over ℂ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Event this mathematics itself poses difficulties of seeing this formalism as representing anything physical in space and time, represented in all physical theories thus far as concepts over ℝ, to which, the formalism relates, with the help of Born’s rule which converts complex quantities into real one, by means of probabilistic predictions of the outcome of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Bohr noted in 1927 (before the Hilbert space version of the formalism was introduced, while the role of ℂ in the formalism was already in place), “the symbolic [rather than representational] character of Schrödinger’s method appears not only from the circumstance that its simplicity, similarly to that of the matrix theory [of Heisenberg], depends essentially upon the use of imaginary arithmetic quantities” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All theories just mentioned, apart from QM, are based in the assumption, defining all “epistemologically classical theories,” as they may be called (which designation would apply to relativity as well), that one can observe the phenomena considered without disturbing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As a result, these phenomena can be identified with the corresponding physical objects and their independent behavior and, ideally, represent this behavior and predict it, in the case of individual or simple systems, ideally exactly, by using this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is no longer possible in dealing with quantum phenomena, regardless of interpretation, and hence also in realist interpretations of QM, or alternative theories, such as Bohmian mechanics, of quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, this situation opens the possibility of RWR interpretations of QM or QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The irreducible role of measuring instruments in the constitution of quantum phenomena grounded Bohr’s interpretation, in all its versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As noted from the outset, Bohr adjusted, sometimes significantly, his interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Bohr argued in the Como lecture, which presented his first interpretation of QM (but the argument was retained in all versions of his interpretation, including the ultimate, RWR, one), in classical physics and relativity “our … description of physical phenomena [is] based of the idea that the phenomena concerned may be observed without disturbing them appreciably” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 53;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' emphasis added].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, “any observation of atomic phenomena will involve an interaction [of the object under investigation] with the agency of observation not to be neglected” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 54;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' emphasis added].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One should keep in mind the subtle nature of this contrast: the interaction between the object under investigation and the agency of observation gives rise to a quantum phenomenon rather than disturbs it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr became weary of the language of “disturbing of phenomena by observation” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='11 Bohr grounded his interpretation (in all its versions) in this role and, in the ultimate version of his interpretation, in the strong RWR concept of reality, as applied to quantum objects, placed beyond conception and thus made invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The behavior of the observable parts of measuring 11 Relativity represented a step in this direction, insofar as, in contrast to Newtonian mechanics, space and time were no longer seen as preexisting (absolute) entities then measured by instruments, such as rods and clocks, but were instead defined by the latter in each local reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Still the interference of observational instruments into the behavior of the objects considered could be disregarded, thus allowing, as in classical physics, identification of these objects with the observed phenomena for all practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, the objects under investigation can be considered independently of their interactions with measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum phenomena preclude this type of idealization, to the point of, in RWR interpretations, making the ultimate nature of the reality responsible for quantum phenomena invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr often reflected on these affinities, as well as differences, between relativity and quantum theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 115, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 41, 1935, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 701-702]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 15 instruments and, with them, quantum phenomena were idealized as representable by means of classical physics, by the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Eventually, Bohr adopted the term “phenomenon” to refer strictly to what is observed in measuring instruments, as effects of their interaction with quantum objects: I advocated the application of the word phenomenon exclusively to refer to the observations obtained under specified circumstances, including an account of the whole experimental arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In such terminology, the observational problem is free of any special intricacy since, in actual experiments, all observations are expressed by unambiguous statements referring, for instance, to the registration of the point at which an electron arrives at a photographic plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Moreover, speaking in such a way is just suited to emphasize that the appropriate physical interpretation of the symbolic quantum-mechanical formalism amounts only to predictions, of determinate or statistical character, pertaining to individual phenomena appearing under conditions defined by classical physical concepts [describing the observable parts of measuring instruments].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' emphasis added] As defined by “the observations [already] obtained under specified circumstances,” phenomena refer to events that have already occurred and not to possible future events, such as those predicted by QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is the case even if these predictions are ideally exact or deterministic, which they can be in certain circumstances, such as those of EPR type experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The reason that such a prediction cannot define a quantum phenomenon is that a prediction for variable Q (for example, that related to a coordinate, q) cannot, in general, be assumed to be confirmable by a future measurement, in the way they can be in classical physics or relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One can always perform a complementary measurement, that of p (the momentum), which will leave any value predicted by using Q undetermined by the uncertainty relations, which in principle preclude associating a physical reality corresponding to a coordinate q when one measures p [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 210-212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As earlier, I use capital vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' small letters to differentiate, as is necessary, Hilbert-space elements, here operators, like Q and P, associated with predicting the values of measured quantities, like q and p, observed on measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, one can never speak of both variables unambiguously, even if they are associated with measuring instruments, while any references, even that to a single property of a quantum object considered independently is ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical physics, this difficulty does not arise because one can, at least in principle, always define both variables simultaneously and unambiguously speak of the reality associated with both variables and assign them to the object itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, in any quantum experiment we always deal with a system containing an object and an instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, in considering quantum phenomena, on the one hand, there is always a discrimination between an object and an instrument, and, on the other, the impossibility of physically separating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This impossibility compelled Bohr to speak of “the essential ambiguity involved in a reference to physical attributes of objects when dealing with phenomena where no sharp distinction can be made between the behavior of the objects themselves and their interaction with the measuring instruments,” as opposed to a reference to what is observed which, as classical by the Bohr postulate, can be unambiguous and communicated as such [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are several reasons for adding the Dirac postulate to the Heisenberg and Bohr postulates, defining Bohr’s ultimate interpretation, in QM and, as noted, even more so in QFT, as considered in [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 273-306, 2021b, 2022b], beginning with the fact that no properties can be assigned to a quantum object apart from observations in Bohr’s interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, in any strong RWR interpretation, nothing at all can be said or even thought about what happens between observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Need one, then, still speak of quantum objects between observation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Also, as explained in detail in Section 5 (although, as just indicated, this point is implied by Bohr’s concept of a phenomenon), in quantum physics in each experimental arrangement defining an observation one must, regardless of interpretation, discriminate “between those parts of the physical system considered which are to be treated as measuring instruments and those which constitute the objects under investigation” [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 701].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The difference between them is, however, not uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is related to the arbitrariness of the “cut,” considered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For the moment, it follows that it is how one sets up an experiment that defines what is “the object under investigation” in this experiment, which invites assuming the concept of a quantum object to be applicable only at the time of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The situation has additional complexity 16 because, while still treatable by means of QM, “the object under investigation” in a quantum experiment may not be strictly a quantum object: it may be partly classical, for example, containing the cat of the cat experiment, which is why Bohr speaks here of “objects under investigation” rather than quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This object, however, can only be only partly classical because it must contain, as its part, a properly quantum object, such as an electron or a photon, or some composite quantum object, to observe quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I shall return to this aspect of the situation, which bears importantly on the cat experiment, in Section 5, merely noting here that, a classical part, such as the cat, of such a combined object is always the same object and hence does not obey the Dirac postulate, which only applies to quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In accordance with Bohr’s concept of a phenomenon, whatever is the object of investigation in a quantum experiment, it cannot be considered independently of its interaction with the measuring instrument, thus, making any quantum experiment and hence quantum theory involve both a combination and a separation of an object and an instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Because, however, the object under investigation in a quantum experiment must contain a properly quantum object, this is also a combination of what is invisible to thought and cannot be communicated unambiguously, and what is visible to thought, via observational instruments, and can be communicated unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, as noted, the Bohr postulate also manifests the transition, via observation, from the ultimate, “quantum,” reality to the classical level of observation, and conversely, in the initial preparation of an experiment, from the classical level of observation to the ultimate, “quantum,” reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' According to Bohr: The essential lesson of the analysis of measurements in quantum theory is thus the emphasis on the necessity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' in the account of the phenomena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' of taking the whole experimental arrangement into consideration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' in complete conformity with the fact that all unambiguous interpretation of the quantum mechanical formalism involves the fixation of the external conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' defining the initial state of the atomic system concerned and the character of the possible predictions as regards subsequent observable properties of that system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any measurement in quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and it is first the combination of measurements of both kinds which constitutes a well-defined phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1938, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 101] One begins an experiment by classically preparing an observational instrument and registering, at time tprep, the data obtained by the interaction between this instrument and a quantum object, thus setting up the workings of the ultimate reality considered, placed beyond representation or even conception, by the Heisenberg postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is the classical to the quantum, the visible to thought to the invisible to thought, conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Then by setting up a new observational device, one makes a new observation at time tobserv registering an outcome of the experiment, possibly as predicted by QM, in which case the observational instrument needs to be prepared accordingly, for, as explained, one can always perform a different type of measurement at this moment in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is the quantum to the classical, the invisible to thought to the visible to thought, conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If, however, in assuming, as I do here, the Dirac postulate, a quantum object is only an idealization defined by an observation, rather than of something that exist independently (vis-à-vis the ultimate reality responsible for quantum phenomena assumed to exist independently), could one still speak of the same quantum object, say, the same electron, in two successive observations, with the second confirming the prediction based on the first of the formalism of QM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The case can be given a strictly RWR interpretation, insofar as all these properties are, physically, those of measuring devices, impacted by quantum objects, rather than of these objects themselves, placed beyond representation or conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Rigorously speaking, if the concept of a quantum object is only applicable at the time of observation, then a prediction based on a given measurement and the new measurement based on this prediction could only concern a new quantum object, and not an object that one measured earlier in making a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Accordingly, one deals with two different quantum objects, two different electrons, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' To consider them as the same electron is, however, a permissible idealization in low-energy QM, or low- energy QFT, regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, speaking of the same electron in successive measurements in high- energy (QFT) regimes is meaningless, because these measurements can register quantum objects of 17 different types, say, in the case quantum electrodynamics (QED) an electron in the initial and a positron or photon in the next measurement [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 279-292, 2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QFT supports adding the Dirac postulate to the Heisenberg and Bohr postulate, in RWR interpretations, but, as I argue here, there are reasons also to do so in low-energy (QM) regimes, including, it may be shown, the complexities involved in the double-slit and related experiments [Plotnitsky 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, there is no difficulty in speaking of the same classical object, even if it is part of the object of investigation by quantum means, such QM, in a quantum experiment, which, again, require a properly quantum object, such as emitted particle in the cat experiment, to be a quantum experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At all stages of the cat experiment, we deal with the same cat, dead or alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The state of a classical object can change in time, just as the state of a measuring instruments does when impacted by a quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If one tosses a coin, its state will change throughout its trajectory before it falls with either head or tail side up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Of course, this “sameness” is an idealization, possible and necessary in classical physics or relativity, or in dealing with classical objects, including measuring instruments, in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Heraclitus famously said, one cannot step in the same river twice because neither the river nor the one who steps into it is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such concepts, however, do not apply to quantum objects, such as electrons, which, while they can change their location, momentum, or energy, are considered as strictly indistinguishable from each other in terms of any invariant characteristics, such as mass, changes, or spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In RWR interpretations, these quantities are still only observable as effects of the interactions between quantum objects and measuring instruments, because no concepts apply to quantum objects, whether one defines them as existing independently, as in Bohr, only at the time of measurement, as here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Two key concepts defining classical physics and relativity, (classical) “measurement” and (classical) “causality,” become no longer applicable in quantum theory in RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The term “measurement” is a remnant of classical physics and the history that shaped it, beginning with ancient Greek thinking and the rise of geometry, geo-metry, there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In Bohr’s and the present view, a quantum measurement does not measure or, in the first place, is not an observation of any property of the ultimate constitution of the reality responsible for quantum phenomena, a property that this reality would be assumed to possess before or even during the act of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concept of observation requires a redefinition as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' An act of observation in quantum physics establishes, creates, quantum phenomena by an interaction between the instrument and the quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This act is a unique event of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='12 This view also gives a new meaning to and gives a central significance to the category of event, as defining a new physical situation each time, akin to important events that transform the situation in life or culture, including politics, except that in quantum physics every event of observation radically transforms the situation and redefines the possibly future vis-à-vis the preceding events, no longer meaningful for predictions concerning the future from this point on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As a result, quantum theory becomes a theory of transition probabilities between events, thus defined by experimental technology and our decisions concerning which experiment to performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Then what is so observed as the data or information can be measured classically, just as one measures what is observed in classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There, however, what is observed or measured could be associated with the object considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In quantum physics, there is a difference between observations, which construct phenomena, and measurements, which classically measure physical properties of the phenomena thus constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In speaking of “quantum measurement,” I refer to this whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It follows that measuring instruments must contain both the visible (even to our immediate phenomenal perception) observable classical stratus and the quantum stratum, which 12 While this formulation echoes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Wheeler’s invocation, inspired by Bohr, of “an elementary act of creation,” the present view and, I would argue that of Bohr, may be different from Wheeler’s view of a “participatory universe” [Wheeler 1983, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 189, 194;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Wheeler 1981].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As existing independently of us, as it is assumed to be in the present and Bohr’s views, (the reality of) the universe is not participatory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' only independent phenomena, as created by us, and the world we experience are participatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is possible that by “a participatory universe,” Wheeler refers to the world of our experience, including of quantum phenomena as our acts of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Wheeler, however, does not qualify his view in this way, and in any event, he never advances, and does not appear to adopt, the idea of reality without realism, which underlies the present and, I argue, Bohr’s view, even though Bohr does not use the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 18 enables their interactions with quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This interaction is quantum and cannot be observed and, in RWR interpretations, be visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is, in Bohr’s language, “irreversibly amplified” to the classical level of observable effects, such as a spot left on a silver screen [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='13 The nature of causality in QM changes as well, as classical causality is no longer possible in RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By “classical causality” I refer to the claim that the state, X, of a physical system is determined, in accordance with a law, at all future moments of time once its state, A, is determined at a given moment of time, and state A is determined by the same law by any of the system’s previous states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This assumption implies a concept of reality, which defines this law, thus making this concept of causality ontological or realist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are several reasons for my choice of “classical causality,” rather than just causality, used more commonly for this type of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The main one is that it is possible to introduce alternative, probabilistic, concepts of causality, applicable in QM, including in RWR interpretations, where classical causality does not apply (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 207-218]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Some, beginning with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Laplace, have used “determinism” to designate classical causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I define “determinism” as an epistemological category referring to the possibility of predicting the outcomes of classically causal processes ideally exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical mechanics, when dealing with individual or small systems, both concepts become equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, classical statistical mechanics or chaos theory are classically causal but not deterministic in view of the complexity of the systems considered, which limit us to probabilistic or statistical predictions concerning their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In quantum phenomena, deterministic predictions are not possible even in considering the most elementary quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is because the repetition of identically prepared quantum experiments in general leads to “different recordings” of the observed data (associated with the kinematic and dynamical variables), and unlike in classical physics, this difference cannot be diminished beyond the limit, defined by h, by improving the capacity of our measuring instruments [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Recordings” refers to both those of the initial measurement, enabling a prediction, and those of the second measurement, which would verify this prediction, a combination that, as explained above, generally defines an experiment in physics, but takes a new meaning in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These recordings will be different either one repeats the whole procedure in the same set of experimental arrangements or if one builds a copy of the apparatus and sets it up in the same way, as we do to separately verify the outcomes of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Either repetition is always possible because the preparations of the instruments could be controlled classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, their interaction with quantum objects (or in the present view, the ultimate reality responsible for quantum phenomena and, at the time of measurement, quantum objects) cannot be controlled, which compelled Bohr to speak of “the finite and uncontrollable interaction between the object and the measuring instruments in the field of quantum theory” [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 700].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The respective probabilities of the first and the second measurements are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The most crucial, however, is the difference in the outcomes of the second (predicted) measurement in repeated setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As noted, one can prepare any given state, say, 13 The physical nature of this “amplification” is part of the problem, commonly, including by this author, seen as unsolved (although there are claims to the contrary, for example, on lines of decoherence or consistent histories approaches), of the transition from the quantum to the classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The subject is beyond my scope here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Fortunately, quantum phenomena and QM allow us to bypass this problem in quantum measurements or predictions, seen here as the transition from the invisible to thought to the visible to thought (or vice versa in a preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Bohr noted, QM is “justified only by the possibility of disregarding in its domain of application the atomic structure of measuring instruments themselves in the interpretation of the results of experiments” [Bohr 1937, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This disregard, as Bohr observed, may lead to new complexities in high-energy physics and QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As he said,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' invoking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' a renunciation of visualization: “For a correlation of still deeper laws of nature involving not only the mutual interaction of the so-called elementary constituents of nature but also the stability of their existence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' this last assumption can no longer be maintained,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as we must be prepared for a more comprehensive generalization of the complementary mode of description which will demand a still more radical renunciation of the so-called visualizations” [Bohr 1937,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As it happens, QFT (in high-energy regimes) still disregards “the atomic structure of measuring instruments,” which may be responsible for the appearance of infinities and the necessity of renormalization and other, still unresolved, complexities there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 19 that of a “spin-up,” as manifested in the corresponding measurement, even though one cannot do so in a single experimental preparation but only by post-selecting the required preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, the outcome of the second (predicted) measurement cannot be controlled at all, only allowing one to predict the probability or, if the experiment is repeated, statistics of the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The statistics of the outcomes of multiply repeated experiments performed in both such experimental settings will be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, an individual quantum experiment cannot be reproduced, as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' is always possible to do so in classical physics, because the interference of measurement can be neglected or controlled, at least in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All data observed in quantum experiments remains classical, by the Bohr postulate, and hence visible to thought (or even to the immediate phenomenal perception) and can be communicated unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Unlike in classical physics, however, this data cannot be recreated by a different system, which combines a quantum object (in the present view, again, a concept only applicable at the time of observation) and an apparatus, the observable part of which is described classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This situation embodies the no cloning theorem [Park 1970, Dieks 1982, Wootters and Zurek 1982].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As noted, the probabilistic or statistical character of quantum predictions must, on experimental grounds, hold in interpretations of QM or alternative theories of quantum phenomena (such as Bohmian mechanics) that are classically causal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM or QFT, in RWR interpretations, are not classically causal because the ultimate nature of reality responsible for quantum phenomena is assumed to be beyond a representation or conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Classical causality would imply at least a partial conception and even representation of this reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These circumstances imply a different reason for the recourse to probability in quantum theory in RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' According to Bohr: [I]t is most important to realize that the recourse to probability laws under such circumstances is essentially different in aim from the familiar application of statistical considerations as practical means of accounting for the properties of mechanical systems of great structural complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, in quantum physics we are presented not with intricacies of this kind, but with the inability of the classical frame of concepts to comprise the peculiar feature of indivisibility, or “individuality,” characterizing the elementary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 34] The “indivisibility” refers to the indivisibility of phenomena in Bohr’s sense, defined by the impossibility of considering quantum objects independently from their interactions with these instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Individuality” refers to the assumption that each phenomenon is individual and unrepeatable, as well as discrete relative to any other phenomenon, and correlatively, to the essential randomness of individual quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Collectively they may not be strictly random by virtue of one or another form of quantum correlations (such as EPR-type correlations, at stake in Bell’s or Kochen-Specker theorem), which are, however, strictly quantum as well and not found in classical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This randomness is not found in classical physics, because even when one must use probability there, at bottom one deals with individual process that are classically causal and in fact deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, in classical physics, randomness does not ultimately exist or is assumed ultimately not to exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' only probability does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In principle, one can isolate an individual constituent of the structurally complex mechanical system, say, a molecule of a gas, something that, as classical, is in its behavior, visible to thought, and predict its behavior ideally exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is, however, the “in principle” that is crucial, because this is never possible in considering individual quantum systems, no matter how elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By the same token, such systems or (since the term “system” is not ultimately applicable either, except at the time of measurement) the ultimate nature of the reality considered can never be made visible to thought, which is, again, the reason why they cannot be assumed to be classically causal or predicted deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, as explained, the possibility of never observing quantum objects as isolated defined Bohr’s concept of a phenomenon, and in the present (more radical) view, quantum objects are only defined, still as invisible to thought, at the time of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum physics, then, contains an essential randomness not found in classical physics, which is at bottom classically causal and, when it comes to the behavior of its elemental individual constituents, deterministic, thus making the recourse to probability a practical, epistemological matter, as Bohr says.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A coin toss is an example of a classical probabilistic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The outcome can, in 20 practice, only be predicted probabilistically due to the mechanical complexity of the process, beginning with the motion of the hand tossing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, this is still a classically causal process, the outcome of which is determined and could, in principle, be predicted ideally exactly with sufficient technical and computational capacities, which is, as explained, the meaning of “classically causal.” The recourse to probability is practical, epistemological, due to our lack of knowledge concerning the underlying behavior of the systems considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the case of any quantum system, no matter how simple, this idealization is not possible: its behavior is not assumed to be classically causal in RWR interpretations, and as invisible to thought, it cannot be so assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='14 Quantum physics, however, only contains this randomness, rather than is entirely random, because it allows for probabilistic or statistical predictions (purely random events do not, which makes it impossible to handle them scientifically) and, more crucially, correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One of the greatest mysteries of quantum phenomena is how random individual events can, under certain circumstances, give rise to an order, even if only a (statistical) correlational order [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 253-256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM predicts these correlations, but at least in RWR interpretations, it does not explain them, any more than it explains how any single outcome of an observation or measurement, comes about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The emergence of either is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I shall now explain Bohr’s concept of complementarity, especially, as it appears in his ultimate, strong RWR interpretation, where it applies to phenomena in Bohr’s sense as outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As defined generally complementarity is characterized by: (A) a mutual exclusivity of certain phenomena, entities, or conceptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and yet (B) the possibility of considering each one of them separately at any given point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and 14 One might further distinguish between indeterminacy, as a more general category, and randomness, as a most radical form of indeterminacy, when a probability cannot be assigned to a possible event, which may also occur unexpectedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Both indeterminacy and randomness only refer to possible future events and define our expectations concerning them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Once an event has occurred, it is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' An indeterminate nature of events may either allow for assuming an underlying classically causal architecture (which may be temporal) of the physical reality responsible for this nature, whether this process is accessible to us or not, or disallow for making such an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The first case, as just explained, defines indeterminacy in classical physics, such as classical statistical physics or chaos theory, or more radically in considering the so-called algorithmic complexity, such as Kolmogorov complexity (also known as Solomonoff-Kolmogorov-Chaitin complexity), which may not be computable, but still for practical, epistemological reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The second is found in QM or QFT in RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' According to Bohr, the idea of indeterminacy (or, again, randomness) apart from a classically causal order has “hardly been seriously questioned until Planck’s discovery of the quantum of action” (Bohr 1938, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As he said on a later occasion (in 1949): “[E]ven in the great epoch of critical [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', post-Kantian] philosophy in the former century, there was only a question to what extent a priori arguments could be given for the adequacy of space-time coordination and causal connection of experience, but never a question of rational generalizations or inherent limitations of such categories of human thinking” (Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Even more radical philosophical questionings of the classical idea or ideal of causality, such as those by David Hume, are those of our epistemological capacity to perceive the underlying classically causal world, which would be presupposed at the ultimate level as inaccessible to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is impossible to ascertain that an apparently random sequence of events, events that occurred apparently randomly, was in fact random, rather than connected by some rule, such as that defined by classical causality, and there is no mathematical proof that any “random” sequence is actually random (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', Aaronson 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 71-92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The sequences of indeterminate events that allow for probabilistic predictions concerning them is a different matter, although there is still no guarantee that such sequences are not ultimately underlain by classically causal connections in the case of quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Experimentally, again, quantum phenomena only preclude determinism, because identically prepared quantum experiments in general lead to different outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It follows that the claim of quantum randomness can, in principle, be falsified, but establishing a classically causal theory or algorithm that reproduces the indeterminate or random data in question, which becomes no longer indeterminate random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This would imply that RWR interpretations, which precludes such connections, does not correspond to the ultimate nature of reality responsible for quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' See [D’Ariano 1922], which establishes the existence of a falsifiable quantum random generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, such a generator cannot be classical, because all classical (or relativistic) theories of individual systems are deterministic, that is, can be so idealized as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 21 (C) the necessity of considering all of them at different moments of time for a comprehensive account of the totality of phenomena that one must consider in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concept was never given by Bohr a single definition of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, this definition may be surmised from several of Bohr’s statements, such as: “Evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena [some of which are mutually exclusive] exhaust the possible information about the objects” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical mechanics, we can comprehend all the information about each object within a single picture because the interference of measurement can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This allows us to identify the phenomenon with the object under investigation and establish the quantities defining this information, such as its position and momentum, in the same experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In quantum physics, this interference cannot be neglected and leads to different, in fact mutually exclusive, experimental conditions for each measurement and their complementarity, in correspondence with the uncertainty relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The situation implies two incompatible pictures of what is observed, as phenomena, in measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, the possible information about a quantum object, the information to be found in measuring instruments, could only be exhausted by the mutually incompatible evidence obtained under different experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, once made, either measurement, say, that of the position, will provide the complete actual information (manifested in measuring instruments) about the object, as complete as possible, at this moment in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One could never obtain the complementary information, provided by the momentum measurement, at this moment in time, because to do so one would need simultaneously to perform a complementary experiment on it, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=" Thus, parts (B) and (C) of the above definition of complementarity are as important as part (A) and disregarding them can lead to a misunderstanding of Bohr's concept, often misleadingly identified with just (A)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s complementarity is not only about a mutual exclusivity of things, but also about performing quantum experiments by human agents, in which a mutual exclusivity becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' That we have a free (or at least sufficiently free) choice as concerns what kind of experiment we want to perform is in accordance with the very idea of experimentation in science, including in classical physics [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 699].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, contrary to the case of classical physics or relativity, implementing our decision concerning what we want to do will allow us to make only certain types of predictions and will irrevocably exclude certain other, complementary, types of possible predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In other words, we have a freedom, at least a sufficient degree of freedom, of choice which experiment to perform in classical and quantum physics alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical physics (or relativity), however, it does not matter in fundamental terms because all variables necessary for defining the future course of reality, in accord with classical causality, can always be determined at any moment in time, as there is no complementarity or the uncertainty relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, by virtue of complementarity, it does matter in quantum physics: By staging, by decision, our experiments in one complementarity way or the other, we define the course of reality, even if only probabilistically, because, while we can control the set-up of the experiment, we cannot control the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such uncontrollable outcomes are no longer a matter of surprise that nature confronts us with but is instead what we expect from nature, or our interaction with nature, in quantum experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It also follows that we always, at any point, have a freedom, in any event, a sufficient degree of freedom to make this choice or to change our choice and thus a future course of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Beyond, as discussed below, the Bohr-EPR debate concerning the EPR experiment, this aspect of complementarity is related in Bell’s and the Kochen-Specker theorem, or the Conway-Kochen free will theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter connections are, however, a separate subject beyond my scope here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For the moment, more immediately complementarity is a reflection of the fact that, in a radical departure from classical physics or relativity, the behavior of quantum objects of the same type, say, electrons, or, again, the ultimate nature of reality responsible for quantum phenomena defined by such objects, is not governed by the same physical law, especially a representational physical law, in all possible contexts, specifically in complementary contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This leads to incompatible observable physical effects in complementary contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, the mathematical formalism of QM offers correct probabilistic or statistical predictions of quantum phenomena in all contexts, in RWR interpretations 22 under the assumption that the ultimate nature of reality responsible for quantum phenomena is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='15 However, as Bohr observed, reiterating his argument concerning the nature of quantum probability considered above: Just in this last respect [of the renunciation in each experimental arrangement of the one or the other of two aspects of the description of the physical phenomena] any comparison between quantum mechanics and ordinary statistical mechanics,—however useful it may be for the formal presentation of the theory,—is essentially irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Indeed we have in each experimental arrangement suited for the study of proper quantum phenomena not merely to do with an ignorance of the value of certain physical quantities, but with the impossibility of defining these quantities in an unambiguous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 699] It might be noted that wave-particle complementarity, with which the concept of complementarity is often associated, had not played a significant, if any, role in Bohr’s thinking, especially after the Como lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr was always aware of the difficulties of applying the concept of physical waves to quantum objects or assuming both types of behavior, particle-like and wave-like, pertain to the same individual entities, such as each photon or electron itself, considered independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s ultimate solution to the dilemma of whether quantum objects are particles or waves was that they were neither, any more than anything else, by the Heisenberg postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Instead, either “picture” refers to one of the two mutually exclusive sets of discrete individual effects, described classically by the Bohr postulate, of the interactions between quantum objects and measuring instruments, particle-like, which may be individual or collective, or wave-like, which are always collective, composed of discrete individual effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' An example of the latter are interference effects, composed of a large number of discrete traces of the collisions between the quantum objects and the screen in the double-slit experiment in the corresponding setup (when both slits are open and there are no means to know through which slit each object has passed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These two sets of effects may be seen as complementary, also when it comes to calculating the probabilities or statistics for each set of events, or, if one takes a Bayesian view, for each event of each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The two types of effects involved are mutually exclusive and require mutually exclusive experimental setups to be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical physics, wave-like (radiation) and particle-like objects or (as they can be identified) phenomena were treated by two mutually exclusive theories, which is not the same as being complementary in Bohr’s sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter must include (B) and (C) part of the concept, applicable to the same (quantum) objects or the ultimate reality responsible for quantum phenomena, but leading two different phenomena by (A), depending on which setup one decided to use, predicted, differently, by the same theory, QM or QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I would like, in closing my discussion of the (strong) RWR view, as defined by the role of the invisible to thought in quantum physics, briefly to reflect, from this perspective, on the EPR experiment and the Bohr-EPR exchange concerning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' My reflections follow [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 227-272], which offers a detailed discussion, although the angle of visible and invisible to thought is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The case is, however, both exemplary and highly significant in this context, as Bohr, as noted above, realized in stressing the significance of the EPR experiment as “suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization” (Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One might give a new angle on and amplify Bohr’s point, by arguing that that in their argument, EPR in effect assume that the independent reality of quantum objects is visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' EPR’s argument is, however, based on disregarding or at least not adequately considered the constitutive role of observational instruments in defining quantum phenomena in the way Bohr argued to be necessary, based on an analysis of this role in his reply [Bohr 1935].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, while EPR do, unavoidably, refer to “measurement,” EPR do not considered or even mention measuring instruments, the constitutive role of which in defining all physical variable concerned would make it difficult or even impossible to assume that the ultimate nature of reality responsible for quantum phenomena can be visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 15 This situation is also responsible for what is known as “contextuality,” which was considered from the RWR perspective in [Plotnitsky 2019, Plotnitsky 2021a], and, along different lines, in [Jaeger 2019, Howard 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' See also Khrennikov’s extended survey [Khrennikov 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 23 EPR advanced the following argument based on the criterion of reality they which, they thought, equally applicable in classical and quantum physics: “If, without in any way disturbing a system, we can predict with certainty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity” [Einstein et al, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While, however, this criterion is unproblematically applicable in classical physics, it is, Bohr contended, “ambiguous” in the case of quantum phenomena, because of the role of measuring instruments in defining all such quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' EPR’s argued that it is possible to ascertain “an element of reality” pertaining to a quantum object, the second, S2, object of the EPR pair (S1, S2), independently of any interaction between S2 and a measuring instrument (thus “without in any way disturbing the system”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This association is made possible by a prediction by means of QM, say, of variable q (like that associated with the position operator Q) with “probability equal to unity,” a prediction based on the measurement performed on S1, as is indeed possible, at least ideally or in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As earlier, I use capital letters, Q or P, to refer to the operators in the Hilbert space considered, and small letters, q or p, to physical variables probabilistically predicted by the formalism by using Q or P, which have no physical connections to q or p apart from these predictions in RWR interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' EPR argue that because this prediction, “with probability equal to unity,” is possible “without in any way disturbing” S2, this property could be ascertained as an element of physical reality pertaining to S2, in accordance with their criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As such, it is in effect assumed to be visible to thought and unambiguously communicable, even though it is not actually observed or measured, and as such is not available to our immediate phenomenal perception at the time of prediction, or at any time, unless a measurement is performed, or even then because we can only perceive what is observed in measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While in the latter case it is in principle possible to associate such a measured quantity with an element of reality pertaining to the object itself, one is, obviously, outside the situation covered by EPR’s criterion, because we no longer deal with a prediction and of course disturb the object by an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, it is the concept of visible to thought that is crucial here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr counterargued that, while EPR’s claim would work in classical physics, the situation was different in considering quantum phenomena, including those of the EPR type, because of the essential role of observational instruments in the constitution of all quantum phenomena and, thus, in any unambiguous application of the concept of reality or of an element of reality in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This role, he argued, must be taken into consideration even in the case of predictions “with probability equal to unity” without a measurement previously performed on the system considered, S2, and instead by using a measurement performed on S1, as is ideally possible in the EPR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As, however, I noted earlier and as Bohr argued in his reply, this prediction is not sufficient for assigning an element of physical reality to S2, contrary to EPR’s claim based on their criterion of reality, assumed by then to equally apply in both classical and quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is, however, not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical physics, where one can, in principle, always measure and define both variables simultaneously, by neglecting the interference of observational instruments, it is possible to speak, at any moment in time, unambiguously of the reality in terms of its physical elements, thus visible to thought, associated with both conjugate classical variables, Q and P (as functions of real variables) and define them as pertaining to the object itself considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Everything, at any point, is always visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Not so, in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Let us assume that by measuring qS1 on S1 and using the formalism, applied to Q (an operator in a complex Hilbert space), one makes a prediction, “with probability equal to unity,” concerning qS2 associated, via a measuring instrument, with S2 at some future time, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If one measures q at time t, one then will indeed obtain the value qS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, one can, instead of q, always measure at time t the complementary variable, p (which would relate to the momentum operator P in the formalism, although one does not use in a measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If one does so, the value of q becomes completely undetermined, ambiguous, by the uncertainty relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, this measurement of p would preclude associating any physical reality with the predicted value qS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, qS2, as defined by this prediction, may be visible to thought, but it can no longer correspond to any element of physical reality that can be associated with S2, or at least there is no way to experimentally ascertain such a correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' S2 is assumed to exist and hence be real, but to assume so is not the same as associating 24 an element of reality with it and thus making it visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='16 This association is only possible if the measurement, confirming the predicted value, qS2, is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Doing so, however, can be in principle precluded by making a complementary measurement and, thus, in contrast to classical physics (where both conjugate variables can always be assigned, corresponding to elements of reality, simultaneously), disabling the association of the predicted value qS2 with S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is so even if one assumes that one can associate an element of reality with S2 as such, rather than only with a classical observed part of a measuring instrument, at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In other words, unless the corresponding measurement is performed, qS2 can correspond to no elements of physical reality, and the possibility of establishing such a correspondence can be denied if one measures p instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This situation is captured by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Peres’s statement that “unperformed experiments have no result” [Peres 1978].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s claim concerning “an essential ambiguity” of EPR’s criterion is defined by this situation, not considered by EPR in advancing this criterion, or by Einstein in his related arguments, based in the same criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Bohr stated in the passage of his reply cited above, in view of complementarity, “we have in each experimental arrangement suited for the study of proper quantum phenomena not merely to do with an ignorance of the value of certain physical quantities, but with the impossibility of defining these quantities in an unambiguous way” (Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 699;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' also Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is, he argues, absolutely no possibility to unambiguously define both “elements of reality” in question for S2, “without in any way disturbing” it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One can only do so for one or other of complementarity quantities, say, q, by making the corresponding measurement on S1, qS1, and predicting the reality of the same type of element, qS2, for S2, still under the assumption that one could in principle perform the corresponding measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' That, however, irrevocably precludes one from predicting the complementary element of reality, pS2, for S2, because any measurement of p on S1 was precluded by measuring qS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, if one instead measures p on S2, which of course would require disturbing S2, one in turn irrevocably precludes ascertaining qS2 as an element of reality pertaining to S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Locating this ambiguity enables Bohr to argue that QM can be seen as both complete within its proper scope (as complete as nature allows our theory of low-energy quantum phenomena to be) and local, insofar it does not entail any physical action at a distance, or at least that EPR, who argued that QM is either incomplete or nonlocal in this sense, did not demonstrate otherwise, as explained in detail in [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 227-272].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='17 The EPR-Bohr exchange was crucial for the development of Bohr’s thinking, leading him to his ultimate, strong RWR interpretation and, correlatively, a deeper understanding of the nature of complementarity as a physical concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It compelled Bohr eventually to adopt the view that no measurable quantity, even a single such quantity (rather than only both complementary quantities, as precluded by the uncertainty relations) and hence no element of reality can be attributed to a quantum object even at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While a quantum object was assumed by Bohr to exist and hence be real independently of observation, any reference to the nature of its reality becomes ambiguous, making Bohr speak of “the essential ambiguity involved in a reference to physical attributes of objects when dealing with phenomena where no sharp distinction can be made between the behavior of the objects themselves and their interaction with the measuring instruments” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such 16 It should be kept in mind that, as Schrödinger was the first to note in defining entanglement, in dealing with entangled systems it is not possible to speak of the properties on each system separately or even (Schrödinger does not appear to go that far, at least not expressly) even of two separate systems [Schrödinger 1935, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 160-161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, once a measurement on S1 is performed (thus also establishing it as quantum object], S1 and S2 are no longer entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This situation, it might be added, gives another justification to using the Dirac postulate, which only defines either system as such at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 17 EPR were aware and assumed in their argument that both quantities cannot be measurement or predicted simultaneously, and their criterion of reality allows for assigning both “elements of reality” to S2 without simultaneously predicting both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' They argued, however, that the only alternative to their argument is the assumption of the nonlocality (an action at a distance) of QM or quantum phenomena [Einstein et al, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr counterargued that this nonlocality, as well as the incompleteness of QM, can be avoided by virtue of the ambiguity, and hence inapplicability, of EPR’s criterion of reality to quantum phenomena, as here discusses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A detailed argument is offered in [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 227-272].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 25 attributes, as elements of reality, can only be unambiguously ascribed (under the constraint of the uncertainty relations) to certain parts, elements, of quantum phenomena, defined by the observable parts of measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This fact makes these elements open to being described by classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While, however, Bohr associated the ultimate, invisible-to-thought, reality responsible for quantum phenomena with quantum objects, in the present interpretation, by the Dirac postulate, the concept of a quantum object is only applicable at the time of observation, still as an RWR concept, which precludes associating any attributes, elements, with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The character of the ultimate reality considered as invisible to thought equally defines both interpretations arising from, and arguably reaching the most radical manifestation of, “the spirit of Copenhagen,” in Heisenberg’s memorable phrase “der Kopenhagener Geist der Quantenheorie,” honoring Bohr’s contribution to our understanding of quantum theory [Heisenberg 1930, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' iv)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The existence, at least a possible existence, of a reality invisible to thought (the Heisenberg postulate), which is, at the same time, ultimately responsible for what is visible to thought in quantum phenomena (the Bohr postulate), is what Bohr saw as “an epistemological lesson of quantum mechanics” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At least, this is an epistemological lesson of his interpretation of quantum mechanics, to which the present interpretation adds the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Perhaps, however, quantum mechanics or physics in general cannot teach us lessons otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is just that there appears now (this has not always been the case) to be more consensus, albeit not an entirely unanimous one either, as concerns our interpretation of classical physics and relativity as realist theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' When it comes to QM or QFT, the proliferation of diverse (and sometimes incompatible) interpretations and the debate concerning them, still overshadowed by the Bohr-Einstein confrontation, continue with an undiminished intensity and no end in sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But then, the stakes are high: the future of our understanding of nature and thought alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The Bohr-Schrödinger exchange on classical concepts in quantum measurement Bohr’s insistence, reflecting (in present terms) the Bohr postulate, on the indispensable role of classical physical concepts in considering measuring instruments is often misunderstood, and the subject is significant in the context of the cat experiment, which provides an additional reason for addressing this insistence in detail in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is instructive to consider in this connection Schrödinger’s comments on this aspect of Bohr’s thinking in Schrödinger’s letter after reading Bohr’s reply to EPR (in a prepublication version), while working on his cat-paradox paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The exchange, relevant to Schrödinger’s overall argument in his paper, might have also affected his comments on the cat experiment, although the origin of the experiment appears to be a suggestion by Einstein [Fine and Ryckman 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger’s (long) letter and Bohr’s (brief) reply in part resume an earlier exchange, in 1928-1929, on the subject among Einstein, Schrödinger, and Bohr [Plotnitsky 2021a, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 32-34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger writes: You [Bohr] have repeatedly expressed your definite conviction that measurements must be described in terms of classical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For example, on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 61 of the volume published by Springer in 1931 [the original German edition of [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1]]: “It lies in the nature of physical observation, that all experience must ultimately be expressed in terms of classical concepts, neglecting the quantum of action” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 94-95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' And ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 74 “the invocation of classical ideas, necessitated by the very nature of measurement” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' And once again [in the reply to EPR] you talk about “the indispensable use of classical concepts in the interpretation of all [proper] measurement” [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 701, where the printed version adds “proper”].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' True enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' shortly thereafter you say: “The removal of any incompleteness in the present methods of atomic physics … might indeed only be affected by a still more radical departure from the methods of description of classical physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' involving the considerations of the atomic constitution of all measuring instruments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which it has hitherto been possible to disregard in quantum mechanics.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This might sound as if what was earlier characterized as inherent in the very nature of any physical observation as an “indispensable necessity”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' would on the other hand after all just be a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' fortunately still permissible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' convenient way of conveying information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' a way we presumably sometime will be forced to give up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If this were your opinion, then I would gladly agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' However, the subsequent stringent and clear comparison with the theory of relativity make me doubt whether, in what I just said, I have understood your 26 view correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Because, if we considered the theory of relativity as a conceptual edifice in itself, without any relationships to quantum mechanics, we would presumably never be able to renounce the sharp separation between space and time in any measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Still, it seems possible that in connection with the unavoidable mutual modification of these two theories, both would be forced to shake off their classical eggshell—and that this is what you mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (Letter to Bohr, October 13, 1935 [Bohr 1972-1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 505]) As Schrödinger admits (“it seems possible”), this may not be and, I would argue, is not what Bohr means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' First, especially given that Bohr’s argued in his reply to EPR that QM is a complete theory within its scope (as complete as nature allows our theory of nonrelativistic quantum phenomena to be), it is clear that “incompleteness” in Bohr’s passage cited by Schrödinger does not refer to QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It refers to the fact that at the time QFT was hardly adequately developed even in the case of QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Yukawa’s meson theory of nuclear forces just introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') QED, too, only worked then to the first order of approximation, beyond which QED led to the appearance of infinities, which were only handled by renormalization fifteen years later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The passage in question was removed from Bohr in the published version of his response to the EPR paper [Bohr 1935], as Bohr explained in his reply to Schrödinger’s letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' He said: “I have left out the reference to the possible significance of the atomic constitution of all measuring instruments for the solution of the still unexplained difficulties of electron theory [QED].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The reason is that together with Rosenfeld I am just about to finish a paper about a measuring problem in electron theory in which this question will be elucidated somewhat more fully” [Letter to Schrödinger, October 25, 1935, in Bohr 1972-1996, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 511].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='18 Schrödinger was aware that Bohr referred to the incompleteness of QED and possibly QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is clear, however, from this comment and related elaborations by Bohr including in [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 89-91, 115], to which Schrödinger refers in his letter, that the point is not that the observable parts of measuring instruments should no longer be described by classical physics in QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Speaking of “a still more radical departure [than in QM] from the method of description of classical physics” only refers to a more radical situation in QFT as concerns a possible necessity, as against QM, of considering the atomic structure of measuring instrument, along with its observable parts, described classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter aspect of quantum measurement would remain in place in QFT in Bohr’s view, for the reasons discussed earlier in the present article and explained in Bohr’s reply to Schrödinger’s letter, while the atomic constitution interaction may need to be considered in a relativistic quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, we still do not have a quantum theory that does so, and as currently constituted, QFT still works in the absence of such an account, which may be responsible for its difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='19 It is not clear either whether such a theory is possible or necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QFT does contain unresolved difficulties (even apart from the absence of a quantum theory of gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It does work, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In works well as a predictive theory or, one might argue, a framework, something sometimes referred to in theoretical physics as “phenomenology” (not to be confused with the used of the term in philosophy or when one speaks of our phenomenal representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QED is now the best confirmed physical theory ever as concerns its predictions, probabilistic or statistical as they are, which predictions are, however, again what quantum experiments allows us, as things stand now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger, by contrast, appeared to think that Bohr believes that such a theory, as well as relativity, “would be forced to shake off their classical eggshells” of the description of measuring instruments, possibly even in QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But, as I argue, this not what Bohr thinks: classical “eggshells” are part of phenomena, and unlike is classical physics, if one shakes them off or breaks them you will only create new eggs with eggshells, without ever exposing, making visible, what is inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One cannot make an omelet out of the eggs of quantum phenomena, only new eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any subdivision of a phenomenon can only result in a new phenomenon or phenomena, still each with classically described “shells,” without ever exposing quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Bohr explained later: “The individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will 18 This paper was not published and only became available in the same volume of Bohr’s collected works [Bohr 1972-1996, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='195-209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 19 See, note 13 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 27 demands a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, Bohr speaks of closed phenomena, or the wholeness or indivisibility of phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr says in his letter to Schrödinger: However, these considerations [of the atomic structure of measuring instruments] do not have any close connection to the Einstein paradoxes and to the question of limitation of the [classically] causal description of quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On this point I must confess that I cannot share your doubts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' My emphasis of [sic: on] the point that the classical description of experiments is unavoidable amounts merely to the seemingly obvious fact that the description of any measuring arrangement must, in an essential manner, involve the arrangement of the instruments in space and their functioning in time, if we shall be able to state anything at all about phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The argument here is of course first and foremost that in order to serve as measuring instruments, they cannot be included in the realm of application proper to quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Letter to Schrödinger, October 25, 1935, Bohr 1972-1996, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 511] In other words, measuring instruments in their observable parts are and must be visible to thought and even to our immediate phenomenal perception, to “be able to state anything at all about phenomena” and thus to unambiguously communicate our findings, along with the mathematics that predicts them, to meet “basic requirements of science,” as Bohr said in his reply to EPR [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 697].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, the ultimate nature of the reality responsible for observed phenomena may be and, in Bohr’s view, is invisible to thought, and hence nothing about it can be communicated unambiguously or at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The same situation is found in high-energy (QFT) regimes, whether we will ever be able to include the atomic constitution of measuring instruments in the theory or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Heisenberg says, following Bohr’s argument, and aware of Bohr’s exchanges with both Einstein and Schrödinger on the subject: Therefore, it has sometimes been suggested that one should depart from the classical concepts altogether and that a radical change in the concepts used for describing the experiments might possibly lead back to a nonstat[ist]ical [sic!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' ], completely objective description of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This suggestion, however, rests upon a misunderstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concepts of classical physics are just a refinement of the concepts of daily life and are an essential part of the language which forms the basis of all natural science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Our actual situation in science is such that we do use the classical concepts for the description of the experiments, and it was the problem of quantum theory to find theoretical interpretations of the experiments on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is no use in discussing what could be done if we were other beings than we are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At this point we have to realize, as von Weizsäcker has put it, that “Nature is earlier than man, but man is earlier than natural science.” The first part of the sentence justifies classical physics, with its ideal of complete objectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The second part tells us why we cannot escape the paradox of quantum theory, namely, the necessity of using classical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Heisenberg 1962, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 56] There is indeed no paradox here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Classical concepts reflect the essential workings of our biological and specifically neurological nature born with our evolutionary emergence as human animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Our thinking, as the product of this machinery, is classical in that it is consistent with and leads to the concepts of classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any concept we form derive from and can only apply to observed phenomena, and quantum phenomena are physically classical as observed phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' They are different from classical phenomena because the data observed in them precludes us from describing how they come about by classical physics (which incapacity led to quantum theory) or in RWR interpretations, any physical theory or even making them available, visible, to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such a conception, which would make the emergence of these data visible to thought, may be precluded by the same evolutionary biological or neurological nature of ours and, thus, by our classical thinking and language, developed in the interaction with (classical) objects consisting of millions of atoms, rather than anything on the atomic scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Heisenberg 1930, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is another manifestation (correlative to classical physics) of the fact that human nature and thus our thought are “earlier than natural science” and limit the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM or QFT, however, allows one to probabilistically predict the data considered, without representing or even without us conceiving of how these data come about, or at least it allows for (RWR) interpretations, according to which QM or QFT does no more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 28 In classical physics we only need one theory for observing (or measuring), representing, and predicting the phenomena considered, which can be identified with the object considered, the interference of observation can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In both relativity and quantum theory (QM and QFT) we need classical theory to observe and measure the phenomena considered and the measuring instruments, but by relativistic and quantum theory, respectively, but with a crucial difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In relativity we can, just in classical physics, still neglect the interference of measurement and, as a result, represent the behavior of the objects considered and predict this behavior, ideally deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In quantum theory this interference cannot be neglected, essentially defining quantum phenomena as different from the objects considered, which makes our predictions, in general probabilistic, regardless of interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In RWR interpretations, quantum objects or in the present view (in which quantum objects are only defined at the time of measurement by the Dirac postulate) the ultimate nature of reality responsible for quantum phenomena is placed beyond representation or conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is true that some classical theories, such as classical statistical mechanics or chaos theory, are probabilistic, but these theories are not fundamental because they do not deal with the ultimate constitution of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As explained,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' fundamental physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as things stand now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' requires three types of theories—classical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which do not consider the roles of both c and h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' relativistic (which are epistemologically classical),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which do not consider the role of h but do that of c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and quantum which must take into account h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and in high-energy regimes c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' with both relativistic and quantum theories still using classical physics in representing the observable parts of measuring instruments and the outcomes of observations or measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These considerations do not imply that new concepts, physical or (which is, however, not the issue at the moment) mathematical, are not possible in quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM and QFT or their understanding and interpretation would not have been possible without the invention of new concepts, with Bohr’s concepts of complementarity and phenomenon, or Schrödinger’s concept of entanglement, among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The question is whether one can avoid classical physical concepts or classical physics or whether new realist concepts, describing the ultimate nature of the reality responsible for quantum phenomena are possible or even necessary, as both Einstein and Schrödinger thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the first question, Bohr’s or the present view is that classical concepts and classical physics cannot be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the second question, Bohr answers or at least that of the present author would be that such new realist concepts or theories may not be possible, which is not the same as are not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' They might be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Complementarity and phenomenon are nonrealist concepts as concerns the ultimate constitution of the reality responsible for quantum phenomenon, but they contain realist components by involving the description of observed phenomena by (“old”) classical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Entanglement, defined as a concept mathematically, could, as concerns the physical reality considered, be understood along RWR lines as well, in accord with Bohr’s view of the EPR experiment, manifesting entanglement and “suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization” [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger himself spoke of the “entanglement of predictions,” defined by the corresponding aspects of formalism, rather than quantum objects [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 161;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' emphasis added].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger was,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' not yet finished in his letter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and asked another question,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which surprised Bohr as revealing something in Bohr’s thinking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' of which Bohr was not entirely aware himself at the time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and which is perhaps the most intriguing part of Schrödinger’s letter: However that may be [as concerns a possible removal of the classical description of observation in relativistic quantum regimes],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' there must be clear and definite reasons which cause you repeatedly to declare that we must interpret observations in classical terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' according to their very nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Whenever you say that, you state it so definitely and clearly, in the indicative, without any reservation like “probably”, or “it might be”, or “we must be prepared”, as if this were the uttermost certainty in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It must be among your firmest convictions— and I cannot understand what it is based upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It could not be just the point (about which you talked so insistently to me already in 1926): that our traditional language and inherited concepts were completely unsuited to describe the phenomena with which we are concerned now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Because, in the course of the development of our science (and mathematics), from its earliest beginning to the situation at the end of the nineteenth century this was certainly the case over and over again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If the break with the old tradition seems greater now than ever before, then we should take into account 29 that a particular time perspective is responsible for forming the impression that developments in which we ourselves take part, stands out as being more important and more essential that earlier ones, which we cite only from history, and whose stages we get to know mostly in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In fact, it is often difficult for us to imagine earlier ways of thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' And although the difficulty of such a historical step back actually speaks most eloquently of how significant [the step] must have seemed to the pioneers of their earlier advances, still now and then we cannot avert feeling: “Incredible that, up to then, people were so narrow-minded!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Here, the underestimation of the time perspective shows itself most clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus I think that the fact that we have not adapted our thinking and our means of expression to the new theory cannot possibly be the reason for the conviction that experiments must always be described in the classical manner, thus neglecting the essential characteristics of the new theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (Letter to Bohr, October 13, 1935 [Bohr 1972-1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 508-509]) Indeed, as Bohr’s reply to Schrödinger, cited above, suggests, this is not the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is not a matter of a going beyond a tradition, say as that of classical physics or even earlier quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is difficult to object, and Bohr would not, to what Schrödinger says on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hence, it would also be difficult to agree that Bohr was ever neglecting the essential characteristics of QM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' quite the contrary, he affirmed them, not the least, as essentially different from classical physical theories, both deterministic or probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s emphasis of the classical description of measuring instruments is itself one of the essential characteristics of quantum theory, given that what is so classically observed can only be predicted by quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is not that “we have not adapted our thinking and our means of expression to the new theory,” because in fact physicists had so adapted their thinking (in terms of physical, mathematical, and even daily concepts), and Bohr was one of the first to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The reason for the conviction that “the experiments must also be described in a classical manner” are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as stated by Bohr in his reply: “the seemingly obvious fact that the description of any measuring arrangement must,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' in an essential manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' involve the arrangement of the instruments in space and their functioning in time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' if we shall be able to state anything at all about phenomena.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' more accurately,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' what is observed in experiments must be so described,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' because,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' as Bohr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' added: “the argument here is of course first and foremost that in order to serve as measuring instruments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' they cannot be included in the realm of application proper to quantum mechanics.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' too,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' is one the most essential features of quantum theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' which brings into our thought a relation to what is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But this relation would not be possible without describing in a classical manner what is visible to thought in quantum experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s “must” in “we must interpret observations in classical terms” is stated “so definitively” and “without any reservation” because, while QM could become obsolete one day (although, remaining in place for a century now, not anytime soon in the author’s view) or the RWR view, possibly in favor of realism, this necessity of interpreting observation in classical terms will remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger was astute to notice Bohr’s “must,” as Bohr didn’t fail to acknowledge this point in his reply: “I found it most amusing that you noticed—which I myself had not at all been aware of—that just on this point, and only on this one, I do not say, ‘it might be’” (Letter to Bohr, October 13, 1935 [Bohr 1972-1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 512]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr will be more aware of this fact from this point on, with its significance even more pronounced in his subsequent writings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s “must be” reflects his assumption of the necessity of unambiguous, and in this sense objective, communication of, along with the logical and mathematical structure of quantum theory, the outcomes of experiments, insured by the classical description of the observable part of measuring instruments, in accordance with the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As he said later (in 1949): It is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanations, the account of all evidence must be expressed in classical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The argument is simply that by the word “experiment” we refer to a situation where we can tell others what we have done and we have learned and that, therefore, the account of the experimental arrangement and the results of the observation must be 30 expressed in unambiguous language with suitable application of the terminology of classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 39]20 Our expectations or probability assignments concerning such outcomes may be different, depending on different information we have pertaining to a given experiment, and in this sense, they are subjective or personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The latter might be a better concept insofar as these assignments are shaped by things in the world, such as measuring instruments or the world itself which is assumed in this article or by Bohr to exist independently and thus to be external to an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Things are rarely, if ever, completely subjective, permitting that such exterior factors are interiorized at the time of an assignment of one or another probability to a future event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is nothing paradoxical or inconsistent with Bohr’s claims in this understanding of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In life, too, we can have different expectations concerning future events given the information we possess (which may be different), although QM, as a mathematical- experimental science, gives us a precise probability calculus to predict quantum events, which is, again, all it does in the present view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Life rarely gives us such means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At the same time (hence the consistency with Bohr’s claims), any measurement, in any quantum experiment that would be performed would give a definitive, visible and informationally communicable outcome, and as such is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' An agent cannot control it but can only predict it probabilistically by means of QM (cum Born’s rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One might be able to decide (although it may not be simply a matter of a free choice by our consciousness or even unconscious) which observation or measurement to perform, for example, one or the other complementary observation, but one cannot control the specific outcome of it as concerns which value one obtains, even if one controls the preparation of the instrument that will register that outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In addition, as noted, one can always perform an alternative, complementary, measurement at the end point of the experiment, which will irrevocably disable the original estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (In classical mechanics, one can, again, always measure and predict, deterministically, all variables necessary for accounting, representationally, for the system considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') This makes measurement objective in this double sense—the lack of control of an outcome and the possibility of an unambiguous communication of an outcome—but in the present view, only in this double sense, rather than objectively attributing anything to nature itself, apart from its existence and, as part of it, human existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Making an observation or measurement is, as stated, a unique act or event of creation with a unique outcome that can be performed by a particular agent or several agents and as such has subjective or, again, personal aspects, including those shaping our decision concerning this action, a decision inherent in the very idea of experiment [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 699].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Once the measurement is performed, however, the outcome becomes fixed as a permanent record, part of the archive of physical data, always classical and visible to thought or even our immediate phenomenal perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It may be unknown to others, but that it is not the same as being or bound to remain subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is true, too, that, as any record, it must still be experienced as such by us or others to be meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, performing an act of observation or measurement is personal (if sometimes determined collectively), but its outcome need not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It can also be experienced differently by different agents, and in this sense, it is always personal and, in the first place, human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Science is a human enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But sharing and communicating our estimates of possible events and experiences is also human and doing so is helpful and even unavoidable in human life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Science capitalizes 20 As indicated above (note 8), it is possible to argue that, while necessary for the description of the observable quantum phenomena and measurements associated with them, classical physics is not a separate theory but rather a limit case of QM, thus eliminating it from the class of fundamental theory, which is, however, not the present view or, I would argue, that of Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, in the present view, if considered by itself, the cat in the cat experiment, is always a classical object that cannot be handled, either representationally or (which is only possibility in RWR interpretations) predictively, by QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM is only applicable in the cat experiment because there is a properly quantum aspect to it: the emission of a particle by the radioactive atomic substance used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As will be seen, the same considerations apply in the Wigner’s friend experiment, sometimes used to argue that everything can be considered as quantum, without any use of classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Most of these arguments, moreover, contend or imply that the observable parts of measuring instruments can also be handled by QM, without, in contrast to Schrödinger’s (subtler) argument that the situation requires new concept beyond both classical and quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 31 on this fact and on the possibility that the communication involved may be made unambiguous, helped by the use of mathematical symbols, central to modern physics, from Galileo on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These symbols, too, or their organization are visible to thought and hence unambiguously communicable, including those of the mathematical formalism of QM or QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Mathematics itself, as a discipline, depends on this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In classical physics and relativity, however, how the outcomes of experiments come about is visible to thought as well, and may be assumed to be independent of observation, for all practical purposes, but, in the present view, still only for all practical purposes, defined by human agents and agencies, such as science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Not so in quantum physics, essentially dealing with and fundamentally shaped by what is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In quantum physics, the role of human agents and experimental technology cannot in principle be neglected, as reflected in the nature of quantum probability, which, as discussed above, is no longer due, as in classical physics when it uses probability, to our insufficient knowledge of how the phenomena considered come about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At stake in RWR interpretations is the impossibility in principle of any knowledge or even conception concerning how this happens, which makes probability fundamentally irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The mathematics of quantum mechanics is visible to thought, and as such is unambiguously communicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But how what this mathematics predicts (in general probabilistically) comes about, as outcomes of quantum experiments, is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' We do not know what Schrödinger thought upon receiving Bohr’s reply, although it appears that he had never have accepted Bohr’s view concerning the irreducible role of classical concepts in quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is not clear, for example, to what degree, if any, the cat paradox or even his paper overall were an attempt to show that new physical concepts may after all be necessary in quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As explained below, it appears that, by saying that “the 𝜓-function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts,” he assumes the cat to be a quantum object [Schrödinger 1938, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This view has a difficulty in the fact that, if we open the box or use a box with glass walls, we will see the cat, the same cat, at any stage of the experiment, while one can never so see a properly quantum object, such as an electron, for detecting which one always needs an instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, moreover, a quantum object is a concept that only applies at the time of the experiment by the Dirac postulate, and hence implies that each observation concerns a different quantum object, although identifying these objects is permissible in low-energy (QM) regimes, but not high-energy (QFT) regimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the cat experiment, we always see the same cat, which can, as noted, change its state, but not its sameness, always visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Be it as it may on that score, Schrödinger thought, as did Einstein, that new concepts associated with quantum objects and their behavior might be necessary received a new support from the EPR experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Such concepts, they thought, would ground a realist alternative to QM, viewed by Schrödinger as “perhaps after all a convenient calculational trick” [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is difficult to assume that Einstein saw it as anything more than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Neither thought that QM was likely to be interpreted on realist lines, although such interpretations have been advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For Bohr, as explained, the EPR experiment confirmed, in accordance with his (strong RWR) interpretation, “how far, in quantum theory, we are beyond the reach of pictorial visualization,” to the point of reaching what is invisible to thought [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger’s cat experiment through the optics of visible and invisible to thought Schrödinger’s paper containing the cat experiment was a response to EPR’s paper, which it discusses at some length, and was, arguably, most important for the concept of entanglement, introduced by Schrödinger, and its overall discussion of QM, its main concern, as reflected in its title “The present situation in quantum mechanics” [Schrödinger 1935].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' His analysis is thoroughgoing and penetrating, even though (or perhaps because( Schrödinger assessed QM, especially as interpreted along the (Copenhagen) RWR lines, as “a doctrine born of distress” [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' He saw QM, if not necessarily as incomplete insofar as concerns its capacity to predict all that was possible to predict (or else nonlocal), as EPR argued, but then as “perhaps after all only a convenient calculational trick” [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' EPR’s experiment gave Schrödinger, as it did to Einstein, new hopes that an alternative realist theory of quantum phenomena might be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat experiment was part of Schrödinger’s overall 32 analysis of QM, a relatively marginal part, which did not appear to have initially received much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Neither did initially the paper itself, even the concept of entanglement, introduced there, a major contribution to QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' During the last half a century or so, however, the cat experiment has been interminably discussed in technical, philosophical, and popular literature, and has even acquired a semi- mythical status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are many reasons for this upsurge of attention to it, such as its role in helping realist or classical causal views of QM or fundamental physics, or countering the Bohr postulate, often resisted as much as the lack of realism or classical causality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', [Plotnitsky 2022b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is of course also a narrative appeal to the experiment, especially in popular accounts, but not only there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' From the present perspective, there is nothing especially remarkable or revealing in the cat experiment, or anything that would challenge Bohr’s or the present interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There does not appear to be any evidence that Bohr ever commented on the experiment or on Schrödinger’s paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The letter exchange, discussed above, between Schrödinger and Bohr concerning Bohr’s emphasis on the classical description of measuring instruments is relevant to the cat experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But, as preceding discussion makes clear, this exchange took place before Schrödinger’s paper was published and was about Bohr’s views, rather than any aspect of Schrödinger’s paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I’d surmise that Bohr would not find anything in the paradox either of much interest or as challenging his views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I’d also surmise that he was likely to have seen, as I do here, the cat as a classical and not a quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As indicated above, while not assumed by Bohr, the Dirac postulate, which only applies to quantum and not to classical objects, lends further support to this view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is because the postulate states that each quantum observation concerns a different quantum object, while only allowing one to assume that successive observations deal with the same quantum objects as a statistically permissible idealization of low energy (QM) regimes but not in high-energy (QFT) regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, the cat is aways the same object (if in a different classical state) at any stage of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At least, as I shall argue, it is difficult to assume otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' According to Schrödinger, then: One can even set up quite ridiculous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The first atomic decay would have poisoned it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The 𝜓-function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 157] In the present interpretation the last sentence would not apply, at least as stated, and the preceding two sentences, which are in accord with the present view, appear to contradict the last sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I shall explain why this is so presently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' First, however, Schrödinger adds an elaboration that is rarely discussed or given proper attention, which provided a further context for his thought experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' He says: “It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' That prevents us from so naively accepting as valid a ‘blurred model’ for representing reality” [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A blurred model is defined by a view of the 𝜓-function as “an imagined entity that images the blurring of all variables at every moment [unless a measurement intervenes] just as clearly and faithfully as the classical model [images] its sharp numerical values” [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In other words, the problem arises if one sees the 𝜓 -function as representing the independent behavior of quantum systems, in this case as blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, the 𝜓-function does not “faithfully” represent the behavior of the quantum object considered or the ultimate reality responsible for quantum phenomena, because it does not represent this reality at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It only provides an (discrete) “expectation-catalog” for possible future experiments, as Schrödinger himself called it [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In developing his wave mechanics, Schrödinger initially aimed for a (wave-like) representation of the ultimate reality responsible for quantum phenomena in his project for his wave mechanics that led him to his famous equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' He 33 had, however, long given up on the idea by this point in view of the difficulties of reconciling his wave mechanics had with observable features of quantum phenomena, in particular their discreteness and the probabilistic nature of predictions concerning them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' His equation itself can of course be and has been (immediately in the Göttingen-Copenhagen circles) interpreted so as to accommodate these features, especially given M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Born’s probabilistic interpretation of the 𝜓-function, eventually part of RWR interpretations of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These interpretations, including the present one, would, however, contrary to Schrödinger’s statement, preclude the claim that “the 𝜓-function of the entire system would express [the situation] by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts,” unless Schrödinger meant that his claim only applies to blurred variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' His claim, however, appears to be more general and applicable when one does not view the variable considered as blurred, as his subsequent reference to the cat experiment indicates [Schrödinger 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In any event, in the present view, the cat, as a classical object, would always be either dead or alive at any stage of the experiment, as Schrödinger’s preceding sentences imply: “If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The first atomic decay would have poisoned it.” Why then claim: “the 𝜓-function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is also, in principle, possible that by his phrasing “the 𝜓-function of the entire system would express this by having in it” (emphasis added) Schrödinger only meant the mixing of amplitudes for these two outcomes so that “the 𝜓-function of the entire system” contains both possible future outcomes, as would be the case in the present view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Schrödinger, however, does not qualify his statement in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, without any conflict with the first two sentences, which only refer to a classical object, what will be “mixed” or superposed are mathematical state vectors in the formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This mixture enables the probabilities of predicting the atomic decay involved, to which, as a quantum process, such terms as “dead” or “alive,” or any other terms, do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is only because of this purely mathematical mixture that one is able to estimate the probability of finding the cat dead or alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The 𝜓-function has no association with the cat apart from these predictions (via Born’s rule), and it never represents the state of the cat, as a classical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The 𝜓-function never represents the physical state of a quantum object either, as would be implied, by suggesting that the cat is seen as a quantum object, by Schrödinger’s formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, the cat is never mixed or smeared in equal parts between the living and the dead cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is either the alive or dead cat at any stage of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One merely does not know (after a certain moment in time, while cat is inside the box) whether it is alive or dead until one opens the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is nothing that can be said or thought of concerning the ultimate reality responsible for quantum phenomena (including, quantum objects, in the present view defined only at the time of observation by the Dirac postulate), including that responsible for the atomic decay in the cat experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By contrast, there are always things we can say about any classical reality, involved in quantum experiment, as part of what is, in principle observable, in them, as is the cat in the cat experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='21 The former reality is invisible to thought, the latter is visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 21 The emphasized phrase deliberately echoes Heisenberg’s famous and much misunderstood, especially along empiricist (Machian-like) lines, opening claim in his first paper of QM to the effect that he aims to ground his new mechanics in “the relationships between quantities which in principle are observable” [Heisenberg 1925, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The quantities in question are empirically observable in measuring instruments, but the relationships in question (the word usually disregarded in empiricist readings of this statement) are the probabilistic relationships established by his new mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Heisenberg said, shortly before completing his paper: “What I really like in this scheme is that one can really reduce all interactions between atoms and the external world .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' to transition probabilities” [Heisenberg, Letter to Kronig, 5 June 1925;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' cited in Mehra and Rechenberg 2001, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' By speaking of the “interactions between atoms and the external world,” this statement suggests that QM was only predicting the effects of these interactions observed in measuring instruments, without representing how these effects come about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As explained, this procedure replaced measurement in the classical sense (of measuring some preexisting properties of 34 One could, however, in the arrangement of the cat experiment, consider the cat as part of an object under investigation, concerning which the prediction in question is made by means of QM, but as explained earlier, only as part of this object because a proper quantum object must be involved in order to have a quantum experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat experiment is a quantum experiment because of the radioactive decay and not because of the cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Therefore, considering the cat as part of an object under investigation does not change the point that the cat is a classical object, always visible to thought or to our immediate sense perception (before and after the experiment, or throughout if, if the box has glass walls), but not a quantum object, which is never available to our phenomenal perception and, in the present view, is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One can at any point see the cat as such, independently of a quantum observational device, by opening the box or, again, using the box with glass walls, but one can never see a quantum object as such or rather (since it cannot be seen) establish its presence without a suitable observational device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A quantum object cannot be observed as separated from the phenomenon considered, which is a result of the interaction between this object and the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It would, accordingly, be more reasonable to see the cat as a classical object, while, within in the overall arrangement of the experiment, being part of the object of investigation by quantum means, enabling one to predict its possible classical state of being dead or alive at the final stage of experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As such the cat is also an object that can be described by ordinary language, as opposed to a quantum object like an electron, which is, in RWR interpretations, merely a name, without a concept attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This fact makes misleading using, as is done sometimes, such notations as |𝜓⟩ = 𝛼|𝑑𝑒𝑎𝑑⟩ + 𝛽|𝑎𝑙𝑖𝑣𝑒⟩, as opposed to the something like |𝜓⟩ = 𝛼|ℎ⟩ + 𝛽|𝑣⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The later refers to state vectors, in a superposition, used to predict definitive classical events (which are never in a superposition), such as, within the chain of events in the arrangement, that of the cat being dead or alive in the cat experiment, but has no other connections to either (physical) state of the cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Technically, QM predicts the effects that quantum objects (or in the present view, the ultimate reality responsible for quantum phenomena) can have on the classical world we experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These effects define quantum phenomena or events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As any observation in quantum physics, opening the box in the cat experiment is the phenomenon that reveals a classical state of the reality, a state in this case already established in advance, which includes the cat, either dead or alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One or another properly quantum object, such as a radioactive atom and a particle it emits is always necessary to have such an effect, even if the object under investigation, as different from a measuring instrument, can contain a classical object, such as the cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Calculating the probability of any such prediction will have to involve h because of the radioactive decay as involving properly quantum objects, while the cat is of no help in estimating such probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' To support the case just outlined more rigorously, I turn to Bohr’s argument in his reply to EPR, concerning “discriminating in each experimental arrangement between those parts of the physical system considered which are to be treated as measuring instruments and those which constitute the objects under investigation,” and the question of the cut thus arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' According to Bohr: This necessity of discriminating in each experimental arrangement between those parts of the physical system considered which are to be treated as measuring instruments and those which constitute the objects under investigation may indeed be said to form a principal distinction between classical and quantum-mechanical description of physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is true that the place within each measuring procedure where this discrimination is made is in both cases largely a matter of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While, however, in classical physics the distinction between object and measuring agencies does not entail any difference in the character of the description of the phenomena concerned, its fundamental importance in quantum theory … has its root in the indispensable use of classical concepts in the interpretation of all proper measurements, even though the classical theories do not suffice in accounting for the new types of regularities with which we are concerned in atomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In accordance with this situation there can be no question of any unambiguous interpretation of the symbols of quantum mechanics other than that embodied in the well-known rules which allow us to predict quantum objects) with establishing, by using measuring instruments, quantum phenomena, which can be treated classically without measuring the properties of quantum objects, a view was adopted and developed by Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 35 the results to be obtained by a given experimental arrangement described in a totally classical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 701;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' second emphasis added] It is important to avoid two common misunderstandings of this and related statements by Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The first concerns measuring instruments, in view of Bohr’s insistence on the classical description of the observable part of measuring instruments, a subject discussed in detail in Sections 2 and 3, beginning with the fact that instruments have quantum parts through which they interact with quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The second concerns quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s statement does not mean that, while observable parts of measuring instruments are described by classical physics, the independent behavior of quantum objects is described by means of the quantum-mechanical formalism, which assumption would be in conflict with the RWR interpretation held by Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This type of (realist) view has been adopted by some, sometimes under the heading of “the Copenhagen interpretation,” beginning, influentially, with Dirac’s and von Neumann’s classic studies, with Dirac’s book originally published in 1930 and von Neumann’s (in German, in 1932 [Dirac 1958, von Neumann 1955].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Both books, moreover, assume a classically causal independent behavior of quantum objects, with probability brought in only by measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This was, however, not Bohr’s view, especially at this stage of his thinking in 1935, or even almost immediately after the Como lecture of 1927, which may be seen as having adopted, still ambivalently, this type of view and which arguably influenced both Dirac and von Neumann in this regard [Plotnitsky 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 198-211].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the passage in question, Bohr only says that classical theories cannot account for how quantum phenomena (physically described classically) come about or predict what is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' He does not say that the independent behavior of quantum objects or objects under investigation (which may not be quantum but must contain quantum objects) is represented by the formalism of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In Bohr’s view, the “symbols” of QM only have a probabilistically predictive role, without, by the Heisenberg postulate, offering a representation of how quantum phenomena come about, while quantum phenomena themselves are represented by classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' So, QM does not represent them either, by the Bohr postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, in Bohr’s interpretation, while predicting, in general probabilistically, the data observed as part of phenomena, the formalism of QM is otherwise dissociated in physical terms from both the ultimate nature of reality responsible for quantum phenomena and these phenomena themselves, which phenomena are described by classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The circumstance that “the place within each measuring procedure where this discrimination is made is … largely a matter of convenience” is related to, although is not quite the same as, the arbitrariness of the cut or the Heisenberg cut, or sometimes the Heisenberg-von-Neumann cut, because Heisenberg and von Neumann favored the term (each giving it a somewhat different meaning), not used as such by Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr qualifies this claim, and this qualification is important, including in the context of the cat experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While “it is true that the place within each measuring procedure where this discrimination is made is … largely a matter of convenience,” it is true only largely but not completely, because “in each experimental arrangement and measuring procedure we have only a free choice of this place within a region where the quantum-mechanical description of the process concerned is effectively equivalent with the classical description” [Bohr 1935, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 701].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='22 Thus, the ultimate constitution of the physical reality or quantum objects and in quantum part of the instruments interacting with quantum objects is never on the measurement side of the event, and by the same token they can never serve as measuring instruments either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As beyond representation or even conception, as invisible to thought, quantum objects cannot be assigned any properties, even at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This impossibility is correlative to their position of always being on the other (than measurement) side of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All observable properties, are, by the Bohr postulate, only those of the observable parts of measuring instruments, described classically, but appearing under the impact of quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM only predicts these visible properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In Bohr’s or the present view, in part by virtue of associating the cut with the discrimination between what is considered the object under investigation and what is considered as the measuring instruments, the cut 22 This situation may be seen as a manifestation of Bohr’s correspondence principle, according to which the quantum-mechanical and the classical descriptions give the same predictions in the classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 36 and its (within certain limits) shifting nature need not imply that the classical theory, say, classical mechanics or classical electromagnetic theory is a special form of QM or (in the case of electromagnetism) QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As emphasized throughout this article, in Bohr or the present view, or that of Heisenberg, these are two different theories that deal with two different types of objects, even though classical objects are still composed of quantum objects or, in the present view (because the concept of a quantum object only applies at the time of observation) of the same ultimate reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' We do not know how classical objects (which could be considered, at least in principle, independently of observational technology) emerge from this this ultimate reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is true that a quantum system with a large number of coherent (quantum) states behaves close to a classical system, close but a) not quite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and b) still it is only a special class of quantum systems, which are still not the same as classical systems, which one uses to describe classical objects, including measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As indicated earlier, the cut, as here understood, reflects the possibility of placing some classical parts of the overall arrangement in a quantum experiment (including the cat experiment) on either side of the cut, as “an object under investigation,” concerning which predictions can be made by QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As I argue, however, the arrangement must include a quantum object (like a particle in radioactive decay in the cat experiment) for the experiment to be quantum, to have quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, the cat is always visible at least to our mind’s eye, even when, while inside the box, not actually available to our sense perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A quantum object is never available to such a perception, and one always needs an experimental device (which could be multi-stage, as in the cat experiment) capable of interacting with this object to have a quantum effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This effect is manifested in classically observed phenomenon or event, such as the cat being alive or dead, in this case in fact, rather than being a properly quantum effects, preceded and made possible by another classically observed event, the breaking of the flask, which is more properly quantum effect, due to the interaction between it and the emitted particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This interaction is quantum, while the rest is the chain of classical events triggered by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' accordingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' does not call a composite object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' containing both classical and quantum objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' on the object side of the cut a “quantum object,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' but the “object under investigation.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' while they can also be objects under investigation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' are only those objects that strictly belongs to the ultimate reality responsible for quantum phenomena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and they are always on the object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' never measurement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' side of the event,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' and hence a quantum object can never be a measuring instrument either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Other parts of objects under investigation in quantum experiments are physically classical objects, such as the cat or everything inside the box, except that part of the flask that can interact with the emitted particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' While it can be part of an object of investigation in a quantum experiment and treated by quantum means, if considered by itself, a classical object cannot be treated as a quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any prediction or measurement associated with a quantum object, elemental (such as an electron) or composite (such as a Josephson device), will always involve h, thus correlative to what is invisible to thought in quantum physics, even though h itself is observed classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Observing the cat in the cat experiment does not require h because the cat is a classical object, which, again, cannot be treated as a quantum object, only a quantum part of it can, like protons in its body in the MRI test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, in certain circumstances, a quantum object could be treated for all practical as a classical object, but without ever being a classical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, as when it is far enough from the nucleus (for large quantum numbers), an electron can be treated as behaving classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is, however, an approximation or idealization which disregards possible quantum effects of this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As Bohr noted, also connecting this situation to “mechanical pictures” and “classical pictures,” as visible to thought: [I]n the limit of larger quantum numbers where the relative difference between adjacent stational states vanishes asymptotically, mechanical pictures of electronic motion [as orbits] may be rationally utilized [by the correspondence principle].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It must be emphasized, however, that this connection cannot be regarded as a gradual transition toward classical theory in the sense that the quantum postulate [as an essential discontinuity and individuality of quantum phenomena] would lose its significance for high quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the contrary, the conclusions obtained from the correspondence principle with the aid of classical pictures depends just upon the assumptions of the conception of stationary and of [discrete] individual transition processes are maintained even in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [Bohr 1987, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 85] 37 By this point (in 1927), Bohr adopts the view, which, following Heisenberg, equally renounced both the classical, orbital “picture” of stationary states, still assumed in Bohr’s 1913 theory, and any classical view of the transitions, “quantum jumps,” between states, already abandoned by Bohr’s 1913 theory, the first instance of the RWR view (even if only partially applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The concept, while it enabled Bohr to account for the stability of atoms (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Rutherford’s preceding view), was nevertheless incompatible with classical mechanics and classical electrodynamics alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Neither the time nor direction of each jump could be explained, although it could be predicted probabilistically or statistically, which, thus, from a classical perspective paradoxically, brought the atomic stability and quantum randomness together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This stability is of course that of a dynamic system, which can change its states, although these changes could only be registered in measuring instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s conceptual framework makes the term “jump” misleading in suggesting some representation of what happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Electrons do not jump;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' quantum states (as physical states) discontinuously change, and no representation of how they do this is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' What was responsible for these changes was assumed to be real, but this reality was assumed to be at least beyond representation, in accord with the weak RWR view, although intimating the strong RWR view, insofar as no concept of how these transitions appeared to be possible to form either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In Heisenberg’s approach leading him to his invention of QM, the same situation defined the case of electrons in stationary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Electrons were not moving in orbits around nuclei: their quantum states (associated with variables other than energy, as the energy remained the same in a stationary state) were changing, with these changes observable as discrete phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In this view, there were only the states of quantum objects, manifested in measuring instruments, and transitions between these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This was a decisive shift in our understanding of the nature of physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='23 One might say that, rather than making any transition to a new energy, an electron in a given stationary state disappears and a new electron is born in this new stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Each corresponding measurement will detect a different electron, in accord with the Dirac postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The wave function for an electron in an atom can be recast in terms of annihilation and creation operators, used in QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr’s statement cited above concerning the behavior of electrons in the case of large quantum numbers confirms his view of classical objects and processes as visible to thought or even our immediate phenomenal perception, and the behavior of quantum objects as, at this point (in 1927), no longer at least to our general phenomenal intuition, and if not yet invisible to thought, as Bohr came to understand the situation by the late 1930s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The behavior in the limit of large quantum numbers can be treated for all practical purposes as that of classical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This treatment is, however, merely a workable approximation of what is the ultimate nature of the reality responsible for what is thus observed, a reality invisible to thought, and one still require a measuring instrument for this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At bottom one still deals with the combination of stationary states and discontinuous quantum jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These states are too close to each other for this combination to be detected, but one would, as it were, register these states (as invisible to thought electrons or photons they emit still cannot be “seen”) by “zooming” on them, if one had an instrument to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any such instrument would, however, need to be able, by interacting with electrons or “emitted” photons, to register properly quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Technically, an “emission,” too, is a classical concept, which cannot represent how photons are “emitted,” which is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' All we can see are traces of photons, or what we assume to be photons, traces manifested, literally visible, in spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Similarly, a macro quantum object (still defined as such by its microscopic quantum constitution), such as a Josephson device, can only be detected as quantum by means of a suitable instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Otherwise, it will be observed as a classical object and as such as something (two superconductors standing in a lab) available our immediate phenomenal perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Thus, in quantum physics, on the one hand, there is always a discrimination between an object and an instrument, and, on the other, their indivisibility in quantum phenomena, or what Bohr calls the wholeness of phenomena or its closed nature, from which one can never extract the object itself at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Any investigation in quantum theory must involve this combination, which is also 23 I am indebted to Laurent Friedel on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 38 that of what is invisible to thought and as such cannot be communicated unambiguously, and what is visible to thought, via observational instruments, and can be communicated unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This situation thus sharply contrasts with that of classical physics or relativity, where the role of measuring instruments can be neglected or controlled and where, as a result, which always deal with what is visible to thought and, as such unambiguously communicable or sharable as information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' If the object under investigation is classical (visible, representable, with its character unambiguously communicable, and so forth), like the cat in the cat experiment, it can always be considered independently apart from quantum experiments and discussed unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is, as noted, never any ambiguity in assessing the cat as an independent object in the cat experiment but only two unambiguously defined possibilities, each visible to our mind’s eye, of the cat being either dead or alive, with the probability defined by the 𝜓- function associated with the atomic decay and only secondarily to the state, always classical, of the cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' A cat, inside or outside the box, is always a cat, dead or alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' As such it can only be seen as a physically classical part of the arrangement, inside the box, before the interaction with the particle emitted by an atom, which can never be observed (and terms like particle or emission cannot apply in any sense we can attribute to these terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat can be on both sides of the event (or the cut), but the radioactive decay or the particle emitted by it can only be on one side, the side of the object, and never the measurement side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This emission occurs or not regardless of the cat in the box, or the box, or the flask, all of which are classical and are parts of the arrangement made by us, while the radioactive atom is prepared by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat could be removed from the box in advance without affecting this possible quantum event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The flask is the only classical object that interacts with the emitted particle, which enable one to register with the presence of the emission, if it occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Technically, one need not see the opening the box as a quantum experiment, as the properly quantum experiment in the arrangement, which is the shattering of the flask, occurs (if it does) before the box is opened, and then the outcome of this experiment leads to the event that classically affects the cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat is more like an “agent,” akin to (although of course not the same as) “Wigner’s friend” in a related famous experiment, than an instrument, and is, again, never a properly quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='24 Neither, again, is anything else in the arrangement, except the flask, inside the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' But, as explained, even if one does see the whole arrangement and an instrument (an arrangement and an instrument made by us as humans), the cat, just as the box, it is still only a classical part of this arrangement, always visible to our mind’s eye 24 Although it has additional complexities, the Wigner’s friend experiment [Wigner 1961] can be considered along the lines of the argument advanced here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In Wigner’s scenario, “the friend” hidden from “Wigner” inside some lab (just as the cat is hidden from the observer in the cat experiment), performs an experiment on a previously prepared quantum system, S, with the outcome, which, unlike the initial preparation, is hidden from “Wigner” as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' “Wigner” leaves the lab after the initial preparation, which enables one to associate with S (which is, in the present view, not the same as assign to S) the wave function |𝜓⟩, known to both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' QM can, then, be used by “Wigner” in estimating this hidden outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is, I would argue, possible while, just as in the present view of the cat experiment, considering “the friend” as a classical object within the overall arrangement, which, as that of the cat experiment, contains a proper quantum object, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The case would require a separate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I might note, however, that most discussions of the Wigner’s friend experiment and the problems and paradoxes found in many of them, beginning with Wigner’s own encounter assume that “the friend” (or sometimes “Wigner”) can be considered as a quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' For more recent treatments, see [Pusey 2018, Bauman and Brukner 2020, DeBrota et al 2020], and further references in these articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is not my aim to assess these arguments (sometimes questioning each other) as concerns their effectiveness in resolving the “paradoxes” of Wigner’s experiment and Wigner’s own initial argument, which, I would argue, in effect suggests, even if against Wigner’s own grain, the difficulty of assuming that the friend is a quantum object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Another, related, feature of some of recent arguments, most especially [DeBrota et al 2020], which is based in quantum Bayesianism (QBism), is their claim of the subjective nature not only of our predictions, a view assumed here as well, but also of the outcomes of quantum measurements, a view not assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, following Bohr, these outcomes are objective in the sense of being unambiguously communicable (with further qualifications given above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This assumption is correlative to that of the physically classical description of the observable part of measuring instruments and quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In the present view, “the friend” is a classical object, just as is the cat in the cat experiment, even though the arrangement considered can be treated by QM as concerns “Wigner’s” estimates of the outcome of the friend’s measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 39 or directly if the box has glass walls, even if the arrangement requires us to use QM to predict what happens once one opens the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat or the friend in the Wigner’s friend experiments is not an instrument either, and either arrangement implies the presence of an observational instrument capable of interacting with quantum objects and registering (classically) the outcome of such interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat does not, of course, consciously observe such an outcome in the way a human agent would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The cat can only manifest this outcome by being dead or alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' One the other hand, the friend in Wigner’s friend experiment does consciously observes it, which fact was central to Wigner’s original argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This difference, however, does not change the fact that the cat or the friend is a classical object or that both experiments essentially depend on the role of properly quantum objects, which are never of the measurement side of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nor could they serve as measuring instruments, which requires to have a classical part in order to be observed by an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nor of course could quantum objects serve as agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Conclusion I return, in closing, to Denmark, first, not to that of Bohr but that of Shakespeare and Hamlet three centuries earlier, and the lines of the play, used as my first epigraph: Hamlet [commenting on his dead father]: My father—methinks I see my father.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Horatio: Where, my lord?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hamlet: In my mind’s eye, Horatio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [The Tragedy of Hamlet, Prince of Denmark, Act 1, Scene 2, ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 183-185] The reason for Horatio’s puzzlement is that he saw the ghost of Hamlet’s father and wondered if perhaps Hamlet has already seen the ghost as well, which Hamlet’s response proves not to be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hamlet’s encounter with the ghost of his father is yet to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' At stake in this scene is Hamlet’s image of his father in his mind’s eye, and thus as something visible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This image shadows Hamlet and the play from beginning to end, and Hamlet’s encounter with the ghost, dramatic and consequential for the play, adds to the power as this image in and over Hamlet’s mind, but this power was already there all along.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I am, however, not concerned here with much discussed psychological, such as psychoanalytic, implications of this power, but instead with the capacity of our thought, conscious and unconscious, to create an image of the world and of objects in the world, on which Shakespeare capitalizes in Hamlet and his other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' No less remarkable, however, is our thought’s capacity to think that which is beyond thought, is invisible to thought, and hence has no image in our mind’s eye, such as the ultimate nature of the reality responsible for quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Shakespeare might have realized this capacity of thought at least to some degree, as suggested by Hamlet’s comment to Horatio after his encounter with the ghost: Horatio: O day and night, but this is wondrous strange!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Hamlet: And therefore as a stranger give it welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' [The Tragedy of Hamlet, Prince of Denmark, Act I, Scene 4, ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 165-166] Some editions have “our philosophy.” “Your philosophy” makes Hamlet more suspicious of philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It makes him more akin to a quantum physicist, who can only estimate the probabilities of future events defined by experiments Hamlet stages at the castle of Elsinore, a prominent aspect of the play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There are, quantum physics may indeed be telling us, things in, or beyond, heaven and earth that we cannot dream of or otherwise see in our mind’s eye, consciously or unconsciously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr is reported to have replied, after the rise of quantum physics but before QM was discovered, to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Høffding’s question “Where can the photon be said to be?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' with “To be, to be, what does it mean to 40 be?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (cited in [Wheeler and Ford 1998, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 131]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bohr might have been echoing the most famous sentence of Shakespeare’s Hamlet, “To be, or not to be, that is the question” (Act 3, ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 1749), realizing that in quantum physics one might want to or even must ask first “What does it mean to be?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (Hamlet’s famous monologue is, too, about much more than merely deciding to live or die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=') Høffding’s and Bohr’s questions are still unanswered and, in Bohr’s ultimate, RWR-type, view, are unanswerable, when it comes to quantum objects, such as photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Even as invisible to thought, quantum objects are idealizations (and hence still products of thought), in the present interpretation, ultimately only applicable at the time of observation, even if Bohr himself did not go that far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Either way, such questions as “Where can something be said to be?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' or “When had something happened?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' can only be asked about quantum phenomena, observed in measuring instruments, and as such visible to thought, to our mind’s eye, or even to our immediate perceptuon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Nature has no photons or electrons, any more than being or reality, including that of the RWR-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Admittedly, as are our thought and hence these concepts are created (we don’t know how either) by our brains, which are part of by biological and neurological constitution, and in this sense still by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This is, however, not the same as saying (as is done sometimes) that nature uses these concepts through us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Rather, nature allows us to create concepts—daily, physical, philosophical, or mathematical—and use them in considering our interactions with nature by means of technology, beginning with that of our bodies, and again, our thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' It is this interaction and only this interaction that enables us to idealize some part of the constitution of nature, even its ultimate constitution, as something that is invisible to thought and hence that cannot appear in our mind’s eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This brings me to my second epigraph “To die for the invisible—this is metaphysics,” courtesy of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Levinas’s book, Totality and infinity: An essay on exteriority [Levinas 2012, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 39] (originally published in French in 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' The book is about ethics (as is, along with its many other themes, Hamlet as well), and as such it might appear distant from quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Levinas’s epistemology, however, advanced in this book not only shares some of the philosophical genealogy, for example, in Kant’s philosophy, with quantum theory, but might have more direct connections with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Quantum theory and its epistemological problems, and possibly Bohr’s ideas, were known to Levinas, as they were widely discussed on the French intellectual scene to which Levinas’s work belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Levinas’s concept of exteriority, expressly associated by him with “the invisible” [l’invisible], has manifested affinities with the idea of invisible to thought and even strictly means, in the ethical domain, that which is invisible to thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' My main interest here is the association, quite dramatic—“To die for the invisible!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' No less!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content='—of the invisible with metaphysics, which I would like to connects to quantum physics and indeed to all modern physics, from Galileo on, as a mathematical-experimental science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Insofar as one means by metaphysics something exterior to nature or, to use the ancient Greek word, physis, especially referring by metaphysics to something theological, modern physics excludes metaphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' There is no metaphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' On the other hand, insofar, because the idea of physics as a mathematical-experimental science or the idea of nature in the first place still belongs to thought, even when nature is invisible to thought, there is only metaphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Modern physics navigates and negotiates between both, “no metaphysics” and “only metaphysics.” How close we come, in modern physics, to understanding nature, including in its ultimate constitution, even if the latter is ultimately invisible to thought, depends on our interactions with nature by means of experimental technologies and mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' These interactions are part of nature, too, but a particular part of it, specific to us, to our thinking and technologies, beginning with that of our bodies and brains, which are responsible for our thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Our thought, however, also has a capacity to reach what is beyond it, is invisible to it, and to affirm it: to die for the invisible—this is metaphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' This paper was in part prompted by the question by Lorenzo Maccone concerning the relationships between the cat paradox and Bohr’s complementarity, and the subsequent exchanges, for which I thank him, even though our views concerning quantum theory are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I am grateful to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Mauro D’Ariano for exceptionally illuminating conversations concerning the subjects considered the article and fundamental physics in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' I am also happy to thank Gregg Jaeger and Andrei Khrennikov for many discussions on quantum foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' 41 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bauman V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', Brukner C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (2020) Wigner’s friend as a rational agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' In: Hemmo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=', Shenker, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (eds) Quantum, probability, logic: The work and influence of Itamar Pitowsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Springer, Cham, Switzerland, 2020, 91-99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Bell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' (2004) Speakable and unspeakable in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFKT4oBgHgl3EQf4S7l/content/2301.11933v1.pdf'} +page_content=' Cambridge, UK: Cambridge 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